[philip hartman] ordinary differential equations ((bookza.org)

Ordinary DifferentialEquations

SIAM's Classics in Applied Mathematics series consists of books that were previouslallowed to go out of print. These books are republished by SIAM as a professionalservice because they continue to be important resources for mathematical scientists.

Editor-in-ChiefRobert E. O'Malley, Jr., University of Washington

Editorial Board

Richard A. Brualdi, University of Wisconsin-MadisonHerbert B. Keller, California Institute of TechnologyAndrzej Z. Manitius, George Mason UniversityIngram Olkin, Stanford UniversityStanley Richardson, University of EdinburghFerdinand Verhulst, Mathematisch Instituut, University of Utrecht

Classics in Applied Mathematics

C. C. Lin and L. A. Segel, Mathematics Applied to Deterministic Problems in theNatural SciencesJohan G. F. Belinfante and Bernard Kolman, A Survey of Lie Groups and LieAlgebras with Applications and Computational MethodsJames M. Ortega, Numerical Analysis: A Second CourseAnthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: SequentialUnconstrained Minimization TechniquesF. H. Clarke, Optimization and Nonsmooth AnalysisGeorge F. Carrier and Carl E. Pearson, Ordinary Differential EquationsLeo Breiman, ProbabilityR. Bellman and G. M. Wing, An Introduction to Invariant ImbeddingAbraham Berman and Robert J. Plemmons, Nonnegative Matrices in the MathematicalSciencesOlvi L. Mangasarian, Nonlinear Programming*Carl Friedrich Gauss, Theory of the Combination of Observations Least Subjectto Errors: Part One, Part Two, Supplement. Translated by G. W. Stewart

Richard Bellman, Introduction to Matrix AnalysisU. M. Ascher, R. M. M. Mattheij, and R. D. Russell, Numerical Solution ofBoundary Value Problems for Ordinary Differential EquationsK. E. Brenan, S. L. Campbell, and L. R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic EquationsCharles L. Lawson and Richard J. Hanson, Solving Least Squares ProblemsJ. E. Dennis, Jr. and Robert B. Schnabel, Numerical Methods for UnconstrainedOptimization and Nonlinear EquationsRichard E. Barlow and Frank Proschan, Mathematical Theory of Reliability

*First time in print.

Classics in Applied Mathematics (continued)

Cornelius Lanczos, Linear Differential OperatorsRichard Bellman, Introduction to Matrix Analysis, Second EditionBeresford N. Parlett, The Symmetric Eigenvalue Problem

Richard Haberman, Mathematical Models: Mechanical Vibrations, PopulationDynamics, and Traffic FlowPeter W. M. John, Statistical Design and Analysis of ExperimentsTamer Ba§ar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, SecondEdition

Emanuel Parzen, Stochastic Processes

Petar Kokotovic, Hassan K. Khalil, and John O'Reilly, Singular PerturbationMethods in Control: Analysis and Design

Jean Dickinson Gibbons, Ingram Olkin, and Milton Sobel, Selecting and OrderingPopulations: A New Statistical Methodology

James A. Murdock, Perturbations: Theory and Methods

Ivar Ekeland and Roger Temam, Convex Analysis and Variational Problems

Ivar Stakgold, Boundary Value Problems of Mathematical Physics, Volumes I and II

J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations inSeveral Variables

David Kinderlehrer arid Guido Stampacchia, An Introduction to VariationalInequalities and Their Applications

F. Natterer, The Mathematics of Computerized Tomography

Avinash C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging

R. Wong, Asymptotic Approximations of Integrals

O. Axelsson and V. A. Barker, Finite Element Solution of Boundary ValueProblems: Theory and Computation

David R. Brillinger, Time Series: Data Analysis and Theory

Joel N. Franklin, Methods of Mathematical Economics: Linear and NonlinearProgramming, Fixed-Point Theorems

Philip Hartman, Ordinary Differential Equations, Second Edition

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Ordinary DifferentialEquationsSecond Edition

Philip HartmanThe Johns Hopkins University

Baltimore, Maryland

Society for Industrial and Applied MathematicsPhiladelphia

Χοπψριγητ ♥ 2002 βψ τηε Σοχιετψ φορ Ινδυστριαλ ανδ Αππλιεδ Ματη

This SIAM edition is an unabridged, corrected republication of the edition publishedby Birkhauser, Boston, Basel, Stuttgart, 1982. The original edition was published byJohn Wiley & Sons, New York, 1964.

1 0 9 8 7 6 5 4 3 2 1

All rights reserved. Printed in the United States of America. No part of this book maybe reproduced, stored, or transmitted in any manner without the written permission ofthe publisher. For information, write to the Society for Industrial and Applied Mathe-matics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication Data

Hartman, Philip, 1915-Ordinary differential equations / Philip Hartman.

p. cm. - (Classics in applied mathematics ; 38)Previously published: 2nd ed. Boston : Birkhauser, 1982. Originally published:

Baltimore, Md., 1973.Includes bibliographical references and index.ISBN 0-89871-510-5 (pbk.)

1. Differential equations. I. Title. II. Series.

QA372 .H33 2002515'.352-dc21 2002017641

siam is a registered t

To the memory

of my parents

To the patienceof Sylvia, Judith, and Marilyn

ContentsForeword to the Classics EditioPreface to the First Edition

Preface to the Second EditionErrata

I. Preliminaries 11. Preliminaries, 12. Basic theorems, 23. Smooth approximations, 64. Change of integration variables, 7

Notes, 7

II. Existence 81. The Picard-Lindelof theorem, 82. Peano's existence theorem, 103. Extension theorem, 124. H. Kneser's theorem, 155. Example of nonuniqueness, 18

Notes, 23

III. Differential inequalities and uniqueness 241. Gronwall's inequality, 242. Maximal and minimal solutions, 253. Right derivatives, 264. Differential inequalities, 265. A theorem of Wintner, 296. Uniqueness theorems, 317. van Kampen's uniqueness theorem, 358. Egress points and Lyapunov functions, 379. Successive approximations, 40

Notes, 44

IV. Linear differential equations 451. Linear systems, 452. Variation of constants, 483. Reductions to smaller systems, 494. Basic inequalities, 545. Constant coefficients, 576. Floquet theory, 60

xvxsvii

xviii

xix

x Contents

7. Adjoint systems, 628. Higher order linear equations, 639. Remarks on changes of variables, 68

APPENDIX. ANALYTIC LINEAR EQUATIONS, 70

10. Fundamental matrices, 7011. Simple singularities, 7312. Higher order equations, 8413. A nonsimple singularity, 87

Notes, 91

V. Dependence on initial conditions and parameters 931. Preliminaries, 932. Continuity, 943. Differentiability, 954. Higher order differentiability, 1005. Exterior derivatives, 1016. Another differentiability theorem, 1047. S- and L-Lipschitz continuity, 1078. Uniqueness theorem, 1099. A lemma, 110

10. Proof of Theorem 8.1, 11111. Proof of Theorem 6.1, 11312. First integrals, 114

Notes, 116

VI. Total and partial differential equations 117

PART I. A THEOREM OF FROBENIUS, 117

1. Total differential equations, 1172. Algebra of exterior forms, 1203. A theorem of Frobenius, 1224. Proof of Theorem 3.1, 1245. Proof of Lemma 3.1, 1276. The system (1.1), 128

PART II. CAUCHY'S METHOD OF CHARACTERISTICS, 131

7. A nonlinear partial differential equation, 1318. Characteristics, 1359. Existence and uniqueness theorem, 137

10. Haar's lemma and uniqueness, 139Notes, 142

Contents xi

VII. The Poincare-Bendixson theory 1441. Autonomous systems, 1442. Umlaufsatz, 1463. Index of a stationary point, 1494. The Poincaré-Bendixson theorem, 1515. Stability of periodic solutions, 1566. Rotation points, 1587. Foci, nodes, and saddle points, 1608. Sectors, 1619. The general stationary point, 166

10. A second order equation, 174

APPENDIX. POINCARÉ-BENDIXSON THEORY ON 2-MANiFOLDS, 18211. Preliminaries, 18212. Analogue of the Poincaré-Bendixson theorem, 18513. Flow on a closed curve, 19014. Flow on a torus, 195

Notes, 201

VIII. Plane stationary points 2021. Existence theorems, 2022. Characteristic directions, 2093. Perturbed linear systems, 2124. More general stationary point, 220

Notes, 227

IX. Invariant manifolds and linearizations 2281. Invariant manifolds, 2282. The maps Tt 2313. Modification of F(ξ), 2324. Normalizations, 2335. Invariant manifolds of a map, 2346. Existence of invariant manifolds, 2427. Linearizations, 2448. Linearization of a map, 2459. Proof of Theorem 7.1, 250

10. Periodic solution, 25111. Limit cycles, 253

APPENDIX. SMOOTH EQUIVALENCE MAPS, 256

12. Smooth linearizations, 25613. Proof of Lemma 12.1, 25914. Proof of Theorem 12.2, 261APPENDIX. SMOOTHNESS OF STABLE MANIFOLDS, 271Notes, 271 L

xii Contents

X. Perturbed linear systems 2731. The case £ = 0, 2732. A topological principle, 2783. A theorem of Wazewski, 2804. Preliminary lemmas, 2835. Proof of Lemma 4.1, 2906. Proof of Lemma 4.2, 2917. Proof of Lemma 4.3, 2928. Asymptotic integrations. Logarithmic scale, 2949. Proof of Theorem 8.2, 297

10. Proof of Theorem 8.3, 29911. Logarithmic scale (continued), 30012. Proof of Theorem 11.2, 30313. Asymptotic integration, 30414. Proof of Theorem 13.1, 30715. Proof of Theorem 13.2, 31016. Corollaries and refinements, 31117. Linear higher order equations, 314

Notes, 320

XL Linear second order equations 3221. Preliminaries, 3222. Basic facts, 3253. Theorems of Sturm, 3334. Sturm-Liouville boundary value problems, 3375. Number of zeros, 3446. Nonoscillatory equations and principal solutions, 3507. Nonoscillation theorems, 3628. Asymptotic integrations. Elliptic cases, 3699. Asymptotic integrations. Nonelliptic cases, 375

APPENDIX. DISCONJUGATE SYSTEMS, 384

10. Disconjugate systems, 38411. Generalizations, 396

Notes, 401

XII. Use of implicit function and fixed point theorems . . . . 404

PART I. PERIODIC SOLUTIONS, 407

1. Linear equations, 4072. Nonlinear problems, 412

Contents xiii

PART II. SECOND ORDER BOUNDARY VALUE PROBLEMS, 418

3. Linear problems, 4184. Nonlinear problems, 4225. A priori bounds, 428

PART III. GENERAL THEORY, 435

6. Basic facts, 4357. Green's functions, 4398. Nonlinear equations, 4419. Asymptotic integration, 445

Notes, 447

XIII. Dichotomies for solutions of linear equations . . . . . 450

PART I. GENERAL THEORY, 451

1. Notations and definitions, 4512. Preliminary lemmas, 4553. The operator T, 4614. Slices of ||Py(t) | | 4655. Estimates for ||y(t)||, 4706. Applications to first order systems, 4747. Applications to higher order systems, 4788. P(B, D)-manifolds, 483

PART II. ADJOINT EQUATIONS, 484

9. Associate spaces, 484

11. Individual dichotomies, 48612. P'-admissible spaces for T' 49013. Applications to differential equations, 49314. Existence of PD-solutions, 497

Notes, 498

XIV. Miscellany on monotony 500

PART I. MONOTONE SOLUTIONS, 500

1. Small and large solutions, 5002. Monotone solutions, 5063. Second order linear equations, 5104. Second order linear equations (continuation), 515

10. The operator T', 486

xiv Contents

PART II. A PROBLEM IN BOUNDARY LAYER THEORY, 51

5. The problem, 5196. The case A > 0, 5207. The case λ < 0, 5258. The case A = 0, 5319. Asymptotic behavior, 534

PART III. GLOBAL ASYMPTOTIC STABILITY, 537

10. Global asymptotic stability, 53711. Lyapunov functions, 53912. Nonconstant G, 54013. On Corollary 11.2, 54514. On "J(y)x • x ^ 0 if x -f(y) = 0", 5415. Proof of Theorem 14.2, 55016. Proof of Theorem 14.1, 554

Notes, 554

HINTS FOR EXERCISES, 557

REFERENCES, 581

INDEX, 607

Foreword to theClassics Edition

Those of us who grew up having this tome as our text or supple-mentary guide often referred to it with a certain reverence as being theultimate authority. Some of my colleagues simply called it "The Bible."This is not a book that trots out the usual topics and examples in aformulaic way. Instead it is written by one of the world's leading practi-tioners, whose purpose appears to have been to give the serious studentall the necessary tools for work at the forefront and to introduce researchtopics which, at the time of the book's original publication in 1964, werehot off the press. Thus, the book is comprehensive in fundamental the-ory but goes far beyond that in its presentation of several importantand interesting topics as well as of some proofs which had never beforeappeared in print.

I am delighted that SIAM has undertaken to reprint Hartman'swonderful text, not only because of its historical significance but be-cause it contains almost everything one would want in a modern ad-vanced course in ordinary differential equations. The topological ideasand techniques of functional analysis that are so important today areclearly explained and applied. The basic theory of ordinary differen-tial equations is given in sufficient generality that the book serves as avaluable reference, whereas many other texts supply only the standardsimplified theorems.

What makes the text outstanding, in my mind, are the chapterson invariant manifolds, perturbations, and dichotomies. The idea offirst establishing the existence of invariant manifolds for maps, andthen applying the theory to solution operators for ordinary differentialequations suitably modified outside a neighborhood of a critical point,has become a standard approach, now also followed for equations ininfinite-dimensional spaces. One finds a complete proof of the Hartman-Grobman theorem on transforming a nonlinear to a linear flow in theneighborhood of a hyperbolic equilibrium. Theorems on smooth equiva-lence and on the smoothness of invariant manifolds are presented—thesebeing important for perturbation and normal form theory. Poincare sec-tions are used to show persistence of hyperbolic periodic orbits under

xvi Foreword

perturbations in the system—again, a technique that is now widely used.All of this has found considerable use in the geometrical theory

of differential equations and dynamical systems, such as in helping toresolve questions of structural stability. But it is also true that invariantmanifold theory has become ubiquitous in both science and engineering.Typically the theory is used to deduce qualitative information aboutsolutions to large, possibly infinite-dimensional, systems by reducing theessential dynamics to a small-dimensional system.

Readers will find many parts of this outstanding book as essentialtoday as they were when the book was first published. So it is that inPhilip Hartman's book we find the ever-useful classical theory and whatcould be considered the modern theory of ordinary differential equationsand dynamical systems.

Peter Bates, Brigham Young University

Preface to the First Edition

This book is based on lecture notes of courses on ordinary differentialequations which I have given from time to time for advanced under-graduates and graduate students in mathematics, physics, and engineering.It assumes a knowledge of matrix theory and, if not a thorough knowledgeof, at least a certain maturity in the handling of functions of real variables.

I was never tempted to scatter asterisks liberally throughout this bookand claim that it could serve as a sophomore-junior-senior textbook, forI believe that a course of this type should give way to basic courses inanalysis, algebra, and topology.

This book contains more material than I ever covered in one year butnot all of the topics which I treated in the many courses. The contents ofthese courses always included the subject matter basic to the theory ofdifferential equations and its many applications to other disciplines (as,for example, differential geometry). A "basic course" is covered inChapter I; §§ 1-3 of Chapter II; §§1-6 and 8 of Chapter III; Chapter IVexcept for the "Application" in § 3 and part (ix) in § 8; §§1-4 of ChapterV; §§1-7 of Chapter VII; §§ 1-3 of Chapter VIII; §§ 1-12 of ChapterX; §§ 1-4 of Chapter XI; and §§ 1-4 of Chapter XII.

Many topics are developed in depth beyond that found in standardtextbooks. The subject matter in a chapter is arranged so that moredifficult, less basic, material is usually put at the end of the chapter (and/or in an appendix). In general, the content of any chapter depends onlyon the material in that chapter and the portion of the "basic course"preceding it. For example, after completing the basic course, an instructorcan discuss Chapter IX, or the remainder of the contents of Chapter XII,or Chapter XIV, etc. There are two exceptions: Chapter VI, Part I, aswritten, depends on Chapter V, §§5-12; Part III of Chapter XII is notessential but is a good introduction to Chapter XIII.

Exercises have been roughly graded into three types according todifficulty. Many of the exercises are of a routine nature to give thestudent an opportunity to review or test his understanding of the tech-niques just explained. For more difficult exercises, there are hints in theback of the book (in some cases, these hints simplify available proofs).Finally, references are given for the most difficult exercises; these serve

xviii Preface

to show extensions and further developments, and to introduce thestudent to the literature.

The theory of differential equations depends heavily on the "integrationof differential inequalities" and this has been emphasized by collectingsome of the main results on this topic in Chapter III and § 4 of ChapterIV. Much of the material treated in this book was selected to illustrateimportant techniques as well as results: the reduction of problems ondifferential equations to problems on "maps" (cf. Chapter VII, Appendix,and Chapter IX); the use of simple topological arguments (cf. ChaptersVIII, § 1; X, §§ 2-7; and XIV, § 6); and the use of fixed point theoremsand other basic facts in functional analysis (cf. Chapters XII and XIII).

I should like to acknowledge my deep indebtedness to the late ProfessorAurel Wintner from whom and with whom I learned about differentialequations, first as a student and later as a collaborator. My debt to himis at once personal, in view of my close collaboration with him, andimpersonal, in view of his contributions to the resurgence of the theoryof ordinary differential equations since the Second World War.

I wish to thank several students at Hopkins, in particular, N. Max,C. C. Pugh, and J. Wavrik, for checking parts of the manuscript. I alsowish to express my appreciation to Miss Anna Lea Russell for the excellenttypescript created from nearly illegible copy, numerous revisions, andchanges in the revisions.

My work on this book was partially supported by the Air Force Officeof Scientific Research.

PHILIP HARTMANBaltimore, MarylandAugust, 1964

Preface to the Second EditioThis edition is essentially that of 1964 (John Wiley and Sons, Inc.)

with some corrections and additions. The only major changes are inChapter IX. A few pertinent additions have been made to the bibli-ography as a Supplement, but no effort has been made to bring it upto date.

I should like to thank Professors T. Butler, K.T. Chen, W.A. Coppel,L Lorch, M.E. Muldoon, L. Nicolson and C. Olech for corrections and/orsuggestions, and Mrs. Margaret A. Einstein for able secretarial assistance.

P.H. (10/73)

Errata

This book has been photographically reproduced from the edition pub-lished by Birkhauser, Boston, Basel, Stuttgart, 1982. The following arecorrections for typographical errors in that edition.

Page 262: In lines 6 and 2 from the bottom, replace ^ K+ — K° by G K~.

Page 264: At the end of the line following (14.8), replace ^ by 6.

Page 264: Replace the last five lines of page by the following:

(d) There is a constant C > 1 such that

This is clear for j — I from the normalizations following (4.1), andthe cases j > I follow by induction.

Page 265: Replace the part beginning at Proof by the following:

Proof. If 0 < r2 < ri,f G K+(r2] - M+ and r, = T~k(^£, then,by the property of ri,Tlr) G S'(ro) for i = 0 , . . . , fc(0- Put

We show by induction that r2 can be chosen so that

for 0 < i < fc(0- Assume (14.14) for 0 < i < j - 1 < fc(£). ThenCO') = (TiTT^-n - T^-1^} + (TiT^r? - TT^-1^), so that

XX Errata

Thus Cax < 1 implies that

If we define r? = ^(A) by r% — minfri, r\/(C\ + 2), 1], then N > X givesboth ||C(j)|| < CAr2

A < Cxr2 < ri and

This completes the proof of the induction for (14.14) and of assertion(14.12). The definition of £(j) and the choice j ~ k(£) in (14.15) give(14.13).

Page 266: Delete the first two lines.

Page 267: Line 4 should read "respectively, for suitably small r^ — (A).Note that"

Page 267: Line 10 should read "In view of (e) and its proof, this makesthe validity of (h) clear."

Page 271G: In Bnf; of the last line, replace £ by 77.

Page 271J: In the third line following (4.6), in {Tn} replace T by r.

Page 272: Add the following to the text:

The translators for the Russian edition noted errors in the original argu-ments of steps (d) and (e) in the proof of Theorem 12.2; see P. Hartman,Obyknovennye Differencial'nye Uravneniya, IzdatePstvo "Mir," Moscow(1970), translated by I. H. Sabitov and Yu. B. Egorov, edited by V. M.Alexeev. Compare the corrections for pages 262-267 in the errata, pagesxix-xx.

Page 501: In the ninth line from the bottom insert a 2 before Im.

Chapter I

Preliminaries

1. Preliminaries

Consider a system of d first order differential equations and an initialcondition

where y' = dy\dt, y — (y1,...,/) and/= (/*,... ,/d) are J-dimensionalvectors, and/(f, y) is defined on a (d + l)-dimensional (/, y)-set E. Forthe most part, it will be assumed that/is continuous. In this case, y ==• y(t)defined on a /-interval J containing t = t0 is called a solution of the initialvalue problem (1.1) if y(t 0) = y0, (t, y(t)) e £, y(i) is differentiable, andy'(i) — /(/, y(/)) for t e /. It is clear that y(t) then has a continuousderivative. These requirements on y are equivalent to the following:y('o) = 2/o. (t, 3/(0) 6 E> J/(0 continuous and

for /eJ .An initial value problem involving a system of equations of /wth order,

where zu) = d'z\dt\ z and F are e-dimensional vectors, and F is definedon an (me + 1 )-dimensional set E, can be considered as a special case of(1.1), where y is a d = me-dimensional vector, symbolically, y = (z, z(1),, z(m~l}) (or more exactly, y = (z1,..., z% z1 ',..., z",. . . , z'*"*-1'));

correspondingly,/(f, y) = (z^,..., a*1"-", F(r, y)) and y0 = (z0, z,...,z*^1'). For example, if e = 1 so that z is a scalar, (1.3) becomes

1

2 Ordinary Differential Equations

The first set of questions to be considered will be (1) local existenc(does (1.1) have a solution y(t) defined for t near ?0?); (2) existence in thelarge (on what /-ranges does a solution of (1.1) exist ?); and (3) uniquenessof solutions.

The significance of question (2) is clear from the following situation:Let y,f be scalars, /(/, y) defined for 0 t 1, \y\ 1. A solutiony = y(t) of (1.1), with (/ 0, y0) = (0,.D), may exist for 0 < t \ andincrease from 0 to 1 as t goes from 0 to £, then one cannot expect to havean extension of y(t) for any / > J. Or consider the following scalar casewhere/(/, y) is defined for all (t, y):

It is easy to see that y = c/(l — ci) is a solution of (1.4), but this solutionexists only on the range — oo < / < 1/c, which depends on the initialcondition.

In order to illustrate the significance of the question of uniqueness, lety be a scalar and consider the intial value problem

This has more than one solution, in fact, it has, e.g., the solution y(i) s 0

Figure 1.

and the 1-parameter family of solutions defined by y(t) = 0 for t ^ c,y(t) = (t — c)2/4 for / ^ c, where c 0; see Figure 1. This situation istypical in that if (1.1) has more than one solution, then it has a "con-tinuum" of solutions; cf. Theorem II 4.1.

2. Basic Theorems

This section introduces some conventions, notions and theorems to beused later. The proofs of most of the theorems will be omitted.

Preliminaries 3

The symbols O, o will be used from time to time where, e.g., /(/) =O(g(tJ) as / -»• oo means that there exists a constant C such that |/(f)l =C |g(f)l for large /, while/(O = o(g(0) as / -> oo means that C> 0 can bechosen arbitrarily small (so that if g(t) 7* 0, f(t)lg(t) -»• 0 as t -* oo).

"Function" below generally means a map from some specified set ofa vector space R* into a space Rd, not always of the same dimension. Rd

denotes a normed, real ^-dimensional vector space of elements y =(y1,..., y*1) with norm |y|. Unless otherwise specified, \y\ will be thenorm

and ||y|| the Euclidean norm.If y0 is a point and £ a subset of Rd, then dist (yc, £), the distance from

y0 to £, is defined to be inf |y0 — y\ for y e E. If £lt £2 arc two subsetsof Rd, then dist (£1} £2) is defined to be inf \yl — y2| for yl E Elt y2 e £2,and is called the distance between El and £2. If £1 (or £2) is compact andElt £2 are closed and disjoint, then dist (£lf £2) > 0.

If £ is an open set or a closed parallelepiped in Rd,/e Cn(£), 0 n <oo, means that/(y) is continuous on £ and that the components of/havecontinuous partial derivatives of all orders k is n with respect toy1,..., y*.

A function /(y, 2) =/(y1,..., y4*, z1 , . . . , 2') defined on a (y, «)-set £,where y e Rd, is said to be uniformly Lipschitz continuous on E with respectto y if there exists a constant K satisfying

withy =1,2. Any constant K satisfying (2.1) is called a Lipschitz constant(for/on £). (The admissible values of K depend, of course, on the norm'sin the/- and y-spaces.)

A family F of functions/(y) defined on some y-set £ cr Rd is said to beequicontinuous if, for every e > 0, there exists a 6 = <5 > 0 such that\f(Vi) -/(y2)l ^ « whenever ylt y2 e £, ^ - y2| (5 and /e F. Thepoint of this definition is that (5 does not depend on / but is admissiblefor all/e F. The most frequently encountered equicontinuous families Fbelow will occur when all/e F are uniformly Lipschitz continuous on £and there exists a K> 0 which is a Lipschitz constant for all/6F; inwhich case, d can be chosen to be d — t\K.

Lemma 2.1. If a sequence of continuous functions on a compact set Eis uniformly convergent on E, then it is uniformly bounded and equicontinuous.

Cantor Selection Theorem 2.1. Let /i(y),/2(y),... be a uniformlybounded sequence of functions on a y-set E. Then for any countable setD c: £, there exists a subsequence fn(l)(y),fn(2)(y),. .. convergent on D.

In order to prove Cantor's theorem, let D consist of the points ylt yz, • • • •Also assume that/n(y) is real-valued; the proof for the case that/B(y) =(fn

l(y)> • • • >/«%)) is a ^-dimensional vector is similar. The sequence ofnumbers /i(y),/2(y),... is bounded, thus, by the theorem of Bolzano-Weierstrass, there is a sequence of integers /^(l) < «1(2) < ... such thatlim/^yj) exists as k -> oo, where n = n^(k). Similarly there is a sub-sequence n2(l) < «2(2) < ... of/ijO), «j(2),... such that \imfn(y^) exists

as k -*• oo for n = nz(k). Continuing in this fashion one obtains successivesubsequences of positive integers, such that if /*,(!) < nf(2) < ... is theyth one, then lim/^) exists on k -*• oo, where n — nf(k) and / = 1,... ,y.The desired subsequence is the "diagonal sequence" «j(l) < «2(2) <«3(3) < ... . Variants of this proof will be referred to as the "standarddiagonal process."

The next two assertions usually have the names Ascoli or Arzelaattached to them.

Propagation Theorem 2.2. On a compact y-set E, /ef/iG/),/2(y),.. . be asequence of functions which is equicontinuous and convergent on a densesubset of E. Thenfi(y),fz(y),... converges uniformly on E.

Selection Theorem 2.3. On a compact y-set E c: Rd, Ietfv(y\f4y\...be a sequence of functions which is uniformly bounded and equicontinuous.Then there exists a subsequence fconvergent on E.

This last theorem can be obtained as a consequence of the precedingtwo. By applying Theorem 2.3 to a suitable subsequence, we obtain thefollowing:

Remark 1. If, in the last theorem, y0eE and/0 is a cluster point of thesequence/^o),/2(y0),..., then the subsequence/n(1,(y),/n(2,(y),... in theassertion can be chosen so that the limit function/(y) satisfies/(y0) =/0.

Remark 2. If, in Theorem 2.3, it is known that all (uniformly) con-vergent subsequences of/i(y),/2(y),... have the same limit, say/(y), thena selection is unnecessary and/(y) is the uniform limit of/i(y),/2(y),....This follows from Remark 1.

Theorem 2.3 and the following consequences of it will be usedrepeatedly.

Theorem 2.4. Let y,fe Rd andf0(t, y^ftt, y),/a(f, y\...bea sequenceof continuous functions on the parallelepiped R : t0 / 5s /0 + a, \y —y0\ ^ b such that

Let yn(t) be a solution of

2

y)f n(1)(y), fn(2)(y), . . . which is uniformlyu

uniformly on R.

Preliminaries 5

on [t0, /o + a]> where n = 1 ,2, . . . , and

(2.5) *n-^><» yn^!/o as n-+co.Then there exists a subsequence yn(\)(f)t yn(u(t), • • • which is uniformly

convergent on [/0, tQ + a]. For any such subsequence, the limit

is a solution of (2.40) on [t0, /0 -f a]. In particular, if (2.40) possesses aunique solution y = y0(t) on [f0, f0 -|- a], //ze«

Proof. Since/i,/2,... are continuous and (2.3) holds uniformly on R,there is a constant K such that \fn(t, y)\ ^ K for n = 0, 1,... and(t, y) E R; Lemma 2.1. Since |yn'(OI = ^» it is clear that £ is a Lipschitzconstant for ylt y2,..., so that this sequence is equicontinuous. It is alsouniformly bounded since \yn(t) — y0\ ^ b. Thus the existence of uni-formly convergent subsequences follows from Theorem 2.3. By (2.3),Lemma 2.1, and the uniformity of (2.6), it is easy to see that

uniformly on [t0, t0 + a] as k ~+ oo. Thus term-by-term integration isapplicable to

where n = n(k) and k -> oo. It follows that the limit (2.6) is a solution of(2-40).

As to the last assertion, note that the assumed uniqueness of the solutiony0(0 of (2.40) shows that the limit of every (uniformly) convergent sub-sequence of yi(i), yz(t),... is the solution y0(f). Hence a selection is un-necessary and (2.7) holds by Remark 2 above.

Implicit Function Theorem 2.5. Let x, y, /, g be d-dimensional vectorsand z an e-dimensional vector. Let f(y, z) be continuous for (y, 2) near apoint (y0, z0) and have continuous partial derivatives with respect to thecomponents of y. Let the Jacobian det (df'ldy*) •£ 0 at (y, z) — (y0»

zo)-Let x0 =/(y0»

2o)- Then there exist positive numbers, e and o, such that ifx and z are fixed, \x — x0\ < d and \z — z0| < d, then the equation x =

f(y, z) has a unique solution y = g(x, z) satisfying \y — y0\ < €. Further-more, g(x, z) is continuous/or \x — x0\ < 6, \z — z0| < 6 and has continuouspartial derivatives with respect to the components ofx.

For a sharper form of this theorem, see Exercise II 2.3.

uniformly

3. Smooth Approximations

In some situations, it will be convenient to extend the definition of afunction/, say, given continuous on a closed parallelepiped, or to approxi-mate it uniformly by functions which are smooth (C1 or C°°) with respectto certain variables. The following devices can be used to obtain suchextensions or approximations (which have the same bounds as/).

Let/(/, y) be defined on R:t0 <; t g tlt \y\ b and let )/(/, y)| M.Let/*(f, y) be defined for t0 <51 /x and all y by placing /*(/, y) =f(ty y)

if |y| b and/*(f, y) = /(>, £y/|y|) if \y\ > b. It is clear that/*(f, y) iscontinuous for t0 / tlt y arbitrary, and that |/*(/, y)| M. In somecases, it is more convenient to replace/* by an extension of/which is 0 forlarge |y|. Such an extension is given by/°(f, y) =/*(/, y)9>°(|y|), where<p°(s) is a continuous function for / 0 satisfying 0 <f°(s) ^ 1 for5^0, <p%y) = 1 for 0 J < b, and ) = 0 for j > 6 + 1.

In order to approximate f(t, y) uniformly on R by functions fe(t, y)which are, say, smooth with respect to the components of y, let <p(s) be afunction of class C°° for s 0 satisfying y(s) > 0 for 0 j < 1 andq>(s) = Q for SiZ. 1. Then there is a constant c > 0 depending only onq>(s) and the dimension d, such that for every e > 0,

wher« ||y||' = (S |y*|2)!/* is the Euclidean length of y. Put

where »? = (^x , . . . , rf), so that

Since/ (/, y) is an "average" of the values of/0 in a sphere \\TJ — y|| ^ efor a fixed t, it is clear that/* ->/° as c -+ 0 uniformly on /0 £! / /lt yarbitrary. Note that |/«| A/ for all e > 0 and that /*(f, y) « 0 for|y| A + 1 + «. Furthermore,/*(?, y) has continuous partial derivativesof all orders with respect to y1,..., y*.

The last formula can be used to show that if f°(t, y) has continuouspartial derivatives of order k with respect to y1,..., y*1, then the corre-sponding partial derivatives of /*(f, y) tend uniformly to those off°(t, y)as € -> 0.

Preliminaries 7

4. Change of Integration Variables

In order to avoid an interruption to some arguments later, it is con-venient to mention the following:

Lemma 4.1. Let t, u, U be scalars; U(u) a continuous function onA u B; u = u(t) a continuous function of bounded variation ona<t^b such that A < u(t) < B. Then, for a t b,

where the integral on the left is a Riemann-Stieltjes integral and that on theright is a Riemann integral.

The point of the lemma is the fact that the change of variables t -> ugiven by u = u(t) is permitted even when u(t) is not monotone (and notabsolutely continuous).

Proof. It is clear that the relation (4.1) holds for a t b if u(t) has acontinuous derivative. For in this case, both integrals in (4.1) vanish att = a and have the same derivative U(u(t))u'(i). If u(i) does not have acontinuous derivative, let ii^r), uz(t),... be a sequence of continuouslydifferentiable functions on a t b satisfying A ±± un(i) ^ B, un(t) -*•w(/) as n -»• oo uniformly on [a, b], and such that the sequence of totalvariations of Mx(0, «z(0» • • • over ta> ^1 ^s bounded. (The existence, offunctions u^(i), uz(t),... follows from the last section.) Then (4.1) holdsif u(t) is replaced by un(i). Term-by-term integration theorems applied toboth sides of the resulting equation lead to (4.1).

Notes

SECTION 2. Theorems 2.2 and 2.3 go back to Ascoli [1] and Arzela [1], [2].

Chapter II

Existence

1. Hie Picard-Lindelof Theorem

Various types of existence proofs will be given. One of the most simpleand useful is the following.

Theorem 1.1. Let y,/eRd; f(t,y) continuous on a parallelepipedR:tQ / /„ + a, \y — y0\ ^ b and uniformly Lipschitz continuous withrespect to y. Let Mbea bound for \f(t, y}\ on R; a = min (a, b/Af). Then

has a unique solution y = y(t) on [t0) t0 + a].It is clear that there is a corresponding existence and uniqueness theorem

if R is replaced by t0 — a t f0, \y — y0\ ^ b. It is also clear fromthese "right" and "left" existence theorems that if R is replaced by\t — 'ol = a-> \y — y<>\ = b, then (1.1) has a unique solution on \t — /0| a,since the solutions on the right and left fit together.

The choice of a = min (a, b\M) in Theorem 1.1 is natural. On the onehand, the requirement a ^ a is necessary. On the other hand, the require-ment a bjM is dictated by the fact that if y = y(t) is a solution of (1.1)on [/0, t0 + a], then \y'(t)\ ^ M implies \y(t) - y0\ < M(t - tQ), whichdoes not exceed b if t — t0 51 bjM.

Remark 1. Note that in Theorem 1.1, \y\ can be any norm on Rd, notnecessarily the norm (2.1) or the Euclidean norm.

For another proof of "uniqueness," see Exercise III 1.1.Proof by Successive Approximations. Let y0(/) = y0. Suppose that

yk(t) has been defined on [t0, t0 + a], is continuous, and satisfies \yk(i) —y0| b for k - 0,. . . , n. Put

Then, since/(/, yn(t)) is defined and continuous on [t0, t0 + a], the sameholds for yn+1(0- It is also clear that

8

Existence 9

Hence y0(t), y^(t),... are defined and continuous on [r0, f0 -j- a], and\yn(t)-y»\^b.

It will now be verified by induction that

n = 0, 1, . . . , where £is a Lipschitz constant for/. Clearly, (1.30) holds.Assume (1.30),..., (l.S^). By (1.2),

for « 1. Thus, the definition of A" implies that

and so, by (U^),

This proves (1.3W).In view of (1.3J, it follows that

is uniformly convergent on [f0, /„ + a]; that is,

Since /(f, y) is uniformly continuous on R, f(tt yn(t)) -*•/(?, y(0) asn -> oo uniformly on [r0, /„ 4- a]. Thus term-by-term integration isapplicable to the integrals in (1.2) and gives

Hence (1.4) is a solution of (1.1).In order to prove uniqueness, let y = z(/) be any solution of (1.1) on

['o, >0 + «]. Then

An obvious induction using (1.2) gives

and n = 0, 1, If n -* oo in (1.7), it follows from (1.4) that \y(t) —z(/)| 0; i.e., y(t) = z(t). This proves the theorem.

Remark 2. Since z == y, (1.7) gives an estimate of the error of approxi-mation

Exercise 1.1. Show that if, in addition to the conditions of the theorem,/(/, y) is analytic on R (i.e., in a neighborhood of every point (t°, y°) E R,f(t, y) is representable as a convergent power series in t — t°, y1 — y01,...,y<* — yM\ then the solution y = y(t) of (1.1) is analytic on (/„, /„ + a].The analogous theorem in which t and the components of y,f are allowedto be complex-valued is also-valid.

Exercise 1.2. If, in Theorem 1.1, v is near to y0, then the initial valueproblem y' = f(t, y), y(t0) = v has a unique solution y = y(t, v) on someinterval [t0, t0 + /?] independent of v. Show that y(t, v) is uniformlyLipschitz continuous with respect to (t, v) for t0 t t0 + /5, v near y0.

2. Peano's Existence Theorem

The next theorem to be proved drops the assumption of Lipschitzcontinuity and the assertion of uniqueness.

Theorem 2.1. Let y,fe R"; /(/, y) continuous on R:t0 ^ / ^ /„ + a,\y - y<>\ ^ b; M a bound for \f(t, y)\ on R; a = min (a, tyM). Then (1.1)possesses at least one solution y = y(t) on [t0, t0 -f a].

In this theorem, \y\ can be any convenient norm on Rd.Proof. Let 6 > 0 and y0(t) a C1 ^/-dimensional vector-valued function

on [/0 - o, t0] satisfying y0(t0) = sU^oO) - 3^ol ^ b, and |y0'(OI ^ M.For 0 < e <5, define a function y€(t) on [/0 — 6, t0 + a]

by putting y€(t) = y&t) on [/0 - d, t0] and

Note first that this formula is meaningful and defines y^f) for /0 = ^ =/„ + aj, ax = min (a, e), and, on this interval,

It then follows that (2.1) can be used to extend yf(f) as a C° function over[/„ — <5, /„ + a2], a2 = min (a, 2e), satisfying (2.2). Continuing in thisfashion, (2.1) serves to define y€(t) over [/0, /„ + a] so that yf(t) is a C°function on [t0 — 6, /„ + a], satisfying (2.2).

Existence 11

It follows that the family of functions, yf(t), 0 < ^ 6, is equicontinuous. Thus, by Theorem 12.3, there is a sequencee(l) > (2 ) > . . . , such that e(/i) -* 0 as n -»• oo and

on [/0 — d, r0 + a]. The uniform continuity of/implies that/(/, y€(n)(t —f(n)) tends uniformly to/(/, y(t)) as n -*• oo; thus term-by-term integrationof (2.1) where e = e(ri) gives (1.5). Hence y(t) is a solution of (1.1). Thisproves the theorem.

An important consequence of Peano's existence theorem will often beused:

Corollary 2.1. Let f(t, y) be continuous on an open (t, y)-set E andsatisfy \f(t, y)\ ^ M. Let £0 be a compact subset of E. Then there existsan a > 0, depending on E, EQ and M, with the property that if(t0, y0) e £0,then (1.1) has a solution on \t — t0\ ^ a.

In fact, if a = dist (£0, d£) > 0, where dE is the boundary of E, thena = min (a, a/M). In applications, when/is not bounded on E, the set Ein this corollary is replaced by an open subset £° having compact closurein E and containing £0.

Exercise 2.1 (Polygonal Approximation). Under the conditions ofTheorem 2.1, define a set of functions y^(t) as follows: Let 2T:/0 <ti < • • • < tm = t0 + a be a mesh on [/„, t0 + a] with a degree of fineness«$(E) = max (tk+l - tk) for k = 0, . . . , m - 1. On [t0, fj, put ys(f) =

yo + (t ~ 'o)/(>o» y<»). If yM has been defined on [/0, /J, /: < m, andlyE(0 - yd *» put y£(0 yE(^ + (f - /»)/(/», yM) on [rfc, /^j.This serves to define yz(t) on [f0, /0 + a] as a continuous piecewise linearfunction. Prove Theorem 2.1 by obtaining a solution of (1.1) as a limit ofa suitable sequence y£{1)(0, yj:(2)(0» • • • , where £(£(«)) -*• 0 as n -* oo.[Note that if the solution of (1.1) is not unique, then not all solutions canbe obtained by this procedure; cf. the scalar problem y' = \y\lA, y(Q) = 0.]

Exercise 2.2 (Another Proof). There exists a sequence of continuousfunctions f^i, y),f2(t, y),... on R which tend uniformly to/(f, y) on R,\fn(t, y)l ^ M for (/, y)16 R and n = 1, 2 , . . . , and fn(t, y) is uniformlyLipschitz continuous with respect to y; cf. § 13. Consider y' =fn(t, y),y('o) — yo and apply Theorems 1.1 and 12.4. [In contrast to Exercise 2.1,all solutions of (1.1) can be obtained by the method of this exercise; cf.the proof of Theorem 4.1.]

Exercise 2.3. [This exercise gives a sharpened form of the ImplicitFunction Theorem 12.5 (with no parameters 2). In the statement of thistheorem, the norm \\A\\ of a d x d matrix occurs. Let Rd be the (real)</-dimensional vector space with any convenient norm \y\. Then ||y4|| isdefined to be \\A\\ = max \Ay\ for \y\ -— 1. This norm ||.«4|| depends on the

f

choice of the norm \y\ in Rd.] Let x = f(y) be a function of class C1 onD:\y\< b, and let/(0) = 0. Let the Jacobian matrix, fv(y) = (df'jdy")for j, k = I , . . . ,* / , be nonsingular on Z) and put M = max ||/,~~%)||,MI = max ||/w(y)|| for y e A where/"1 is the inverse of the matrix/,, and||/J, H/J"1!! denote the norms of the respective matrices. Let D^lyl <b/MM^ (Note that MMl ^ 1. Why?) Then there exists a domain DQsuch that D! <= D0 c £) and a; = /(?/) is a one-to-one map of [the closureof] D0 onto [the closure of] the ball /5°: |#| < £/M; see Figure 1. Assum-ing the Implicit Function Theorem 12.5, deduce the result just stated from

Figure 1.

Peano's Existence Theorem 2.1 by writing the equation x = f(y) for y inthe form £t = f(y), where | 7* 0 is a constant vector, differentiating withrespect to t to obtain the differential equation y' =/^1(y)l, and consider-ing the solution satisfying the initial condition y(Q) = 0. (It is possible toavoid the use of the Implicit Function Theorem by using the results of§V6.)

3. Extension Theorem

Let/(f, y) be continuous on a (/, y)-set E and let y — y(t) be a solution of

on an interval J. The interval / is called a right maximal interval ofexistence for y if there does not exist an extension of y(i) over an interval/! so that y = y(t) remains a solution of (3.1); / i,s a proper subset ofj^;J, /! have different right endpoints. A left maximal interval of existencefor y is defined similarly. A maximal interval of existence is an intervalwhich is both a left and right maximal interval.

Theorem 3.1. Letf(t, y) be continuous on an open (t, y)-set E and lety(t) be a solution of (3.1) on some interval. Then y(t) can be extended (as a

1

Existence 13

solution) over a maximal interval of existence (co_, <w+). Also, if(u>_, o>+) isa maximal interval of existence, then y(t) tends to the boundary dE of E as/->&>_ and t-> o>+.

The extension of y(t) need not be unique and, correspondingly, a>±depends on the extension. To say, e.g., that y(t) tends to dE as / -> a>+ isinterpreted to mean that if £° is any compact subset of £, then (t, y(t)) $£° when / is near a>+.

Proof. Let Elt £2,... be open subsets of £ such that £ = U£n; theclosures £lt £z,... are compact, and £„ <= En+l (e.g., let En = {(t, y):(t, y) e £, \t\ < H, \y\ < n and dist ((/, y), dE) > !/«}). Corollary 2.1implies that there exists an en > 0 such that if (?0, y0) is any point of En,then all solutions of (3.1) through (/0, j/0) exist on \t — r0| en.

Consider a given solution y = y(f) of (3.1) on an interval J. If J is not aright maximal interval of existence, then y(t) can be extended to an intervalcontaining the right endpoint of J. Thus, in proving the existence of aright maximal interval of existence, it can be supposed that y(t) is definedon a closed interval a t b0 and that y(t) does not have an extensionover a t < oo.

Let «(1) be so large that (60, y(60)) e nu)- Then y(t) can be extended

over an interval [b0, b0 + «n(i)]. If (b0 + en(1), y(b0 + n(1))) e EnW, theny(0 can be extended over another interval [bQ + n(1), bQ 4- 2en(1)] oflength n(1). Continuing this argument, it is seen that there is an integer/(I) 1 such that y(t) can be extended over a t bl} where b± —bo +y(l)^n(i) and (blt y(M ^ £B(1).

Let n(2) be so large that (blt yfa)) e ^n(2)- Then there exists an integerj(2) > 1 such that y(t) can be extended over a t bz, where bz =*i +./(2K(2> and (ba, y(b£) $ £n(2).

Repetitions of this argument lead to sequences of integers «(1) <«(2) < . . . and numbers b0 < b± < ..-. such that y(t) has an extensionover [a, to+), where <w+ = lim bk as /: -»• oo, and that (6fc, y(Afc)) ^ £„<*>.Thus (!>!, yC^)), (fc2, y(bz)),... is either unbounded or has a cluster pointon the boundary dE of £.

To see that y(t) tends to dE as / -> o>+ on a right maximal interval[a, a>+), it must be shown that no limit point of a sequence (tlt y(tj),(t2, y(t%)),..., where tn -*• co+, can be an interior point of £. This is aconsequence of the following:

Lemma 3.1. Letf(t, y) be continuous on a (t, y)-set £. Let y = y(t) bea solution o/(3.1) on an interval [a, 6), 6 < oo, for which there exists asequence tlt t2,.. . such that a tn-+8 as n-*-oo and y0 = lim y(tn)exists. If f(t, y) is bounded on the intersection of E and a vicinity of thepoint (d, y0), then

If, in addition, f (8, y0) is or can be defined so that f(t, y) is continuous at(<5, y0)> then y(t) £ Cl[a, 6] and is a solution of (3.1) on [a, d].

Proof. Let e >0 be so small and Af > 1 so large that \f(t, y)\ ^ M€for (/, y) on the intersection of E and the parallelepiped 0 d — t e,\y - yQ\ <€. If«is so large that 0 < d - tn e/2M and \y(Q - y0\ £/2 , then

Otherwise, there is smallest tl such that /„ < tl < <5, |y(/x) — y(tn)\ =M((5 - /„) K Hence |y(» - y0\ <: fr + |y(fn) - y.| < « for rw ^f < f1; thus |y'(OI ^ Me for tn^t£ t1. Consequently, \y(tl) - y(tn)\ <M€(tl - tn) < Mf(8 — tn). This proves (3.3), hence (3.2). The last partof the lemma follows from y'(t) = f(t, y(t)) —*/(<$, y0) as / -* <5.

Corollary 3.1. Let /(/, y) be continuous on a strip t0 :£ t r0 + a(<°°)> yeR d arbitrary. Let y = y(t) be a solution of (I.I) on a rightmaximal intervalJ. Then either J = [t0, t0 + a] or J = [/0, d), d t0 + a,and \y(t)\ -> oo as t ~> d.

More generally,Corollary 3.2. Let f(t, y) be continuous on the closure E of an open

(t, y)-set E and let (1.1) possess a solution y = y(t) on a maximal rightinterval J. Then either J = [t0, oo), or J = [t0, d] with d < oo and(6, y(d)) E BE, orJ= [f0, t5) with 8 < oo and \y(t)\ -+ oo as t -> «5.

A somewhat different, but very useful, result is given by the followingtheorem.

Theorem 3.2. Let /(/, y) andf^t, y), f2(t, y),... be a sequence of con-tinuous functions defined on an open (t, y)-set E such that

holds uniformly on every compact subset of E. Let yn(t) be a solution of

(tn, yn0) E E, and let (o>w_, oon+) be its maximal interval of existence. Let

Then there exist a solution y(i) of

having a maximal interval of existence (o>_, cu+) and a sequence of positiveintegers «(1) < »(2) < . . . with the property that if <o_ < t1 < t2 < eu+,then can_ < t1 < tz < a)n+for n — n(k) and k large, and

t

Existence IS

uniformly for tl ^ / ^ t*. In particular,

(3.9) lim sup wn_ ^ o>_ < w+ ^ lim inf o>n+ as H = «(&) -»> oo.

Proof. Let £1? £2,... be open subsets of E such that £ = U£M, theclosures Elt £2,... are compact and En c: En+1. Suppose that (/„, y0) e £t

and hence that (tn, yn0) e for large n.In tfce proof, y(t) will be constructed only on a right maximal interval of

existence [/0, o>+). The construction for a left maximal interval is similar.By Corollary 2.1, there exists an clt independent of n for large n, such

that any solution of (3.5) [or (3.7)] for any point (tn, yn0) e £x [or (?0, y0) e£J exists on an interval of length centered at t — tn [or t = f0]. ByArzela's theorem, it follows that if «(1) < n(2) < ... are suitably chosen,then the limit (3.8) exists uniformly for t0 t 5J t0 +et and is a solution of(3.7). If the point (/0 + el5 y(t0 + e^) e Et the sequence n(l) < n(2) < ...can be replaced by a subsequence, again called n(l) < n(2) < . . . , suchthat the limit (3.8) exists uniformly for /0 +

i = ' = 'o + 2«i and is asolution of (3.1). This process can be repeated/ times, where (f0 -f /wel5

y(t0 -f we!)) 6 Ev for m = 0, . . . ,y — 1 but not for w —j.

In this case, let tl = t0 + yei and choose the integer r > 1 so that(fi> y('i))e ^r- Repeat the procedure above using a suitable cr > 0(depending on r but independent of n for large «) to obtain y(t) on aninterval tl f fi + ;'i r»

where ('i + m r» y(/! + w r)) 6 £r for m =0, . . . _ / ! — 1 but not for m =/!. Put /2 = 'i + ji€

r-

Repetitions of these arguments lead to a sequence of f-valuesf0 < 'i < - • « and a sequence of successive subsequences of integers:

such that (3.8) holds uniformly for /„ < t ^ /m if n(k) = nm(A:). Puta>+ = lim /m (^ oo). Since (/m+1, y(/m+1)) ^ £TO, for m = 1,2,. .., [t0, co+)is the right maximal interval of existence for y(t). The usual diagonalprocess supplies the desired sequence «(1) < «(2) < . . . . This provesthe theorem.

4. H. Kneser's Theorem

The following theorem concerning the case of nonunique solutions ofinitial value problems will be proved in this section.

Theorem 4.1. Letf(t, y) be continuous on R:tQ / < /0 + a,\y — y0\ ^b. Let |/(/, y)| < M,'y. = min (a, b/M) and t0 < c tQ + a. F/«flr//7, /e/

Sc be the set of points ycfor which there is a solution y = y(t) of

on [f0, c] such that y(c) = yc; i.e., yc e Sc means that ye is a point reache

connected set.Exercise 4.1. If y is a scalar, Theorem 4.1 has a very simple proo

even without the assumption t0 c t0 + a. The conclusion is that S

is either empty, a point, or a closed y-interval. Prove Theorem 4.1 in thicase by first showing that if ylf y2 e Sc, so that (4.1) has solutions y/r) o[/„, c] such that y,(c) = yj for/ =1,2, and if yx < y° < y2, then y° e Se

Proof. Let 2 denote the set of solutions of (4.1). These exist o[t0, c] and Sc is the set of points y(c), where y(t) e £. To see that the setSe is closed, let yne -* yc, H -+ oo, and ync e Sc. Then ync = yn(r)'for someyn(i) e S. By Theorem 12.4, y^f), y2(0> • • - has a subsequence which isuniformly convergent to some y(t) E £ on [t0, c]. Clearly, yc = y(c) e E.

Suppose that the assertion is false, then Sc is not connected and istherefore the union of two nonempty, disjoint closed sets 5"°, S1. Since Sccis bounded, 6 = dist (5°, S1) > 0, where dist (S°, S1) = inf |y° - y1! fory° E 5°, y1 E S1. For any y, put e(y) = dist (y, 5°) - dist (y, S1), so thate(y) ^ <5 > 0 if y e S1 and e(y) ^ -^ < 0 if y 6 5°. The function <?(y) iscontinuous and e(y) 5^ 0 for y 6 5C.

Let e > 0 and y(f) e S. There exists a continuous function g(t, y)depending on e and (the fixed) y(/), defined for J0 f ^ c and all y suchthat (i) |g(f, y)| + e; that (ii)

that (iii) g(t, y) is uniformly Lipschitz continuous with respect to y; andthat (iv) y = y(t) is a solution of

In order to see this, let g*(t, y) be a function with the properties (i)-(iii),but with M + e replaced by M in (i) and by |e in (4.2); cf. § 13. Letg(t, y) = g*(t, y) +/(/, y(0) - g*«, l<0). Then \g(t, y) - £*(>, y)| !/('»2/(0) — *(^ (0)1 ie. so that conditions (i), (ii) follow. Condition(iii) is clear and (iv) follows from g(t, y(f)) =f(t, y(t)) = y'(t).Let y = y0(/), i(0 e S, and y0(c) e 5°, yx(c) 6 S1. For a given €>Q,

let £0(f, y), gi(f, y) be functions with the properties (i)-(iv), when y(/) =0(f), yj(/), respectively. Consider the 1-parameter family of initial valuproblems:

at t — c by some solution of (4.1). Then Se is a continuum, i.e., a close

Existence 17

where 0 6 1 an

Since ge(t, y) is uniformly Lipschitz continuous with respect to y, (4.4)has a unique solution y = y(t, 0); see Theorem 1.1. As a consequenceof Theorem 1.1 (where b > 0 is arbitrary), this solution exists on [f0, c]since gd is bounded, \ge(t, y)\ M + e, on the strip t0^ t c and all y.

Note that \ge(t, y)| M + e, c f0 + cc imply that \y(t, 0) - y0| (M + c)cc for f0 <S / < c. Theorem I 2.4 implies that y(t, 6) -» y('> #0),6 -*• 00, uniformly on [f0, c]. In particular, y(c, 0), hence e(y(c, 0)), is acontinuous function of 0. Since y(c, 0) = y0(c) e 5°, y(c, 1) = y^(c) G S1,so that e(y(c, 0)) -<5 < 0, e(y(c, 1)) > d > 0, there exists a 0-value,0 = ,7, 0 < r) < 1 such that e(y(c, ??)) = 0.

If 2/o(0> 2/i(0 are fixed, a choice of an r\ depends only on , say r\ = ??(e).Let e = !/«,«> 1, and let gn(t, y) = ge(t, y\ where 0 = »?(!/«). Thus(4.2) and (4.5) show that

and, by the choice of 0 = r),

has a (unique) solution y = y(n)(t) on [/0, c] such that e(y(n)(c)) = 0. Thesequence y(1)(r), y<2)(0, • • . has a subsequence which is uniformly con-vergent, say to y = y(t\ on [/„, c]. Since |y(n)(/) — y0\ ^ * for t0^t^min (c, f0 + 6/(M + !/«)) and min (c, t0 + bl(M + l//i)) -*• c as « -»• oo,Theorem 12.4 implies that y(t) is a solution of (4.1) on [t0, c]. Alsoe(y(c}) = lim e(y(n)(c)) = 0 as n -+ oo. But then y(c) e Sc and c(y(c)) = 0.This contradiction proves the theorem.

Exercise 4.2. Show by an example that Sc need not be a convex set if

boundary of a circle.Exercise 4.3. (a) Let/(f, y) be continuous for t0 t ^ f0 + a and all

y. Let /0 < c /0 + a and assume that all solutions y(i) of (4.1) exist ont0 ^ / ^ c. Then Sc is-a continuum. (6) Show, by an example, that Scneed not be connected if d— 2 and not all solutions of (4.1) exist fortQ^t<t0 + c.

Exercise 4.4. Let f(t, y), c be as in Theorem 4.1 or in part (a) ofExercise 4.3 and Tc = {(/, y): t0 / ^ f0 + c, y = y(f) for some solutionof (4.1)}. In particular, Sc = {y: (c, y) E Tc}. (d) Let y* be a boundarypoint of Sc. Show that (4.1) has a solution y = y*(t) such that (/, y*(tj) ison the boundary of Te for t0 f ^ ?0 + c and that y*(c) = y*. This is atheorem of Fukuhara; see Kamke [2]. (b) Let (tlt y^ be a point of the

d > 1, where y is a </-dimensional vector; e.g., if d =*= 2, Sc cam be the

boundary of rc, where /0 < 'i < c- Show that there need not exist asolution y(t) of (4.1) such that y(t^) = yl and (t, y(t)) is on the boundaryof Te on any interval /„ / ^ tv + c, e > 0. This is a result of Fukuharaand Nagumo; see Digel [1].

5. Example of Nonuniqueness

In order to illustrate how bad the situation as to uniqueness can become,it will be shown that there exists a (scalar) function U(t, u) continuous onthe (t, w)-plane such that for every choice of initial point (f0, w0), the initialvalue problem

has more than one solution on every interval [/„, t0 + e] and [/0 — , t0]for arbitrary e > 0.

Let S0 be the set of arcs,

* = 0, ±1,. . ., considered to be made up of subarcs defined on theintervals of length 1, k t k + 1, and k = 0, ±1,

For every n = 0, 1, 2 , . . . , there will be constructed a set Sn of twicecontinuously differentiate arcs

The symbol Sn will denote either the set of arcs (5.3) or the set of points onthese arcs. The set Sn of arcs (5.3) will have the properties that (i)

(ii) the arcs u = uik(t) and u = ui+ltk(t) have exactly one endpoint incommon; (iii) for any pair j, k, there is at least one index h such thatuh,k-i — WA+I,*-I = ujk at f — kft" and an index i such that uitk+l =tii+itk+i = uik at / as (k 4- l)/2n; (iv) any two arcs of Sn which have apoint in common have the. same tangent at that point; hence (v) anycontinuous arc u = u(t), say, on a / b, which is made up of arcs of Sncan be continued over — oo < f < oo, not uniquely, so as to have thesame property and any such continuation is of class C1 (and piecewise ofclass C2); also, (vi) if Un(t, u) is defined on the point set Sn to be theslope of the tangent at the point (r, w) e Sn, then Un(t, u) is uniformlycontinuous on Sn and arcs of (v) constitute the set of solutions of

Existence 19

Figure 2. The heavy curved lines represent arcs of S0. Heavy and light curved linesrepresent arcs of Si if m0 = 3. The construction of the arcs of Si, not in S0, is indicatedabove: the arcs u(t) = «0(f), «i(f), ii,(r), «8(0 = »(') are defined on [a, b] = [0,1], thearcs v0(t), Vi(t), vt(t) on [c, b] = [J, 1]. The sketch makes it clear how 50 or 5t divides

the plane into sets <7.

(vii) the sets S0, Slt... satisfy 5n c Sn+1, So that t/n+1(f, «) is an extensionof £/„(/, w); (viii) 5 = U-^n and, in fact, the set of end points (fc/2n,ujk(kl2

n)\ for j, k = 0, ±1 , . . . and « = 0, 1 , . . . , is dense in the plane;

finally, (ix)

which is defined on S = \JSn has a (unique) continuous extension overthe plane. Condition (ix) is the only nontrivial condition. (The con-struction of S is indicated in Figure 2.)

Let 7T2 = e0 > j > . . . ,

Suppose that Sn has already been constructed so that the functions (5.3)


satisfy

and that if

then

and, if n > 0 and no arc of Sn^ lies between u — uik(t), u = uhk(t\ then

The set of arcs Sn+l will be obtained from those of Sn by inserting oneach interval, [£/2B, (2k + l)/2n+1] and [(2k + l)/2n^, (k + l)/2n], afinite number of arcs between the arcs u = uik(i)t u = uj+lk(i) of Sn. Thearcs of Sn and these inserted arcs will constitute the set Sn+1.

For convenience, let u(t) — uik(t\ v(t) = ui+lk(t\ a = kj2n, b =(k + l)/2n, andc = \(a + b). Suppose that u(a) = v(d); the constructionin the case u(b) = v(b) is similar. Then u(t), v(t) are defined on [a, b]tb-a = 2~n;

Let m = mn > 0 be an integer to be specified below. For / = 0, 1, . . . ,m, put

on [a, 6]. Then i/0(0 = "(0, "TO(0 = y(0. and

It is clear from (5.14) that

Existence 21

For / = 0, 1,.. ., m — 1 and c = \(a + b), put

on [c, b], so that c — a = b — c — l/2n+1 implies that

The relations in (5.23) involving derivatives follow from

From (5.24) and (5.18)-(5.20),

Also, by (5.21),

so that (5.18H5-19) give

Finally, let m = mn be chosen so large that

In order to obtain Sn+1 from Sn, let the arcs u = u^t), i = 0,. .., m,on [a, c] and the arcs u = ut(t), i = 0 , . . . , m, and u = vh(t), h = 0 , . . . ,m — 1, on [c, b] be inserted between u — u(t), u = v(t). It is clear from(5.19)-(5.20), (5.25)-(5.26), and (5.27) that the analogues of (5.8) and(5.10) hold if n is replaced by n + 1. Also the analogue of (5.11) followsfrom (5.19), (5.26), (5.27).

This completes the construction of the sequence 50 c 5t c .... it isclear that S = \JSn is dense in the (f, tt)-plane.

The continuity of U(t, u), given by (5.6), will now be considered. Letp ^ n 0. The set of arcs Sn divide the plane into closed sets G of theform G = {(/, u):y ^ / <! 6, un(t) ^ u 51 vn(t}}, where no point of Sn isinterior to G; M = un(t) and M = un(f) on [y, i], 6 — y = 2/2", are arcseach made up of two arcs of Sn; un = vn at / = y, <5; and un < yn on

(y,«).

Let (/„, MP) e G n 5^ and let (f1, w1) be any point of the boundary of G.The difference

will be estimated. Consider first the case that p = n. Then (t0, up) is onthe boundary of G, say uv = wn('o)- Since Un(t, un(t}) = wn'(0. it is seenby (5.8) that

Thus, in the case that/? = w, An ^ 4M/2n.Let p> n. It can be supposed that (/„, wp) 6 5,, — S^. Let «n =

Mn(/0), and wn ^ wn+1 fs • • • up, where (f0, w^) is the highest point of thesegment t — f0, u^ ^ u <j uv which in 5",, j = n + 1,...,/;. Then, by(5.11),

Hence

If this is combined with AB < 4Af/2n, it follows that

Consider now two points (/t-, wt), / = 0, 1, in Sv,p «. Each of thesepoints (/,, M,) is contained in a region G = Gi of the type just considered.There exist points (/% «*) on the boundary of Gf such that

(where, e.g., (/°, «°) = (/», i/1) if G0 = Gj). Thus the above estimate forA, implies that

Since £/„(/, M) is uniformly continuous on 5n, it follows from the last threeformula lines that U(ty u) is uniformly continuous on S. Hence U(t, u) hasa continuous extension, denoted also by U(t, u), on the (/, w)-plane.

It will now be verified that (5.1) has the asserted property. It is clearthat any continuous arc u = u(t) on an interval [c, d] made up of subarcsof S is a solution of (5.1). The Extension Theorem 3.1 or the case d = 1 ofTheorem 4.1 shows that if (f0, w0) is any point of a set G of the type justconsidered, then (5.1) has a solution u — u(t) over [y, d] satisfyingu = un = vn at / as= y, 6. Such a solution can be continued to the left oft =* Y [right of t — d] in a nonunique manner by using arcs of Sn. If n is

Existence 23

sufficiently large, the interval [y, d] containing f0 can be made arbitrarilysmall. This completes the verification.

Notes

For references and discussion of the history of existence theorems, see Painleve [1],Vessiot [1], Muller [3], and Kamke [4,1, pp. 2, 33].

SECTION 1. Theorem 1.1 goes back to Cauchy (see Moigno [1]) and Lipschitz [1].Its proof by successive approximations is due to Picard [1] and Lindelof [1], althoughthis method had been used earlier in special cases by Liouville and by Cauchy. Theexistence theorem in Exercise 1.1 is often associated with the names of Cauchy andPoincare who used the method of majorants. A proof by successive approximationswas given by von Escherich [2].

SECTION 2. Theorem 2.1 is due to Peano [2]. Simplifications in the proof have beenmade by Mie, de la Vallee Poussin, Arzela, Montel, and Perron. The proof in the textutilizes a device of Tonelli [1]. The polygonal approximations in Exercise 2.1 go backto Cauchy (and the method of Cauchy-Lipsehitz).

The result of Exercise 2.3 is due to Wazewski [4] but the proof in the Hints may benew; cf. Nevanlinna [1] and Heinz [3]. Results of this type often involve a "degreeof continuity" for/vft/) at y — 0; cf. Yamabe [1], Bartle [1], and Sternberg andWintner [IJ.

SECTION 4. Theorem 4.1 is a theorem of H. Kneser [1 ]; the proof in the text is that ofMuller [2]; for related results, see Pugh [1]. The suggestion given in Hints for Exercise4.2 is due to C. C. Pugh (unpublished); cf. also Fukuhara and Nagumo [1]. Exercise4.3(<z) is a result of Kamke [2]. Exercise 4.4 is due to Fukuhara [1 ]; see also Kamke [2],Fukuhara [2], and Fukuhara and Nagumo [1]. A related example is given in Digel [1].

SECTION 5. The first example of this type was given by LavrentiefT [1]; the examplein the text is that of Hartman [27].

*For a generalization and another proof, see Stampacchia [SI ].

Chapter III

Differential Inequalities and Uniqueness

The most important techniques in the theory of differential equationsinvolve the "integration" of differential inequalities. The first part of thischapter deals with basic results of this type which will be used throughoutthe book. In the second part of this chapter immediate applications aregiven, including the derivation of some uniqueness theorems.

In this chapter u, v, U, V are scalars; y, z,/, g are /-dimensional vectors.

1. Gronwall's Inequality

One of the simplest and most useful results involving an integralinequality is the following.

Theorem 1.1. Let u(i), v(t) be non-negative, continuous functions on[a,b]; C =± 0 a constant; and

Then

in particular, if C = 0, then v(i) = 0.For a generalization, see Corollary 4.4.Proof. Case (i), C> 0. Let V(t) denote the right side of (1.1), so that

v(t) ^ V(t\ V(t] ^ C> 0 on [a, b}. Also, V'(t) = u(t)v(t] <| u(t)V(t).Since V > 0, V'\V ^ «, and V(a) = C, an integration over [a, t] gives

V(t) ^ C exp \u(s) ds. Thus (1.2) follows from v(t) V(t).Jo,Case (ii), C = 0. If (1.1) holds with C = 0, then Case (i) implies (1.2

for every C> 0. The desired result follows by letting C tend to Exercise 1.1. Show that Theorem 1.1 implies the uniqueness

of Theorem II 1.1.

24

Differential Inequalities and Uniqueness 25

2. Maximal and Minimal Solutions

Let U(t, u) be a continuous function on a plane (f, «)-set E. By amaximal solution u = «°(f) of

is meant a solution of (2.1) on a maximal interval of existence such thatif u(t) is any solution of (2.1), then

holds on the common interval of existence of w, w°. A minimal solutionis similarly defined.

Lemma 2.1. Let £/(/, u) be continuous on a rectangle R:tQ 1= t tQ + a,|w-i/0,| < b\ let \U(t, «)| < M and a = min (a, b\M\ Then (2.1) /uua solution u = u°(t) on [t0, tQ + a] vwY/i /Ae property that every solutionu = «(0 o/V = U(t, M), w(f0) ^ M0 satisfies (2.2) on [f0, *0 + «].

In view of the proof of the Extension Theorem II 3.1, this lemma impliesexistence theorems for maximal and minimal solutions (which will bestated only for an open set E):

Theorem 2.1. Let U(t, u) be continuous on an open set E and (t0, «0) 6 E.Then (2.1) has a maximal and a minimal solution.

Proof of Lemma 2.1. Let 0 < a' < a. Then, by Theorem II 2.1,

has a solution u — un(t) on an interval [/0, /0 + a'] if n is sufficiently large.By Theorem 12.4, there is a sequence n(l) < «(2) < • • • such that

exists uniformly on [/„, /0 + a'] and is a solution of (2.1).It will be verified that (2.2) holds on [f0, t0 + a']. To this end, it is

sufficient to verify

for all large fixed n. If (2.5) does not hold, there is a t = tlt t0 < ^ <f0 + a' such that wfo) > wn(/i). Hence there is a largest f2 °

n ['o> ?i)>where «(/2) = «n(/2)5 so that w(f) > «„(/) on (tz, /J. But (2.3) implies that"TI'(^) = "'('2) + I/", so that wn(0 > «(/) for /(> r^ near r2. This con-tradication proves (2.5). Since a' < a is arbitrary, the lemma follows.

Remark. The uniqueness of the solution u = u°(t) shows that un(i) -+u°(t) uniformly on [f0, t0 + a'] as n -> oo continuously.

s

3. Right Derivatives

The following simple lemmas will be needed subsequently.Lemma 3.1. Let u(t) e Cl[a, b]. Then KOI has a right derivative

DR KOI/of & ' < b, where

and DR KOI = "'(0 sgn w(0 #XO 5* 0 andDR |«(/)|. I«'(OI '/«(') = 0.In particular, \DR \u(t)\ \ = |«'(OI-

The assertion concerning DR \u(t)\ is clear if u(t) ^ 0. The case whenM(0 = 0 follows from u(t + h) = h(u'(t) + o(l)) as h -» 0, so thatK/ + A)| = /KI«'(OI + o(l)) as 0 < h -* 0.

Lemma 3.2. Let y = y(t) e CJ[a, 6], TAe/i |y(/)| AaJ a right derivativeDR ItfOI «^ | >« ly(OI | \y'(t)\for a^t<b.

Since |y(OI = max (\yl(t)\,..., |y*(OI)» there are indices k such thatI#*(OI = ly(OI- 1° tne following, k denotes any such index. By the lastlemma, |y*(OI has a right derivative, so that

For small h > 0, \y(t + h)\ = max* \yk(t + A)|, so that by taking themax,, in the last formula line,

Thus DR \y(t)\ exists and is maxfc DR |y*(OI- Since \DR \yk(t)\ \ = \yk'(t)\ <|y'(OI» Lemma 3.2 follows.

Exercise 3.1. Show that Lemma 3.2 is correct if |y| is replaced by theEuclidean length of y.

4. Differential Inequalities

The next theorem concerns the integration of a differential inequality.It is one of the results which is used most often in the theory of differentialequations.

Theorem 4.1. Let U(t, u) be continuous on an open (t, u)-set E andu — u\i) the maximal solution of (2.1). Let v(t) be a continuous function on[*o> 'o + °] satisfying the conditions v(t0) ^ MO, (/, 0(0) e E, and v(t) hasa right derivative DRv(t) on t0 t < /0 + a such that

Then, on a common interval of existence ofu°(t) and v(f),

Remark 1. If the inequality (4.1) is reversed and v(t9) ^ MO, then the

conclusion (4.2) must be replaced by v(t) ^ «„(/), where u = u0(t) is theminimal solution of (2.1). Correspondingly, if in Theorem 4.1 the functionr(0 is continuous on an interval f0 —

a = * = *o with a left derivativeDLv(t) on (/„ — a, t0] satisfying DLv(t) ^ U(t, v(t)) and v(t0) ^ w0, thenagain (4.2) must be replaced by y(f) ^ M0(0-

Remark 2. It will be clear from the proof that Theorem 4.1 holds ifthe "right derivative" DR is replaced by the "upper right derivative"where the latter is denned by replacing "lim" by "lim sup" in (3.1).

Proof of Theorem 4.1. It is sufficient to show that there exists a 6 > 0such that (4.2) holds for [/0, t0 + d]. For if this is the case and u°(t), v(t)are defined on [fc, /„ + /?], it follows that the set of /-values where (4.2)holds cannot have an upper bound different from /?.

Let n > 0 be large and let d > 0 be chosen independent of n such that(2.3) has a solution u = «„(/) on [/„, t0 + 6]. In view of the proof ofLemma 2.1, it is sufficient to verify that v(t) ^ «„(/) on [t0, t0 + <5], butthe proof of this is identical to the proof of (2.5) in § 2.

Corollary 4.1. Let v(t) be continuous on [a, b] and possess a rightderivative DRv(t) ^Qon [a, b]. Then v(t) ^ v(d).

Corollary 4.2. Let U(tt u), u°(t) be as in Theorem 4.1. Let V(t, u) becontinuous on E and satisfy

Let v — v(t) be a solution of

on an interval [t& t0 + a]. Then (4.2) holds on any common interval ofexistence ofv(t) and u°(t) to the right of t — t0.

It is clear from Remark 1 that if v(t) is extended to an interval to theleft of / = t0, then, on such an interval, (4.2) must be replaced by r(/) M0(f) where w0(f) is a minimal solution of (2.1) with MO v(t0).

Corollary 4.3. Let u°(t) be the maximal solution of u — U(t, u), w(/0)= wo; u = M0(0 the minimal solution of

Let y = y(i) be a C1 vector-valued function on [t0, t0 + a] such that u0 ^l9('a)|£ u°,(t,\y(t)\)eEand

on [^0,^0 + *]• Then, the first [second] of the two inequalities

holds on any common interval of existence ofu0(t) and y [u°(t) and y].

This is an immediate consequence of Theorem 4.1 and Remark 1following it, since |y(OI has a right derivative satisfying — |y'(OI DR I*/(OI ^ \y'(t)\ by Lemma 3.2. (In view of Exercise 3.1, this corollarremains valid if \y\ denotes the Euclidean norm.)

Exercise 4.1. (a) Let/(f, y) be continuous on the strip S:a ^ / ^ b,y arbitrary, and let/*(f, yl,..., y*) be nondecreasing with respect to eachof the components y', i k, of y. Assume that the solution of the initialvalue problem y' =f(t, y), y(d) — y0 is unique for a fixed y0, and that thissolution y = y(t) exists on [a, b]. Let z(t) = (z\t),..., zd(t)) be contin-uous on [a, b] such that zk(t) has a right derivative for k = 1, . . . , d,z\a) ^ y0* and Dtf?(t) <>fk(t, z(0) for a < t^ b [or z*(a) > y0* andZ)^fc(0 £/*(/,z(0) for a < / < ft]. Then z*(0 y*(0 [or z*(0 £ /(/)]for a t b. (This is applicable if g(t, y) is continuous on S, z(t) is asolution of z' = #(/, z) and z*(a) y0*, gk(t, y) ^f(t, y) on S [or z*(a) yf,g\t,y) ^/*(/, y) on S].) See Remark in Exercise 4.3.

(b) If, in part (a), all initial value problems associated with y' =f(t, y)have unique solutions, /*(/, y) is increasing with respect to y*t i 5* k and

3

then z*(0 < Vo*(0 [or z*(0 > y0*(01 for a < / 6, Jk = 1,..., rf.(c) If, in addition to the assumptions of (a), there is an index h such that

/*(/, y) is nondecreasing with respect to y*, then y0h(t) — zA(0 is non-

decreasing [or nonincreasing] on a t b.(d) If the assumptions of (b) and (c) hold, then y0

A(/) — z*(/) is increasing[or decreasing] on a t b.

(e) Let M, J7 denote real-valued scalars and y = (y1,.. ., y*) a real */-dimensional vector. Let U(t, y) be continuous for a / ^ ft and arbitraryy such that solutions of w(d) = U(t, u,u',..., M(d~l)) are uniquely deter-mined by initial conditions and that U(t, y1,.. ., y**) is nondecreasingwith respect to each of the first d — 1 components yj,j= 1, .. . ,d — 1,of y. Let «i(0> w2(0 be two solutions of u(d) = U on [a, ft] satisfying«V>(fl) 4%i) for j = 0, . . . , d - 1. Then Mw>(0 «J»(0' for j =0, . . . , < / — 1 and a^t^b; furthermore, u(

2i}(t) — u(S)(t) is non-

decreasing for/ = 0, . . . , d — 2 and a t ft.Exercise 4.2. Let /(f, y), #(/, y) be continuous on a strip, a t b

and y arbitrary, such that/*(f, y) < £*(f, y) for £ = 1, . . . , d and that,for each £ = ! , . . . , < / , either /*(f, y1,..., y") or g*(f, y1,..., yd) isnondecreasing with respect to y1, / 5^ A:. On a / ft, let y = y(0 be asolution of y' = /(/, y), y(a) = y0 and z = z(0 a solution of z' = g(t, z),z(a) = z0, where y<* z0

fc for k = 1, . . . , d. Then y*(0 zfc(0 for« t ^.b.

Exercise 4.3. Let/(/, y) be continuous for f0 ^ r ^ /0 + a, \y — yc| ft such that /*(f, y1,..., y*) is nondecreasing with respect to each y',

fc = !,...,•</, and z'(a) < y fc = !,...,•</, and z'(a) < y fc = !,...,•</, and z'(a) < y

/ 5* k. Show that y — /(/, y), y(/0) = y0 has a maximal [minimal] solutiony0(0 with the property that if y = y(t) is any other solution, then yk(t) ^yo*(0 [yfc(0 = Vok(t)] holds on the common interval* of existence. Remark:The assumption in Exercise 4.1(a) that the solution of the initial valueproblem y' =/(?, y), y(a) = y0» ls unique can be dropped if y(t) is replacedby the maximal solution [or minimal solution] y0(t).

Exercise 4.4. Let/(/, y), g(t, y) be linear in y, say/*(r, y) = ^a/*(f) and $*(/, y) = Sfefc,(rV + g*(0, where akj(t\ bkj(t\f\t\ g\t) arecontinuous for a t b. Let y(/), 2(0 be solutions of y' =f(t, y\y(a) = yQ and z' = g(t, z), z(a) = z0, respectively. (These solutions exist on[a,b]i cf. Corollary 5.1.) What conditions on ajk(t\ bik(t\f\t), gk(t\y0, z0 imply that |z*(OI ^ /(O on [a, 6] for A: = 1 , . . . , dl

Theorem 4.1 has an "integrated" analogue which, however, requiresthe monotony of U with respect to M. This theorem is a generalization ofTheorem 1.1:

Corollary 4.4. Let U(t, u) be continuous and nondecreasing with respectto u for /0 t tQ + a, u arbitrary. Let the maximal solution u = iP(t)of (2.1) exist on [t0, tQ + a]. On [?„, t0 + a], let v(t) be a continuous functionsatisfying

where v0 < MO. Then v(t) ^ u\t) holds on [/„, t0 + a].Proof. Let V(t) be the right side of (4.8), so that v(i) < V(t\ and

K'(,) j/^ !,(,)). By the monotony of 17, V\t) ^ f/(r, HO)- HenceTheorem 4.1 implies that V(f) ^ «°(/) on [/0, t9 + d\; thus v(t) < u°(t)holds.

Exercise 4.5. Corollary 4.4 is false if we omit: U(t,u) is non-decreasing in u.

Exercise 4.6. State the analogue of Corollary 4.4 for the case that theconstant v0 in (4.8) is replaced by a continuous function v0(t).

Exercise 4.7. Let'y,/, z be ^/-dimensional vectors; /(/, y) continuousfor f0 ^ / ^ t0 + a and y arbitrary such that /*(/, y1,..., t/*) is non-decreasing with respect to each y\j= 1 , . . . , d. Let the maximal solu-tion y0(t) of y' =/(r, y), y(t0) = y0 exist on [f0, /0 + a]; cf. Exercise 4.3.Let z(f) be a continuous vector-valued function such that zk(t) ^ y0

k +

/fc(5, 2(5)) ds for r0 < / ^ /0 + a. Then z*(0 < y/(0 on [r0, ?0 + a].<0

5. A Theorem of WinrnerTheorem 4.1 and its corollaries can be used to help find intervals of

existence of solutions of some differential equations.Theorem 5.1. Let U(t, u) be continuous for t0 t r0 + a, u 0, and

let the maximal solution of (2.1), where u0 si 0, exist on [tQ, t0 + a], e.g.,

JUS*

1

let U(t, M) = V(M), where y(u) is a positive, continuous function on u 0such that

Let /(/, y) be continuous on the strip t0 / ^ /„ + a, y arbitrary, andsatisfy

Then the maximal interval of existence of solutions of

where \y0\ ^ MO, is [/0, /„ + a].Remark 1. It is clear that (5.2) is only required for large \y\. Admis-

sible choices of y>(w) are, for example, y(w) = CM, CM log M, ... for largeu and a constant C.

Proof. (5.2) implies the inequality (4.6) on any interval on which y(t)exists. Hence, by Corollary 4.3, the second inequality in (4.7) holds onsuch an interval and so the main assertion follows from Corollary II 3.1.

In order to complete the proof, it has to be shown that the functionU(t, u) = y(w) satisfies the condition that the maximal solution of

exists on [/0, /0 + a] by virtue of (5.1). Since y > 0, (5.4) implies that for-any solution M = M(/),

Note that y> 0 implies that i/'(0 > 0 and u(t) > 0 for t > t0. ByCorollary II 3.1, the solution u(t) can fail to exist on [/0, /0 -f- a} only if itexists on some interval [t0, <5) and satisfies M(J) -»• oo as / —»• ^ ( a). Ifthis is the case, however, / ->• <5 in (5.5) gives a contradiction for the left sidetends to 6 — t0 and the right side to oo by (5.1). This completes the proof.

Remark 2. The type of argument in the proof of Theorem 5.1 suppliesa priori estimates for solutions y(t) of (5.3). For example, if V>(M) is the sameas in the last part of Theorem 5.1, let

y>(|y|) implies that a solution y(t) of (5.3) satisfies

rf. (5.5).

and let M = <f>(y) be the function inverse to v = *¥(u). Then \f(t, y)\ ^

Exercise 5.1. Let /(/, y) be continuous on the strip /„ = t ^ t0 + a,y arbitrary. Let |/(/, y)\ <p(t)v(\y\), where y(t) ^ 0 is integrable on[/0, t0 + a] and y(«) is a positive continuous function on u 0 satisfying(5.1). Show that the assertion of Theorem 5.1 and an analogue of Remark2 are valid.

Corollary 5.1. If A(t) is a continuous d x d matrix function and g(t) acontinuous vector function for /„ < t ^ t0 -f a, then the (linear) initial valueproblem

has a unique solution y = y(t), and y(t) exists on t9 f= t t0 + a.This is a consequence of Theorem II 1.1 and Theorem 5.1 with the

choice of y(w) = C(l + u) for some large C.In a scalar case, Theorem 5.1 can be "read backwards":Corollary 5.2. Let U(t, u), V(t, u) be continuous functions satisfying (4.3)

on tQ / ^ /0 + a, u arbitrary. Let some solution v = v(t) of (4.4) on[/0, d), 6 t0 + a, satisfy v(t) -*• oo as t -> d. Then the maximal solutionu — w°(0 of(2.1) has a maximal interval of existence [a, w+), where a>+ 6,and u°(t) -»• oo as t -> co+.

6. Uniqueness Theorems

One of the principal uses of Theorem 4.1 and its corollaries is to obtainuniqueness theorems. The following result is often called Kamke's generaluniqueness theorem.

Theorem 6.1. Letf(t, y) be continuous on the parallelepiped R: '0 = ' ='o + a» \y ~~ y*\ = °- Let w('» ") be a continuous (scalar) function on^o-'o < ' = 'o + «» 0 = u = 2^> w^ ' e properties that co(t, 0) = 0 andthat the only solution u = u(t) of the differential equation

on any interval (t0, t0 + e] satisfying

w w(/) = 0. For (t, t/j), (/, yz) e R with t > r0, let

Then the initial value problem

has at most one solution on any interval [t0, t0 + «].

In Theorem 6.1, we can also conclude uniqueness for initial valueproblems y =f(t, y), y(O = y^ for /t 5^ t0. Theorem 6.1 remains validif Euclidean norms are employed.

Exercise 6.1. Show that Theorem 6.1 is false if (6.2) is replaced byw(0, u'(t) -* 0 as t -> /0 + 0.

Proof. The fact that

implies of course that u(t) = 0 is a solution of (6.1).Suppose that, for some e > 0, (6.4) has two distinct solutions y — yr(t),

y2(t) on /0 t^ t0 + . Let y(t) = &(/) - y2(0- By decreasing «, ifnecessary, it can be supposed thaty(t9 + c) 0 and \y(t0 + e)| < 2b,Also t/(O = y'('o) = 0. By (6.3),\y\t)\ ^ a>(t, \y(t)\) on (/„, t0 + e]. Itfollows from Corollary 4.3 (and theRemark 1 following Theorem 4.1) thatif u — «„(/) is the minimal solution ofthe initial value problem u' = a>(t, u),u(t0 + «) = \y(t0 + e)|, where 0 <\y(tQ + e)l < 2b, then

(6.6) |j<OI «o(0on any subinterval of (t0, t0 + e] onwhich M0(r) exists; see Figure 1.

By the proofs of the Extension Theorem II 3.1 and Lemma 2.1, w0(f) canbe extended, as the minimal solution, to the left until (/, w0(0) approachesarbitrarily close to a point of dR0 for some /-values. During the extension(6.6) holds, so that (f,«o'(0) comes arbitrarily close to some point (d, 0) edR0 for certain f-values, where d 5: t0. If d > t0, then (6.5) Shows thatHO(/) has an extension over (/„, t0 + e] with u0(t) — 0 for (/0, <$]. Thus,in any case, the left maximum interval of existence of w0(f) is (/0, t0 + «].It follows from (6.6) that w0(/) -> 0 and w0(0/(' - ^o) -»• 0 as/-> /0 + 0- By the assumption concerning (6.1), MO(/) = 0. Since thiscontradicts u0(t0 + e) = \y(t0 + e)| 5^ 0, the theorem follows.

Corollary 6.1 (Nagumo's Criterion). If t0 = 0, //ie/i <y(r, w) = w// wadmissible in Theorem 6.1 (/.<?., r^e conclusion, of Theorem 6.1 /zo/ds if(6.3) w replaced by

/or (/, #1), (/, «/2) e /? with t > /„).Exercise 6.2. The function w(/, M) = u\t in Corollary 6.1 cannot be

replaced by o)(t, M) s= CM// for any constant C> 1. Show that if C> 1,

Figure1.

then there exist continuous real-valued functions f(t, y) on 0 t 1,|yl <* I with the properties that

but that y' — f(t, y), y(Q) = 0 has more than one solution.Corollary 6.2 (Osgood's Criterion). I f t 0 = Q, then a>(t, u) = <p(0v(

is admissible in Theorem 6.1 ify(t) ^ Q is continuous for 0 < t a; y(u)

is continuous for u^.Q and y(0) = 0, y(«) > 0 ifu > 0; and <p(t) dt <oo, dujy>(u) — co.J+o

Actually, the continuity condition on <p(/) in this corollary can beweakened. The analogous uniqueness theorem can be proved directly if<p(/) is only assumed to be integrable over 0 < / ^ a.

Exercise 6.3 [Generalization of Corollaries 6.1 and (6.2)]. Let f0 = 0.(a) If <p(t) ^ 0 is continuous for 0 < t a, show that a>(t, «) = y>(t)u

r f « "1is admissible in Theorem 6.1 if and only if lim inf <p(s) ds + log t\ < oo

LJt Jas t -> +0. (b) Let <p(t) ^ 0 be continuous for 0 < t a; v>(") continuous

for 0 M 2Z>, y<0) = 0, y(«) > 0 for 0 , and I £/M/V(M) = oo.J+o

Show that a)(t, u) == 9?(/)v(w) is admissible in Theorem 6.1 if, for every

C> 0, lim sup t-1®! C + q>(s) ds\ > 0 as / -» 0, where u = O(v) is thewfunction inverse to *V(u) = I dsjif)(s).Ju

Exercise 6.4. Let y(") be continuous for |w| ^ 1, y>(0) = 0. Show thatthe initial value problem u' = y>(«), w(0) = 0 has a unique solutionu(t) = 0 unless there exists an e, 0 < e 1, such that either y(") 0for 0 M e and l/y>(w) is (Lebesgue) integrable over [0, e] or vK") = 0for — e <j u 0 and l/v(«) is (Lebesgue) integrable over [—e, 0].

Exercise 6.5. Let/, o> be as in Theorem 6.1. Show that there existsa function a>0(t, u) which is continuous on the closure of /?0[ is nondecreas-ing with respect to u if (f,«);thus coQ(t, 0) = 0; the only solution of u' = o>0(f, u) and u(tQ) = 0 on anyinterval [t0, tQ + e] is u(t) = 0; and |/(f, yj -/(/, yj| ^ w0(f, | - yt|).(Note that, since a>0 is continuous on the closure of /?0, any solution ofu' — o>0(f, M) on (tQ, /0 + e] satisfying (6.2) is necessarily continuouslydifferentiable and is the usual type of solution on {r0, f0 + «].)

Exercise 6.6. (a) Let e0 , . . . , d_i be non-negative constants such thateo + -" + *d-i == I- Let U(t, y)= U(t, y\ ..., y*) be a real-valuedcontinuous function on R:Q £1 r ^ a and |t/*| 6 for k = 1,..., d

J+o

2

such that \U(t, y,) - U(t, y2)\ ^ J €J_1(d -k + l)!r<«-^" |yx* - yt*| if*=i

t > 0. Show that the dth order (scalar) equation w(d) = t/(/, M, «',...,M«*-D) nas at most one solution (on any interval 0 / e a) satisfyinggiven initial conditions w(0) = w0, w' = MO', . .., w^"1' ) = «{f ~1), wherew0, w 0 ' , . . . , u(d~1} are rf given numbers on the range |«| A. (A) Note thatpart (a) remains correct if the constants e0 , . . . , €A_^ are replaced bycontinuous non-negative functions «0(0, • • . , «d-i(0 such that «•„(/) +• • • + ««-i(0î-

Exercise 6.7. (a) Let/(f, y) be continuous for /?:0 / ^ a, |y| A.On R0:Q < f ^ a, |w| ^ 2/>, let co^r, w), <u2(/, M) be continuous non-nega-tive functions which are nondecreasing in u for fixed t, satisfy *>,(/ , 0) = 0,and

Let there exist continuous non-negative functions a(/), /?(/) for 0 r asatisfying a(0) = 0(0) = 0, 0(0 > 0 for 0 < / < a, and a(0/£(0 -> 0 as/ -*• 0. Suppose that each solution u(i) ofu' == w^t, u) for small / > 0 withthe property that u(t) -> 0 as / -*• 0 satisfies u(t) ^ a(0 on its interval ofexistence. Finally, suppose that the only solution of v — a)2(t, v) for small/ > 0 satisfying v(t)jft(t) -»• 0 as / -» 0 is v(t) =. 0. Then the initial valueproblem y' =f(t, y), y(0) == 0 has exactly one solution, (b) Prove that<*>i(t, u) = Cu\ a>z(t, u) = kujt are admissible if k > 0, 0 < A < 1,k(\ - A)< 1 with a(/) = C(l - A)/1"1-^, /5fr) = t*.

The following involves a "one-sided inequality" and gives "one-sideduniqueness."

Theorem 6.2 Letf(t, y) be continuous for t0 t ?0 + a, \y — y0| /?.Considering y, f to be Euclidean vectors, suppose that

for /„ / ^ t0 + a and \yt — y0\ ^ b, i = 1, 2, w/rere fte <fo/ denotesscalar multiplication. Then (6.4) /las af rno^/ one solution on any interval[to, t0 + « ] , > 0.

When it is desired to obtain uniqueness theorems for intervals [/0 — e, /0].it is necessary to assume the reverse inequality in (6.8).

Corollary 6.3. Let U(t, u) be a continuous real-valued function for/0 / t0 + a, \u — w0| b which is nonincreasing with respect to u(for fixed /). Then the initial value problem u' = U(t, u), u(t0) = «0 nas at

most one solution on any interval [/0, t0 + e], e > 0.Proof of Theorem 6.2. Let y = y^(t\ y2(0 be solutions of (6.4) on

[/0, /0 + e]. Let (5(r) = \\yz(t) - ^(Oll2 = &2 - î) - (y2 - î) be thesquare of the Euclidean length of yz(i) — y^(t\ so that (5(/0) = 0, d(t) ^ 0.

But (5'(0 = 2(y2' - ft') • (y2 - yj ^ 0 by (6.8). Hence 6(t) = 0 on[to, *o + «] as was to be proved.

Exercise 6.8 (One-sided Generalization of Nagumo's Criterion and ofTheorem 6.2). Theorem 6.2 remains valid if condition (6.8) is relaxed to

for f0 < r ^ r0 + a [or if the right side is replaced by ifc>(/,||y2 — yj*)or || yx - y21| eo(r, ||.ft - y21|) with o> as in Theorem 6.1 ].7. van Kampen's Uniqueness Theorem

In the following uniqueness theorem, conditions are imposed on afamily of solutions rather than on/(r, y) in

Theorem 7.1. Letf(t, y) be continuous on a parallelepiped R: tQ t *o + a, |y — y0| == *• £*' there exist a function rj(t, tlt ft) on t0 t,^^to + a, |ft — y0| < )5« 6) w/Y/i the properties (i) that, for a fixed (t^ ft),y = »?('> 'i» yO is a solution of

(ii) r/rar »y(f, fj, yx) is uniformly Lipschitz continuous with respect to yx;finally, (iii) f/wf no /vw> solution arcs y — r)(t, tlt y^, y = ?y(r, r2» yz) /"»*through the same point (t, y) M/i/ew »?(r, tlf yj & rj(t, /2, y2) for tQ^t^,t0 + a. Then y = r)(t, t0, y0) is the only solution of (7,1) for ?0 = ^i =>o + a, |ft - y.| /?.

Exercise 7.1. Show that the existence of a continuous »?(/,/i, yi)satisfying (i) and (iii) [but not (ii)] does not imply the uniqueness of thesolution of (7.1).

Exercise 7.2. When /(/, y) is uniformly Lipschitz continuous withrespect to y, it can be shown that a function y = rj(t, t^ y^ satisfying theconditions of the theorem exists (for small ft > 0); e.g., cf. Exercise II 1.2.Show that the converse is not correct, i.e., the existence of »y(f, /1} yOsatisfying (i)-(iii) does not imply that f(t, y) is uniformly Lipschitz con-tinuous with respect to y (for y near y0).

Proof. Let y(t) be any solution of (7.1). It will be shown that y(f) ="n(t> >o> y<>) for small t — t0 > 0.

Condition (ii) means that there exists a constant K such that

for /0 /, /! f0 + a and - y0| j3, |y2 - y0| ^.Let |/(r, y)| <ZM on R. Then any solution y = y(t) of (7.1) satisfies

|y(0 - yol ^ W - ^o) ^ ij8 if >o '^ 'o + 0/2M. Thus »y(r, 5, y(*)) is

denned and fo(f, s, y(s)) - y(s)\ £ M \t - s\ < tf if tQ /, s t0 +

/8/2M. Hence

where y = min (a, fij2M). Condition (iii) means that any point on anyof the arcs y = ^(t, tlt yj can be used to determine this arc. Thus (7.3),

Figure 2. The case d — dim y is 1.

with yv = y(t^ and y2 = r](tlt s, y(s)), implies that

if t0 t, tlts t0 + y; cf. Figure 2.Let / be fixed on t0 / /0 + y. It will be shown that

To this end, put

so that a(t0) = 0 and a(r) = r(i). Then (7.5) and (7.7) imply that

Since y = rj(t, s, y(sj) is a solution of y' =/ through the point (s, y(s)),

t

it is seen that »?ft, s, y(s)) = y(s) + ft — s) [f(s, y(sj) + 0(1)] as t± -+ s.Also, t/ft) = y(s) + ft - s) [f(s, y(sj) + 0(1)] as /! -* j. Hence (7.8)gives trft) — (7(5) = /fo(l) |r4 — s| as /! -* 5; i.e., </(;/<& exists and is 0.Thus a(s) is the constant a(?0) = 0 for t0 5 f. In particular, T(?) =a(t) satisfies (7.6), as was to be proved.

Exercise 7.3 (One-sided Analogue of Theorem 7.1). Let /ft y) becontinuous on R:t0^ t t0 + a,\y — y0\ ^ b. Let there exist a function»?(A tlt yj) on /„ < ^ < f ^ /0 -|- a, |yt — yQ\ ^ p« b) with the proper-ties (i) that, for fixed ft, yj, y = »/(/, /x, y^ is a solution of (7.2) and (ii)that there exists a constant K such that for max ft, /2) t* < t t0 + a.

Then i/ = t^(r, rl5 y^ is the only solution of (7.2) for sufficiently smallintervals ft, /x + e], e > 0, to the right of ^ (but not necessarily to theleft of /J.

8. Egress Points and Lyapunov Functions

Let/ft y) be continuous on an open (/, y)-set Q and let Q0 be an opensubset of Q. Let 3Q0 and H0 denote the boundary and closure of £10,respectively. A point ft, y0) £ 9Q0 n Q is called an egress point [or aningress point] of Q0 with respect to the system

if, for some solution y = y(t) of (8.1) satisfying y(t0) = y0, there exists an > 0 such that (t, y(tj) e D0 for t0 - e < t < /„ [or for t0 < t < /„ + «J.If, in addition, ft y(0) $ Q0 for t0 < t < t0 + e [or for J0 - e < / < /0]for a small e > 0 for eivry such y(t) , then (/0, y0) is called a ^rn'cregress point [or strict ingress point]. A point (/„, y0) £ 5Q0 n O will bereferred to as a nonegress point if it is not an egress point.

Lemma 8.1. Letf(t, y) be continuous on an open set O and O0 an opensubset of Q, such that dQ0 n ii is either empty or consists of nonegresspeints. Let y(t) be a solution of (S.I) satisfying (t°,y(t°y) GQ0/or somet°. Then (/, y(/)) £ Q0 on a right maximal interval of existence [t°, co+).

If the conclusion is false, there is a least value f0(> *°) °f t, where(t0, y(toy) £ 9O0 O H. But then ft, y(/0)) is an egress point, which contra-dicts the assumption and proves the lemma.

Let wft y) be a real-valued function defined in a vicinity of a pointft, yj) £ ft. Let y(t) be a solution of (8.1) satisfying yft) = y^. If «ft y(/))is differentiate at t = tlf this derivative is called the trajectory derivativeof u at ft, i^) along y = y(t) and is denoted by lift, yx). When wft y) hascontinuous partial derivatives, its trajectory derivative exists and can be

)

calculated without finding solutions of (8.1). In fact,

where the dot denotes scalar multiplication and grad u = (dufdy1,...,Bit/dy*) is the gradient of u with respect to y.

Let (t0, y0) e 9O0 n Q and let u(t, y) be a function of class C1 on aneighborhood AT of (t0, y0) in Q such that (t, y) E Q0 n N if and only ifw('» y) < 0. Then a necessary condition for (t0, y0) to be an egress pointis that u(t0, y0) ^ 0 and a sufficient condition for (J0, y0)

to °

e a strict

egress point is that u(t0, y0) > 0. Further, a sufficient condition for('o> ^o) t° be a nonegress point is that u(t, y). 0 for (/, y) e Q0.

When the system under consideration

is autonomous (i.e., when the right side does not depend on t), definitionsare similar. For example, let/(y) be continuous on an open y-set Q, Q0

an open subset of Q, and y9 e B£1Q n Q. The point y0 is called an egresspoint of Q0 with respect to (8.3) if, forsome solution y(t) of (8.3) satisfying#(0) = y<>> there exists an e > 0 such that y(t) e Q0 for — e < t < 0. If,in addition, y(t) $ &0 for 0 < / < c for some c > 0, then y0 is called astrict egress point. A lemma analogous to Lemma 8.1 is clearly valid here.

For an application of these notions, consider a function f(y) definedon an open set containing y = 0. A function V(y) defined on a neighbor-hood of y s= 0 is called a Lyapunov function if (i) it has continuous partialderivatives; (ii) V(y) ^ 0 according as \y\ =£: 0; and (iii) the trajectoryderivative of V satisfies V(y) < 0.

Theorem 8.1. Let f(y) be continuous on an open set containing y = 0,/(O) = 0, and let there exist a Lyapunov function V(y). Then the solutiony = 0 q/"(8.3) is stable (in the sense of Lyapunov).

Lyapunov stability of the solution y = 0 means that if e > 0 is arbitrary,then there exists a <5 > 0 such that if \y0\ < <5 , then a solution y(t) of(8.3) satisfying the initial condition y(0) = yQ exists and satisfies \y(t)\ < efor / ^ 0. If in addition, y(t) -> 0 as t -> oo, then the solution y = 0 of(8.3) is called asymptotically stable (in the sense of Lyapunov). Roughlyspeaking, Lyapunov stability of y s 0 means that if a solution y(t) startsnear y = 0 it remains near y = 0 in the future (t 0); and Lyapunovasymptotic stability of y = 0 means that, in addition, y(t) -*• 0 as t -> oo.

Proof. Let e > 0 be any number such that the set \y\ e is in the openset on which/and V are defined. For any r\ > 0, let d(rf) be chosen sothat 0 < d(rj) < € and V(y) < y if \y\ < flfo).

Reference to Figure 3 will clarify the following arguments. Since V(y)is continuous and positive on \y\ = e, there is an 17 = rje > 0 such that

V(y) > r\ for \y\ = *. Let Q0 be the open set {y: \y\ < e, V(y) < ??}. Theboundary 9Q0 is contained in the set {y: \y\ < «, V(y) = rj}. The functionu(y) = V(y) — 17 satisfies u(y) < 0 at a point y, \y\ < €, if and only ify e Q0. Clearly li = V 5j 0. Hence, no point of 3Q0 is an egress point.Consequently, by the analogue of Lemma 8.1, a solution y(t) of (8.3)satisfying y(0) e Q0 remains in Q0 on its right maximal interval of existence[0, o>+). Since Q0 is contained in the sphere \y\ in Q, it follows thato>+ = oo; Corollary II 3.2.

Figure 3.

Finally, put 6€ = d(r)€) > 0, so that K(y) < t\ if \y\ < d€ < e. Thus\y(G)\ < ^ implies that t/(0) 6 Q0, hence y(i) exists and y(r) e Q0 for ? 0.In particular, \y(t)\ < c for / 0. This proves the theorem.

Exercise 8.1. Let/(y) be continuous on an open set containing y = 0and let/(0) = 0. Let (8.3) possess a continuous first integral V(y) [i.e., afunction which is constant along solutions y = y(i) of (8.3)] such thatV(y) has a strict extremum (maximum or minimum) at y = 0. Then thesolution y = 0 of (8.3) is stable.

Theorem 8.2. If, in Theorem 8.1, V(y) ^ 0 according as \y\ 0, thenthe solution y = 0 o/(8.3) is asymptotically stable (in the sense ofLyapwov).

Proof. Use the notation of the last proof. Let y(t) be a solution of(8.3) with |y(0)| < <56. Since V 0, it follows that K(y(0) is nonincreasingand tends monotonically to a limit, say c j> 0, as f -*• oo.

Suppose first that c — 0. Then Xr)-*0 as t^- oo. For otherwise,there is an e0 > 0 such that 0 \y(t)\ ^ e for certain large f-values. Butthere exists a constant mQ > 0 such that V(y) > m0 for c0 = Ivl = e; thusF(y(/)) > m0 > 0 for certain large t-values. This is impossible; hence,y(t) -* 0 as t -> oo.

Suppose, if possible, that c > 0, so that 0 < c q and K(y) < ^c if\y\ < ^(zc) < c- Hence |y(OI ^ ^(ic) for large /. But the assumption on

P implies that there exists an m > 0 such that V(y) ^ — m < 0 if 6(%c) ^\y\ ^ e. In particular, V(y(t)) ^ — m < 0, for all large /. This is impos-sible. Hence c = 0 and y(t) -> 0 as t -*- ao. This proves the theorem.

A result analogous to Theorem 8.1 in which the conclusion is that thesolution y = 0 is not stable is given by the following:

Exercise 8.2. Let f(y) be continuous on an open set E containingy — 0 and let /(O) = 0. Let there exist a function V(y) on £ satisfyingV(Q) = 0, having continuous partial derivatives and a trajectory derivativesuch that P(y) ^ 0 according as \y\ 0 on £. Let V(y) assume negativevalues for some y arbitrarily near y = 0. Then the solution y = 0 is not(Lyapunov) stable.

Theorems 8.1 and 8.2 have analogues for nonautonomous systemswhich depend on a suitable modification of the definition of Lyapunovfunction: Let/(f, y) be continuous for t T, \y\ b and satisfy

A function K(/, y) defined for t T, \y\ b is called a Lyapunov functionif (i) V(t, y) has continuous partial derivatives; (ii) V(t, 0) = 0 for f 2: Tand there exists a continuous function W(y) on \y\ ^ b such that W(y) ^ 0according as \y\ 0, and V(t, y) W(y) for t ^. T; (iii) the trajectoryderivative of V satisfies l/(t, y) 0.

Theorem 8.3. Let f(t, y) be continuous for t T, \y\ b and satisfy(8.4). Let there exist a Lyapunov function V(t, y). Then the solutiony = 0 q/"(8.1) is uniformly stable (in the sense of Lyapunov).

Here, Lyapunov stability means that if e > 0 is arbitrary, then thereexists a de > 0 and a t€ ^ T such that if y(t) is a solution of (8.1) satisfying|2/('°)I < <5 for some f° fe, then «/(f) exists and \y(t)\ < € for all t ^ /°.If, in addition, y(t) ->• 0 as r —>• oo, then the solution y = 0 is calledLyapunov asymptotically stable. The modifier "uniform" for "stability"or "asymptotic stability" means that t€ can be chosen to be T for all£ > 0 .

Theorem 8.4. Let f(t, y), V(t, y) be as in Theorem 8.3. In addition,assume that there exists a continuous W£y)for \y\ b such that Wfy) ^ 0according as \y\ 5: 0 and that V(t, y) — W^for t T. Then the solu-tion y = 0 of (S.I) is uniformly asymptotic stable (in the sense of Lyapunov).

ExerciseA.l. (a) Prove Theorem 8.3. (b) Prove Theorem 8.4.

9. Successive Approximations

The proof of Theorem II 1.1 suggests the question as to whether or nota solution of

can always be obtained as the limit of the sequence (or a subsequence) ofthe successive approximations defined in § II 1. That the answer is in thenegative is shown by the following example for a scalar initial valueproblem

where U(t, u) will be defined for t 0 and all «.Consider the approximations w0(f) = 0 and

Let U(t, 0) =- 2f, hence ii^r) =-/*; put U(t,-t2) =0, hence w2(r) = 0.Then uZn(t)=Q and u2n+1(0=-'2 for « 0. It only remains tocomplete the definition of U(tt u) as a continuous function to obtain thedesired example.

One possible completion of this definition is to let U(ttu) =^2/if« = 0, U(t, u) = 0 if u ^ — tz, and to be a linear function of u when~* ta= u< 0, t > 0 fixed. In this way, we obtain an example in whichU(t, u) is nonincreasing with respect to u (for fixed t 0). In this case, thesolution of (9.2) is unique (Corollary 6.3) although no subsequence of thesuccessive approximations converge to a solution.

If the solutions of (9.1) are unique by virtue of Theorem 6.1 witha monotone o>, then successive approximations converge to a solution.

Theorem 9.1.* Let R, R0,f, o> be as in Theorem 6.1. Let \f(t, y)\ ^ Mon R and a = min (a, b\M). Let o>(/, u) be nondecreasing with respect to u.Then y0(0 = y0 and

are defined and converge uniformly on [f0, /0 + a] to the solution y = y(t)of (9.1).

Proof. By Exercise 6.5, it can be supposed that <o(t, u) is continuousoh the closure of R0 and is nondecreasing with respect to u for fixed t.

The sequence of approximations (9.3) are uniformly bounded and equi-continuous on [t0, t0 + a] and hence possesses uniformly convergentsubsequences. If it is known that yn(t) — yB_i(0 -> 0 as n -*• oo, then(9.3) implies that the limit of any such subsequence is the unique solutiony(t) of (9.1). It then follows that the full sequence y0, y l s . . . convergesuniformly to y(t)\ cf. Remark 2 following Theorem 12.3. Thus, in orderto prove Theorem 9.1, it suffices to verify that A(/) = 0, where

*See Notes, page 44.

Since |/| ^ M on R,

The right side is a.t most A(f2) + f + 2Af (^ — t2\ for large « if e > 0.Hence A(/x) ^>l(/2) + e + 2M | - /2|. Since e >0 is arbitrary and/!, /2 can be interchanged, |A(^) — A(/2)| 2M\t: — tz\. In particular,A(r) is continuous for /0 / t0 + a.

By the relation (9.3),

Hence, by (6.3),

For a fixed r on the range r0 < t /0 4- a, there is a sequence of integers«(1) < «(2)< ... such that \yn+l(t) - yn(t)\ -> A(r) as n = «(A:) -* oo andthat Ai(j) = lim \yn(s) — ywli(^)| exists uniformly on t0^s t0 + at. as« = n(k) -> oo. Thus,

Since ^(s) ^ lim sup |yn(j) — yn-i(s)\ = A(J) and co(t, u) is monotone in",

By Corollary 4.4, A(/) < «„(/), where MO(/) is the maximal solution of

Since this initial value problem has the unique solution u0(t) = 0, itfollows that A(f) = 0. This proves the theorem.

Exercise 9.1. Show that under the conditions of Exercise 6.7(a), thesuccessive approximations y0(t) = 0 and (9.3), where t9 — 0 and y0 = 0,converge uniformly on 0 < f min (a, bjAf) to the solution of y' =/(/,2/),y(0) = 0.

Exercise 9.2. For two vectors, y = (y1,..., y*) and 2 = (z1,..., 2*),use the notation y ^ 2 if y* ^ 2* for A: = 1,..., d. Let/= (f\ ... ,/*)and y = (y1,..., y1*). Assume that f(t;y) is continuous on R:Q <st^ a, |y| ... onO ? a =»min (a, £/M), where y0±(0 = ±M(l,..., \)t and

Show that y0+(0 > y1+(t) > ... and y^_(t) ^ y^f) <* ... and thatboth sequences converge uniformly to solutions of y' =f(t, y), y(0) = 0.(b) Show that y0±(0 can be replaced by continuous funct/0±(0 on 0 t <; a satisfying |y0±(OI = * and

(e.g., 2/o-(0 = y<> is admissible if /(/, y0) ^ 0).Exercise 9.3. (a) Using the notation y 5: z introduced in Exercise 9.2

let/(f, y) be continuous for / ^ 0 and all y and satisfy/(/, yx) 5=/(f, y^if yi ^ ya- Let y(f) be a solution of y' = —f(t, y) satisfying y(f) ^ y(0)for t 0; cf., e.g., §XIV2. Consider the successive approximations

y0(0, yi(0, • • • defined by y0(0 = y(0), yn(t) = y(0) - [*/(*, y^*)) AJofor « = 1, 2, Let 2n(0 denote the "error" zn(0 = yn(0 — y(/). Showthat (- l)"«n(0 0 for n = 0,1, . . . and (- l)nzn'(0 0 for « = 1, 2,...and f ^ 0. (Convergence of the successive approximations is not asserted.)

n(b) Let En(t) = 2 (—l)m/m/w! be the nth partial sum of the MacLaurin

OT-O

series for er*. Show that (-l)n(£n(r) - e~0 > 6 for n — 0, 1,... andr ^ O .

Exercise 9.4. Let £/(r, M) be real-valued and continuous for t 2: 0 andarbitrary M and (7(r, M) nondecreasing with respect to u for fixed /. Let«0, MO' be fixed numbers and u(t) a solution of u" — — U(t, M). Definesuccessive approximations for u(t) by putting w0(f) = u0 + MO'^ and

Then M0(0, «i(0, • • • are defined for / ^ 0. (a) Suppose that u(i) satisfiesu(t) ^ «o + Mo'? °n its right maximal interval of existence [0, <o+). Show thatco+ = oo and that the "error" vn(t) = un(t) — u(t) satisfies (— l)nt>n(0 0, (-l)wrn'(0 0 for n - 1, 2,. . . and t 0. (Convergence of the

nsuccessive approximations is not asserted.) (b) Let Cn(f) = 2 (~~ Owf2m/

(2w)! and Sn(t) = 2 (-l)*"^^1 /" + 1)! be the nth partial sums ofm=0

the Maclaurin series for cos / and sin t, respectively. Show that (~l)n

[Cn(0 - cos t] 0 and (-1)B [Sn(t) - sin t] 0 for n = 0,1,... and/ 0. (c) Let U(t, u) = q(i)u, where ^(/) 0 is continuous and non-decreasing for / ^ 0. Using Theorem XIV 3.1« and the remarks followingit, show that (a) is applicable if MO j> 0 and «0' > 0 [i.e., show that u(t) ^«0 + u0't for t 0].

Notes

SECTION 1. Theorem 1.1 goes back essentially to Peano [1]. A special case wasstated and proved by Gronwall [1]; a slightly more general form of the theorem (whichis contained in Corollary 4.4) is given by Reid [1, p. 290]. The proof in the text is thatof Titchmarsh [1, pp. 97-98].

SECTION 2. Maximal and minimal solutions were considered by Peano [1]; seePerron [4].

SECTION 4. Differential inequalities of the type (4.1) occur in the work of Peano [1]and of Perron [4]. Theorem 4.1 and its proof are taken from Kamke [1] and areessentially due to Peano. Exercises 4.2 and 4.3 are results of Kamke [2]; see Wazewski[7]. A special case of Corollary 4.4 is given by Bihari [1]. Exercise 4.6 is a result ofOpial [1].

SECTION 5. Results of the type in Theorem 5.1 and Exercise 5.1 were first given byWintner [1], [4].

SECTION 6. Theorem 6.1 is due to Kamke [1]. An earlier version, in which it isassumed that (o(t,«) is continuous also for / = 0, was given by Perron [6]. (Exercise6.5, due to Olech [2], shows that, in a certain sense, Perron's theorem is not less generalthan Kamke's.) For the case d = 1, earlier results of the type of Perron's were givenby Bompiani [1] and lyanaga [1]. For Exercise 6.1, see Szarski [1]. For Corollary 6.1,see Nagumo [1 ]; a less sharp form was first proved by Rosenblatt [1 ] with a>(t, u) = Cujtand 0 < C < 1. An example of the type required in Exercise 6.2 was given by Perron[8]. For Corollary 6.2, see Osgood [1]. For Exercise 6.3(a), see Levy [1, pp. 46-47].For Exercise 6.4, see Wallach [1]. For Exercise 6.5, see Olech [2]. For a particular caseof Exercise 6.6, see Wintner [22]. For Exercise 6.7, part (a), see F. Brauer [1], whogeneralized the result of part (b) due to Krasnosel'skil and S. G. Krein [1].

For other uniqueness theorems related to those of this section, see F. Brauer andS. Sternberg [1]. These involve estimates for a function V(t, \y2(t) — yi(t)\) instead of|j/i(0 — y»(0|- For earlier references on the subject of uniqueness theorems, see Mtiller[3] and Kamke [4, pp. 2 and 33].

SECTION 7. Theorem 7.1 is a result of van Kampen [2].SECTION 8. The terminology "egress point" arid "ingress point" is that of Wazewski

[5]. Exercise 8.1 is due to Dirichlet [1]; it was first given by Lagrange [1, pp. 36-44]under the assumption that V(y) is analytic and that the Hessian matrix (d^y/dy* By')of V at y — 0 is definite. This result is the forerunner of Lyapunov's Theorem 8.1.Theorems 8.1 and 8.2, Exercise 8.2, and Theorems 8.3 and 8.4 are due to Lyapunov [2](and constitute the basis for his "direct" or "second" method); cf. LaSalle andLefschetz [1]. For references and recent developments on this subject, see W. Hahn [1],Antosiewicz [1], Massefa [2], and Krasovskil [4].

SECTION 9. The example of nonconvergent successive approximations is due toMiiller [1]. Theorem 9.lf as stated, is due to Viswanatham [1]. Earlier versions andspecial cases are to be found in Rosenblatt [1], van Kampen [3] (cf. also Haviland [1]),Dieudonne [1], Wintner [2], LaSalle [1], Coddington and Levinson[1], and Wazewski [8]. The reduction of the proof of Theorem 9.1 to the verificationthat A(/) s 0 is due to Dieudonne (and independently to Wintner) and is used by theauthors following them. Exercise 9.1 is a result of F. Brauer [1] and generalizesLuxemburg [1]. Exercise 9.2(o) is due to Muller [1]; cf. LaSalle [1] for part (b). ForExercise 9.3(a), cf. Hartman and Wintner [16]. For Exercise 9.4, cf. Wintner [16].

*If we omit " w is nondecreasing in u", Theorem 9.1 remains valid whend = dim ^>l ,but not if d = 1; Evans and Feroe [SI].

Chapter IV

Linear Differential Equations

In this chapter, u, v,p are scalars; c, y, z,/, g are (column) ^-dimensionalvectors; and A, B, Y, Z are matrices. The scalars, components of thevectors, and elements of the matrices will be supposed to be complex-valued.

1. Linear Systems

This chapter will be concerned with some elementary facts about linearsystems of differential equations in the homogeneous case,

and in the inhomogeneous case,

Throughout this chapter, A(i) is a continuous d x d matrix and/(/) acontinuous vector on a /-interval [a, b]. Recall the following fundamentalfact stated as Corollary III 5.1.

Lemma 1.1. The initial value problem (1.2) and

a < /0 5: b, has a unique solution y = y(t) and y(t) exists on a t b.The fact that the elements of A(t) and components of y are complex-

valued does not affect the applicability of Corollary HI 5.1. For example,(1.2) is equivalent to a real linear system for a 2</-dimensional vector madeup of the real and imaginary parts of the components of y. Actually, thesimplest proof of Lemma 1.1 is a direct one employing the standardsuccessive approximations:

Exercise 1.1. Prove Lemma 1.1 by using successive approximations.(This proof also gives the majorization \y(t)\ ^ eK|<-'0' \y0\ if K denotesa constant such that \A[t)y\ ^ K\y\ for all vectors y and a t ^ b; cf.(4.2) below.)

45

Exercise 1.2. Let A(f) = (aik(t)) be (not necessarily continuous but)integrable over [a, b]; i.e., let the entries afk(t) be Lebesgue integrableover [a, b]. Show by successive approximations that Lemma 1.1 remainscorrect. Here a solution y(t) is interpreted as a continuous solution of theintegral equation

or, equivalently, y(t) satisfies (1.3) and is absolutely continuous on [a, b]with its derivative y\t) satisfying (1.1) except on a null set (i.e., a set ofLebesgue measure 0).

The uniqueness of solutions of (1.1), (1.3) implies thatCorollary 1.1. Ify = y(t) is a solution of (I.I) andy(t^ = Qfor some

/0, a<t0<b, then y(t) = 0.For the solutions of (1.1) and (1.2), there is the obvious theorem:Theorem 1.1 (Principles of Superposition) (i) Let y = yt(t), yz(i) be

solutions of (1.1), then any linear combination y = c^y^t) + c^(t) withconstant coefficients clt c2 is a solution of (1.1), (ii) Ify — y^t) and y =y0(t) are solutions of (1.1) and (1.2), respectively, then y = y0(t) + y:(t)is a solution of (1.2); conversely, ify = y0(0> 3 (0 ore solutions of (1.2), theny = £/o(0 — y°(0 w a solution of (1.1).

The vector equation (1.1) can be replaced by a matrix differentialequation,

where Y is matrix with d rows and k (arbitrary) columns. It is clear that amatrix Y = Y(i) is a solution of (1.4) if and only if each column of Y(t\when considered as a column vector, is a solution of (1.1).

Corollary 1.1 and the principle of superposition imply that if Y = Y(t)is & d x k matrix solution of (1.4), then rank Y(t) does not depend on t.That is, if yi(0» • • • » y r ( f ) are r solutions of (1.1), then the constant vectorsyi(t0),.. •, yr(to) are linearly independent for some f0 if and only if they arelinearly independent for every /„ on a /0 o.

Below, unless otherwise specified only d x d matrix solutions Y = Y(t)of (1.4) will be considered. In this case, either det Y(t) s 0 or det Y(t) ^ 0for all t. This fact can be strengthened as follows:

Theorem 1.2 (Liouville). Let Y = Y(t) be a d x d matrix solution of

For a square matrix A = (aik), the trace of A is defined to be the sum ofits diagonal elements, tr A =* S %.

Linear Differential Equations 47

Proof. Let A(t) = (ajlc(t)), j,k = 1 , . . . , d. The usual expansion forthe determinant A(f) = del 7(0, where Y(i) = (yk'(t)), and the rule fordifferentiating the product of d scalar functions show that

where r,-(0 is the matrix obtained by replacing the yth row (yi(t),...,y4'(0) of Y(t) by its derivative (y>'(/),..., y£(0). Since y>'(0 = S a^by (1.1), it is seen that the/th row of F,(0 is the sum of afi(t) times theythrow of 7(/) and a linear combination of the other rows of Yf(t). Hencedet F,.(0 = a,XO det F(f) and so, A'(/) = (tr v4(0) A(r). This gives (1.5).

By a. fundamental matrix Y(t) of (1.1) or (1.4) is meant a solution of (1.4)such that det Y(t) ^ 0. In order to obtain a fundamental matrix Y(t)t letY(t) be a matrix with columns y^f)»• • • , y<t(0» where y = y/f) is a solutionof (1.1) belonging to a given initial condition y/f0) — y*o> where y10,...,yd0 are (constant) linearly independent vectors. It is clear that all funda-mental matrices Y(t) can be obtained in this fashion. Let Y(t) = Y(t, t0)denote the particular fundamental matrix satisfying

Exercise 1.3. Let A(t) be a continuous d x d matrix for t 0 suchthat every solution y(f) of (1.1) is bounded for / ^ 0. Let Y(t) be afundamental matrix of (1.1). Show that r-1(0 is bounded if and only if

Re I I tr A(s) ds is bounded from below.

If y(/) is a solution of (1.4) and c is a constant vector, the principleof superposition states that

is a solution of (1.1). Furthermore, if Y(t) is a fundamental solution of(1.4), then every solution of (1.1) is of the form (1.7) with c = I^CoM'o);that is,

In particular, if Y(t) — Y(t, /0), then

More generally, if F = Y0(t) is a matrix solution of (1.4) and C is aconstant d x d matrix, then Y(t) — Y0(t)C is a solution of (1.4). WhenY0(t) is a fundamental solution of (1.4), all d x d matrix solutions of (1.4)are obtained in this fashion and all fundamental solutions are obtained inthis way with a choice of C, det C &• 0.


Lemma 1.2. Let Y(t) = Y(t,t0) be the fundamental solution of (I A)satisfying (1.6). Then, for t0, t e [a, b],

Proof. By the remarks just made, where C = Y(s, f0), the right side is afundamental matrix and reduces to Y(s, f0) at / — J. Since the left side of(1.10) is a matrix with the same properties, the relation (1.10) is clearfrom uniqueness (Lemma 1.1).

2. Variation of Constants

A linear change of dependent variables in (1.1) or (1.2) will often beused.

Theorem 2.1. Let Z(t) be a continuously dijferentiable, nonsingulard x d matrix for a / b. Under the linear change of variables y-+z,where

(1.2) is transformed into

In particular, ifZ(t) is a fundamental matrix for

where B(t) = Z'(t)Z~\t) is continuous for a t b, then (2.2) becomes

The equation (2.2) is clear, for (2.1) implies that y' = Z'z + Zz', sothat z' = Z~V - Z'z) and y1 = Ay +/from (1.2).

In the particular case where A(t) — B(t), Z(t) == Y(t), and the latter isa fundamental matrix for (1.1), the change of variables (2.1) is called a"variation of constants," i.e., the replacement of the constant vector c in(1.7) by a variable vector z. In this case, (2.4) reduces to z' = Y~l(i)f(t),so that its solutions are given by a quadrature

where c — z(/0) is a constant vector. In view of (2.1), this gives the firstpart of the following corollary. The last part follows from (1.10).

Corollary 2.1. Let Y(t) be a fundamental matrix of (\.\). Then thesolutions of '(1.2) are given by


In particular, if Y(t) — Y(t, /0)> (2.5) becomes

Formula (2.5) or (2.6) shows that the solutions of (1.2) are determinedby a quadrature if the solutions of (1.1) are known. For arbitrary c,Y(t)c in (2.5) is merely an arbitrary solution of (1.1).

Exercise 2.1. Let A(t) be continuous on some ^-interval (not necessarilyclosed or bounded), Y(t) a fundamental matrix for (1.1), and Z(t) acontinuously differentiable, nonsingular matrix. Under the change ofvariables, y = Z(i)z, let (1.1) become 2' = C(/)z; cf. (2.2) wheref(t) = 0.For any matrix A, let A* denote the complex conjugate transpose of Aand AH = \(A + A*), the Hermitian part of A. (a) Show that if Z(f) isunitary [i.e., Z*(0 = Z~\t)l then CH(t) = Z*(t)AE(t)Z(t) [since thederivative of Z*(t)Z(t) = I is 0]. (b) Let Z(t) = Y(t)Q(t\ so that Q =Y~1Z is continuously differentiable and nonsingular. Show that C(t) =— Qrl(t)Q'(t). In particular, C(t) is triangular [or diagonal] if Q(t) istriangular [or diagonal].

Exercise 2.2 (Continuation), (a) Show that there exists a unitary Z(i)such that C(t) is triangular; in this case, C(t) is bounded if A(t) is bounded.(b) Show that there exists a bounded Z(0 such that C(f) is diagonal. It isnot claimed that Z(t) can be chosen so that Z~\f) is bounded.

3. Reductions to Smaller Systems

If a set of r linearly independent solutions of (1.1) is known, the deter-mination of all solutions of (1.1) can be reduced essentially to the problemof determining the solutions of a linear homogeneous system of d-rdifferential equations. The simplest formulae giving this reduction are,however, "local," i.e., applicable only on subintervals of [a, b] and varyfrom subinterval to subinterval.

Let Y = Yr(t) be a d x r matrix solution of (1.4). Corresponding to agiven point t = tQ of [a, b], renumber the components of y so that ifn(>) = (!fc'(OW - 1, . . . , « / and * - 1,.. . . r, a

where Yrl(t) is an r x r matrix, then det Yrl(t0) j& 0. Let [y, d] be anysubinterval of [a, b] containing r0 on which det Yrl(t) j& 0. Define thed X d matrix

where Id_r is the unit (d — r) x (d — r) matrix. Then det Z(f) =det rrl(/) 9* 0 on [y, d]. A simple calculation shows that Z~\t) is amatrix of the form

where Z^(t) is the (d — r) x r matrix

Note that Z'(0 = (r/(r),0) = A(t)(Yr(t\ 0); i.e.,

Introduce the change of variables y = Z(t)z into (1.1), then the case/(/) = 0 of (2.2) gives the resulting differential equation for 2. Writingthe right side of (2.2), with/ = 0, as Z~\t)[A(t)Z(t) - Z'(t)]z gives, byvirtue of (3.2) and (3.5),

Let Au(t),^2z(0 be square r x r, (</ — r) x (d — r) matrices such that

and let

where zx is an r-dimensional vector, z2 a («/ — r)-dimensional vector. Then(3.3) and (3.8) show that (3.6) splits into

Note that (3.10) is a linear homogeneous system for the (d — -dimen-sional vector z2 and that zl is given by a quadrature when z2 is known. In(3.10), Za(i) is given by (3.4). The reduction of (1.1) to (3.10) is of courseonly valid on an interval [y, 6] where Yrl(t) is nonsingular. The result canbe summarized as follows:

Lemma 3.1. Let Y = Yr(t) be a d X r matrix solution of (I A) on [a, b]such that if Yr(t) is written as (3.1), then det Yrl(t) = det(yfc'(f)), wherek,j=* 1,..., r, does not vanish on [y, d] [a, b]. Then the change of

variables (2.1) in terms of (3.2) reduces (1.1) to (3.9)-(3.10) on [y, (5] ww/r/c/j -412, A2Z and Zr2 are defined by (3.7) a/ia* (3.4), respectively.

An application. Consider a system (1.1) in which A(t) — (ayfc(f)) satisfies

(3.11) «w+i(0 * 0 and (0 = 0 for k y + 2

on [a, b]. It is readily verified that a solution y = y(t) of (1.1) is known assoon as its first component u = y\t) is known.

Corollary 3.1. Let A(t) satisfy (3.11) on [a, b] and let (1.1) possess dsolutions yv(t\ . . . , yd(t) with the property that, for k — 1,. . . , d anda^t<b,

(3.12) Wk(t) = det (y/(0) * 0 for /,;=!,..., *.

Then (1.1) « equivalent to the single differential equation ofdth order foru = yl of the form

where

andW_^= W0= I,a0i= 1-Proof. This will be seen to follow by d — 1 successive applications of

the reduction process described in Lemma 3.1 with r = 1. Introduce thenotation

Then, by a standard formula for minors of the "adjoint determinant!'

In order to verify this, let a symbol (e.g., W^_^ or Wfy denote either amatrix or its determinant. Let yTOM denote the cofactor of the (n,m)thelement in the (a + 1) x (a 4- 1) determinant W*?1. In particular,y« - w*» y«,«+i = -W& /.+!.« = -WJUand y«+li.+i - » . Con-sider the product of the determinants Wf^lT, where

On the other hand, by matrix multiplication

The last two displays give (3.17).In order to systematize notation, write y(1), A1 = (ajt) fory, A, respec-

tively. Thus Wv = yfin(f) 7* 0. Introduce new variables y(z} by thevariation of constants associated with (3.2), r = 1; namely,

Consider y(2) to be a (</ — l)-dimensional vector y(2) = (yf 2>, . . . ,yf2>)»so that y^ is not considered a component of y{2). Then (1.1) reduces, by(3.9)-(3.10)and(3.4), to

so that solutions of (D2°) are determined by quadratures when solutionsof (Da) are known. Using (3.18) and the known solutions y(in,..., y(1)d

of (1.1), we obtain d—\ solutions y(2W = (yf2) j ( r ) , . . . ,y f2) j (0) of(Djj) fory = 2, . . . , </; namely,

In particular, y[f)2(i) = ^2/ i 5^ 0> and this procedure can be repeatedto reduce the order of (D2). Suppose that the successive changes ofvariables 7\,..., ra_! have been defined, each of the form

where y(a+1) = (y(aaVi), • • •, 3/?a+i)) is a (</ — a)-dimensional vector

and yfa+i) is not considered a component of t/(a+1). Suppose that there

Linear Differential Equations S3

results a system of differential equations for y(aa+1), y(a+i) of the form

where j, k = a + ! , . . . ,< / , and suppose that d — a + 1 solutions y(a)r =(3&)r(0,..., 3/?«)r(0) of (/)«) are given by

In particular, since W^ = W^

It is readily verified from ra and (3.17) that (Da+i) hasrf — a solutionsy(*+»t = W«¥i)j(0» • • • , <+i)j(0) for y = a + 1,.. . , d given by(3.21a+1). For, by (3.21a) and (3.22), the relations in (3.20) show thaty<a«+i) = tf«)^-i/^and

The replacement of yfa), yfa) by #fa)j, y*a)j given in (3.21 J and the use ofW. = Wi and (3.17) give (3.21a+1).

Note that yw is a 1-dimensional vector yw) = t/fd) and that (Dd) whichis a linear homogeneous equation has, by (3.22), the general solutionifid)(t) — cWdIWd_i, where c is an arbitrary constant. Thus (Dd) isequivalent to

The assumption (3.11), which has not been used so far, and an inductionon a show that a*k — ajk if d k > j = a , . . . , d — 1. Hence

so that, by (3.22), (D*+l) reduces to

Note that, by virtue of (3.22), the first equation for T, in (3.20) gives

y<a«


Hence, by (3.25),

or, by (3.14),

(3.27)

By (3.18), u — yl1} determines y[Z) — W^u\W^ — a0u which satisfies(a0uY = yfgj/fl! by (3.27). Thus yf3) = ajfafi)' satisfies fofa,")']' =yf4)/tf2 by (3.27). Repetitions of this argument give

Note that (3.25), with a = d — 1, shows that yff is

Hence the desired result (3.13) follows from (3.23).

4. Basic Inequalities

Let the norm \\A\\ of the matrix A be defined by

This norm of A depends on the norm \y\ of the vector y. For the choiceof either the norm \y\ = max (jy1!,..., |y|") or the Euclidean norm, thereis the following estimate for solutions of (1.2):

Lemma 4.1. Let y — y(t) be a solution of (1.2) and t, tQ E [a, b]. Then

Proof. By (1.2), the inequality \y'\ \\A(t)\\ • \y\ + f/(/)| holds and isan analogue of (III 4.6). Thus, if «°(f) is the (unique) solution of

satisfying w(f0) = w0 with UQ = |y(/0)l, i.e.,

Corollary III 4.3 and the remark following it imply that |y(f)| ^ «°(/) fort0 t b. This gives (4.2) for /0 t. Similarly, if u = «0(r) is thesolution of

satisfying w(fc) = w0(= |y('o)l)» i.e.,

then \y(t)\ ^ M0(r) for t0 t. Interchanging t and t0 in the inequality soobtained gives (4.2) for / ^ /„.

Corollary 4.1. Let A0(t); A^i), Az(t),,.. be a sequence of continuousd X d matrices andf0(t); fi(t),f&)t... a sequence of continuous vectorson [a, b] such that An(t) -»• A0(t) andfn(t) -+f0(t) as n -> oo uniformly on[a, b]. Let y — yn(t) be the solution of

where a^tn^b and (tn, yB) -> (t0, y0) as « -*• oo. Then yn(t) ->• y0(t)uniformly on [a, b] as n -*• oo.

Proof. It is clear that 11 (011, M8(OII, • • • and l/KOU/zCOl, • • . areuniformly bounded on [a, b]. Thus Lemma 4.1 implies that ly^OL\yt(t)\,... is uniformly bounded, say \yn(t)\ ^ c for n = 1,2,..., anda^t^b. The right side /„(/, y) - ^n(f)y +/n(/) of the differentialequation in (4.4n) tends uniformly to/(/, y) — A0(t)y +f0(t) as n -* oofor a t < A, |y| c. Thus Corollary 4.1 follows from Theorem 12.4.

For a matrix A, let 4* denote the complex conjugate transpose of Aand AH = \(A + A*\ the Hermitian part of A. Let

denote the scalar product of the pair of vectors y, z (so that in the case ofvectors with complex-valued components, y - z is the complex conjugateof 2 • y). In particular,

(4.5)

Finally, let /MO, /i° be the least-and* greatest eigenvalue of A , i.e., the leastand greatest zero of the polynomial det (AH — A/) in A. (The fact that AH

is Hermitian implies that its eigenvalues are real.) Equivalently, if ||y||denotes the Euclidean norm, then /*„, /*° are given by

or

where "Re a" denotes the real part of the complex number a.When y(0»2(0 are differeniiable vector-valued functions, then

(y(0-z(0)' = y'(0-*(0 + rtO-^W; in particular, (||y(OII2)' -/(O'riO +y(0' y\t) = 2 Re [y'(0 • KOI- This implies

Lemma 4.2. Let \\y\\ denote the Euclidean norm of y; ft0(t), ju°(t) theleast, greatest eigenvalue of the Hermitian part AH(t) of A(t); and y(t) asolution of (1.2). Then

(where \\y\\' denotes either a left or right derivate of ||y||). Hence, fora^t0^t^b,

Proof. The inequalities (4.8) follow from (1.2). For, on the one hand,!</) 0 implies that l|y(OII' = Klly(OII2)'«OII = Re 1 (0 • KOl/IIKOII;on the other hand, y(t) = 0 implies that \D \\y(t)\\ \ = ||/(r)|| for D = DRor D = DL. The continuity of/1(0 implies that of fi0(t) and /*°(/)» so that

(4.9) and (4.10) are consequences of (4.8) by Theorem III 4.1 and Remark 1following it.

Consider a function y(t) for t 0. A number r is called a (Lyapunov)order number for y(t) if, for every e > 0, there exist positive constantsC0(e), C(e) such that

for all large t.

for some arbitrarily large t.

When y(f) ^ 0 for large r, an equivalent formulation of (4.11) and (4.12)is

Lemmas 4.1 and 4.2 show that if A(t) is continuous for t 0, then asufficient condition for every solution y(f) ^ 0 of the homogeneoussystem (1.1) to possess an order number r is that

or, more generally, that

in which case


in which case

5. Constant Coefficients

Let £ be a constant d x d matrix and consider the system of differentialequations

Let «/! 7* 0 be a constant vector, A a complex number. By substituting

into (5.1), it is seen that a necessary and sufficient condition, in order that(5.2) be a solution of (5.1), is that

i.e., that A be an eigenvalue and that yt 5* 0 be a corresponding eigen-vector of /?. Thus to each eigenvalue A of R, there corresponds at least onesolution of (5.1) of the form (5.2). If R has simple elementary divisors(i.e., if R has linearly independent eigenvectors ylt..., yd belonging to therespective eigenvalues A l f . . . , Ad), then

is a fundamental matrix for (5.1).In the general case, a fundamental matrix can be found as follows:

Successive approximations for a solution of the initial value problem(5.1) and y(0) = c are

so that an induction shows that

This suggests the following definition: For any d X d matrix B, let eB

denote the matrix defined by

where the matrix series on the right can be considered as d2 (scalar) series,one for each of the elements of eB. If B = (£,fc), then (4.1) shows that


\bih\ \\B\\ fory, k = 1,. . . , d. Since (4.1) clearly implies \\Bn\\ ^ \\B\\n,it follows that if Bn = (£jj?>), then \b$\ ^ \\B\\n. Thus each of the d1

series for the elements of eB is convergent. The standard proof for thefunctional equation of the exponential function shows that

This makes it clear that (5.6) converges uniformly on any bounded/-interval to y = eRtc, which is then a solution of (5.1) by (5.5). In otherwords,

is a fundamental matrix for (5.1) and Y(Q) = /.Consider the inhomogeneous equation

corresponding to (5.1). In this case formula (2.6) for the general solutionof (1.2) reduces to

for the general solution of (5.19).Let Q be a constant, nonsingular matrix. The change of variables

transforms (5.1) into

This has the fundamental matrix Z = e*7*. It is readily verified fromR = QJQ~l and the definition (5.7) that

Since a fundamental matrix, say em, can be multiplied on the right by aconstant nonsingular matrix, say Q, to obtain another fundamentalmatrix, it follows that

is also a fundamental matrix for (5.1).Let Q be chosen so that / is in a Jordan normal form, i.e.,

where /(/) is a square h(j) x h(j) matrix with all of its diagonal elementsequal to a number X — A(j) and, if h(j) > 1, its subdiagonal elements

Linear Differential Equations

equal to 1 and its other elements equal to 0:

59

where A = A(y'), h — h(j), and Kn is the nilpotent square matrix

and h(\) + • • • + h(g) = d. J(j) is just the scalar A = A(y) and Kh = 0 if/»(;)=!.

From J = diag [/(I),..., /(£)] follows Jn = diag [J"(l),..., J"(g)];hence

In view of (5.8) and (5.15), exp J(j)t = eu exp Kht where A = A(/), A =A(y). Note that Kh

z is obtained from Kh by moving the 1's from the sub-diagonal to the diagonal below this; A^3 is obtained by moving the 1's tothe next lower diagonal; etc. In particular, Kh

h = 0. Hence

The formulae (5.12), (5.17) and (5.18) completely determine the funda-mental matrix (5.9) of (5.1).

It follows from these formulae that if y(t) is a solution of (5.1), then itscomponents are linear combinations of the exponentials e*(1)t,..., e*(a)t

with polynomials in / as coefficients. These polynomial coefficientscannot, of course, be chosen arbitrarily.

Thus the problem of determining the solutions of (5.1) is the algebraicone of determining the Jordan normal form J of R and a matrix Q suchthat J = Q~1RQ. The simple case, at the beginning of this section,leading to (5.4) corresponds to a situation where h(j) — 1 fory = !,...,#and g = d, so that exp J(j)t is the 1 x 1 matrix (scalar) eu, A = A(y).

Note that, in any case, if the eigenvalues of R are Ax, A 2 , . . . , Ad, then(5.1) has d linearly independent solutions yt(t),..., yd(t) such that theorder number of yt(t) is T = Re fory = 1,.. . , d\ cf. (4.13).

6. Floquet Theory

The case of variable, but periodic, coefficients can theoretically bereduced to the case of constant coefficients. This is the essence of thefollowing.

Theorem 6.1. In the system

let P(t) be a continuous d x d matrix for — oo < / < oo which is periodicof period p,

Then any fundamental matrix Y(t) of (6.1} has a representation of the form

and R is a constant matrix (and Z(t), R are d X d matrices).If y0 is an eigenvector of R belonging to an eigenvalue A, so that e

my0 =

y0e**, then the solution y = Y(t)y0 of (6.1) is of the form 2i(0^ where thevector zx(/) = Z(t)y0 has the period p. More generally, it follows, fromthe structure of em discussed in the last section, that if y(t) is any solutionof (6.1), then the components of y(t) are linear combinations of terms ofthe type a(/)/V, where a(/ + p) — a(/), k is an integer 0 k ^ d — 1,and A is an eigenvalue of R.

Neither the matrix R nor its eigenvalues are uniquely determined by thesystem (6.1). For example, the representation (6.3) can be replaced byY(t) = Z(Oe~27nV*+2wtJ)<, i.e., Z(t) by Z(t)e~2irU and R by R + 2niLOn the other hand, the eigenvalues of eR are uniquely determined by thesystem (6.1). This will be clear from the argument below which shows

that eR is determined by the fundamental matrix Y(t), while if 7(0 isreplaced by the arbitrary fundamental matrix 7(/)C0, where C0 is anonsingular constant matrix, then eR is replaced by C^~leRCQ. Theeigenvalues cr^ . . . , ad of C = eRp are called the characteristic roots ofthe system (6.1). If A 1 ? . . . , Ad are eigenvalues of R, then e*1,.. ., e** arethe eigenvalues of eR so that if the numeration is chosen correctly, al =e*lJ),..., ad = e***. The numbers A j , . . . , Ad which are determined by(6.1) only modulo 2-rri/p are the characteristic exponents of (6.1). It is seenfrom (6.3) that (6.1) has d linearly independent solutions y^{i),..., yA(t)such that the order number of yt(t) is Re A,. = p-1 Re log cr, for j =! , . . . ,< / .

Proof of Theorem 6.1. Since Y(t) is a fundamental matrix of (6.1), itfollows from (6.2) that Y(t + p) is also a fundamental matrix of (6.1).Hence, by the remark preceding Lemma 1.2, there exists a constantnonsingular matrix C such that

It will be shown that det C 5* 0 implies that there is a (nonunique)matrix R such that

i.e., C has a logarithm /?/?. If this is granted, (6.4) can be written as

Define Z(t) by (6.3), i.e., by Z(t) = Y(t)e~m\ cf. (5.8). Then Z(/ + p) =y(r + p)e-R(t+p) = [7(f + p}e-^\g-K = 7(0e-*f by (6.6). ThusZ(f + /?) = Z(r), as claimed.

In order to complete the proof, it is necessary to verify the existence ofan R satisfying (6.5). Since (6.5) is equivalent to QCQ~l = exppQRQ~l,it is sufficient to suppose that C is in a Jordan normal form. In fact, theconsiderations of the last section show that it is sufficient to consider thecase that C is a matrix of the form / = AT + K, where the elements of Kare 0 except for those on the subdiagonal which are 1; cf. (5.14) and(5.17). Also det .7 = A" 5* 0 implies that X 0. Writing / = A(7 + KjK)and noting that log (1 + / ) = * / — f2/2 + /3/3 , we are led toexpect that a logarithm of/ is given by

i.e., J = le8 or equivalently, / + K/A. = es. This can be readily verifiedas follows:

Note that the series in (6.7) is, in fact, a finite sum since Kd = 0. Theformal rearrangement of power series to obtain

i.e., exp [log (1 + /)] = 1 + /, is valid for |f | < 1. It is clear that the sameformal calculation gives 6s = 1 + KjL Furthermore, this formalcalculation is permissible, since the powers of K commute and there isobviously no question of convergence to be considered.

7. Adjoint Systems

Consider again the system (1.1). If A* is the complex conjugate trans-pose of A, the system

is called the system adjoint to (1.1). The corresponding inhomogeneoussystem is

There are several results relating (1.1) and (7.1). The first of these isLemma 7.1. A nonsingular, d x d matrix 7(0 is a fundamental matrix

for (1.1) if and only if (Y*(t))~l = (Y~l(t))* is a fundamental matrix for(7.1).

This follows from the fact that if 7(0 has a continuous derivative Y',then (r-»(/))' = -Y-\t)Y'(t)Y-\i) as can be seen by differentiatingY(i) y-J(0 = I. Thus, if 7(0 is a fundamental matrix for (1.1), so thatY' = A Y, then (Y'1)' = — Y~1A and taking the complex conjugatetranspose of this relation gives (7*"1)' = — A*(Y*~*). The converse isproved similarly.

Exercise 7.1. Show that (1.1) has a fundamental matrix 7(0 which isunitary, 7= Y*~\ if and only if (1.1) is self-adjoint; i.e., if and only ifA(t) is skew Hermitian, A = — A*. In this case, if y = y(0 is a solution of(1.1), then ||y(OII is a constant.

Lemma 7.2 (Green's Formula). Let A(t), f(t), g(t) be continuous fora ^ / < b; y(t) a solution of (1.2); z(t) a solution of (7.2). Then, fora^t<b,

where the dot denotes scalar multiplication.This relation is proved by showing that both sides have the same

derivative, since Ay - z = y -A *z.

8. Higher Order Linear Equations

In this section let />0(0>/>i(0»... ,pA-\(t\h(t) be continuous, real- orcomplex-valued functions for a t b. The linear homogeneousdifferential equation,

and the corresponding inhomogeneous equation,

will be considered. The treatment of these equations reduces to that of(1.1) and (1.2) by letting y = (i/(0), w ( 1 > , . . . , M<*-»), where u = «(0),

and/(f) = (0,..., 0, /*(/))• It seems worthwhile, however, to summarizethe essential facts for this important special case:

(i) The initial value problem «(r0) = «0, w'(f0) = w0', . . . , u(*~1}(t0) =w|)

d~1) belonging to (8.2), where MO, w0', . . . , w^~1} are darbitrary numbers,has a unique solution u = «(/) which exists on a t ft. In particular,if (8.2) is replaced by (8.1) and MO = u0' = - - - = M^-1) = 0, then w(r) =0; hence no solution u(t) ^ 0 of (8.1) has infinitely many zeros on a t^b.

(ii) (Principle of superposition) (a) Let u — Ui(t), »2(0 be two solutionsof (8.1), then any linear combination u = Ci"i(0 + czu(t) with constantcoefficients clf c2 is also a solution of (8.1); (b) if w = M(/) and M = «x(r)are solutions of (8.1) and (8.2), respectively, then u = u(f) + ut(t) is asolution of (8.2); conversely, if u — «o(/), M^/) are solutions of (8.2), thenu = «0(f) — Wj(r) is a solution of (8.1).

When the functions U j [ t ) , . . . , wfc(0 possess continuous derivatives oforder k — 1, their Wronskian or Wronskian determinant, W(t) —W(t, «!,..., «*), is denned to be det (i^~l\t)) for i,j = 1, . . . , A:,

A set of k continuous functions u^(t),..., uk(t) on a t b is said tobe linearly dependent if there exists constants clt..., ck, not all 0, suchthat CjWjXO + * * • + ckuk(t) = 0 for a / ^ 6. Otherwise, the functionsMi(0, • • • , «fc(0 are called linearly independent. It is clear that if ult..., ukhave continuous derivatives of order k — 1, then a necessary condition forult..., uk to be linearly dependent is that W(t\ ult..., wfc) = 0 fora ^ t b. It is also clear that the converse is false (e.g., u^(t)t w2(f) can belinearly independent on 0 t 1 with u^t) = 0 for 0 5! t H andi/2(0 = 0 for H t^ 1 so that W(/; ult u£ = 0 for 0 <j / £ 1). Fork = d solutions of (8.1), however, the following holds:

(iii) Let ut(t),..., ud(t) be solutions of (8.1) and W(t) = W(t;u^..,t«,j). Then

and MJ, . . . , ud are'linearly dependent if and only if W(t) vanishes at onepoint, in which case W(t) = 0 for a / ^ b.

Formula (8.4) is a particular case of (1.5) in view of (8.3). The last partof (iii) is a consequence of uniqueness (i) and the superposition principle00-

(iv) Let Mr(0,. • • , «<*-i(0 be d — 1 linearly independent solutions of(8.1). Then equation (8.1) is equivalent to the equation

for the unknown w. When d = 2, this equation is equivalent to

If d linearly independent solutions ult..., ud of (8.1) are known, we canparticularize (2.5) to find a formula giving solutions of (8.2) in terms of aquadrature. It is easier to verify the following directly:

(v) For a fixed s, a s f= b, let u = u(t; s) be the solution of (8.1)determined by the initial conditions

and let w0(0 oe an arbitrary solution of (8.1). Then

is the solution of (8.2) satisfying u(k)(a) = u(f\a) for k = 0, . . . , < / — 1.It is easy to deduce from (1.7) and Lemma 1.2 that u(t; s) and its d

derivatives «',..., u(d} with respect to / are continuous functions of (t, s)


for a /, s b. In particular, the integral in (8.7) exists and is a functionpossessing d continuous derivatives with respect to t which can be calcu-lated formally. A direct verification substituting (8.7) into (8.2), showsthat (8.7) is the solution of (8.2) satisfying the specified initial conditions,

(vi) Consider a differential equation,

where a0, alt..., ad^ are constants. The associated characteristicequation is defined to be

If y is the vector y = (u(d~1},..., u', u), then (8.8) is equivalent to thesystem (5.1) where R is the constant matrix

Note that the components of y are written in the order reverse to thatconsidered in connection with (8.1).

It is readily verified that u = eu is a solution of (8.8) if and only if (8.9)holds. Actually, (8.9) is identical with the characteristic equationdet (U — R) = 0 for R. In order to see this, consider the relation Ry = tywhere y ^ 0. It is seen that this relation holds if and only if A satisfies(8.9) and y = c(hd~l,..., A2, A, 1) for some constant c. Thus A is aneigenvalue of R if and only if A satisfies (8.9). It follows that (8.9) is thecharacteristic equation of R when the roots of (8.9) are distinct. If theroots are not distinct, the coefficients a0t.. ., ad_^ in (8.9) [and correspond-ingly in (8.10)] can be altered by arbitrarily small amounts so that theresulting polynomial has distinct roots and hence is the characteristicpolynomial of the altered R. The desired conclusion follows by lettingthe arbitrarily small alterations tend to 0.

Exercise 8.1. By induction on d, give another proof that (8.9) is thesame as the equation det (A/ — R) = 0. (Still another proof follows fromExercise 8.2.)

If A is a root of (8.9) of multiplicity m, 1~ fs m ^ d, then u = ***,te**,..., tm~le** are solutions of (8.8). In order to see this, let L[u]denote the expression on the left of (8.8) if w(f) is any function with dcontinuous derivatives. Thus (8.8) is equivalent to L[u] — 0. Let F(A) be


the polynomial on the left of (8.9), so that F = dFjdX = • • • = dm~lF/fit.™-* = 0 at the given value of L Note that L[e"] = F(X)e^ and, sincethe coefficients of (8.8) are constant, L[/V] = L[&e»ldtf} = a*L[ ]/dtf = &(F(X)e*)ld# = 0 for k = 0,.. . , m - \ at the given value of L

Thus if A( l ) , . . . , %) are the distinct roots of (8.9) and if /t(y) is themultiplicity of the root A(j), then d solutions of (8.8) are given by u —figKM for y = 1, . . . ,£, and A: = 0,.. . , h(j) - 1, where h(\) + • • • +%) - d.

Exercise 8.2. (a) Show that the functions u — tke^i)l, where/ = 1,. . . ,£, and k — 0 , . . . , h(j) — 1 are linearly independent, (b) Let Y(t) be thefundamental matrix for y' — Ry in which the successive columns aresolution vectors y = (u(d~1},..., H(O)) corresponding to u (or w(0)) =t*e***lk! in the following order: first, y = 1 and k = h(\) - 1, /»(!) -2 , . . . , 0; then y = 2 and k = /r(2) — 1,..., 0; etc. Let J =diag [J(\\ ...,%)] in the notation of § 5. Show that Y-*(t)RY(t) = J;i.e., Y(t)J=RY(t) or, equivalently, Y(t)J - Y'(t), In particular, / =Y~\Q)R Y(Q) is a Jordan normal form for R. (c) For another proof of (a)and for use in the proof of Theorem X 17.5, show that

(vii) When the coefficients p0(t),... ,pd-i(t) of (8.1) are periodic ofperiod p, the corresponding linear system (1.1) of first order has periodiccoefficients of period p, by (8.3), and § 6 is applicable.

(viii) (Adjoint equations) Consider a dih order differential equation

with complex-valued coefficients on an interval a t b, where pk(t)has k continuous derivatives for k = 0,1,... ,d. The adjoint equationof (8.12) is defined to be

Note that an integration by parts gives

and repetitions of this give

Thus if w(0, v(t) have continuous dth order derivatives and

then

This is called Green's formula. The differentiated form

is called the Lagrange identity.A corollary of (8.15) is the fact that if u, v are solutions of (8.12) and

(8.13), respectively, then

(ix) (Frobenius factorization) Suppose that (8.1) has d solutions« x ( / ) , . . . , ud(t) such that

Then (8.2) can be written as

where a, = WflW^W^ and WQ = W_l = Wd+1 = 1. This is a con-sequence of Corollary 3.1.

Exercise 8.3 (Polyd). Let p0(t),... ,/J<j_i(0 m (8-1) be continuous andreal-valued for a < t < b. Consider only real functions. Put

so that (8.2) takes the form L[u] — h(t). If a function v(t) has k — 1(2: 0) continuous derivatives, a point / = t0 will be called a zero of v(t) of amultiplicity at least k if i>(?0) = v'(tQ) = •••• = v(k-l}(to) = 0. Equation(8.1) will be said to have property (W} on (a, b) if (8.1) has d solutionsMi(/), .. . , ud(t) satisfying (8.18) for a < t <b; actually this condition fork = d is trivial by (iii). (d) Zeros and property (W). Show that if nosolution u(t) ?£ 0 of (8.1) has d zeros counting multiplicities on [a, b)then (8.1) has property (W) on (a, b). In the remainder of this exercise,assume that (8.1) has property (W) on (a, b). (b) Generalization of Rollerstheorem. Let v(t) have d continuous derivatives on (a, b) and at leastd + 1 zeros counting multiplicities on (a, b). Then there exists at leastone point / = 0 of (a, b) where L[v](6) = 0. (c) Partial converse for (d).Show that no solution u(t) ^ 0 of (8.1) has d zeros counting multiplicitieson (a, b). (d) Interpolation. Let k ^ 1; ml + ' • • + mk

= d, where mj(>. 1) is an integer; tl < • • • < tk points of (a, b) and uf\ i = 1 , . . . , w3

andy = 1, . . . , fc, arbitrary numbers. Then (8.1) has a unique solutionM = u(t) satisfying w(t)(/,) = «jf) for * =0,.., m,-1 and j = 1,..., k.(e\ Equation L[u] = 1. Same as (</) with (8.1) replaced by L[u] = 1.(f) Mean value theorem. Let k; w1} . . . , mk and f . . . , tk be as in (d).Let v(t) have J continuous derivatives on (a, b); u(t) the unique solution of(8.1) satisfying uw(t,) = v(i\t}) for i =0,.., w,-l and j = 1,... , k;u = w0(0 the unique solution of L[u] = 1 satisfying u^(t^ = 0 fori =0,.., mj -1 and j = 1, . . . , £; and a </0 < 6. If [y, d] c (a, &)contains f1} rfc, and f0, then there exists at least one point t = B of (y, d)such that v(t0) = «(/„) + «o(/0)£[*>](0)- (The assertions (b)— (/) reduceto standard theorems if (8.1) reduces to the trivial equation «(d) = 0 withthe solutions u— 1, ? , . . . , td~l.)

Exercise 8.4. Let/?0(r),... ,pd-i(t) in (8.1) be continuous for a < / <£. Show that no solution u(t) 0 0 of (8.1) has d zeros counting multi-plicities on {a, b) if and only if no solution u(t) =^ 0 has d distinct zeros on(a, b). See Hartman [15].

9. Remarks on Changes of Variables

This section contains remarks which will be referred to in later chapters.(i) If R is a constant d x d matrix, the usual Jordan normal form

/ = Q~1RQ for R under similarity transformations is described by(5.14)-(5.16). It will often be convenient to note that the 1's on thesubdiagonal of Kh in (5.16) can be replaced by an arbitrary e y^ 0. Thisis a consequence of the following formula in which/(y) is given in (5.15):

if fi = diajg(l,r1 , . . . , «*-*)•(ii) Consider a real nonlinear system of differential equations of the

form

where J? is a constant matrix. Under a linear change of variables withconstant coefficients,

(9.1) becomes

Although R is a matrix with real entries, its eigenvalues need not be real.Correspondingly, there need not exist a real Q such that / is in a Jordannormal form. For a Q with complex entries, f(t, Qz) may not be defined.

The point to be made, however, is that for many purposes the formalchange of variables (9.2) using a Q with complex entries is permissible if(9.2) and (9.3) are suitably interpreted. Formal operations with (9.3)are then legitimate.

If Q is chosen so that / = Q~1RQ is in a Jordan normal form, then thecolumns of Q are eigenvectors of R or a power of R. Thus, if R has a realeigenvalues (counting multiplicities), 0 rSs a ^ d, then the other eigenvaluesof R occur in pairs of complex conjugate numbers. Let d — a = 2/3.Correspondingly, it can be supposed that the first j3 columns of Q are,respectively, complex conjugates of the next /? columns and that the last acolumns are real. Thus, if Q0 is the matrix

where lh is the unit h x h matrix, then QQ0 is a matrix with real entries.The change of variables


which is equivalent to

The differential equations (9.5) are real equations; the differentialequations in (9.6) result by taking linear combinations of one or twoequations in (9.5) with constant coefficients 1 or ±/.

Below, equation (9.3) is to be interpreted as (9.6). This is equivalent tosaying that, in (9.3), z*+/> = zk for k = 1 , . . . , / ? and z*+2/} is real for k = 1,. . . , a. Thus we can consider the variables in (9.3) to be w = (w1 , . . . , w*),where w* = |(z* + z*) and H>*+^ = — \i(zk — z*) for k = 1, . . . , ft, andw* = z* for 2p<k^d.

Exercise 9.1. Let R be a constant d x d matrix with eigenvaluesAj , . . ., Xd such that A x , . . . , Xk are simple eigenvalues for some k,1 ^ k ^ d. Let G(t) be a continuously differentiable d X d matrix for

/•oo

t ^ 0 such that G(t) -> 0 as t -* oo and || (?'(/)II A < ». (a) For large /,

show that R + G(t) has fc simple eigenvalues Ay(f) such that A,(/) -*• A,- as

f°°f->oo; A/0 is continuously differentiable and \Xf'(i)\dt < oo for

j = 1 , . . . , & . (6) Let Q0 be a constant, nonsingular matrix such thatQo~lRQ0 = diag [*i, • • • , * » , £oL where E0 is a (d — k) x (d — k)matrix (e.g., suppose that QolRQQ is a Jordan normal form for R). Letef = (er

l,..., erd), where e/ is 1 or 0 according asj=r ory r. Show

that, for large /, R + G(t) has an eigenvector y-(t\ [R + G(0]yXO =A/Oy/0, such that y,(/) -* (?0e3 as / ->• oo, and yf(t) has a continuous deriv-

fooative satisfying ||y/(f)ll dt < co fory = 1,..., k. (c) Show that, for large

/, there exists a continuously differentiate, nonsingular matrix Q(t) such

/•oothat g(oo) = lim Q(t) as / -* oo exists and is nonsingular, || Q'(t)\\ dt <

oo, and Q~l(t)[R + G(t)]Q(t} has the form diag [^(f), .. - , A*(0, (01,where E(t) isa(d — k) x (f/ — A:) matrix.

APPENDIX: ANALYTIC LINEAR EQUATIONS

10. Fundamental Matrices

This appendix deals with a linear system of differential equations

in which / is a complex variable and A(t) is a matrix of single-valued,analytic functions on some open set E in the /-plane. "Analytic" is usedhere in the sense of "regular analytic." In a small neighborhood of apoint t0eE, (10.1) has a fundamental matrix Y(t) which is an analyticfunction of t (i.e., which has elements that are analytic functions of r).This follows from a modification of the proof by successive approximationsof the existence theorem, Lemma 1.1; cf. Exercise II 1.1. Hence, if E issimply connected, the monodromy theorem implies that Y(t) exists and isa single-valued, analytic function of t on E.

Most of this appendix deals with the case when E is not simply connectedbut is a punctured disc 0 < |/| < a.

Lemma 10.1. Let A(t) be a matrix of single-valued, analytic functionson the disc 0 < \t\ < a and suppose that A(t) [i.e., at least one of theelements ofA(t)] is not analytic at t = 0. Then (10.1) cannot have a funda-mental matrix Y(t) which is single-valued, analytic on 0 < \t\ < a andcontinuous at t = 0 with det 7(0) y* 0.

Proof. If this is false, then Y(i), Y~l(t) are analytic on \t\ < a and so isA(t) = Y\t) Y~l(t) by Riemann's theorem on removable singularities.

Theorem 10.1. Let A(t) be a matrix of single-valued, analytic functionson 0 < \t\ <a. Then any fundamental matrix Y(t) of (10.1) (which need

not be single-valued) has a representation of the form

where Z(t) is a matrix of single-valued, analytic functions on 0 < \t] < a,R is a constant matrix, and

If T is a constant nonsingular matrix, then Y(t)T is a fundamentalmatrix and

T can be chosen so that T~1RT is in a Jordan normal form. The form ofthe matrix tT~lRT = exp (r-^KT'log t) can be seen by replacing t by log tin (5.18).

Proof. Let Y(t) be a fundamental solution of (10.1) determined locallynear a point t = /0, 0 < |/0| < a, and continued analytically, possiblymultiply valued. If the point /makes a circuit around t — O inO < |/| < a,then the matrix returns with values, say Y0(t), for / near /0. Since A(t) issingle-valued, r0(r) is also a fundamental matrix for (1.1), thus, thereexists a nonsingular constant matrix C such that

By analyticity, (10.4) holds for analytic continuations of 7(/), 70(/)-For a fixed r, 0 < r < a, consider the matrix function Y(0, r) — Y(reid)

of 6 for - oo < B < oo. Then (10.4) means that Y(d + 2ir, r) - Y(6, r)Cor that 7(0 + 2-rr, r) = 7(0, r)e**R if 27T/7? is a logarithm of C; cf. § 6.It is readily seen that Z0(0, r) = 7(0, r)e~im is of period 2ir in 0; see theargument following (6.6).

Note that Z0(0, r)r~R = 7(0, r)e~imr-R = r(r)r* is an analytic func-tion of t, say Z(t), and is single-valued since Z0(0, r) is of period 2-n in 0.This proves Theorem 10.1.

If (10.2) is a fundamental matrix for (10.1), properties of Z(f) and Rwill be investigated. To this end, a differential equation satisfied by Z willbe calculated. Since R commutes with tR and (tR)' = RtRlt, it followsfrom (10.1) and (10.2) that

Hence

The equation (10.5) is not the type of matrix differential equation con-sidered above since Z occurs as a factor on the left of ZR and on the rightof A(i)Z. In order to deal with (10.5), it is convenient to arrange the d2

elements zjk of the matrix Z = (zjk) in some arbitrarily fixed order andto consider (10.5) as a linear, homogeneous system for a </*-dimensionalvector Z; say

where A (t) is a d2 x d2 matrix. An element of A(t) is a linear combinationof elements of A(t) and R/t with constant coefficients. Hence A (/) issingle-valued and analytic for 0 < |/| < a. In particular, if the elements ofA(t) are analytic or have a simple pole at / = 0, then the same is true ofA(t).

In order to avoid an interruption to the arguments below, a simplealgebraic lemma will be stated and proved here. Let B be a constantd x d matrix, X a variable d x d matrix and Y the commutator

Consider the d2 elements of X and Y arranged in a fixed (arbitrary)manner and (10.7) as a linear transformation from the </2-dimensionalJf-space into itself. Thus (10.7) can be written as

where £ is a. d2 x d2 matrix and X, Y are ^-dimensional vectors.Lemma 10.2. Let the d eigenvalues of B be A 1 ? . . . , Xd counting multi-

plicities. Then the dz eigenvalues of S are A, — Xk for j, k = !,...,</.Proof. Let T be a nonsingular d x d matrix and let C = T~*BT, so

that B, C have the same eigenvalues. Let C be related to C as S is to B.The matrices C and S have the same eigenvalues. In order to see this,note that (10.8) is equivalent to T~1YT= C(T~1XT), for (10.7) can bewritten as

Thus, e.g., BX = XX implies that (CT-*XT) = MT-^XT).Suppose first that the eigenvalues of B are distinct and choose Tso that

C = diag [Aj , . . . , AJ. Then if X = (xik) and Y = (yik), it is seen thatY = CX'— XC is equivalent to yik = (A7 — Ak)xik for j\ k — I,... ,d.Thus C is a diagonal matrix with the d2 diagonal elements A, — At forj, k = ! , . . . ,» and the lemma is proved in this case.

If the eigenvalues of B are not distinct, let Blt Bz,... be a sequence ofmatrices, each having distinct eigenvalues such that Bn -*• B, n -> oo. (Theexistence of Blt B%,... is clear; it is sufficient to suppose that B is in aJordan normal form and to change the diagonal elements by a small

)

amount to obtain Bn.) Then the eigenvalues of Bn can be ordered^in> • • • > ^d» so at A,-B -*• Ay, n ->• co, fory = 1, . . . , d. Correspondingly,the eigenvalues of Sn tend to those of B. Hence the general case of thelemma follows from the special case treated above.

11. Simple Singularities

In (10.2), Z(t) has a Laurent expansion about t = 0. The point t = 0is called a regular singular point for (10.1) if (10.1) has a fundamentalmatrix (10.2) in which the elements of Z(t) do not have an essentialsingularity at t = 0 (i.e., are analytic or have a pole at / = 0). In this case,we have

Corollary 11.1. Let A(t) be analytic and single-valued for 0 < |/| < aand let t — 0 be a regular singular point for (1.1). Then (1.1) has a funda-mental matrix Y(t) of the form

where C is a constant matrix and Z0(t) is analytic for \t\ < a.In fact, if Z(T) in (10.2) has at most a pole at / = 0, then (10.2) can be

written as Y(t] = Z(t)tntR~nI = ZQ(t)tc, where Z0(0 = Z(t)tn is analyticfor some choice of the integer n 0 and C = R — nl.

If the singularity of A (t) [i.e., of each element of A(t)] at t = 0 isat most a pole of order one, then t = Q will be called a simple singularityof (10.1). In this case (10.1) can be written as

where 5, Alt A2, . . . are constant matrices and

is convergent for |/| < a.If a new independent variable defined by s = log / is introduced, (11.1)

becomes

(For a treatment of this system for real s, see Chapter X.) If At = AI —• • • = 0, equation (11.3) is dyjds — By and has the solution y = eBs =tB; cf. § 5 and Corollary 11.1.

Theorem 11.1 (Sauvage). Let t = 0 be a simple singularity for (10.1),so that (10.1) is of the form (11.1) where (11.2) is convergent for \t\ < a.Then t = 0 is a regular singular point for (11.1).

This is an immediate consequence of the following lemma.

Lemma 11.1. Let t = 0 be a simple singularity for (10.1) and let y(i)be a single-valued, analytic solution of(\QA)for 0 < [/| < a. Then y(t) isanalytic or has a pole at t — 0.

For if this lemma is applied to the system (10.6), it follows that-Z(f) in(10.2) is analytic or has a pole at / = 0.

Proof of Lemma 11.1. Let 6 be fixed, 0 6 < 2ir, and / = rei0, sothat dt = eie dr and y(reie) is a solution of

for 0 < r < a. The assumption on A (t) implies that there exists a constantc such that M(reie)y|| < c \\y\\fr for 0 < r \a, where \\y\\ is theEuclidean norm of y. It follows that any solution y •£ 0 satisfies

for 0 < r \a. Hence \\y(rei6)\\ < C/r° for 0 < r \a and a suitableconstant C. Thus if n is a positive integer, n c, then f ny(0 is boundedfor small \t\ and hence is analytic at / = 0. Thus y(t) has at most a pole oforder n. This proves the lemma.

The converse of Theorem 11.1 is false. For it is readily verified that thebinary system

has the fundamental matrix

which is of the form (10.2) with Z(f) = F(0 and R = 0. Thus / = 0 is aregular singular point but is not a simple singularity. However, Theorem11.1 has a partial converse.

Theorem 11.2. Let Q(t) be ad X d matrix of functions which are single-valued and analytic for 0 < |f | < a and such that t = Q is a regularsingular point for

Then there exists a matrix P(t\ which is a polynomial in t and satisfiesdetP(0 = 1, and a diagonal matrix D = diag [a(l),..., a(</)], wherea(j) > 0 is an integer, such that the change of variables

transforms^(ll.4) into the form (11.1)-(!!. 2) for which t = Q is a simplesingularity.

Proof. Since t = 0 is a regular singularity of (11.4), there exists afundamental matrix W(/) of the form

where R is a constant matrix and X(t) is analytic for \t\ < a.Suppose first that det X(Q) ^ 0, then A^O is analytic for \t\< a and

Q(t} = ^'(O^-HO is given by

Hence Q (/) has at most a simple pole at / = 0, so that / = 0 is a simplesingularity for (11.4) and the theorem follows with P(t) = /, D = 0.

Consider the case that det X(G) — 0. Suppose, for a moment, that thereexist matrices P(t\ D of the type specified, such that

where Z(f) is analytic for \t\ < a and det Z(0) ^ 0. Thus

and so (11.5) transforms (11.4) into a system y' = Q0(t)y for whichY(t) — Z(t)tR is a fundamental matrix. By the analogue of (11.7),

Consequently, t = 0 is a simple singularity for the new system y' —Q0(t)y, so that Theorem 11.2 will be proved if the following lemma isverified.

Lemma 11.2. Let X(t) be a matrix of functions analytic for \t\ < a suchthat det A^f) 0. Then X(t) has a representation of the form (11.8), whereP(t) is a matrix which is a polynomial in t and detP(/) si, D —diag [a(l),..., a(</)] wM a(y') 0 an integer, and the matrix Z(t) isanalytic for \t\< a with det Z(0) 0.

Remark 1. The relations (11.8), detP(f) = 1, and detZ(0) * 0 showthat

where a 0 is the order of the zero of det X(t) at f = 0.Proof. The equation (11.8) will be considered in the form

From the usual construction of the inverse of a matrix in terms of minors,divided by the determinant, it is clear that both P(t) and P~\t) have the

properties just specified. Instead of constructing P(t), it will be simpler toobtain P~l(t). Also, the proof will give a matrix P~\i) such that det P~\t)is a constant (?£ 0), not necessarily 1. The normalization detP(f) = 1 isthen obtained in a trivial way.

The matrix P~\f) will be constructed as a product of a finite number ofelementary matrices N of one of the following three types: (i) multipli-cation of a matrix on the left by N interchanges the yth and kth rows,e.g., multiplication by the matrix

on the left interchanges the first and second rows; (ii) multiplication byN on the left multiplies the yth row by a number A 5* 0, e.g., N =diag [1,..., 1, A, 1,. .., 1]; and (iii) multiplication by N on the leftreplaces the yth row by the sum of the yth row andXO times the £th row,where p(f) is a polynomial and k <y"; e.g., fory = 2 and k = 1,

Each of these elementary matrices N satisfies det N — const. j& 0.Let X(t) = (xjk(t)) and let a ^ 0 be the order of the zero of det X(t)

at / = 0. With they'th row of X(t), associate an integer a(y) 0 such thatxik(i) = ta(i}yik(t), where yjk(t) is analytic at / = 0 and at least one of thefunctions y^(0,..., yid(t) does not vanish at t = 0. In particular,

so that det X(t) — tn det Y(t\ where n = a(l) ^ + a(d). Hence

If equality holds in (11.11), then det 7(0) 5* 0 and the lemma is trivialwith P (t)-I and D = E.

If inequality holds in (11.11), it will be shown that X(t) can be multi-plied on the left by a finite number of the matrices of the type N to obtaina matrix X0(t), such that if a(l),..., a(</) belong to XQ(t) as a(l),..., a(d)

belong to X(t\ then a(l) + • • • + a(rf) = a. Thus X&) = P~l(t)X(t)has the form tDZ(t), where detZ(O) 7* 0.

After multiplying X(t) on the left by matrices of type (i), it can besupposed that a(l) < • • • a(d). Let eik = y,fc(0), so that x,fc(r) =tau)(fik -f • ' * )• Then not all of the numbers en , . . . , eld are zero.Suppose that elm, 1 m rf, is the first of these elements which is notzero. Then after multiplying X(t} on the left by a matrix of type (ii), itcan be supposed that elm = 1. Also, by replacing theyth row of X(t) bythe sum of theyth row and/>(0 = —€jmta(i}~a(1) times the first row, it canbe supposed that eim = 0 for j — 2 , . . . , d. Thus the matrix of the newelements €jk = y#(0) is of the form

if, e.g., m = 2. This procedure does not decrease the integers a(l),.. .,a(d).

If not all elements 21, • • • > €zd °f the second row are zero, it can be

supposed that if c2n ls the first different from 0, then e2B = 1 and e,n = 0

for; = 3, ...,d.This procedure can be applied to the second row, then the third row,. . . ,

unless all of the elements efl,..., e,d in theyth row are zero. In this case,a(j) can be replaced by a larger number, again called a(j). If this occurs,the rows are again permuted to obtain the order a(l) ^ • • • < a(d) andthe entire procedure repeated.

After a finite number of repetitions, the procedure of introducing a onein each of the d rows with zeros to the left and below it succeeds because,in view of the limitation a(\) + • • • -f- a(d) ^ a, it is only possible toincrease an index a(j) a finite number of times.

The first column cu, 2 1 , . . . , dl contains at least one element differentfrom zero. Otherwise, the ones occur in the last d — 1 columns, withonly zeros below the ones. This implies that there is a row eilt..., eidwith all zeros, contradicting the construction. If enl is the first element ofthe first column such that enl ?£ 0, then, by the construction, enl = 1 andfml = 0 for n 76 m. Move the nth row to the first position withoutdisturbing the order of the other rows.

In the new matrix, the d — 1 elements 2a> 32> • • • . d2 of the secondcolumn are not all zero, in fact, exactly one (say, the wth) is one and theothers are zero. This llows by the argument of the last paragraph.

)

jk

1

Move the mth row to the second position. Continuing this procedureleads to a matrix X0(t) for which the corresponding (eik) has ones on theprincipal diagonal and zeros below it, thus det (ert) = 1.

Thus the construction gives an X0(t) = P~l(t)X(t) of the form tDZ(i),with Z(0) = (, fc) having detZ(0) = 1 and D = diag [a(l),..., a(</)]where a(l), . . . , a(</) belong to X0(t) as a(l),... , a(d) belong to X(t).This proves the lemma.

Exercise 11.1. Prove that if Q(t) is a d x d matrix of single-valued,analytic functions on 0 < |/| < a, then a necessary condition for t = 0to be a regular singularity for (11.4) is that the elements of Q(t) have atmost a pole (not an essential singularity) at / = 0.

Exercise 11.2. Let A0(s) be a matrix of functions analytic for a < |s\ <oo. The point s = oo is called a simple singularity [or a regular singularpoint] for the system

if t = 0, where / = 1/j, is a simple singularity [or a regular singular point]for y' = A(t)y, where A(t) = —r*A0(\lt). (a) Necessary and sufficientthat s = oo be a simple singularity for (11.12) is that A0(s) -*• 0 as \s\ -»• oo.(£) Let A (t) be analytic for all t tlt..., rn, oo. Necessary and sufficientconditions that / = f l t . . . , /„, oo be simple singularities is that A (/) beof the form A (t) = (/ - /i)"1^ H H (/ - tjr1!^, where RltR2,. . ., Rn are constant matrices.

Theorem 11.3. Let (11.2) Ae convergent for \t\ < a and let

£e a "formal" power series which satisfies (11.1) /« //«e je«5e /Aa/ //reformulae

n » 1, 2 , . . . , AoW. 27K7I (11.13) w convergent for \t\ < a.Proof. Since (11.2) is convergent for |f| < a, there exist constants

c > 0, /> > 0 such that

in the sense that \A^i\ ^ cpk \y\ for all vectors y. (For example, p can bechosen to be any number such that p > I/a.) Choose r > 0 so large that

)

)

Let m be an integer such that \By\ m \y\ for all vectors y. ThenB — nl is nonsingular for n > m, in fact, \By — ny\ (n — m) |y| 7* 0for y T£ 0. Thus if n > m, the equation By — ny = z has a solution y,for any given 2, and

Let y > 0 be chosen so large that

holds for n = 0 , . . . , m. It will be verified by induction that (11.19)holds for all n. Thus assume (11.19J for n = 0,... ,j — 1 and/ — 1 >m. Then

By (11.16) and the induction hypotheses,

so that |z,| ^ yr* by (11.17). Hence (11.18) implies that \yt\ ^ |z,| < yr*.This completes the induction.

Consequently (11.13) is convergent for |/| < 1/r and is a solution of(11.1) for small \t\ > 0. But then it is convergent for |/| < a and is asolution of (11.1) for 0 < |/| < a as no solution of (11.1) has a singularityon 0 < |/| < a.

Exercise 11.3. Show that Theorem 11.1 is false if (11.1) is replaced by(10.1) and it is not supposed that / = 0 is a simple singularity.

Corollary 11.2. Let (11.2) be convergent for \t\ < a and suppose that ifA j , . . . , Ad are the eigenvalues of B, then A, — Afc = 0 or A, — Xk is notan integer for j\ k = !, . . . ,</. T/ien (11.1) /HW a fundamental matrix ofthe form

is convergent for \t\ < a,Proof. The matrix (11.20) is a solution matrix for (11.1) if and only if

Z satisfies the differential equation

cf. (10.5). A formal series

satisfies (11.21) if and only if the formulae

hold for n = 1,2,Consider (11.21) and (11.23)-(11.24) as systems of equations for d2-

dimensional vectors, rather than fordxd matrices. For any constantmatrix D,

has a unique solution if p 5* A, — Xk for/, k = 1 , . . . , d. For this set ofequations can be viewed as BZn — pZn — D, where S is a d2 x dz

matrix with the ^eigenvalues A, — Afc fory, k — 1,..., dby Lemma 10.2.Thus Z0 = /satisfies (11.23) and Zl5 Z2,... can be obtained recursively

from (11.24). The resulting series (11.22) is convergent for \t\ < a byTheorem 11.3. This proves Corollary 11.2.

When the eigenvalues ,..., Ad of B do not satisfy the conditions ofCorollary 11.2, it is possible to change the dependent variable y,

so that (11.1) becomes a system

to which Corollary 11.2 is applicable. In this case, (11.1) has a funda-mental matrix of the form

where U(t) is a polynomial in t and Z(t) is of the same form as in (11.20).The precise result to be proved is the following:

Lemma 11.3. Let (11.2) be convergent for \t\ < a. Then there exists amatrix U(i) with the properties that U(t) is a polynomial in t; thatdet U(t) 9*Qfort?*Q; and that (11.25) transforms (11.1) into a system(11.26) in which the factor ofr\ is a convergent power series for \t\ < a andif plt. .., fjid are the eigenvalues of C, then (ij — fj,k = 0 or fij — ftk isnot an integer for j, k =• ! , . . . ,</ .

Remark 2. It will be clear from the proof that if A x , . . . , Ad are theeigenvalues of B in (11.1), then /Alt..., fid can be ordered so thatXj — fJLf = ny 0 is an integer.

The proof below will not involve a knowledge of the solutions of (11.1)[and gives an algorism for the determination of a C in (11.26), i.e., of a Cin Corollary 11.1]. Another proof of Lemma 11.3, but not of Remark 2

following it, can be obtained if one knows a fundamental matrix for (11.1)as in Corollary 11.1:

Exercise 11.4. Deduce Lemma 11.3 from Lemma 11.2.Proof of Lemma 11.3. Suppose first that B is in a Jordan normal form

B = diag [/(I),... ,J(g)], where/(y) is a Jordan block of the form (5.15)with A = A<j), h = h(j) for j = 1,. . . , g. Let B2 = diag [/(2),..., J(g)]so that B2 is an e X e matrix, where e = d — h(l).

Make the change of variables

and 7A(1), Ie are the unit matrices of the specified dimensions. Then (11.1)becomes

Rearranging the coefficient matrix according to the powers of /, this is ofthe form

where C1, Akl are constant matrices and C1 is given by

n is an A(l) X A(l) matrix and A22 is an e x e matrix. Thus if theeigenvalues of B are A(l),. .., A(l) and A f t (1)+1,..., /td, then those of c!"1

are A(l) - 1 , . . . , A(l) - 1 and Awl)+1 , . . . , Ad.If ^ is not in a Jordan normal form, the same result is achieved by a

variation of constants y = T^V^i)^ where T~1BTV is in a Jordan normalform. It is clear that the lemma follows with U(t) of the form U(t) =TiVi(t)TtV&)... TjVfc), where Tk is a constant nonsingular matrix andVk(t) is of the same type as V(i) for k = 1,... ,y.

Theorem 11.4. Lef (11.2) fo? convergent for \t\ < a. Let A be a fixedcomplex constant and n^ (^ 0) the number of linearly independent vectors ysatisfying By — Xy. Then the number N^ (^ 0) of linearly independentsolutions of(l\.I) of the form

satisfies

and A

)

In particular, the number N0 of linearly independent solutions y(t) of (II.I)which are analytic at t = 0 satisfies

It is not assumed that y0 0 in (11.31). For a generalization of Theorem11.4, see Theorem 13.1.

The proof of this theorem will depend on the proof of Lemma 11.3 andon the following lemma.

Lemma 11.4. Let (11.2) be convergent for \t\ <a. Then the numberNA (= 0) of linearly independent solutions of (11.1) of the form (11.31)satisfies

//"A is an eigenvalue of B and A + 1, A + 2 , . . . are not eigenvalues, then

and y0 ^ 0 /'« any solution (11.31) of (11.1).Proof of Lemma 11.4. It can be supposed that A = 0, otherwise the

change of variables y = /AT? replaces (11.1) by

and A + k, ni+k by fc, «fc.A function (11.13) is a solution of (11.1) if and only if (11.14) and

(11.15J, it si, 2 , . . . , hold. The equation (11.14) has n0 linearlyindependent solutions and if (11.14), and (11.15^,..., (11.15 ) hold,then the solutions of (11.15k) are of the form z0 -f zk, where yk = z0 is aparticular solution of (11.15fc) and zk varies over the wfc-dimensional linearmanifold of solutions of Bz — kz = 0. This proves (11.34).

In the last part of the lemma, it is supposed that A = 0 is an eigenvalueof B, but A = 1,2,... are not. Then (11.14) has «0 linearly independentsolutions y0 and if (11.14) and (11.150,..., (H.15A_j) hold, then (11.15fc)has a unique solution. Thus for a given y0, the vectors ylf y2,... areuniquely determined. The corresponding series (11.13) is convergent for\t\ < a and is a solution of (11.1) by Theorem 11.3. This proves (11.35)and completes the proof of Lemma 11.4.

Proof of Theorem 11.4. In view of (11.34) in Lemma 11.4, only thefirst inequality in (11.32) remains to be proved. Since N^ s£ N^+l ^ • • • ,this inequality will be proved if it is shown that

As in the last proof, there is no loss of generality in supposing thatA = 0 in (11.35). It can be supposed that «0 > 0 and, in view of (11.35)in Lemma 11.4, it can also be supposed that B has an eigenvalue which isa positive integer.

The proof of Lemma 11.3 makes it clear that there exists a change ofvariables (11.25) such that U(t) is a polynomial in / and that (11.1)becomes (11.26) in which the eigenvalues of C are the same as those of Bexcept that any integral eigenvalue A = n > 0 of B has been replaced byA = 0. Let «A(C ) have the same meaning for C as n^ does for B. It will beshown that

To this end, it suffices to re-examine the proof of Lemma 11.3 and showthat the analogue of (11.37) holds at each step. Thus, if the first step leadsfrom (11.1) to (11.29), it suffices to show that

Let B be in a Jordan normal form, B = diag [/(I),... ,/(#)] and letA(l) > 0 be an integer. Make the change of variables (11.28) transforming(11.1) into (11.29), where (11.30) holds. Since A(l) * 0 and B2 is in aJordan normal form, it is clear that «0 rows of B2 contain only zero ele-ments. Hence the rank of C1 in (11.30) is at most d — «0, so that (11.38)holds. Consequently (11.37) follows.

Since A = 1, 2 , . . . are not eigenvalues of C, the last part of Lemma 11.4implies that (11.26) has n0(C ) linearly independent solutions ??(/) which areregular at / = 0. Since U(t) is a polynomial in t, (11.25) shows that (11.1)has at least n0(C) linearly independent solutions y(t) regular at t = 0.In view of (11.37), the inequality (11.36) follows for A = 0. This completesthe proof of Theorem 11.4.

Exercise 11.5. (a) Let A(t) = (aik(t)) be a. d x d matrix of functionsanalytic for |/| < a. Let a(/) = 0 or 1 and a < d if a = a(l) + • • • +a(</). Then the system

has at least d — a linearly independent solutions analytic at t = 0.(b) Let a0(r), flj(0 be analytic for |/| < a, then the differential equation

has at least one solution u(t) & 0 analytic at / = 0. For a generalizationof this exercise, see § 13.

12. Higher Order Equations

Consider a differential equation of dlh order for a function w,

in which the coefficient functions are single-valued and analytic on apunctured disc 0 < \t\ < a. Instead of writing (12.1) as a first ordersystem in the standard way, transform it into a system for the vectory = (y\..., yd), where

Then

Thus t = 0 is a simple singularity for this system if td~kpk(t) is analytic att — 0 for k = 0,... ,d — 1; i.e., if pd^(t) has at most a pole of the firstorder, />d_2(0 has at most a pole of the second order,..., p0(t) has atmost a pole of the dth order. In this case, let

where bk and pkn are constants and the series in (12.4) is convergent for|r| < a. Then (12.1) is of the form

where 00(0> • • • > fl<i-i(0 are analytic at / = 0. Correspondingly, (12.3) isof the form (11.1), where B is the constant matrix:

The coefficient matrix on the right of (12.3) reduces to the constant

matrix B if pkn — 0 for k = 0,. . ., d — 1 and n = 1, 2 , . . . . This is thecase when (12.1) is the differential equation

in which b0,. . ., bd_± are constants. This is called Euler's differentialequation. The solutions of (12.7) are easily determined from the fact that afundamental matrix for the corresponding system (12.3) is Y(t) — tB;cf. the remark following (11.3). Thus the solutions of (12.7) are linearcombinations of functions of the form ?A(log /)*.

The numbers A and permissible values of k are determined by theJordan normal form of B. The equation (12.7) obviously has a solution ofthe form u = t*~ if and only if A is an eigenvalue of B. Substituting U — IK

into (12.6), it is seen that this is the case if and only if F(X) = 0, where

The equation F(A) = 0 is called the indicia! equation for (12.7).Let A( l ) , . . . , A(g) be the distinct solutions of F(A) = 0 with the

respective multiplicities A ( l ) , . . . , h(g\ where /z(l) + • • • + h(g) = d.Then a linearly independent set of solutions of (12.7) is f*0)(log t)k, wherej = 1,.. . , g and k = 0,..., h(j) - 1.

Exercise 12.1. (a) Verify this last statement by the type of argumentin § 8(vi) following Exercise 8.1. (b) The remarks concerning (11.1) and(11.3) show that the change of variables t = e8 reduces the system (12.3)belonging to (12.7) to one with constant coefficients. Verify directly thatthe substitution t = e' reduces (12.7) to an equation with constantcoefficients and, hence, that (a) follows directly from § 8(vi).

Returning to the general equation (12.1) and its corresponding system(12.3), the following theorem will be proved:

Theorem 12.1 (Fuchs). Let /7fc(f) be single-valued and analytic for0 < \t\ < a. Then t = 0 is a regular singular point for (12.3) if and onlyiff = 0 is a simple singularity for (12.3) (i.e.t if and only if (12.4) holds withconvergent series on the right).

It is clear that t — 0 is a regular singular point for the system (12.3)if and only if the solutions of (12.1) are linear combinations of functionsof the form /A(log 0*a(f), where a(f) is analytic for |/| < a.

Proof. The "if" portion of the theorem is a consequence of Theorem11.1. Thus it is sufficient to prove the "only if" portion.

It follows from Theorem 10.1 that (12.1) has at least one solution of theform t^O) = ^(f), where a^f) is analytic for 0 < |r| < a. If it isassumed that / = 0 is a regular singular point for (12.3), then a^f) has at

most a pole at / = 0. In fact, by changing A, it can be supposed that aj(/)is analytic at / = 0. In particular a^/) 5^ 0 for small |/j > 0.

The proof proceeds by induction on the order d. Consider first anequation of order d = 1,

with a solution «x(/) = /^(f), «i(0 analytic at / = 0. It is clear thatpQ(t) = —Ui'WIu^t) has at most a pole of order d = 1 at t = 0.

Assume d > 1 and that the theorem is correct for equations of orderd — 1. Let u = u^(t) be the type of solution described above and introducethe new dependent variable v — u\u^ for small t > 0. Then (12.1) istransformed into an equation of the form

which is an equation order d — 1 for v. It is readily seen that

and since u = u^t) is a solution of (12.1),

where Cjfc —j\/kl(J — k)l are binomial coefficients.The equation (12.9), as an equation of order d — 1 for y', has the

solutions v' = («/M!)' for arbitrary solutions u of (12.1). Consequently,/ = 0 is a regular singular point for the system associated with (12.9).Hence, by the induction hypotheses, td~l~kqk(t) is analytic at t = 0 fork — 0, . . . , < / — 2. Also, M^'/WI has a pole of at most the order k at/ = 0. It follows from (12.10) and (12.11) that/^ has at most a pole oforder 1 at / = 0, pd_2 has a pole of order at most 2, etc. This provesthe theorem.

Exercise 12.2. Let p0(t),...,pd-i(t) be analytic for a < \t\ < oo.The point / = oo is called a simple singularity [or regular singular point]for (12.1) if it is a simple singularity [or regular singular point] for (12.3);cf. Exercise 11.2. (a) Necessary and sufficient that t = oo be a simplesingularity for (12.1) is that f*-*-1/^/)-* 0 as |/| -» oo for k = 0,...,d - 1. (b) Let />0(f), • . . ,/>d-i(0 be analytic for / ?« / l t . . . , tn, oo.Necessary and sufficient that t = fj, . . . , /„ , oo be simple singularitiesfor (12.1) is that/?d_fc(0 be of the form/>d_*(0 - (/ - >i)~*...(/- / J~*«*(/),

for k = 1. . .. ,d, where ak(t) is a polynomial of degree at most k(d — 1).(Differential equations with this property are said to be of Fuchs' type.)

For a second order (d = 2) equation (12.5) having a regular singularpoint at f = 0, it is comparatively simple to discuss the behavior of a setof fundamental solutions. We attempt first to find a solution of the form

by the method of undetermined coefficients. If the roots Ax, A2 of theindicial equation F(A) = 0 [cf. (12.4) and (12.8)] are such that Ax — A2

is not an integer, then we obtain two solutions (12.12) with A = Ax, A2 inthis way; see Corollary 11.2. If A! — A2 0 is an integer, we can stillobtain in this way a solution u(t) of the form (12.12) with A = Ax; seeLemma 11.4. A second linearly independent solution v(i) can be obtainedfrom the fact that, by § 8(iv),

Exercise 12.3. Discuss the nature of the solutions at the (finite)singular points of the equations: (a) /V + tu' + (f2 — fiz)u — 0 (Bessel);(b) (1 - /V - 2tu' + n(n + l)w = 0 (Legendre); (c) (1 - t*)u" -2tu +[«(/! + 1) - m\\ — t2)~l]u = 0 (associated Legendre).

13. A Nonsimple Singularity

In this section, we shall prove a theorem about the number of analyticsolutions of a particular type of linear homogeneous system for which/ = 0 is a singularity, but not necessarily a simple singularity.

Theorem 13.1 (Lettenmeyer). Let A(i) = (a^(0) be ad X d matrix offunctions analytic at t = 0. Let a(j) 0 be an integer for j = 1,.. . ,dand let a < d, where a = a(l) -f • • • + «(</)• Then the system

has at least d — a linearly independent solutions analytic at t — 0.This theorem generalizes Theorem 11.4 which corresponds to the case

<x(/) = 0 or 1; cf. Exercise 11.5.Proof. The solutions y(f) of (13.1) analytic at t = 0 will be determined

by the method of undetermined coefficients. Let the function aik(i) havethe expansion

)

Consider a solution y(t) = (yl(t),..., y*(tj) of (13.1) with convergentexpansions

Then the numbers yj satisfy

for y = ! , . . . ,< / and m = 0, 1, It is understood that yTOJ = 0 if

m < 0. Conversely, if a set of numbers ynj satisfies (13.4) and if (13.3) is

convergent for small |/| and j = I , . . . ,* / , then y(t) = (y*(0» • • •»2^(0)is a solution of (13.1) analytic at / = 0.

Let N be a large fixed integer to be specified below. Divide the systemof equations (13.4,TO) into two systems

Since it is assumed that the series in (13.2) is convergent, there existpositive numbers, c and p, such that

Let 0 < 0 < 1 and define

The last set of relations is equivalent to

Thus the system ]£2 °f equations can be written as

where h — m — a(y) + 1, or in the form

or, finally, as

It is understood that the inner sum on the left over N ^ n w + a(/) — 1is 0 if m + a(/) = N.

Let p, q be fixed integers satisfying 1 p ^ d and Q q ^ N — 1.Instead of (13.10) consider the set of equations

for j — 1 , . . . , d and m^.N. It will be shown that if N is sufficientlylarge, then (13.11) has a solution zjm = zf£ satisfying

To this end, note that, by (13.7) and (13.9),

Thus, if c', c" are suitable constants (independent of p, q, N), then

Hence, if N is sufficiently large,

for all choices of/?, ^.From now on, it is supposed that N is fixed so large that (13.13) and

(13.14) hold. Let p, > N denote a fixed integer and replace the infinitesystem (13.11), where y = ! , . . . , < / and m N, by the finite system ofequations,

f o r j = 1,..., d and N m /*, where v in (13.15) is

The system (13.15) can be written in the form

where £ is 3 vector of dimension e — d(ju — N + 1) with components

z?m for j — ! , . . . ,< / and m = N,..., /u; 77 is a vector of dimension ewith components — cimtPq for y = !, . . . ,</ and m = N,..., ft; and Cis an e x e matrix. Using Euclidean norms, it is clear from (13.13) and(13.14) that

Hence (13.17) has a unique solution £ satisfying ||£|| 2, since ||?y|| =lie + Clll ^ Hell - IIceil ^ I lieil . Thus the finite system (13.15) has aunique solution z?n satisfying

Then by the Cantor Selection Theorem I 2.1, there exists a sequence ofintegers (N <) ^(1) < /*(2) < • • • such that

Note that v in (13.15) and (13.16) satisfies v(j, m, /*) -> m + oc(y) — 1 as// -> oo. Hence, by letting p = fj,(K) -* cx> in (13.15), it follows that zj*is a solution of (13.11) fory = 1, . . . , d and m^.N satisfying (13.12).

Consequently, if ZOT are any dN numbers for p = 1,. .. ,d and q —0, ..., N — 1, then a solution of (13.10) fory = ! , . . . , < / and m TV isgiven by

fory" == 1 , . . . , dand m ^ TV. In other words, the equations of the system22 in (13.6) [i.e., the equations in (13.10) for j = 1,.. ., d and m N]are satisfied if (13.18,J hold fory = 1, . . . , d and m N.

Thus the original system of equations (13.4,TO) for the ymj or, equi-

valently, for the zim are satisfied if 2i anc^ (13.18) hold. If a* =max [a(l), .. ., a(*/)], the system Ji involves the d(N + a* — 1) unknownz,w for y = 1,. . ., d and m = 0, . .., N + a* - 2; cf. (13.4,J fory = 1,. . ., </and Q m N + a(y) — 2. Also the system i consists of[N - 1 + a(l)] + • • • + [N - 1 + a(</)] = d(N - 1) + a equations,where a = a(l) + • • • + <x{d). Add to the system ]£i> tne (possiblyvacuous) set of equations

2,: (13.18,J for j=l,...,d jmd T V ^ / « ^ ^ + a * - 2 .

The system 3 involves the same set of d(N + a* — 1) variables that occurin 2i and consists of d(o.* — 1) equations. Thus, the combined systems,2i and 2», has at least d(N + a* - 1) - [d(N - 1) + a + </(a* - 1)] =d — a linearly independent solutions.

Corresponding to any solution of 2i and ^3> tne equations (13.18,m)f o r / M > 7 V + a * — 2 (together with the equations of Jg) give a solution

of 22- The resulting set of numbers ymj = z,m/(0/>)"*» f°r / — 1, . • • , < /

and m — 0, 1,. . . , is a solution of and £2- In view of (13.12) and(13.18), there is a constant c0 such that \zim\ < c0; hence \ym'\ <* c0/(0p)w

for j = ! , . . . , < / and m ^ 0. Consequently, (13.3) is convergent for|/1 < 6p. This proves Theorem 13.1.

Exercise 13.1 (Perron). Let fl/f) be analytic for |f| < a,j = 0,..., d.Let a be an integer, 0 a < d. Show that

has at least d — a linearly independent solutions analytic at t = 0.Exercise 13.2 (Lettenmeyer). (a) Let ,4(0, (0 be d x d matrices of

functions analytic at t = 0. Let det X(t) ^ 0 and det X(t) have a zero oforder a, 0 a < d, at / = 0. Then the system

has at least d — a linearly independent solutions analytic at / = 0.(b) Let A'(r), /4 (/) be analytic in a simply connected /-domain E such thatdet X(t) & 0 and det X(t) has exactly a zeros in E, counting multiplicities.Let a < d. Then (13.19) has at least d — a linearly independent solutionsy(t) analytic on £.

Exercise 13.3. Let X(t) be as in Exercise 13.2(o). Let /(/, y) =f(t, y1,..., y4) be a ^-dimensional vector, each component of which isan analytic function of (i.e., a convergent power series in) t, y1,..., y*.Then

has a d — a parameter family of solutions y(f) analytic at / = 0. SeeBass [1].

Notes

SECTION 1. Equation (1.5) in Theorem 1.2 as well as the special case (8.4) are givenby Liouville [2] in 1838, although some authors refer to (8.4) as Liouville's formula and(1.5) as Jacobi's. Theorem 1.2 was also given by Jacobi [1, IV, p. 403] in 1845; for theparticular case when the system (1.1) is replaced by one linear equation of the secondorder, the corresponding formula (8.4) occurs in a paper (1827) by Abel [1,1, p. 251].The notion of a fundamental set of solutions is due to Lagrange (circa 1765); see[2, I, p. 473]. The term "fundamental set of solutions" was introduced by Fuchs[4, p. 117] in 1866.

SECTION 2. The method of variation of constants (and Corollary 2.1) is essentially dueto Lagrange (1774, 1775); see [2, IV, p. 9 and p. 159]. The result of Exercise 2.2 goesback to Perron [11]. It has also been proved by Diliberto [1], [2]. The proof given inHints, using Exercise 2.1, is that of Reid [4].

SECTION 3. The possibility of reducing the order when a solution is known goes back(1762-1765) to'd'Alembert for a linear equation of order d. Corollary 3.1 when (1.1)


is replaced by a single equation of order d [cf. (8.2) and (8.19)] is implicit in a result(1833) of Libri [1, pp. 185-194] (who also makes remarks about the possibility ofextending the result to systems). Corollary 3.1 for the case of an equation of dth orderis given explicitly (1874) by Frobenius [1].

SECTION 5. For the analogous case of a linear dth order equation, the general solution[§ 8 (vi)] is due to Euler (1743).

SECTION 6. Theorem 6.1 for the case of a rfth order equation was given (1883) byFloquet [1].

SECTION 7. The adjoint of a linear dth order equation was given by Lagrange about1765, see [2, I, p. 471]. The adjoint system (7.1) was defined in 1837 by Jacob!; cf.[1, IV, p. 403]. The term "adjoint" was introduced by Fuchs [1, p. 422] in 1873.

SECTION 8. See comments on §§ 1-7. Exercise 8.3 contains results*of P61ya [1]; cf.comments of Hartman [15].

SECTION 9. The type of formalism discussed in connection with (9.1)-(9.6) was usedextensively by G. D. Birkhoff [3]. Exercise 9.1 is based on considerations used byCesari [1]; see also Levinson [3].

APPENDIX. For a historical survey and references, see Schlesinger [2j, Forsyth [1],and the encyclopedia article of Hilb [1]; for later references, see A. Schmidt [1].

SECTION 10. Problems arising out of (10.2) were initiated by Riemann in work dated1857 (cf. [1, pp. 379-390]) and independently by Fuchs in 1865 (see [1, p. 124]). Theorem10.1 is essentially due to M. Hamburger [1]; the proof in the text follows Coddingtonand Levinson [2]. Although the paper by Wintner [5] contains many misstatements,including the main theorem, it has a number of good ideas including the suggestion toview (10.5) as a system (10.6) of d2 equations rather than as a matrix equation; forapplications, see § 11 [cf. the proof of Theorem 11.1 which may be new ].

SECTION 11. Theorem 11.1 is due to Sauvage [1]; cf. Hilb [1] for references to Hornand Schlesinger. The Lemma 11.1 is a result of Birkhoff [1]. The partial converse,Theorem 11.2, was given by Sauvage and Koenigsberger; the first complete proof is dueto Horn [1]. The proof in the text is that of Schlesinger [1, pp. 141-162], who attributesthe arguments in the proof of Lemma 11.2 to Kronecker. For similar proofs of thislemma, see Lettenmeyer [1] and Moser [2]; generalizations have been given by Hilbert,J. Plemelj, and G. D. Birkhoff, see Birkhoff [2] for references. Corollary 11.2 and Lemma11.3 are in Rasch [1, p. 113]. Their use in connection with (11.25H11.27) is that ofA. Schmidt [1 ]. Exercise 11.5 is a special case of Lettenmeyer's [1 ] result, Theorem 13.1.

SECTION 12. Theorem 12.1 is due to Fuchs in 1868 [1, p. 212]. It was first proved inspecial cases by Riemann [1, pp. 379-390]. The proof in the text for the "only if"part is that of Thome [1]; see Hilb [1] for references to Frobenius.

SECTION 13. Theorem 13.1, which is due to Lettenmeyer [1], generalizes the result ofPerron [1] in Exercise 13.1 dealing with one equation of dth order. The proof in thetext is a modification of Lettenmeyer's which, in turn, is based on Hilb's proof [2] ofPerron's theorem. For Exercise 13.2, see Lettenmeyer [1].

*For related results, see Mammana [SI].

Chapter V

Dependence on Initial Conditions and P ameters

1. Preliminaries

Let/(y, y) be defined on an open (/, y)-set E with the property that ifOo» ^o) e ^» ^en the initial value problem

has a unique solution y(t) = ??(f, t0, yQ) which is then defined on a maximal/-interval (a>_, o>+), where a)± depends on (t0, y0). In this chapter, theproblem of the smoothness (i.e., of the continuity or differentiabilityproperties) of rj(t, r0, y0) will be considered.

Often, a more general situation is encountered in which (1.1) is replacedby a family of initial value problems depending on a set of parametersz = (z\...,z°),

where for each fixed z, (1.2) has a unique solution y(t) = r)(t, t0, y0, z). Inmost cases, the question of the dependence of solutions of (1.1) on t andinitial conditions can be reduced to the question of the dependence on/, z of solutions of a family of initial value problems (1.2) fa* fixed initialconditions y(t0) = y0', conversely, the question of the dependence ofsolutions of (1.2) on t, t0, y0, z can be reduced to the question of smooth-ness of solutions of initial value problems in which extra parameters z donot occur. The first reduction is accomplished by the change of variablest, y -*• t — t0, y — ya which changes (1.1) to

in which z = (t0,y0) = (t0, y0l,..., y0

d) can be considered as a set ofparameters (and the initial condition y(Q) = 0 is fixed). The secondreduction is obtained by replacing (1.2) by an initial value problem for a(d + e)-dimensional vector (y, z), in which no extra parameters occur:

93

where (/0, yQ, z0) denotes any of the possible choices for (/, y, z). For thisreason, some of the theorems which follow will be stated for (1.2) butproved for (1.1).

Below x, y, r),fe R* and z e Re, where d, e 1.

2. Continuity

The assumption of uniqueness implies the continuity of the generalsolution y — ??(/, /„, y0, z) ofy' -f(t, y, z):

Theorem 2.1. Letf(t, y, z) be continuous on an open (t, y, z)-set E withthe property that for every (f0, y0, z) e £, the initial value problem (1.2), withz fixed, has a unique solution y(t) = ??(/, /0, y0, z). Let co_ < t < co+ bethe maximal interval of existence of y(t) = rj(t, t0, yQ, z). Then a>+ =»ft>+(/0, y0, z) [or a>_ = co_(t0, y0i z)] is a lower [or upper] semicontinuousfunction of (t0, y0, z) E E and rj(t, /„, y0, z) is continuous on the set (o_ <t < co+, (t0, y0, z) E E.

It is understood that co+[(o_] can assume the value -f oo [—00]. Thelower semicontinuity of co+ at (tlt y^ z^) means that if t9 < (o+(tlt ylt za),then a>+(t0, y0, z) t° for all (f0, y0, z) near (tlt ylt Zj); i.e., w+fa, ylt zx) lim inf eo+(/0, y0, z0) as (/„, y0, z0) -*• (tlt ylt zt). The upper semicontinuityof (o_ is similarly defined.

It is easy to see that co±(t0, y0, z) need not be continuous. For supposethat (fj, ylt zj e E and tt < t° < co+fo, ylf zj. If E is replaced by the setobtained from E by deleting the point (t, y, z) = (f°, 7?(f°, /lf ylt zx), Z!),then o)+(/!, yl3 Zj) now takes the value f°, but eo+ is not altered for allpoints (/„, i/o, z) near (/lf ylt Zj).

Remark. Letf(t, y, z), »;(/, ?0, y0, z) be as in Theorem 2.1. For fixed(/, A>» 2)» the relation y — rj(t, t0, y0, z) can be considered as a map carryingy0 into y. The assumption that the solution of (1.2), for (/„, y0, z) e E, isunique implies that this map is one-to-one. In fact the inverse map isgiven by y0 = rj(t0, t, y, z). A consequence of Theorem 2.1 is that themap y0 -* y is continuous.

Proof. Since (1.2) can be replaced by (1.4) and (y, z) by y, there is noloss of generality in supposing that/does not depend on z. Thus, it willbe supposed that/(/, y) is defined on an open (/, y)-set E and that (1.1)has a unique solution y(t) = ri(t, r0» ^o) on a maximal interval of existenceo>_ < / < a>+, where w± = co±(t0, y0). It will be shown that, in this form,Theorem 2.1 is merely a corollary of Theorem II 3.2 for the case/n(/, y) =/(/ ,y),»-l ,2, . . . .

In order to verify that a)+(t0,y0) is lower semicontinuous, choose asequence of points (/lf ylo), (/2, y2o)>... in £ such that (tn, ynQ) -* (^0, y0) eE and co+(/n, yn0) -> c(^ oo) as n -*• oo, where c = lim inf co+(t*, y*) as

Dependence on Initial Conditions and Parameters 95

(**, y*) -»• ('o» 2/0)- Since the solution of (1.1) is unique, it follows fromTheorem II 3.2 that c o>+ = co+(/0, y0); i.e., ct>+(/0, y0) is lower semi-continuous. The proof of the upper semi-continuity of co_(t0, y0) is thesame.

Note that, for the case where /„ =/, « = 1 ,2 , . . . , and the solutionof (1.1) is unique, a selection of a subsequence in Theorem II 3.2 is un-necessary. Thus it follows that r)(t, r0, y0) is a continuous function of('0,2/0) f°r every fixed t, cu_('o> y0) < ' < "M/o.^o); m fact» this ccm-tinuity is uniform for tl < t f2, a>_(f0, y0) < /* < r» < «+(*„, y0)- Inother words, if e > 0, there exists a dfQ > 0, depending on (/0, y0, t1, f2),such that

for f J ^ f ^ r2. But since »?(/, /0, t/0) 's a continuous function of / for

fixed (r0, «/o)» there is a <5 1 > 0, depending on (/0, y0, r1, /2), such that

Hence, if 6€ = min (<5eo, <5C1) and |; - s\, |/0 - /il, |y0 - 2/il < <5£> then

This completes the proof of Theorem 2.1.

3. Differentiability

If it is assumed that/(f, y, z) is of class C1, it follows that the generalsolution y = rj(t, t0, y0, z) of (1.2) is of class C1. In fact, even more iscontained in the following theorem.

Theorem 3.1 (Peano). Letf(t, y, z) be continuous on an open (t, y, z)-setE and possess continuous first order partials df/dyk, df/dz* with respect tothe components ofy andz: (i) Then the unique solution y — r](t, /0, y0, z) of(1.2) is of class C1 on its open domain of definition co_ < t < eo+, (/„, y0, z) EE, where a>± = co±(t0, y0, z). (ii) Furthermore, if /(/) = J(t, f0, y0, 2) istheJacobian matrix (dffdy) off(t, y, z) with respect toyaty — r)(t, t0, y0, z),

/Ae» a; = dr)(t, /0, y0, «)/9y0* w f/» solution of the initial value problem,

M>Aer£ efc = (^j.1,.. ., ef) with ekj = 0 y" j j£ k and ek

k = 1; a; =<fy(f, '<» yo> z)/9zj w /Ae solution of

k

where £,(/) = g^t, t0, y0, z) is the vector df(t, y, z)/dz'" at y = T](t, tQ, y0, z);and Brj(t, tQ, y0, z)/d/0 is given by

The uniqueness of the solution of (1.2) is assured, e.g., by TheoremII 1.1. Note that the assertions concerning drjldy0

k and (3.2) or dyjdz'and (3.3) result on "formally" differentiating both equations in (1.2),i.e., both equations in

with respect to y0k or z'. Similarly, differentiating these equations formally

with respect to t0 shows that x = dr)/dt0 is also a solution of x' = J(t)xsatisfying the initial condition x(t0) = — rj'(t0, t0,y0,z) = -f(t0 , yQ, z).Writing f(t0, yQ, z) — S/*(/0, yQ, z)ek, it follows that (3.4) is a formalconsequence of (3.2) and the principle of superposition (Theorem IV 1.1)for the linear system x' = J(t)x.

More generally, if y(t, j) is a 1-parameter family of solutions of y' =f(t, y, *) for fixed z, if y(tQ, J0) = y0, and if y(t, s) is of class C1 in (/, s), thenthe partial derivative x — dy(t, s)/ds at s = s0 is also a solution of thesystem x' = J(t)x. For this reason x' = J(i)x is called the equation ofvariation of (1.2) along the solution y = rj(t, t0, y0, z).

The assertion concerning x = dy/dyf and (3.2) shows that the Jacobianmatrix (cfy/dy0) is the fundamental matrix for x' = J(t)x which reducesto the identity for / = /0- In particular, Theorem IV 1.2 implies

Corollary 3.1. Under the conditions of Theorem 3.1,

for CD_ < / < W+, where the argument of the integrand is (s, r)(s, t0,y0, z), z).Remark. By (3.5), det (<ty/<?y0) 0- Thus, the continuous one-to-one

map y0-^y = rj(t, t0, y0, z) for fixed (r, t0, z), considered in the Remarkafter Theorem 2.1, is of class C1 and has an inverse of class C1 with respectto (t, t0, y, z). This statement about the inverse is also clear from theexplicit formula y-+yQ = q(t0, t, y, z).

Exercise 3.1 (Liouville). Letf(y) be of class C1 on an open set E andlet y = r](t, y0)be the unique solution of the initial value problem y' =/(y),y(*o) = Vo f°r (*o» 3fo)e E. Show that the set of maps y0^-y defined byy = y(t, y0) for fixed t are volume preserving if and only if div/(y) =sap/a/iso.

Note that the assertion that x = d»?/dy0* is a solution of (3.2) impliesthat the iterated derivative d(dr)jdy^)ldt exists and is /(/) drildy*. The

k

t

last expression is a continuous function of t, t0, y0, z; hence, Schwarz'stheorem implies that d(drjldt)ldy0

k exists and is 3(3j?/3y0*)/9r. A similarremark applies to dyjdz'. Also, note that on the right side of (3.4) thevariable t only occurs in drjjdy0

k.Corollary 3.2. Under the conditions of Theorem 3.1., the second mixed

d2r)fdt dtQ exist and are continuous.In order to avoid an interruption to the proof of Theorem 3.1, the

following simple lemma will be proved first. This lemma is a convenientsubstitute for the mean value theorem of differential calculus when dealingwith vectors, for it avoids some awkwardness in the fact that 0fc depends onk in y(b) - y(a) = (b - aW(OJ,..., yd'(Bd)\ where a < Ok < b.

Lemma 3.1. Let /(f, y) be continuous on a product set (a, b) X K,where K is an open convex y-set, and let f have continuous partials dfjdyk

with respect to the components ofy. Then there exist continuous functionsfk(t, y\-> y*)> k — 1, . . . , < / , on the product set (a, b) x K X K such that

and that if(t, ylt yz) e (a, b) x K x K, then

Infact,fk(t, ylt y2) is given by

Proof. Put F(s) =f(t, sya + (1 — s)yd for 0 s 1. The convexityof K implies that F(s) is defined. Then dF/ds = 2 (yz

k - y f ) df(t, syz +(1 - s^ldy". Hence F(l) - F(G) is the right side of (3.7) if/fc is definedby (3.8). Since F(l) =/(/, y2) and f(0) =f(t,y^ the lemma follows.

Proof of Theorem 3.1. Since (1.2) can be replaced by (1.4) and (y, z)by y, there is no loss of generality in supposing that/does not depend onz when proving the existence and continuity of the partial derivatives of»?.Thus the initial value problem (1.1) having the solution y = r)(t, f0, y0)

on

<u_ < t < co+ is under consideration.In order to simplify the domain on which the function rj must be con-

sidered, let a, b be arbitrary numbers satisfying ai_ < a +,a)

± ~ a)±(ti* yd- Then, by Theorem 2.1, r)(t, t0, y0) is defined and con-tinuous for a t b and (/0, y0) near (flf 3^). In-the following only suoh(t>to>yo) will be considered. Since the assertions of Theorem 3.1 are"local," it clearly suffices to prove the assertions on the interior of such a0» *o> y<»)-set.

k

derivatives dzrj/dyderivatives dzrj/dyderivatives dzrj/dyt

a

(a) In order to prove the existence of dy/dyf, let h be a scalar, evector in (3.2) and, for small \h\,

This is defined on a r 6 and, by Theorem 2.1,

uniformly on [a, A]. By (1.1), (yh(t) - y0(f))' =/(>,</A(0) -/(>,3/o(0).Applying Lemma 3.1 with y2 = yA(/), ft = y0(0>

Introduce the abbreviation

The existence of Br)(t, t0, y0)/ôfc is equivalent to the existence of lim xh(t)as A -» 0.

By (1.1) and (3.9), yA(/0) = y0 + /«?*, and so a:A(/0) = ek. Thus, by (3.11)and (3.12), x = xh(t) is the solution of the initial value problem

where J(t; h) is a d x */ matrix in which the &th column is the vectorfk(t,yo(t), yh(t))> By (3.6), the continuity off^y^yj and (3.10), itfollows that J(t;h) -» /(/; 0) as A -> 0 uniformly on [a, b], where /(/; 0) =J(0 is the matrix defined by (3.1).

Consider (3.13) to be a family of initial value problem depending on aparameter A, where the right side./(f; h)x of the differential equation is con-tinuous on the open-set a < t < b, \h\ small, x arbitrary. Since the solu-tions of (3.13) are unique, Theorem 2.1 implies that the general solutionis a continuous function of h [for fixed (t, f0)]. In particular, x(t) =lim xh(t), h -*• 0, exists and is the solution of (3.2) on a < t < b. Hence<ty(', 4). ôVô* exists.

In order to verify that this partial derivative is continuous with respectto all of its arguments, rewrite (3.2) as

a family of initial value problems depending on parameters (/0, y0). SinceJ(t> to, 2/o) is a continuous function of (t, J0, y0) and initial value prob-lems associated with linear differential equations have unique solutions,Theorem 2.1 implies that the solution x = dr)(t, t0, y0)jdy0

k of (3.14) is acontinuous function of its arguments.

y2

ko

(b) The existence and continuity of drj(t, t0t y0)ldt0 will now be con-sidered. Put

The solution of the initial value problem y' = /, y(t0) = ?/(f0, J0 4- h, y0)is the same as the solution of y' =/, y(/0 + h) = y0; i.e., ??(/, /0 -f A, y0) =*

*?('> 'o> »?('(» >o + h, y0)). Thus

Since ?y(/, r0, y0) has continuous partial derivatives with respect to thecomponents of y0 and r/(/0, ?0 + A, y0) -»• ?7(/0, 'o, Vo) = J^o as h -» 0, itfollows that

as A -+ 0. By the mean value theorem of differential calculus and y0 =**?( o + h, t0 + h, y0), there is a 0 = Bk such that

Note that ^'(r0 + 0/i, /0 + h, j/0) =/**(/„ + 6/r, rj(t0 + Oh, /0 + h, y0)) is/*('o, 0) + o(0 as A -> 0. Thus, as h -> 0,

This shows that dr)ldt0 = lim xh(t) exists as h -> 0 and satisfies the ana-logue of the relation (3.4). This relation imfSlies that d^/df0 is a con-tinuous function of (r, t0, y0).

(c) Returning to (1.2), so that/can depend on z and »? = »?(f, t0, y0, z),it follows that -r\ is of class C1. By applying the results just proved for (1.1)to (1.4), it is readily verified that the assertions concerning (3.2), (3.3),and (3.4) hold. This verification will be left to the reader.

The proof of Theorem 3.1 has the following consequence.Corollary 3.3. Let f(t, y, z, z*) be a continuous function on an open

(t, y, z, z*)-set E, where z* is a vector of any dimension. Suppose that f hascontinuous first order partial derivatives with respect to the components ofy and z. Then

has a unique solution »y = ??(/, f0, y0, z, z*) for fixed^z, z* with (/0, y0, z, z*)"eE; ?? has first order partials with respect to t, f0, the components of y andof z, and the second order partials 32^/9r 9f0, d^/df <W> ^nl^t dz';finally, these partials of r) are continuous with respect to (f, f0, y0, z, z*).

Proof. Since the partial derivatives of r\ involved in this statement arecalculated with 2* fixed, their existence follows from Theorem 3.1. Theircontinuity with respect to (/, /„, y0,2,2*) follows, as in the proof ofTheorem 3.1, by the use of analogues of (3.1), (3.2), Theorem 2.1, and ananalogue of (3.4).

4. Higher Order Differentiability

The question of higher order differentiability of the general solution iseasily settled by the use of Theorem 3.1 and Corollary 3.3.

Theorem 4.1. Let f(t, y, z, 2*) be a continuous function on an open(t, y, 2, z*)-set E such that f has continuous partial derivatives of all ordersnot exceeding m, m 1, with respect to the components ofy and z. Then

has a unique solution 77 = r)(t, t0, y0, z, z*), for fixed z, z* with (/<>, y0, z, 2*) EE, and rj has all continuous partial derivatives of the form

where i 1, /„ 1 and /'„ + 2/Sfc + See, m.Proof. The proof will be given first with i'0 = 0 by induction on m.

The case m = 1 is correct by Corollary 3.3 for 2* of any dimension.Assume the validity of the theorem if m is replaced by m — 1(^ 1).

Consider the analogue of (3.2),

where J = (dffdy) at y = r)(t, t0, y0, z, 2*). By the assumption on / andby the induction hypothesis, the right side, /(/, t0, y0, z, z*)y, of thedifferential equation in (4.3) has continuous partial derivatives of order•^m — 1 with respect to the components ofy, y0, and 2. Hence, by theinduction hypothesis, the solution x = Br](t, t0, y0, z, z*)jdy* of (4.3) hascontinuous partial derivatives of all orders ^ m — 1 with respect to thecomponents of y0 and each of these partials has a continuous partialderivative with respect to t. Similarly, the analogue of (3.3) shows thatdrjjdz* has continuous partial derivatives of all orders :5s m — 1 withrespect to the components of y0, and z, each of these partials has a con-tinuous partial derivative with respect to t. This completes the inductionand shows that ??(/, t9, y0, z, 2*) has continuous partial derivatives of theform (4.2) with /„ = 0, i 1, Sa* -f 2/3,. m.

The existence and continuity of derivatives of the form (4.2) with/„ = 1, / 1, Sot*. + £/?,. m — 1 follows from the analogue of (3.4).This completes the proof.

Corollary 4.1. Letf(t, y, z) be of class Cm, m^ 1, on an open (t, y, z)-set. Then the solution y = r)(t, t0, y0, z) of (1.2) is of class Cm on its domainof existence.

The proof of this useful corollary will be left as an exercise.

5. Exterior Derivatives

Several useful concepts will be introduced in this section. All conceptsare of a "local" nature.

By a (piece of) 2-dimensional surface S of class Cm, m^. 1, in a Euclid-ean y-space Rd is meant a set S of points y in Rd which can be put intoone-to-one correspondence with an open set D of points (M, v) in a Euclidean plane by a function y = y(u, v) of class Cm on D such that the twovectors dy/du, dy(dv are linearly independent at every point of D. Thefunction y = y(u, v) is called an admissible parametrizatibn of S.

If y = y(u, v) is any given function of class Cm, w ^ 1, on an open(M, i?)-set D such that dy/du, dyfdv are linearly independent at a point,hence near a point (w0, i>0) of D, then the set of points y = y(u, v) for(M, v) near (MO, #„) is a piece of surface. For by the implicit functiontheorem, the map (M, v) -*• y is one-to-one for (M, v) near (w0, v0).

Consider a piece of surface 50 of class C1 with an admissible parametriza-tion y = y(u, v) defined on a simply connected, bounded open set Z>0 and apiecewise C1 Jordan curve C in D0 bounding an open subset D of D0. LetS, J be the y-image of D, C, respectively. This situation will be describedbriefly by saying "a piece of C1 surface 5" bounded by piecewise C1 Jordancurve 7."

A differential r-form on an open set £ is a formal expression

with real-valued coefficients defined on E, where p{ ir(y) = ±pfi . ir(y)according as (jlt... ,jr) is an even or odd permutation of (ilt..., if).In particular, / > , - . < (y) = 0 if two of the indices / l s . . . , ir are equal.The form to is called continuous [or of class Cm or 0] if its coefficients arecontinuous [or of class Cm or identically 0] on £. A sequence of differen-tial r-forms on E is said to be uniformly bounded [or uniformly conver-gent] if the sequences of the corresponding coefficients are uniformlybounded [or uniformly convergent]. Differential r-forms can be addedin the obvious way. Differential r- and j-forms can be multiplied to give

v

an (r + .s)-form by the usual associative, distributive laws and anti-commutative law dyi A dy' = — dyj A dy^\ see § VI 2.

A continuous linear differential form (1-form or Pfaffian)

;>n E is said to possess a continuous exterior derivative da> if there exists acontinuous differential 2-form

on £" such that Stokes' formula

holds for every piece of C1 surface S in E bounded by a C1 piecewiseJordan curve J in E.

It is clear that if S is the image of D on the surface 50: y = y(u, v) for(u, v) e D, and J is the image of the Jordan curve C, then (5.3) means that

with the usual convention as to orientation of C around D.If the coefficients p.j(y) of (5.1) are of class C1, then at has a continuous

exterior derivative with da) = S dpfaj) A <fy* or

There are cases, however, where (5.1) has a continuous exterior derivativewhen the coefficients of (5.1) are only continuous. Consider, for example,the case that there exists a real-valued function U(y) of class C1 such thatCD = dU [i.e.,pJ(y) = dUJdy'], then o> has the exterior derivative da) = 0.

The fundamental lemma about the existence of continuous exteriorderivatives is the following:

Lemma 5.1. Let (5.1) be a continuous linear differential \-form on anopen set E. Then (5.1) has a continuous exterior derivative (5.2) on E ifand only if, on every open subset E° with compact closure £° c E, there existsa sequence of \-forms (o1, o> 2 , . . . of class C1 such that con -*• to as n —> oouniformly on E° and da)1, dco*, ... is uniformly convergent on E° (in whichcase, d(jon -> da) uniformly on E° as n -*• oo).

Proof. If a sequence w1, o>2 , . . . of the specified type exists on E°and if the case eo = <wn of (5.3) is written in the form (5.4), an obvious

term-by-term integration gives (5.3). Thus, it is seen that the existenceof sequences to1, o> 2 , . . . is sufficient for the existence of a continuous do>.

Conversely, if (5.1) is a continuous 1-form on E with a continuousexterior derivative (5.2), approximate the coefficients of w, dot by themethod in §13: Let <p(t) be as in § I 3 and put, for e — l/n,

Since the integrals are actually integrals over spheres \\r)\\ ^ e, p^n) andpW are defined on the sets En consisting of points y whose (Euclidean)distance from the boundary of E exceeds e = 1/w. In particular, they aredefined on £° for large n and tend uniformly to pit piit respectively, on£°, as n —*• oo.

Define the C°° forms con = £/><.n)(y) dy> and <xn = SSp{f> (y) dyi A dy*on £"° for large n. Let S be a piece of C1 surface in E° bounded by a C1

piecewise Jordan arc J in E°. Then if « = l[n is sufficiently small and\\i)\\ ^ e, the translation S(ri) of S" by the vector ~TJ is in E and (5.3) isvalid if S is replaced by S(rj). This can be written in a form analogous to(5.4),

Let this relation be multiplied by ce~*V(e~2 ||»?||2) and integrated over1171| ^ e with respect to drf ... drf. An obvious change of the order ofintegration shows that the result can be interpreted as the Stokes' relation

Thus con has the continuous exterior derivative dot71 — ocn

in £Q. This completes the proof.Remark. In deciding whether or not a continuous 1-form (5.1) has the

continuous exterior derivative (5.2), it suffices to verify Stokes* formula(5.3) for rectangles S on coordinate 2-planes yi = const, for / & /, k,where 1 j < k ^ d. This is a consequence of the following exercise.

Exercise 5.1. A continuous differential 1-form (5.1) on an open set £has a continuous exterior derivative if and only if there exists a continuousdifferential 2-form (5.2) such that for every pairy, k (1 j < k ^ d) andfixed y\ with i y^j, i kt the 1-form pfy) dy' + pk(y) dy* has the con-tinuous exterior derivative pjk dy' A dy* + pki dyk A dyj; in fact, if andonly if Stokes' formula (5.3) holds for all rectangle 5 on 2-planes yi =•const, for / ^ jt k with S <=• E.

Exercise 5.2. Let the continuous differential 1-form (5.1) possessa continuous exterior derivative and let/^y) have a continuous derivativewith respect to y* for a fixedy 5* 1. Show that/j/y) possesses a continuouspartial derivative with respect to y1.

Exercise 5.3. Let />,(/), where y = 1,. .., d, be continuous functionson E such that /?/y) has continuous partial derivatives with respect to thecomponents y*, k ^ j, of y. Show that (5.1) has a continuous exteriorderivative.

For the sake of brevity, "vector" and "matrix" notation will be usedin connection with 1-forms and their exterior derivatives. For example,an ordered set of e 1-forms cy 1 ? . . . , cae will be abbreviated CD = (<olf...,cog); analogously if these forms have continuous exterior derivatives,dco denotes the ordered set of 2-forms dot = (dat^ ..., da)e). Finally,if A = (a^yj) is an e x d matrix function on E, by co = A(y) dywill be meant the ordered set of 1-forms co = (<u1 , . . . , coe), where to, =

d2 <*M W'for i = 1, ...,*.

j=i

6. Another Differentiability Theorem

The main result (Theorem 3.1) on the differentiability of general solu-tions has the following generalization.

Theorem 6.1. Let f(t, y, z) be continuous on an open (t, y, z)-set E. Anecessary and sufficient condition that the initial value problem

have a unique solution y = rj(t, t0, y0, z) for all (t0, y0,2) £ E which is ofclass C1 with respect to (t, /„, y0, z) on its domain of definition is that everypoint of E have an open neighborhood E° on which there exist a continuousnonsingular d x d matrix A(t, y, z) and a continuous d X e matrix C(t, y, zsuch that the d differential l-forms

in the variables dtt dy1,..., dy*, dz1,..., dze have continuous exteriorderivatives on E°.

In contrast to Theorem 3.1, the conditions of Theorem 6.1 are invariantunder C1 changes of the variables /, y, z.

It is understood that if A — («„(/, y, z)) and C — (cik(t, y, z)), then (6.2)represents an ordered set of l-forms, the /th one of which is

/ = ! , . . . , < / . If this has a continuous exterior derivative, the latter is adifferential 2-form of the type

where a,,t = —aiw, /9tJ, yifc, diik, eiik = — eifc?- are continuous function of(/, y, z). In this case, define a d x d matrix F(t, y, z) — (fti(t, y, z)) andad x e matrix N(t, y, z) = («„('. y, z)) by

Theorem 6.1 will be proved in § 11 below. The proof of Theorem 6.1will have the following consequence.

Corollary 6.1. Letf(t, y, z) be as in Theorem 6.1, let A(t, y, z), C(t, yy z)exist on E° as specified, and consider only (t, y, z) e E°. Then x = dr}(dy0

k

is the solution of

and x = drjjdz* is the solution of

where ct(f, y, z), «f(/, y, 2) fl/'e the ith columns of C(t, y, z), N(t, y, z), re-spectively, for i = 1, .. ., e am/»; = »/(/, r0, y0, z).

Note that a solution x = a;(/) of (6.7) does not necessarily have a de-rivative, but A(x, rj, z)x(t) has a derivative satisfying (6.7). An obviouschange of variables reduces the linear equations in (6.7) and (6.8) to thetype considered in Chapter IV; cf. (11.1H 11.3).

The statements concerning (6.7) and (6.8) can be written more con-veniently as matrix equations

where d^dy0, drjfdz denote Jacobian matrices.

Exercise 6.1. Let f(t, y, y') be a continuous ^/-dimensional vectordefined on an open (/, y, t/')-set £. Let (ilt..., id) be any set of d integers,0 /',- ^ d, such that no integer except 0 occurs more than once. Lett = y° and suppose that fk(y°, y, y') has continuous partial derivativeswith respect to each of its arguments except possibly yik. Then the initialvalue problem

where y'* 0 if ik y± 0, has a unique solution y = rj(t, t0, y0, y0') andn(t, *0, y0, y0'), ri'(t, tQ, yQ, y0') are of class C1.

The following two exercises are applications of Theorem 6.1 andCorollary 6.1 to differential geometry.

Exercise 6.2. (a) Let (&*(*)), where x = (a:1,..., a^), be a d x dnonsingular symmetric matrix with real entries which are functions ofclass C1 for small \\x\\ and let (gik(x)) be the inverse matrix. Consider theinitial value problem

for the geodesies of dsz = 2S </a^ dfc*, where rjfc = Tj a;) are theChristoffel symbols of the second kind defined by

Assume that "<&2 has a continuous Riemann curvature tensor" in the sensethat each of the d* differential 1-forms co/ = Sfc Tjk dx* has a continuousexterior derivative. Show that (6.11) has a unique solution x = £(t, XQ, x0')for small \t\, \x0\ and arbitrary x0' and that £(f, a;0, ar0'), f'(/, a;0, «„') are ofclass C1 as functions of their 2d -f 1 variables. (Z>) Let z = (z1,..., 2')and a: = (a;1,..., x*), where e ^/, and let z = z(z) be a function of classC2 for small ||a;|| with a Jacobian matrix (dz'jdx*) of rank </. Show that (a) isapplicable to ds2 = ||</z||2 = US gik dxj dx*, where ,-fc is the scalar product(3z/a*0 • (32/ax*). (c) Show that ds* = [1 4- 9(a;2)4^] [(dx^ + (flfo2)2],where d = 2, has more than one geodesic through the point x0 = 0 inthe direction XQ' = (1,0).

Exercise 6.3. Let (hjk(y)}, where y = (y1, y2), be a 2 X 2 symmetricmatrix of real-valued functions of class C1 for ,small ||y|| such that det(hjk) < 0- Let a denote the indefinite quadratic form

Then a has factorizations a = 2co1a)2, where ft>ls to2 are linearly indepen-dent differential 1-forms. (Note that a is not a differential 2-form and


a = 2a>iQ)z is an ordinary, not exterior, product.) (a) There exists a uniquecontinuous differential 1-form o>12 such that

(b) Assume that a has a continuo curvature K = K(y) in the sense thatthe factors o>x, o>2 can be chosen so that o>12 has a continuous exteriorderivative, in which case, K is defined by

Show that there exist functions y(u) = y(ul, «2) of class C1 for small ||M||such that y(0) = 0 and y = y(ti) transforms a into the form

a = 27W du\ where T = T(u) > 0

is of class C1 and has a continuous second mixed derivative such that

(It can be shown that y = y(u) is of class C2; Hartman [16].) (c) Showthat (b) is applicable if a = 0 is the differential equation for the asymptoticlines on a piece of surface of class C3 of negative curvature in Euclidean 3-space. (d) Show that (b) is applicable if a = 0 is the differential equationfor the lines of curvature on a piece of surface of class Cs without umbilicalpoints in Euclidean 3-space.

7. S- and £-Lipschitz Continuity

The proof of sufficiency in Theorem 6.1 falls into two parts: uniquenessand differentiability. It turns out that the necessary and sufficient condi-tion of Theorem 6.1 can be lightened considerably for "uniqueness" alone.Consider the case in which no parameters z occur,

Correspondingly, A = A(t, y) and the analogue of (6.2) is

Here, co — (tolt..., tuj), where

and when dco exists, it is of the form

108 Ordinary Differential Equ ions

where <x.iik = —aiJW, /?„ are continuous functions of (/, y). Correspondingly

The notion of a differential 1-form co with a continuous exterior deriva-tive generalizes the notion of a C1 form co. Lemma 5.1 suggests thefollowing generalization of a 1-form co with uniformly Lipschitz continuoucoefficients:

A continuous linear differential form (5.1) on a domain E will be saidto be S-Lipschitz continuous on E if there exists a sequence of 1-formsco1, co2,... of class C1 on E such that con -> co as n —*• oo uniformly on Eand dco\ dco\ ... are uniformly bounded on E.

Exercise 1.1. Show that if the coefficients of (5.1) are uniformlyLipschitz continuous on E, then (5.1) is S-Lipschitz continuous on E.

Exercise 7.2. Consider the case of dimension d — 2. Let

be continuous on a simply connected, open (y1, y2)-set E with the propertythat there exists a bounded, measurable function/>12(y) on E such that forany subset S of E bounded by a C1 piecewise Jordan curve J in £",

Show that co is S-Lipschitz continuous on every open subset E0 withcompact closure £0 c: £. (It is understood that "measurable" meansmeasurable with respect to plane Lebesgue measure.) One might say thatco has a "bounded exterior derivative." This notion does not generalizereadily to arbitrary dimensions d, for a piece of (2-dimensional) surfaceS is of ^/-dimensional measure zero if d > 2.

The condition that each of the d forms of (7.2) is S-Lipschitz continuouscan be generalized as follows: Letf(t,y) be continuous and A(t, y) becontinuous and nonsingular on E. The form (7.2) is said to be L-Lipschitzcontinuous on E if there exists a sequence of forms co1, co2 , . . . of the type

of class C1 on E such that con -> co uniformly on E as n ->• oo, and thatthere exist constants c0, c satisfying

on E. Here Fn = Fn(t, y) is the matrix belonging to (7.6) defined by theanalogue of formulae (7.3)-(7.5). The inequalities in (7.7) have the follow-ing meaning: if B, C are two d X d matrices, B 51 C means that the

corresponding quadratic forms satisfy £ • Bl; I • C£ for all real d-vectors £, where the dot indicates scalar multiplication. B :g C is equiv-alent to BH ^ Cff, where BH = £(5 + B*) is the Hermitian part of B.

The form (7.2) is called upper L-Lipschitz continuous on E if (7.7) isreplaced by

Consider these conditions when A,f are of class C1. In this case,the exterior derivative of (7.2) can be calculated formally from do) =(</4) A (dy — fdi) — A df t\ dt which gives

Hence, (7.5) shows that

If 5 = A*F = (&„) and G = A*A, then

where // = (Aw) is skew-symmetric. This can be seen as follows: thepartial derivative of G — A*A with respect to I is G' = A*A' -f A*'A,so that the Hermitian part of A*A' is \G'. This accounts for the replace-ment of the first term of (7.10) by that in (7.11). The same remark appliesto the third terms involving differentiation with respect to yk instead of t.For other applications of (7.11), see § XIV 12.

Exercise 7.3. Let /(/, y) be continuous on E and satisfy [/(/, y2) —/('»2/i)l' (#2 — 1) 0- Show that at = dy — f(t, y) dt is upper L-Lipschitz continuous with An = / in (7.6) and c = 0 in (7.8).

8. Uniqueness Theorem

A generalization of the uniqueness theorem contained in Theorem 6.1is the following:

Theorem 8.1. Let f(t, y) be continuous on an open (t, y)-set E and letthere exist a continuous, nonsingular matrix A(t, y) on E such that thel-forms (7.2) are S-Lipschitz continuous on E or, more generally, that (7.2)is L-Lipschitz continuous on E. Then (7.1) has a unique solution y —»?(>, 'o> 2/o) f°r <dl (to, y<>) 6 E. Furthermore, r}(t, t0, y0) -is uniformly Lip-schitz continuous an cbmpact subsets of its domain of definition.

,

This theorem will be proved in § 10 below. It is possible to formulate aone-sided uniqueness theorem analogous to Theorem III 6.2. This isgiven by the following exercise.

Exercise 8.1. Let/, A be as in Theorem 8.1 except that (7.2) is supposedto be only upper L-Lipschitz continuous on E. Then (7.1) has a uniquesolution y = r)(t, t0, y0) to the right (/ ^ t0) of /„ for all (/„, y0) e E.Furthermore, an inequality of the type

holds for / ^ /* max (tlt /g) on compact subsets of the domain ofdefinition of rj(t, /„, y0)-

Exercise 8.2 (Another One-Sided Generalization of Corollary HI 6.1).Let/(f, y) be continuous on R: Q t a,\y\ ^ b. On R, let there exista continuous, nonsingular 4(r, y) such that (7.2) is upper L-Lipschitz con-tinuous on R€: (0 <) e < f ^ a, |y| < 6 with (7.8) replaced by

for n = 1, 2, . . . . Then (7.1) has a unique solution for t0 = 0, |y0| < b.

9. A Lemma

The proof of Theorem 8.1 will depend on the Uniqueness TheoremIII 7.1 and the following lemma.

Lemma 9.1. Let f(t, y) be of class C1 and let A(t, y) be a nonsingularmatrix of class C1 on an open set E such that

where fi(t) is a continuous function. Let y = rj(t) — r\(t, t0, t/0) be a solutionof (7.1) and J(t, y) the Jacobian matrix (dfjdy). Then a solution x(i) ofthe linear "equations of variation"

satisfies, for t t0,

Proof. A differentiation of A(t, r)(t})x(t), using (9.2) and (7.10), showsthat (9.2) implies that

where the superscript 0 indicates that the argument is (/, y) = (/, T?) with77 = 77(0 = r)(t, /0, y0). Thus

since A°x • F°x = x • (A°)*F°x. Note that \\A°x\\z = A°x • A°x = x • (A0)*A°x. Hence (9.1) and (9.5) imply that d \\A°x\\*ldt ^ 2p(t) \\A°x\\* and soa quadrature gives (9.3); cf. Lemma IV 4.2.

Note that if

then — cm A* A ^ A*F cm A* A. In this case, (9.3) and the correspond-ing inequality for / ^ /„ show that

where rj(t), hence x(t), is defined. In particular, since x = drj(t, t0, y0)ldy0k

is a solution of (9.2) by Theorem 3.1, (9.6) and (9.7) imply that

Estimates for dr}(t, f0, y0)ldt0 follow from the analogue of (3.4),

10. Proof of Theorem 8.1

Consider first the case that/, A are of class C1. Let (flt yj) E E and let£° be a convex open neighborhood of (tlt yj with compact closure £° <=• E.Then, by Peano's existence theorem (Corollary II 2.1), there exists an openset £"0 c E° and numbers a, b (> /t > a) such that if (f0, y0) e £„, then»?('> 'o» y<>) exists for a / ^ 6 and (/, rf) e £"°. Note that £0, a, fc dependonly on £° and a bound for |/| on J?0.

Suppose, in addition, that (9.6) and (9.7) hold on £°. Then (9.9) and(9.10) show that there exists a constant AT (depending only on £°, a boundfor |/1 on £° and on m, c) such that

foTa^t<b, (/0, y0) 6 £0, (r, y°) 6 £0;

for a r ^ 6, ((0, y0) e £0> (^°, 2/o) 6 A; and

for a<t,t*^b, (/0, y0) e £0. Finally, interchanging / and t0 in (10.1)gives

provided that a^t^b, (t, rj(tt t0, y0)) 6 £0, (/, ??(/, /„, y0)) e £0. There isa neighborhood £,„, <= £"„ of (t1} yj such that the last proviso [for (10.4)]is satisfied if the interval [a, b] is sufficiently small and (f0, y0), (t0, y°) e Ew.

It must be emphasized that [a, b], £"0, Ew and the constant K in (10.1)-(10.4) depend only on E°, a bound for |/| in £°, and the validity of (9.6)-and (9.7) on E*.

Return to the case that .4,/are not necessarily of class C1; instead A,fare only continuous, A is nonsingular and (7.2) is L-Lipschitz continuouson every open set E° with compact closure £° <=• E. Then there exists aseque of d ordered 1-forms

on E° with C1 coefficients such that An-+ A, hn-+ Af uniformly on E°as n —»• co and (7.7) holds on £°. Since A is nonsingular, /4n(f, y) is non-singular on E° for large n, say, for all n, and o>(n) can be written as

where/n = A~lhn -^/uniformly on £°as n -+• oo. It can be supposed that(7.7) holds on £° with c0 = — c 0; also that there exists an m > 1 suchthat

on£°.For (/!, yj) e E, let £0 in the last paragraph be any convex open

neighborhood of (tlt y^ with compact closure E° <=• E. Since f » f i , f * ...are uniformly bounded for (/, y) e E°, there is an open neighborhood E0 of('i» &i) sucn that any solution y = y(r) of (7.1) or of

for (t0, y0) 6 £0 exists and (/, y(tj) GE°fora^t<b, where a, 6 (>f t > a)are independent of « and the solution y = y(r). Thus there exists a A,independent of «, such that if y = »/„(/, /0> yo) ^s tne solution of (10.7),then (10.1)-(10.4) with r\ = v\n. In particular, the sequence rj^t, t0, y0),rj2(t, t0, y^,... is uniformly bounded and equicontinuous for a t b,('o> 2/o) 6 o- Thus there exists a subsequence, which after renumberingcan be taken to be the full sequence, such that

exists uniformly for a / b, (/0, y0) e 0.

By Theorem I 2.4, y = r\(t, f0, y0) is a solution of (7.1) for a / ^ b.Also, (10.1) and (10.4) hold under the conditions specified on /, /0, y0, y°.The inequality (10.4) implies that if (f, f0, y0) is sufficiently near to (tlt tlt yj,then no two distinct arcs y =* »?(/, t0, y0) pass through the same point(t, y). Hence, by Theorem III 7.1, y = »/(/, /t, y^ is the only solution of(7.1) with (/0, y0) = (/t, yj on small intervals [/! — *,/J, [/i,/i + «].Since (/,, y^ is an arbitrary point on £, Theorem 8.1 follows.


Sufficiency. It is assumed that there exist continuous A, C such that(6.2) has a continuous exterior derivative and det A ^ 0 in a vicinity of apoint of £. It will first be shown that it is sufficient to consider the casethat/does not depend on z. To this end, write (6.1) as (1.4) and let(y, z) be replaced by y,,, so that correspondingly (/, 0) is replaced by /„,.Let Aj.(t, y+) be the matrix

and a)+ — A+(dy+ — /„ dt). Then &>„ = (o>, dz), where w is given by (6.2).Hence co* has continuous exterior derivatives. Thus, if the asterisks areomitted from y*,/#, A+, it is seen that it is sufficient to consider the casethat/ = /(f, y) does not depend on z.

Let (/n yj), £°, £o» fl> > (10.5), »7w(f, J0, y<>) De as 'n tne proof of Theorem8.1. Then (10.8) holds uniformly for a / ^ b, (/„, y0) e£0. Also, inobvious notation, z = 9^n/^o* 's tne solution of

cf. the derivation of (9.4) from (9.2). Introducing the new variables

shows that

Since An(t, r}n)-+A(t, y), Fn(t, yj -+ F(t, 77), /4-J(/, »?B) -^ ^~l(^,»?) asn-+ <x> uniformly for a r < A, (f0, y0) e £0, Corollary IV 4.1 shows that,for fixed (f0, y<>) e o* l'm ^«('» '/n) fyjdyo* exists uniformly for a / < bas n—> <x> and is the solution of

Actually this limit is uniform for a ^ t !>, (?0, y0) 6 J?0. This can be seen,e.g., by Theorem 2.1, by constructing a family of linear differentialequations x*' — ff(t, t& y0, «)»*, where H(t, t0, y0, t) is a matrix continuous

in (/, /0, y0, > which becomes Fn(t, r]^A~l(t, r)n) for e = l//i and F(t, rj)A~\t, rj) for e = 0.

If follows that

for a t b, (t0, y0) e £0 and is the solution of

Consequently, a standard term-by-term differentiation theorem impliesthat drfldy0

k exists for a / b, (t0, y0) e £0 and is the limit (11.5). Asin the proof of Theorem 3.1, it is seen that dr)/dt0 exists and is given by theanalogue (9.10) of (3.4).

This proves that»?(/, /0, y0) is of class Cl if (/, t0, y0) is sufficiently near to('i» 'i> yi)» where (tlt yj is an arbitrary point of E. If now a and £ arechosen arbitrarily, subject only to the condition that»?(/, tlt yx) exists fora ^ t b, then a finite number of applications of formulae of the typey(t, /„, y0) = rj(t, t*, rj(t*, /0, y0)) shows that ??(/, /0, y0) is of class Cl onits domain of existence.

All assertions of Corollary 6.1 except that concerning (6.8) also havebeen verified. The verification of this assertion will be left as an exercise.

Necessity. Assume that (6.1) has a unique solution r) — q(t, t0, y0, z)which is of class C1. Let (tlt ylt Zj) e £ be fixed. Then there is a neigh-borhood E° of (f,, yl5 Zj) such that rj(t, /0, y0, z) exists on an intervalcontaining 70, /j if (/0, y0, z) e E°. Also, since the Jacobian matrix drjldy0is the identity matrix at (/, /„, y0, z) = (/1} tlt ylt Zj), it can be supposedthat £° is so small that det (etyfo, tQt y0, z0)/3y0) 0 for (/„, y0, z0) e £°.

Consider the function ^(/lt /, y, z) of (/, y, z) e £° for fixed t^ Then

where (3.4) has been used with (/0, y0) replaced by (/, y). Thus, if A(t, y, z)= (fyfiyo) at (/, t0, y0, z) = (/lf /, y, z) and C(t, y, z) = (dT/fo, /, y, z)/3z),then CD in (6.2) becomes <o — dr)(tlt t, y, z) which has the continuousexterior derivative d(o = 0. Also, det A ^ 0. This completes the proof.

12. First Integrals

Consider a system of differential equations

in which/is continuous on an open set E. A real-valued function tt(t, y)defined on an open subset E0 of E is called & first integral of (12.1) if it is

exists uniformly

constant along solutions of (12.1). i.e., if y — y(t) is any solution of (12.1)on a ^-interval (a, b) such that (t, y(t)) e £0 for a < t < b, then u(t, y(t)) isindependent of /.

Lemma 12.1. Let u(t, y) be a function of class C1 on an open set £0 <=• £.Then u(t, y) is a first integral of (\2.\) if and only if it is a solution ofthe linear partial differential equation

In fact, (12.2) is equivalent to du(t, y(t))ldt = 0 for all solutions y —y(t) (12.1) such that (/, y(t)) e E0.

Th orem 12.1. Let u = rf(t, y), rf(t, y),..., r/d(t, y) be first integralso/(12.1) of class C1 on an open subset £0 c: Esuch that theJacobian matrix(dy/dy) is nonsingular, where r\ = (77*, . . . , rf). Let (/0, y0) e £o> »?o =iK*o» y<>)> and y = y(t, rj) the function inverse to rj = rj(t, y) for (t, rj) near(^o» %)• Then, for fixed 17, y = y(f,»?) w a solution of(12.1). Furthermore,if(*o, Vo) e £0 °nd u(t, y) is of class C^for (t, y) near (/0, y0), then u(t, y) is a

first integral for (12.1) if and only if there exists a function U — U(rj) ofclass C1 for r\ near i)0 such that «(/, y) = U(r)(t, yfifor (t, y) near (/0, y0).

Proof. Since u = *y'(f, y) for i = 1,..., d is a first integral of (12.1),it follows from (12.2) that

Hence (dr)/dy)-1(dr)ldt') -f/= 0. Since y = y(t, rj) is inverse to rj = rj(t, y\it is seen that y' = dyjdt = —(dr)ldy)-l(dr}ldt), so that y — y(tt rj) is asolution of (12.1) for fixed rj. This proves the first part of the theorem.

If U(rj) is of class C1, it follows readily from the criterion (12.2) that"(A y) = U(ri(t, y)) is a first integral. Conversely, let u(t, y) be a firstintegral for (/, y) near (f0, y0) and put U(rj, t) = u(t, y(t, 77)). Clearly,w(A y) = U(rj(tt y), t). Thus it sufficies to verify that U(r), t) is independentof /. But dU/dt = du/dt + "LtfulWW which is 0 by (12.1) and (12.2).This proves the theorem.

Theorem 12.2. Letf(t, y) be continuous on an open set E. Then, for an('o> Vo) e E, (12.1) has d first integrals r\ = rj(t, y) of class C1 on a neighbor-hood of (t0, y0) satisfying det (dy/dy) 5^ 0 // and only if the initial valueproblem y' —f, y(fc) = y0 has a unique solution y = r](t, t0, y0) of class C1

(with respect to all of its variables.)Proof. If y = rj(t, t0, y0) exists and is of class C1, put rj(t, y) = 7?(f0> t, y)

for (/, y) near (/0, y0) and fixed tQ. Then each component of r)(t, y) is afirst integral, fo* rj(t, y(t)) is the constant y(t0). Also (dr)(t, y)ldy) is theunit matrix at t — /„, hence nonsingular for / near /„.

Conversely, if the components of r) = rj(t, y) are first integrals of classC1 on a neighborhood of a point (/0, y0) of E, det (drjldy) -^ 0 and y =y(t, rj) is the inverse function, put rj(t, tsolution of the initial value problem y' =/, y(f0) = y0 and is of class C1.For fixed /0, it follows that drj(t0, t, y) = (Br)(t0, t, y)ldy0)[dy -f(t, y) dt\\cf. (11.7). Thus Theorem 6.1 implies that the solution y = rj(t, t0, y0) of

y' = /» yC'o) = 2/o is unique and of class C1.

Notes

SECTION 3. Theorem 3.1 was first proved for d = 1 by the method of successiveapproximations independently by Picard (see Darboux [1, p. 363]) and Bendixson [1].(It had been proved earlier by Nicoletti [1] assuming an additional Lipschitz conditionon the partial derivatives of/.) Theorem 3.1, without the parameters z, was proved byPeano [3] using a method similar to that in the text (except that, instead of usingTheorem 2.1, he employed an estimate of the type |x»(/)| exp c \t — ta\ for (3.12) wherec is related to bounds for Idffdy'l; cf. Lemma IV 4.1). The result was rediscovered byvon Escherich [2], using the method of successive approximations, and by Lindelfif [2],using a method similar to Peano's. In [1], Hadamard indicates a proof similar to thatin the text; Lemma 3.1 is given in Hadamard [3, pp. 351-352] with a different proof.

SECTION 5. The definition of a continuous exterior derivative was given by E. Cartan[1, pp. 65-71 ], Lemma 5.1 is due to Gillis [2] (and, in a more general form, was used byhim to answer affirmatively the question of E. Cartan whether a>T A <o, has a continuousexterior derivative if the r-form tor and s-form a>, have continuous exterior derivatives).The proof of Lemma 5.1 in the text is that of H. Cartan [1, pp. 62-63].

SECTION 6. Theorem 6.1 is in Hartman [17]; the proof in the text follows Hartman[26]; cf. [14] for the case d = 1. For Exercise 6.2 and generalizations to extremals,see Hartman [17]. For Exercise 6.3, see Hartman [141

SECTIONS 7 AND 8. See Hartman [26].SECTION 9. See Hartman [26]. Lemma 9.1 is essentially in Lewis [2] (cf. Opial [8]);

for a generalization see Lewis [3].SECTION 12. Theorem 12.1 goes back to Lagrange's work in 1779; cf. [2, IV pp.

624-634].

Chapter VI

Total and Partial Differential Equations

This chapter treats certain problems involving partial differential equationswhich can be solved by the use of the theory of ordinary differentialequations. Parts I and II are independent of each other.

PART I. A THEOREM OF FROBENIUS

1. Total Differential Equations

Let H(y, z) be a continuous d x e matrix on a (d + e)-dimensional openset £, say H = (//„). Consider the set of total differential equations

and initial condition

for some (y0, z0) 6 E. Equation (1.1) is an abbreviation for the set ofpartial differential equations

If e = 1, then (1.3) is a set of ordinary differential equations; existenceand uniqueness theorems for corresponding initial value problems aresupplied by earlier chapters. If e > 1, then in general we cannot expect(1.1) to have solutions. For example, suppose that H(y, z) is of classC1. If a C1 solution y = y(z) of (1.1) exists, then (1.3) makes it clearthat y(z) is of class C2 [since the right side of (1.3) is of class C1]. Butthen dy/dz3 dzm = dy/dz"1 dz' and this leads to the condition

for / = 1, . . . , </andy, w = 1 , . . . , e, which must hold along the solution(y, z) = (y(*\ z). Thus, in this case, a necessary condition that (1.1)-(1.2)

117

have a solution y(z) = y(z, z0, y0) for arbitrary points (y0, z0) of E, is thatthe "integrability conditions" (1.4) hold as an identity on E. When H is ofclass C1, it will be seen that this necessary condition is also sufficient for theexistence of y = y(z, z0, y0) and, furthermore, in this case the solution of(!.!)-(1.2) is unique, and y(z, z0, y0) is of class C1 with respect to all of itsarguments.

Instead of dealing with (1.!)-(!.2) directly, it is more convenient toconsider

where A(y, z) is a continuous, nonsingular d X d matrix, and to pose thefollowing problem: When does there exist a function

of class C1 on a (d + e)-dimensional neighborhood of a point (^0, z0)such that

and that (1.6) transforms (1.5) into a differential for

in dr\ = (drj1, . .., </7yd) with coefficients depending, of course, on (77, z).(The insertion of ^4(y, z) in (1.5) is for convenience only and does notaffect the problem.) When y = y(r], z) exists, the system (1.5) will be saidto be completely integrable at (y0, z0).

If y = y(rj, z) is of class C1 and dy = (dy/drf) dr\ + (dyjdz) dz is insertedinto (1.5), it is seen that the result is of the form (1.9) if and only if

in which case

In particular, (1.8) implies that /)(??, z) is nonsingular.The reduction of (1.5) to a form (1.9) is equivalent to satisfying (1.10),

i.e., (1.1) or (1.3). Hence, the complete integrability of (1.1) or (1.5) isequivalent to the existence of a family of solutions y = y(*?»2) °f 0-0depending on parameters 77 = (q1,..., rf) and satisfying (1.7), (1.8).

The question of the existence of (1.6) can be viewed from a slightlydifferent point of view: When does eo in (1.5) possess a local "integrating

Total and Partial Differential Equations 119

factor", i.e., when does there exist a nonsingular continuous matrix £(y, 2)such that £(y, z)a> is a total differential drj(y, 2), £(y, z)<u = dr\. It is clearthat E(y, z) exists if and only if the problem just posed has a solution; inwhich case, E(y, z) = D~l(r)(y, z),z) where rj(y, z) is the inverse function ofy = y(n, z).

Finally, the question of the existence of (1.6) can be dealt with in stillanother way. Consider the problem of finding real-valued functionsu = u(y, z ) o f d + e independent variables satisfying e simultaneous linearpartial differential equations

where H(y, z) is a continuous d x e matrix on an open set E. The system(1.12) is called the system adjoint to (1.1).

The system (1.12) is said to be complete on E if there exist d solutionsu = (y, 2) , . . . , rj*(y, z) on £ such that the rank of the Jacobian matrix(dy/dy, drjjdz) is maximal (i.e., d) at every point. Actually, (1.12) showsthat this rank condition holds if and only if (dy(y, z)fdy) is nonsingular.If d such solutions r)1,..., rf exist, the system (1.12) can be written as

Multiplication by (drjjdy)'1 shows that this is equivalent to

The condition del (dy/dy) •£ 0 implies that r\ = -r\(y, z) has a local inverseV — y(n, z) of class C1 and so, dy/dz = —(dr)ldy)~l(dr)jdz). Thus (1.14) isequivalent to (1.10), i.e., (1.3). This argument can be reversed. Hence thecomplete integrability of (1.1) or (1.5) at (y0,z0) is equivalent to thecompleteness of (1.12) on a neighborhood of (y0, z0). In particular, whene > 1 and H(y, z) is of class C1, (1.4) is therefore necessary and sufficientfor completeness of (1.12) on a vicinity of every point (y0,20) e £. Thecondition (1.4) can be written as

by virtue of (1.12)The arguments of § V 12 show that if (1.1)-(1.2) has solutions for all

(yo» zo) e £, then a function w(y, 2) of class C1 on £(1.12) if and only if it is a "first integral" of (1.1), i.e., u(y(z\ z) = const,for every solution y = y(z) of (1.1) for which (y(z\ 2) e £„.

A system (1.1) or equivalent system (1.10) for which the integrability

).

i

conditions (1.15) are satisfied is called a Jacobi system. Theorem 3.2 is anexistence and uniqueness theorem for such systems. In Theorem 3.2, itis not assumed that H(y, z) is of class C1, so that integrability conditionstake a form different from (1.15), in fact, a more convenient algebraicform [so that no calculations involve the rather formidable relations (1.4)].This theorem will be deduced from Theorem 3.1 which treats the questionof the complete integrability of (1.1) or (1.5).

Exercise 1.1. (a) Let blk(y\ ..., bdk(y) f o r k — 1, 2 be 2</functions ofclass C1. Define the partial differential operators Xk[u] = S, bjk(y) dujdyj

for k = 1,2 and the operator Xn[u] = S, (Jr,[6w] - Xv[bv]) du/dy*. Showthat Xu is the commutator of X^ Xt in the sense that if u(y) is of class C2,then Xlt[u] = X^Xû)] - X^Xû)]. (b) Show that if u(y) is of class C1

and Xk[u] = 0 for k — 1,2. Then A^fw] = 0. See Exercise 8.2(£) for ageneralization.

Exercise 1.2. Let 5(ar) = (bti(x)), where / — 1,... ,d + e and y =1,... , e, be a (</ + e) x * matrix of rank e, continuous on an open setE in the (d + e)-dimensional z-space, x = (x1,..., a^+'). Define thedifferential operators Yû] — bti(x) dujdx4 for j — 1,..., e. Thesystem (*) Y}[u] = 0,y = 1, . . . , e, is called complete on E if there exist dsolutions u = r)l(x),..., if(x) of class C1 such that the (d + e) x dJacobian matrix (efy*/^) is of rank d. Using Theorem 3.1 show that ifB(x) is of class C1, then the system (*) is complete on a neighboorhood ofevery point x0 e E if and only if the commutator Yik of Yit Yk (cf. Exercise1.1) is a linear combination of Ylt..., Ye [i.e., if and only if there existfunctions cjkm(x), fory, k, m = 1, . . . , e, such that Yik = 2m c>fcmrm or,equivalently, Yk[bti] - Y^] s ETO cikm(x)bim(x) for;, A: = 1, . . . , e and/=! , . . . ,< /+*] .

2. Algebra of Exterior Forms

In order to be able to write integrability conditions in a convenient form,some simple facts about exterior forms will be recalled.

Let 0 < r is */and Crd = </! /r! (d — r)! Consider a vector space ff overthe real field of dimension Crd with basis elements et ,, where»i1 < / ! < / , < • • • < /r . Thus any vector CD in the space W has aunique representation

where c,x . . . {f are real numbers. Introduce the symbols efi... /r forji, • - •, 7r = 1» • • • » r f » where ^ . . . *r - 0 if two indices 7;,...,/ areequal and e^ . . . tf = ±eit... if according as (/,,... ,yr) is an even or oddpermutation of (ilt..., /r), 1 / ! < • • • < /r </. Then any vector o> has

a unique representation of the form

subject to the conditions

where (jlt... ,yr) is an even or odd permutation of (d,. . . , /r) and1 ^ / ! < • • • < ir d. In particular, c, ir = 0 if two of the indicesji, . . . ,y r are equal.

Change the notation for the "basis elements" efi if again, writingef j = dy*1 A • • • A dy*r. Correspondingly, a vector is a differentialr-form

with constant coefficients subject to (2.1).As in the last chapter, multiplication of a differential r-form a/ and an

5-form o>* is defined as a differential (r + sHorm obtained by the usualassociative and distributive laws and the anticommutative law dyi A dy* ——dyj A d^\ so that of A <o* = (— l)r'<w* A tor. This type of multiplicationwill be referred to as "exterior" multiplication.

We can obtain a "change of basis" for the vector space W of differentialr-forms in the following way: Let T = (ftf) be a nonsingular d x dmatrix and let

and, more generally, let the "basis" elements be the exterior product

Then (2.2) becomes a differential r-form of the type

where the convention analogous to (2.1) is observed.In order to see that this is the usual type of change of basis for the space

W, it is necessary to prove the following:Lemma 2.1. Let the "change of basis" (2.3) transform (2.2) into (2.4).

Then (2.2) is 0 (i.e., all c, , = 0) if and only if (2.4) is 0 (/.*., all yj . . . ,= 0).

This follows form the fact that "changes of basis" (2.3) are associative,i.e., if

r

1

then

transforms (2.2) into (2.5). Thus the choice (%) = (/„)"* gives Lemma 2.1.If certain of the dyk are linear combinations of the others (e.g., if

dyh+l,..., dy* are linear combinations of dyl,..., dyh with constantcoefficients), then (2.2) becomes a differential /--form in dyl

t..., dy*.This notion is involved in the following lemma:

Lemma 2.2. Lei a)lf..., cos be linearly independent differential l-formsand a) a differential r-form with constant coefficients. Then the exteriorproduct ft>! A • • • A tog A a) is 0 if and only if the relations atl = • • • = cos = 0imply that eu = 0.

Let the forms W j , . . . , cos be given by

The assumption of linear independence means that a> l s . . . , <os consideredas vectors are linearly independent; i.e., rank (%) = s where / = 1 , . . . , sand j = 1 , . . . , d. The relations wx = • • ' = cos — 0 mean that s of thedyi are expressed as linear combination of the other d — s dyk, in whichcase co becomes an r-form in the latter. The statement of the lemma isto the effect that this r-form is 0 (i.e., all coefficients are 0) if and onlyif Wi A • • • A (os A o> = 0.

The proof will show that the conclusion of Lemma 2.2 can be stated asfollows: Wj A • • • A fos A a) = 0 if and only if there exist s differential(r — l)-forms aj, . . ., a, such that o> = o^ A o^ + • • • + ag A a)s.

Proof. Join d — s rows to the d x s matrix (stj) to obtain a nonsingularsquare matrix. Consider the change of basis (2.3), where (?„) = (ty)"1.Then, with respect to the new basis, co, = drf for 1 i 51 s and, say, tois given by (2.4). The product co^ A • • • A cos A co = 0 if and only ifevery nonzero term of CD contains a factor o>f = dif for 1 / s, i.e.,if and only if CD = 0 when o^ = • • • = cos = 0. Thus the assertion iscorrect with respect to the (drf->..., ^d)-basis and, by Lemma 2.1, withrespect to the (dyl,..., </y*)-basis.

3. A Theorem of Frobenius

The following theorem of Frobenius is the main theorem concerning(1.5).

d d

)


Theorem 3.1. Let H(y, z) be a continuous d x e matrix on an open setE. A necessary and sufficient condition for the complete integrability ofMO = dy — (H(y, z) dz at (y0, z0) is that there exists a continuous, nonsingulard X d matrix A(y, z) on a neighborhood of(y0, z0) such that if at = (a)lt...,eod) is defined by w — Aw0, then CD has a continuous exterior derivativesatisfying

Conditions (3.1) represent the integrability conditions. The expressionon the left is the exterior product of d differential 1-forms colt..., cod andthe differential 2-form dwt. Condition (3.1) will be used in the equivalentform (Lemma 2.2): to = 0 (i.e., to^ — • • • = a>d = 0) implies that dco = 0(i.e., d^ = • - • = da>d = 0).

Exercise 3.1. Show that conditions (3.1) reduce to (1.4) if A(y, z) = /and H(yt z) is of class C1.

Theorem 3.1 should be completed by the following:Lemma 3.1. Let H(y, z) be a continuous d X e matrix on an open set E.

Let Q = Q(£) be the (possibly empty) set of continuous, nonsingulard X d matrices A(y, z) on E such that at — A[dy — Hdz] has a continuousexterior derivative. Then the integrability conditions (3.1) hold for allA e Q. or for no A 6 Q.

For example, if H(y, z) is of class C1, so that dy — Hdz has a continuousexterior derivative, and if (1.4) does not hold, then (3.1) does not hold forany choice of continuous nonsingular A.

Exercise 3.2 (A Simplified Version of Lemma 3.1). Let A(y, z) andH(y, *) be continuous, det A 5* 0, and co = A[dy — H(y, z) dz] have acontinuous exterior derivative satisfying the integrability conditions (3.1).Let A0(y, z) be a C1, nonsingular d x d matrix. Show that A0(y, z)eo hasa continuous exterior derivative given by d(A0w) = A0 dco + (dA0) A co,and hence that the form A0a) satisfies integrability conditions analogous to(3.1).

Exercise 3.3. Let x, y, z be real variables; P(x,y,z), Q(x,y,z),R(x> y, 2) real-valued functions of class C1; and P2 + Qz + R2 5* 0.Show that the integrability condition CD A dot = 0 for the existence of localintegrating factors for eo = P dx + Qdy + Rdz is that P(Ry — Qt) +Q(P, - Rx) + R(QV - Px) = 0.

Exercise 3.4. Show that if H(y, z) is continuous on E and there exists acontinuous nonsingular A(y, z) on E such that eo = A[dy — H(y, z) dz]has a continuous exterior derivative satisfying the integrability conditions(3.1), then every point (y0, z0) 6 E has a neighborhood E0 on which thereis defined a sequence of C1 1-forms co1, eo2 , . . . such that con-*-o> anddcon -> dca uniformly as n -*• oo and con satisfies the integrability conditions.

Exercise 3.5. Let H(y, z) be continuous on E. Show that to = dy —H(y, z) dz has a continuous exterior derivative dco satisfying the integra-bility conditions (3.1) if and only if H(y, z) has continuous partial deriva-tives with respect to the components of y (cf. Exercise V 5.1) and, forl^j<m^e and / = ! , . . . ,</ ,

for all rectangles S with boundaries /, S <= E, in the 2-planes y = const,and zk = const, for k y^j, m; cf. (1.4).

Theorem 3.2. Let H(y, z) be continuous on an open set E. A necessaryand sufficient condition for (1.12) to be complete on a neighborhood of a point(y0, z0) e E is the existence of a continuous nonsingutar matrix A(y, z) on aneighborhood of (y0, z0) as in Theorem 3.1. In particular, if e > 1 andH(y, z) is of class C1, then (1.4) [or (1.15)] is necessary and sufficient for the(local) completeness of (1.12).

The concept of completeness leads at once to:Corollary 3.1. Let H(y, z) be continuous on a neighborhood of a point

(y0, z0) and let (1.12) be complete [i.e., possess d solutions u = r]l(y, z), . . .,rf(y, z) such that r) = (rf-,. . ., rf) satisfies det (dy/dy) ^ 0], Put r)0 =??(y0, z0). Let u(y, z) be a real -valuedfunction of class Cl on a neighborhoodof(y0,z0). Then u(y, z) is a solution of (1.12) if and only if there exists afunction U(rj) of class C1 for rj near 7y0 such that u(y, z) = U(rj(y, z)) for(y, z) near (y0, z0).

The proof of Theorem 3.1 will be given in §4 and that of Lemma 3.1in §5. Theorem 3.2 follows from Theorem 3.1 and the considerations of§ 1. The proof of Corollary 3.1 is similar to that of Theorem V 12.1 andwill be omitted.


Necessity. Let there exist a function y = y(r], z) of class C1 in a vicinityof a point (?/0, z0) satisfying (1.7), (1.8) and transforming

into a form

Thus DQ(TI,Z) — (dyfirj) is nonsingular; cf. (1.11). By condition (1.8),(1.6) has an inverse r\ = r)(y, z) of class C1 on a vicinity of (y0, z0). LetA(y, z) = D0~\rj(y, z), z); so that A is continuous and nonsingular,furthermore, (o = A(dy — H dz) = dr\ has the continuous exteriorderivative dco = d(drj) = 0. This proves the "necessity."


The proof of "sufficiency" will depend on the following lemma whichexhibits the role of the integrability conditions (3.1).

Lemma 4.1. Let /(/, y, z) be continuous on a neighborhood of a point(to, ^o» zo) ana tet there exist a continuous, nonsingular d X d matrixA(t, y, z) and a continuous d x e matrix C(t, y, z) such that

has a continuous exterior derivative

satisfying (3.1). Let y = r)(t, /„> y\, z) be the solution of

Then the matrix

is independent oft.Proof of Lemma 4.1. By Theorem V 6.1, (4.4) has a unique solution

y = r}(t, /„, ylt z) of class C1. In the notation of Corollary V 6.1 [cf.(V 6.9)-(V 6.10)3, Y = A(t, rj, z) (drj/dyj is a fundamental matrix for thelinear system

and Y — A(t, r/, z)[(cty/dz) — C(t, ??, z)] is a solution of

Note some differences in notations here and in §V6; here y^ plays therole of y0, and -AC in (4.3) that of C in (V 6.2).

For tQ fixed, the change of variables (t, y, z) -> (t, ylt z), where y ="n (*> 'o> ^i» 2)> transforms (4.3) into

Since the coefficients in (4.8) possess continuous derivatives with respectto t, the proof of Lemma V 5.1 shows that

where the omitted terms involve dy-l A dy^, dy^ A dzk, and1 dzj A dzk. Inview of the remarks concerning (4.6) and (4.7), this is

where the argument of A, C, Ft N is (t, t], z) and rj = rj(t, t0, y^ z).


At a fixed (/, ylt z), choose dyl to make eo = 0, i.e.,

then (4.9) has the form dco = {• • •} dt A dz + (terms in dz'A dz*), where{• • •} is given by

Since (4.10) makes o> = 0, Lemma 2.2 and the integrability conditions(3.1) imply that do) = 0. In particular {•••} = 0, i.e., N + FC = 0. Inthis case, (4.7) reduces to (4.6). Thus Y = A (ety/etyi) is a fundamentalsolution of (4.6) and Y = A[(dr]fdz) — C] is a matrix solution of the samesystem (4.6). Consequently there is a matrix C° independent of / such that

cf. § IV 1. Since the matrix (4.5) is C°, the lemma is proved.Proof of "Sufficiency" in Theorem 3.1. By assumption, there is a

continuous, nonsingular A(y, z) satisfying the conditions of the theorem.Change notation as follows: let / = z1; if e — 1, let za = 0 and ife > 1, let z2 be the (e — l)-dimensional vector (z2 , . . . , ze). Then (1.1) canbe written as

where/is the first column of the matrix H and C2 is the matrix consistingof the other columns of H. Let y = r)(t, z2, yx) be the solution of y' —f(y, t, 22). KV) = y\ fa fixed z0

l. The change of variables (y, t, z2) -*(yi, t, z2) transforms «> into the form

This can be written as

where //2 does not depend on / = z1 by Lemma 4.1; cf.-(4.8) and (4.5).The form co, of course, satisfies the integrability conditions in the newvariables (yl5 z).

If e •=• 1, so that z2 = 0, the theorem is proved. If e > 1, changenotation by letting z2 = / and z3 = 0 or z3 = (z3,..., z') according ase = 2 or e > 2 and write


where/! = fi(y\, t, z3) is the first column of H2 and C3 is the matrix con-sisting of the other columns. Let yr — rj^t, z3, y2) be the solution ofy\ —f\(y\^ t> 23)> yi(2o2) — Vz- Thus rj! does not depend on z1. The changeof variables (ylt z1, f, z3) -»• (yz, z1, t, z3) transforms co into a form of thetype

where, by Lemma 4.1, H3 = H3(yz, z3) does not depend on z2.If e = 2, so that z3 = 0, the theorem is proved. It is clear that the

process can be continued to obtain a proof of the theorem for any e > 2.

5. Proof of Lemma 3.1

Since the integrability conditions are local conditions, it can be supposedthat E is the neighborhood of a point (y0, z0). Suppose that D(£) is notempty and that the integrability conditions are satisfied for some elementof Q(E). Then, by Theorem 3.1, there exists a y = y(r), z) satisfying (1.7)-(1.8) and transforming

into

Let A(y, z) be an arbitrary element of £l(E). Then

is transformed into

Since the property of having a continuous exterior derivative is not lostunder a C1 change of variables y, z -> 77, z [see the definition of continuousexterior derivative in § V 5], A e Q(.E) implies that (5.4) has a continuousexterior derivative. It is clear from the proof of Lemma V 5.1 that dcois of the form

where Ajk — —Aki, Bik are continuous ^-dimensional vectors.As Z>(^, z) is nonsingular, CD = 0 is equivalent to dy = 0, in which case

dco = 0. Thus (5.4) satisfies the integrability condition. But then (5.3),which results from (5.4) by a C1 change of variables 77, z -> y, z, alsosatisfies the integrability condition. This proves the lemma.

6. The System (1.1)

The theorems of § 3 give necessary and sufficient conditions for theexistence of local solutions of initial value problems (1.1)-(1.2).

Theorem 6.1. Let H(y, z) be a continuous d X e matrix on an open setE. A necessary and sufficient condition that (1.!)-(!.2) have a uniquesolution y = y(z, z0, y0)for z near z0 and all (yQ, z0) e E such that y(z, z0, y0)is of class Cl is the complete integrability of (1.5) at every (y0, z0) 6 E, i.e.,the existence of an A(y, z)for each (y0, z0) e E as in Theorem 3.1. In thiscase, if the product of the two Euclidean spheres R: (\\y — y0\\ ^ b) x(||z — zj < d) is in E and if \f}' H(y, z)£| ^ M for all Euclidean unitd-dimensional vectors rj, e-dimensional vectors £ and (y, z) e R, theny(z) — y(z, 2o> yo) exists on the sphere \\z — z0|| f£ min (a, bfM).

It follows, as remarked in § 1, that if e > 1 and H(y, z) is of class C1,then (1.4) is necessary and sufficient for the existence of local solutions(1.1H1.2) with (y0,z0) arbitrary.

Particularly important cases of (1.1) are those in which H(y, z) is linearin y. Let H0(z), Hj(z),..., Hd(z) be continuous d x e matrices on anopen z-set D and let

so that (!.!)-(1.2) becomes

Corollary 6.1. Le/ ^fc(2) — (fyw,(z))» w'/jere A: = 0,. . . , d, be continuous

d x e matrices"(i — 1,..., d and j = 1,. . . , e) on an open z-set D.Necessary and sufficient for (6.2) to have a solution y = y(z, z0, y0) for allz0 e D, y0 arbitrary is that for 1 j<m^d, and r = 0, ! , . . . ,</ ,

for every rectangle S with boundary J, S c D, on coordinate 2-planesz* = const, for h -^ j, m. In this case, the solution y(z, z0, t/0) of (6.2) isunique and of class C1 with respect to all of its variables. When H0(z),...,Hd(z) are of class C\ conditions (6.3) are equivalent to

This corollary is a consequence of Theorem 6.1 and the followingexercises.

Exercise 6.1. (a) Let //0, . . . , Hd be of class C1, show that conditions(6.4) are the integrability conditions for o> = dy — H(y, z) dz in the case(6.1). (b) Let H0,...,Hd be continuous. Show that to = dy — H(y, z) dzin case (6.1) has a continuous exterior derivative satisfying the integrabilityconditions if and only if the conditions on (6.3) of Corollary 6.1 aresatisfied.

Exercise 6.2 (Continuation). Show that if (6.1) is continuous, then thereexists a continuous nonsingular A(y, z) such that A(dy — H dz) has acontinuous exterior derivative satisfying (3.1) if and only if the same istrue ofdy — H dz.

The proof of Theorem 6.1 combined with the usual arguments of themonodromy theorem will be used to show that, in the linear cases (6.1), thesolutions exist in the large.

Corollary 6.2. Let H(y, z) be continuous for ZE D and all y, where D isa simply connected open set. Assume that the sufficient conditions ofTheorem 6.1 for the local solvability of (1.1M 1.2) are satisfied. For everycompact subset D0 of D, let there exist a constant K —. K(D0) such that\\H(y, z)|| ^ K(\\y\\ + 1) for z e D0 and all y. Then the solution y(z) =y(z, «o> ^o) 0/U.1H1 -2) exists for all zeD.

It will be clear that the condition ||#(y, z)|| ^ K(\\y\\ + 1) can be refinedalong the lines of Theorem III 5.1.

Proof of Theorem 6.1. Necessity. Let (1.1)-(1.2) have a unique solu-tion y = y(z, 20, yc) for z near z0 of class C1. The Jacobian matrix (dyjdy0) isthe identity matrix at z = z0 and hence is nonsingular for z near z0. Forfixed Zo, put y(z, rj) = y(z, z0, 77). This is of class C1 near the point(*7o> *o) = (yo>zo); it satisfies (1.7), (1.8) and transforms (4.1) into a form(4.2) since, for fixed r), y(z) = y(z, rj) is a solution of (1.1), i.e. of (1.10).Thus "necessity" in Theorem 6.1 follows from that in Theorem 3.1.

Uniqueness. Let y = y(z) be a solution of (!.!)-(1.2), say on theEuclidean sphere ||z — z0ll ^ a. For a fixed e-dimensional vector £ of(Euclidean) length one, consider the values y^t) = y(z0 + t£) of y(z) onthe line segment z => z0 + /£, 0 < t a. By (1.1H1.2),

If //(y, z) is smooth (e.g., is uniformly Lipschitz continuous in y), then theinitial value problem (6.5) has a unique solution which is necessarily^i(0 = y(zo + '0 for 0 = t a. The same can be concluded under theconditions imposed here; namely, that A, H art continuous, det A ^Q,and (1.5) has a continuous exterior derivative. For then the forms

in dyi, dt for fixed z0, £ have continuous exterior derivatives. (This followsat once from the definition of "exterior derivative" in § V 5.) HenceTheorem V 6.1 implies the uniqueness of the solution of (6.5).

Existence. By Theorem 3.1 there exists a y(r), z) of class C1 satisfying(1.7), (1.8) and transforming (1.5) into (1.9). For 77 = r)0, y(rj0, 2) is asolution of (1.1)-(1.2); i.e., y(z, z0, yc) = y(rj0, z) exists. It only remainsto verify that y(z, z0, y0) is of class C1 in all of its variables. By (1.8), (1.6)has a C1 inverse rj = r](y, 2) for (y, z) near (y0, z0)- But ?OXyi» zi)>2) for

fixed (yl5 z^, hence fixed rj = rj(y^ zj, is a solution of (1.1) reducing toyl when 2 = 2^ In other words, y(z, zlt yj = y(rj(yi, zj, z) which showsthat y(z, Zj, yj is of class C1.

Domain of Existence. It is readily verified that the conditions on H inthe last part of Theorem 6.1 imply that the solution y^t) — y^t, £) of(6.5) exists for 0 55 / < a, where a = min (a, bfM), for all unit vectors £.In fact, if lift ||' is a right or left derivative, (6.5) and the conditions on Himply that | HyJ'l ^ Af; cf. the proof of Lemma III 4.2. The existenceand uniqueness of solutions of (1.!)-(!.2) for all (/0, y0) imply that y^t, £)is a function y(2) of 2 = z0 + /£ and that y = y(z) is a solution of (1.1 }-(\ .2).This proves Theorem 6.1.

Exercise 6.3. Assume that H(y, z) is of class C1 and that (1.4) holds.Prove the existence of a solution of (1.!)-(!.2) by the use of (6.5).

Exercise 6.4. Assume that H(y, z) is of class C1 and that (1.4) holds.Let 20 =s 0. Prove the existence of a solution of (l.l)-0-2) in the followingmanner: Let h^y, z) be the (/-dimensional vector which is theyth columnofff(y, z). Define y = t/^z1) as the solution ofdy/dz1 = hj^y, z1, 0, . . . , 0),^(0) = y0. If y^_1(«1,.. ., zy-1) has been defined, let yj(z\ ..., zj) be thesolution of dy\dzj = /z^y, z1,. . . , 2', 0, .. ., 0), y(0) = (z1,.. ., z*-1).Show that y = ye(2x,..., ze) is the desired solution of (1.!)-(!.2).

Proof of Corollary 6.2. It is clear from Theorem III 5.1 and theconditions of Corollary 6.2 that, for a fixed £, the solution yl = yv(t) of(6.5) exists on any /-interval J containing t = 0 on which z = z0 -H /£ is inZ>. By the proof of uniqueness of solutions y(z) = y(z, z0, y0) of (1.1H1.2),a pair of solutions y(z, z01, y0i), y(z, 202, y02) of (1.1) on z-spheres about z01,202, where 20( = z0 + /,£, y0l = y^/,), and /!, /2 eJ, coincide on any commondomain of definition. Consequently, the solution y = y(z, 20, y0) can bedefined on an open subset of D containing the line segment 2 = z0 + t£,teJ.

These arguments show that the same is true if the line segment on^ =s z0 + f£ is replaced by any polygonal path P in D which begins at20 and has no self-intersections. If two such polygonal paths Plt P2 whichbegin at z0 and end at zl are considered, the two solutions y(z, z0, y0)defined on neighborhoods of Plt P2 agree at z === zlf First, this is clear if

1

the paths Plt Pz are sufficiently near to each other. Second, it then followswithout this nearness condition on P^ P2 by virtue of the simply connected-ness of D. Hence the solution y(z) = y(z, «0, y0) can be defined (as asingle-valued function of z) on D, as was to be proved.

PART H. CAUCHY'S METHOD OF CHARACTERISTICS

7. A Nonlinear Partial Differential Equation

Consider a partial differential equation

for a real-valued function u = u(y) of d independent real variables, whereF(u, y, p) is a real-valued function of 1 + d + d variables on an open-set^M+I- A solution of (7.1) is a function u = u(y) of class C1 on an openy-set EA such that («O), y, uy(y)) e E^+l for y e Ed and (7.1) becomes anidentity in y. Here uy(y) — ( d u f d y 1 , . . . , du/dy*) is the gradient of u.

In general, solutions of initial value problems are sought, i.e., solutionsof (7.1) which take given values on a piece of a hypersurface S. To be moreexplicit, let S be a piece of C^-hypersurface in y-space, i.e., let S be a set ofpoints

£(y) is of class C1 in a vicinity of y = y0 and rank (d£/9y) = d — 1. Let9> be a given function on S or, equivalently, let <p = q?(y) be a given func-tion of y for y near y0. Then the "initial condition" is the requirementthat the solution u = u(y) reduces to 9? on S, i.e.,

The existence theorems to be obtained are local in the sense that onlysolutions u = u(y) defined for y near y0 = £(y0) will be obtained. Themethod to be used is that of Cauchy which reduces the problem to thetheory of ordinary differential equations and is called Cauchys methodof characteristics. There is no analogue of this method for systems offirst order partial differential equations.

The following abbreviations Fu = dFjdu, Fv = (dF/dy1,..., dFjdy*)and Fp = (dFfdp1,..., dFjdp*) will be employed. A dot denotes the usualscalar product of rf-dimensional vectors.

In order to motivate the method to be employed, consider the followingheuristic arguments in the special case of (7.1) which is a linear partialdifferential equation of the form

a

i.e., F(u,y,p) does not depend on u and is of the form F(u,y,p) =2/*(y)/>* = ^M ' P- If u — u(y) is a solution of (7.4) and y = y(t] is asolution of the system of ordinary differential equations

then (7.4) shows that u(y(t)) is a constant. Solutions y — y(t) of (7.5) arecalled characteristics of the partial differential equation (7.4).

Figure 1.

Suppose that no characteristic is tangent to S. This condition can beexpressed by

(7.6)

if S is given by (7.2). In (7.6), [(d£/dy), Ff] is a d x d matrix, the firstd — 1 column of which constitute the d x (d — 1) Jacobian matrix(d£(y)/dy) and the last column is the vector F,,.

In this case, S is said to be noncharacteristic and the characteristicswith initial points on 5" fill up a small piece Ed of y-space. The valueu(y) of a solution u at a point y e Ed must be the same as the given valueof <p(y) at the initial point £(y) on 5 of the characteristic through y; seeFigure 1. Conversely, it is to be expected that a function u(y) defined onEd in this fashion is a solution of (7.3)-(7.4). Under suitable smoothnessconditions on Fthis turns out to be the case; cf. § V 12 for the relationshipbetween solutions of (7.4) and first integrals of (7.5).

Instead of a linear equation (7.4), consider a somewhat more complicatedequation, say, a quasi-linear equation, i.e., an equation in which the highestorder (=first) partial derivatives occur linearly,

If a solution u == u(y) is known and we consider a solution y(t) of (7.5),


where the right side is /(M, y) = f(u(y), y), then the equation F = 0implies that du(y(t))ldt = - U(u(y(tJ), y(t)). This leads to the set ofordinary (autonomous) differential equations

in which the right sides are functions of u and y, but not of the independentvariable t. It will turn out that problems for the quasi-linear equationF — 0 can be reduced to problems for this system of ordinary differentialequation.

Returning to the general nonlinear case ,(7.1), characteristics will bedefined. These are not generally level curves of a solution as in the linearcase. Assume that F(u, y, p) is of class C1 on some open (M, y, /?)-setand assume that u = u(y) in (7.1) is of class C2. We can reduce the"nonlinearity" of (7.1) by differentiating (7.1) with respect to a fixedcomponent ym of y to obtain a second order partial differential equationfor u which is quasi-linear, i.e., linear in the second order partials of u.This equation can be formally written as a first order, quasi-linear equationfor/" = dufiy™,

Thus, in analogy to the above, we are led to the ordinary differentialequations

to which we can add du/dt = 'Lfiufdy'W or

The differential equations for yj and u do not depend on m. Lettingm = I,... ,d gives a set of autonomous ordinary differential equationsfor u, y, p which can be written as

where the argument of Ftt, Fy, F9 is (u, y,p) and the independent variable/ does not occur. A solution y — y(t), p = p(t), u = u(t) of (7.9) is calleda characteristic strip and the projection u(t), y(t) of a solution into the(u, y)-space is called a characteristic. A condition of the type

prevents a characteristic from reducing to a point. The "derivation" of(7.9) will be stated as a formal result in Lemma 8.2.

A solution of the initial value problem (7.1), (7.3) cannot exist unlessthere exists a function p = p(y) on S which is the gradient of «(y) at y =£(y) and satisfies

The last condition results upon differentiation of (7.3) with respect to y*.In particular, there is a vector pa = p(y0) such that

Assume that the "initial data is noncharacteristic at y = y0," i.e., that

where the first d— 1 columns of the matrix in (7.15) are the vectorsd£(yo)/dy*» i « 1, . . . , < / — 1, and the last is Fp(u0, y0,/?0). Then, by theimplicit function Theorem I 2.5, (7.13), (7.14), (7.15) and y = £(y) 6 C2,Fe C1, 9? e C2 imply that (7.11)-(7.12) has a unique solution/; = p(y) ofclass C1 in a vicinity of y = y0 which satisfies p(y0) = p0.

Note that in the nonlinear case, we cannot speak of "S being non-characteristic" but only of "the initial data being noncharacteristic."The initial data consists of S: y — £(y), the function <p(y), the vector p0,and, when (7.15) holds, the implicitly determined p(y). By continuity,(7.15) implies that

for y near y0. When (7.16) holds, the initial data is called noncharacteristic.Exercise 7.1. Suppose that F in (7.1) does not depend on u. What is

the form of the system (7.9) for the characteristic strips? Note that thedetermination of u in (7.9) reduces to a quadrature.

The initial value problem (7.1), (7.3) is often considered in another form,to be obtained now. This form of the problem will be used in § 10. Supposethat 5 in (7.2) is of class C1. Without loss of generality, it can be supposedthat y0 = 0 and that S in (7.2) is given in the form t/* — y(y\ . •., y*"1),where y is of class C2 for (y1,..., y*1"1) near 0 and y(0) = 0. If

are introduced as new coordinates, again called y, then in the new coordi-nates S is a piece of the hyperplane y* =• 0 near y = 0. The partial differen-tial equation (7.1) is transformed into another of the same type, although if,e.g., the original Fis of class C2, the new Fhas continuous first and secondderivatives except possibly for those of the type 32F/dy <?y*. The condition(7.15) in the new coordinates system becomes BF(u0, y0,p0)/dpd ^ 0. Thus,if (7.13) holds, the equation F(w, y, p) = 0 can be solved forpd in terms ofw, y,pl,. •. ,pd~l, say/?d = —H(u, y,pl,.'.. >pd~l\ and (7.1) is equivalent to/ + H(u, y,pl,... ,pd~l) - 0 for (w, y,p) near (w0, y0,pQ).

Thus, if the notation is changed by replacing d — 1 by d and y by(y, t), then the initial value problem takes the form

where u — u(t, y) is the unknown function, <p(y) is the given initial function,ut = dufdt, and uv = (dufdy1,..., dufdy*).

Exercise 7.2. (a) Write H as H(u, t, y, q\ where y = (y1,..., ya) andq = (q1,..., q*). Find the differential equations for the characteristicstrips for the partial differential equation (7.17), using / as the independentvariable, (b) Simplify the result of part (a), assuming that H(t, y, q) isindependent of u. [Note that in this case, when H is a. Hamiltonianfunction, (7.17) is the Hamilton-Jacobi partial differential equation andthe nontrivial parts of the equations for the characteristic strips are theequation of motion in the Hamiltonian form.]

8. Characteristics

The relationships between solutions of (7.1) and characteristic strips orcharacteristics will now be determined.

Lemma 8.1. Let F(u, y, p) be of class C1. Then F(ut y, p) is a firstintegral of the system (7.9); i.e., F is constant along any solution of (7.9).

Proof. It suffices to verify that if y(f), p(t), u(t) is a solution of (7.9),then the derivative of F(u(t), y(t),p(f)) is 0. This is equivalent to

Lemma 8.2. Let F(u, y, p) be of class C1 on an open set £2d+i and letu = u(y) be a solution of(l.\)of class C2 on an open set Ed. For y0 E Edtthere exists a characteristic strip y(t),p(t), u(t) for small \t\, such that«(0 = u(y(t)),p(t) = w,(y(0) andy(Q) = y0.

In particular, in the (u, y)-space, the arc (w, y) = (u(t), y(i)) lies on thehypersurface u — u(y) and (—!,/?(/)) is a normal vector to this hyper-surface at the point (u, y) = (u(t), y(t)). Thus if u = u(y) is a solution of

d

d

d

(7.1) of class C2, the hypersurface u = u(y) can be considered to be madeup of characteristics.

Corollary 8.1. If the solutions of initial value problems associated with(7.9) are unique (e.g., if Fe C2) and u = Wi(y), uz(y) are two solutions of(7.1) of class C2 which "touch" at y = y0 [i.e., u^^ = «2(y0), «lv(y0) =w«v(2/o)l» then they "touch" along a characteristic arc y — y(t).

Proof of Lemma 8.2. [This is a repetition of the "derivation" of (7.9).]Consider a solution y = y(t) of the initial value problem

Differentiating (7.1) with respect to ym gives

PutXO = M»(y(0)- Then (8.1) implies that (8.2) can be written as the mthcomponent of Fup + Fv + p' = 0, where the argument of FM, Fv is(«(y(0), y(t\ XO)- Also, if u = M(y(0), then u' = uv(y(t)) -y'isp-F, by(8.1). Thus y = y(t),p = wv(y(0)>« = «(y(0) is a solution of (7.9). Thisproves the lemma.

Remark. It remains unknown whether Lemma 8.2 is valid if it is onlyassumed that u(y) E C1 and d > 2. In this direction, there is a partialresult for d > 2 and a complete result for d — 2:

Exercise 8.1. Let F(u, y, p) be of class C1, u(y) of class C1 in a neighbor-hood of y0 and a solution of (7.1). (a) Let m be fixed, 1 m </. Showthat there exists a solution y(0 of the initial value problem

such that/?m(/) = [du(y)ldym]vscv(t} has a continuous derivative with respectto t satisfying pm> = —dFjdy™ — Fupm, where the argument of 9F/9ym, Fuis [w(y(/))» y(0> w»(y(0)J- (b) In particular, if the solution of the initial valueproblem y' — Fv(u(y), y, uv(y)\ y(0) = y0 is unique [so that y(i) in part (d)does not depend on m], then the conclusion of Lemma 8.2 is valid. (This isapplicable, e.g., if F(u, y,p) depends linearly on /?.) (c) If Fv 7* 0 at("> y>p) = («(yo)> ^o» M»(yo)) and d—2, then the conclusion of Lemma 8.2is valid (under the assumptions F e C1, u E C1).

Exercise 8.2. Let /XM» y»/0 afld G(w> y,p) be of class C1 on an open(w, y,/?)-domain. Define the function H(u, y,p) by

(Iff, G are linear in/? and independent of M, H corresponds to the "com-mutator" off, G defined in Exercise 1.1.) Let u(y) be a solution of class C1

of both F — 0 and G — 0 on some y-domain D. (a) Show that if, in

addition, u(y) is of class C2, then u(y) is a solution of H = 0. (b) Show thatif u(y) e C1 and if through each point (u, y, p) = (w(y0)» y«» «v(y<>))» therepasses a characteristic strip for F = 0 as in Lemma 8.2 [e.g., if Exercise8.1(Z>) is applicable], then w(y) is a solution of tf = 0.

9. Existence and Uniqueness Theorem

The main theorem on (7.1) is the following.Theorem 9.1. Let F(u,y,p) be of class C2 on an open domain E^+l.

Let (UQ, y0, p0) E EM+l. Let (7.2) be a piece of hypersurface of class Cz

defined for y near y0 and £(y0) = &>• Let <p(y) be a function of class C2for ynear y0 and <p(y0) = «„. Finally, let (7.13), (7.14), and (7.15) hold. Then,on a neighborhood Edof y — y0, there exists a unique solution u = u(y) ofclass C2 of the initial value problem (7.1)-(7.3).

Note that (7.15) implies that Fp(u0, y0,/?0) 7* 0 and that rank (d£(y)/3y)is d — 1. The condition Fp(u0, y0,/?0) 0 implies the existence of hyper-surfaces S satisfying (7.15). For example, if dFfdp* 9* 0 at (M0»yo»/'o)>then the hyperplane y* = y0

d is an admissible 5.Proof. By the argument in §7, there is a unique function p = p(y) of

class C1 for y near y0 satisfying (7.11 )-(7.12) and p(y0) = p0. Let y = Y(t, y),p = p(t, y), u = U(t, y) be the solution of (7.9) satisfying the initialcondition

By Theorem V3.1, this solution is unique and Y(t, y), P(t, y), U(t, y),7'('» y). '('» 7\ U'(t> y) are of class C1 for small |f | and y near y0. Since Fis a first integral for (7.9),

In fact, the function in (9.2) does not depend on / by Lemma 8.1 and, at/ = 0, (9.2) reduces to (7.11).

By the first part of (7.9) and by (9.1), at (/, y) = (0, y0), the JacobiandY(t, y)/8(/, y) at / = 0, y = y0 is

Thus assumption (7.15) shows that this Jacobian determinant is not 0.Hence, there exists a unique map

for y near y0 inverse to


The map (9.3) is of class C1. Put

for y near y0. Then (9.2) becomes

Thus the existence assertion will be-proved if it is shown that

i.e., du(y) = P(t(y), y(y)) • dy. Under the change of variables (9.4),y -*• (r, y), with nonvanishing Jacobian, this is equivalent to

or to

The relation (9.8) follows from (7.9). Thus only (9.7) remains to beverified.

For a fixed y, let Ay(/) denote the expression on the left of (9.7). Thus itsuffices to show that A/f) = 0. Note that, by (7.12) and (9.1),

In what follows, let F = F(u, y, p) and u = U(t, y), y = Y(ty y), /? =P(t, y). Differentiating the left side of (9.7) with respect to / gives

The change of order of differentiation is permitted since y',p', u' are ofclass C1. Using (7.9), the last relation becomes

If (9.2) is differentiated with respect to y>, it is seen that the sum of the firsttwo terms on the right is — Fu du/By*. Hence,

Since A,(r) satisfies a linear homogeneous differential equation and theinitial condition (9.9), it follows that A,(0 = 0.

Hence (9.5) is a solution of (7.1)-(7.3). Also u(y) is of class C2 since itsgradient (9.6) is of class C1. Finally, when Fe C2, so that solutions of(7.9) are uniquely determined by initial conditions, the uniqueness of C2

solutions of (7.1)-(7.3) follows from Lemma 8.2, the remarks following it,and the existence proof just completed. (For another uniqueness proof,see the next section.)

It should be mentioned that when the initial data is not "noncharacter-istic," in general there will not exist a solution.

In a sense, the existence Theorem 9.1 is unsatisfactory for it producessolutions of class C2 when it is natural to ask for solutions only of class C1.It is reasonable to inquire as to whether or not the differentiability con-ditions can be lightened in Theorem 9.1 and still obtain solutions of classC1. To some extent, this question can be answered in the negative.

Exercise 9.1. Let x, y be real variables and f(x) a continuous nowheredifferentiable function ofx. Show that ux — uv +f(x + y) = 0, u(0, y) =0 has no C1 solution. Thus continuity of F is not sufficient to assure theexistence of solutions.

Exercise 9.2. Even FeC1 and analytic initial data is insufficient toassure existence of solutions. Let x, y, q be real variables. Letf(q) be areal-valued function of class C1 for s,mall \q\ such that df/dq is not Lip-schitz continuous at q = 0. Show that, on the one hand, the procedurein the proof of Theorem 9.1 does not lead to a solution of ux =/(«,,),"(0, y) = \yt. (The difficulty arises from the fact that the analogue of themap (9.4) has no inverse.) On the other hand, Exercise 8.1(c) implies thatif a solution exists, then it is obtainable by such a procedure. Hence thereis no C1 solution.

Exercise 9.3. The last exercise shows that, in a sense, the following isthe "best" theorem: Theorem 9.1 remains correct if "F, £(y), <p, u(y) e C2"is replaced by "F, £(y), <p, u(y) are of class C1 with uniformly Lipschitzcontinuous partial derivatives." See Wazewski [3], This can be provedby a suitable modification of the proof of Theorem 9.1 using, e.g., the factthat uniformly Lipschitz continuous functions possess total differentialsalmost everywhere. For a different proof, see Digel [2].

10. Haar's Lemma and Uniqueness

Let (7.1), (7.3) be replaced by (7.17) and (7.18), i.e., by

It follows from Theorem 9.1 that if H(u, t, y, q) is of class C2 on an openset E^zd containing the point (u, t, y, q) = (y(0), 0,0, <pv(OJ) and <p(y) is ofclass C2 for y near y = 0, then (10.1), (10.2) has a unique solution u —u(t, y) of class C2 for small \t\, \y\. The situation as to uniqueness forsolutions of (10.1)-(10.2) is very simple.

Figure 2. The case d = dim y is 1.

Theorem 10.1. Let H(u, t, y, q) be defined on an open set E2+2d containingthe point (u, t, y,q) = 0 and satisfy a uniform Lipschitz condition withrespect to (u, g). Let <p(y) be a function of class C1 satisfying <p(Q) = 0,(pv(0) = 0. Then (10.1)-(10.2) has at most one solution of class C1 on aneighborhood Edofy = 0.

This follows by applying the following lemma (with C = N — 0) to thedifference v = uz(t, y) — u^t, y) of two solutions u^t, y), uz(t, y).

Lemma 10.1. Let v = v(t, y) be a real-valued function of class C1 on aset R: 0 t a«a), |y*| ^ L(a - t)for i=l,...,dand satisfy

where L, M > 0 and C,N>Q are constants. Then, on R,

See Figure 2.

Proof. Let C", N' be arbitrary constants satisfying

and put

so that

It will be shown that

on R, so that letting C" -* C and N'-> N gives

Replacing v by — v in this argument gives (10.5).It is clear from (10.3), (10.6), and (10.7) that u - v > 0 for small t > 0.

If (10.9) does not hold on R, there is a point (t0, y0) of R such that 0 </0 a, (10.9) holds on that portion of R where 0 t < t0, and equalityholds at (r0, y0).

For any of the 2d choices of ±, the points of the line segments

are in R for 0 < ? /0, for \±L(t0 - t) + yQk\ ^ L(a - /) since |yc

fc| ^(fl — 'o); see Figure 2. The difference u — v at the point (10.11) ispositive for 0 t < f0 and 0 at t = t0. Consequently, the derivative ofM — v along the line (10.11) is nonpositive at t = /„. This gives

From (10.8), wt = Mu + A^', so that ut = Afy + N' = M |y| + TV" at(^» y) = (>o> yo)- This fact and «„ = 0 give

If ± is chosen so that ±dv/dyk — \dvldyk\ at (f 0, y0)» the resulting inequalitycontradicts (10.4) since N' > N. This implies (10,9) on R and proves tjielemma.

Exercise 10.1. (a) Let B be the (/,y)-set B = {(t,y) : 0 g / < a,cfc + Lfcf ^ yfc ^ <4 —. Lfc? for A: = 1,..., d}, where Lfc ^ 0, ck < affc, and

2Ljfl ^dk — ck. Let u(t, y) be a real-valued function of class C1 on B andlet m(s) = max u(s, y) taken over the set Ba = {(/, y):(f, y) e 5, / = s}.Then m(t), 0^.t<a, has a right derivative D^n(t) and there exists a point

<*(/, y0 £, such that m(t) = w(f, y0) and D^n(t) — dujdt — 2 |<?M/dy*| Lk

t=ievaluated at (t, y0). (6) Let o>(/, w) be continuous for 0 < / < a, M 0and such that the only solution of M' = eo(/, M) defined for 0 < t e (<a)and satisfying u(t) -»• 0 and w(f)// -*• 0, as f -> +0, is w(/) = 0. LetH(u, t, y, q) be continuous for (u, t, y, q) near (u, t, y, q) = 0 andl#(«i, >, ?i) - H("» t, y, q2)\ < 2 Lfc |ft* - ?2*| + co(f, |Wl - w2|). Let<p(y) be of class C1 for small |y| and satisfy y(G) = 0, 9?v(0) = 0. Then(10.1), (10.2) has at most one solution on the set B.

Exercise 10.2. (a) Let B denote a bounded (t, y)-set defined by in-equalities: 0 t < a, bj(t,y) ^ 0 for y = !, . . . , /», where &,-(/, y) is areal-valued function of class C1. It is assumed that every boundary pointof B lies on either t = 0, / = a, or k of the m hypersurfaces £,(/, y) = 0,1 k ^ m; also, if A: of the hypersurfaces bj(t, y) = 0, sayy —j\,... ,/ft,ahave a point (/, y) in common, then the k differential 1-forms 2(8^/

»=idyi)dyi

t where y =y'i,... ,yfc, are linearly independent at (f, y). Let#(">'» y> ?) be defined ona( l + 1 + d + f/)-dimensional domain E witha projection on the (t\ y)-space containing B. Let u(t, y), v(t, y) be real-valued functions of class C1 on B such that (M, r, y, wv) e E, (v, t, y, vv) e E.Suppose that ut > H(u, t, y, wy), vt ^ H(v, t, y, vv) on B and that M(O, y) >v(Q, y). Finally, suppose that, at every boundary point (t, y) of B commonto k hypersurfaces b^t, y) = 0 for j =j\,... ,jk, we have the inequality

for all non-negative numbers A a , . . . , A4 such that (M, /, y, [M — S A, ]t) B. Then «(/, y) > v(t, y) on 5. See Nagumo [3]. (b) Let J? be the same asthe (/, y)-set in Exercise 10.1 (a). Let H(u, t, y, q) be defined on a (1 + 1 +d + </)-dimensional set E with a projection on the (f, y)-space containingB and satisfying \H(u, t, y, ft) - #(M, f, y, ^2)| S Lk \qf - qz

k\. Letu(t, y), y(/, y) be C1 functions on B such that (u, t, y, MV) e E, (v, t, y, vy) eE\ that ut > H(u, t, y, uy) andvt ^ H(v, t, y, vv) on E; and that w(0, y) >u(0, y). Then u(t, y) > y(/, y) on 5. (c) Deduce Lemma 10.1 from part (b).

Notes

SECTION 1. Connections between the systems (1.1) and (1.12) were considered by Boolein 1862; see E. A. Weber [1] for historical remarks and early references. Discussionsof integrability conditions and existence theorems for Jacobi systems go back (1862)


to Jacobi [1, V, p. 39] and to Clebsch [2] who introduced the concept of "completesystems" (1.12). (See also the reference to A. Mayer [1] in connection with Exercise 6.3.)

The result in Exercise !.!(/>) is due to E. Schmidt. For references to Schmidt, Perron,and Gillis and for generalizations of part (b) see Ostrowski [1 ]. Further generalizations,based on Plis* [2], are given by Hartman [13]; see Exercise 8.2.

SECTION 2. See E. Cartan [1, pp. 49-64].SECTION 3. Under analyticity assumptions, Theorem 3.1 is due to Frobenius [2].

The statement of the theorem in the text, avoiding differentiability assumptions, is dueto Hartman [17]. The proof in the text is adapted from E. Cartan [1, pp. 99-100]. Fora related but somewhat less general theorem on Jacobi systems, see Gillis [1].

SECTION 6. The arguments in the "sufficiency" proof and the existence proof for(1.1 HI-2) suggested in Exercise 6.3 go back to A. Mayer [1]; cf. Caratheodory [1,pp. 26-30]. The proof outlined in Exercise 6.4 was used by Weyl [2, pp. 64-68]. Stillanother proof, using successive approximations, is given by Nikliborc [1]; cf. alsoGillis [1].

SECTIONS 7-9. The initial value problem considered in § 7 is the simplest example ofthe type called "Cauchy's problem" in the theory of partial differential equations.The differential equations (7.9) for the characteristic strips, Theorem 9.1, and its proofare due to Cauchy (about 1819) under conditions of analyticity; see, e.g., [1, pp.423-470]. Actually, a few years earlier, Pfaff [1] had considered the problem of findingsolutions of (7.1) and also introduced, in a cumbersome manner, the system (7.9) forthe characteristic strips in order to reduce the problem to the theory of ordinarydifferential equations. A treatment of nonanalytic equations (7.1) awaited, of course,a knowledge of Theorem V 3.1. In fact, bothPicard's and Bendixson's work in 1896,mentioned in connection with Theorem V 3.1, were written from the point of view ofsolving a nonanalytic linear partial differential equation. A theorem similar to Theorem9.1 was given by Gross [1 ]. For fuller treatments of this problem, see Caratheodory [1 and Kamke [5]. For a theorem involving slightly less differentiability conditions, seeExercise 9.3 and the reference to Wazewski [3] and Digel [2]. Regions of existence forthe solutions were investigated by Kamke and also by Wazewski [2]. For Exercise 8.1,see PliS [2]. The example in Exercise 9.1 was given by Perron [1].

SECTION 10. Lemma 10.1 was given by Haar [1] for the purpose of proving theuniqueness Theorem 10.1. (This paper contains a wrong proof for Lemma 8.2 underthe assumption that u£ C1.) For Exercise 10.1, see Wazewski [1] (and Turski [1] fora generalization which is contained in Exercise 10.2). For Exercise 10.2, see Nagumo [3].

Chapter VIIThe Poincare-Bendixson Theory

The main part (§§ 2-9) of this chapter deals with the geometry of solutionsof differential equations on a plane (d = 2). The restriction to a planeappears essential since the arguments will make repeated use of the Jordancurve theorem. In § 10, the results obtained are applied to certain non-linear second order differential equations.

Recent extensions of the Poincare-Bendixson theory from planes to2-dimensional manifolds are presented in the Appendix in § 12. The lastsection (§ 14) concerns the behavior of solutions of differential equationson a torus.

1. Autonomous Systems

A system of differential equations in which the independent variable /does not occur explicitly,

is called autonomous. A trivial but important property of such systems isthe fact that if y = y(t), a < / < / ? , is a solution of (1.1), then y = y(t + /0)is also a solution for a — t0 < t < ft — t0 for any constant t0. An orbitwill mean a set of points y on a solution y = y(t) of (1.1) without referenceto a parametrization.

Any system y' =/(/, y) can be considered autonomous if the dependentvariable y is replaced by the (d + l)-vector (t, y) and the system y' =/isreplaced by t' = 1, y' =/(/, y), where the prime denotes differentiationwith respect to a new independent variable. For most purposes, however,this remark is not useful.

A point y0 is called a stationary or singular point of (1.1) if/(y0) = 0and a regular point if/(y0) ^ 0- The stationary points y0 are characterizedby the fact that the constant y(t) = y0 is a solution of (1.1). When solutionsof (1.1) are uniquely determined by initial conditions, /(y0) — 0 andy('o) — #o f°r some t0 imply y(t) = y0. This need not be the case ingeneral.

144

The Poincare-Bendixson Theory 145

If (1.1) has a solution C+:y — y(t) defined on a half-line / ^ /„, its setH(C+) of co-limit points is the (possibly empty) set of points y0 for whichthere exists a sequence /„ < tv < ... such that tn -*- oo and y(/n) -> y0 asn -*• oo. Correspondingly, if C~:y = y(f) is defined for r ^ /0, we definethe set A(C~) of a-limit points and if C:y = y(t) is defined for — CD < / <oo, its set of limit points is defined to be A(C) U Q(C).

Remark 1. Q(C+) is contained in the closure of the set of pointsC+:y = y(t\ t t0.

Theorem 1.1. Assume that f(y) is continuous on an open y-set E andthat C+:y = y+(t) is a solution of (I A) for t ^ 0. Then Q(C+) is closed. IfC+ has a compact closure in E, then n(C+) is connected.

Proof. The verification that £1(C+) is closed is trivial. In order to provethe last part, note that by Remark 1, Q(C+) is a compact set. Supposethat Q(C+) is not connected, then it has a decomposition into the union oftwo closed (hence, compact) sets Clt C2 such that dist (Clt C2) = 6 > 0.It is clear that there exists a sequence 0 < ^ < /2 < . . . of /-valuessatisfying dist (y+(tZn+1), Q) -> 0, dist (y+(t2n), C^ -> 0 as n -* oo. Hence,for large n, there is a point / = tn* such that tn < tn* < tn+1, dist (y+(/B*),C,) ^/4 for i = 1,2. The sequence y+(/i*), y+(^*)» • • • has a clusterpoint y0, since C+ has compact closure. Clearly, y0 e H(C+) and dist(y<» Q) = ^/4 for i = l , 2 . This contradiction proves the assertion.

Theorem 1.2. Le//, C+ 6e aj w Theorem l.landy0eE r\ Q(C+). TVien

Aos a/ /eaj/ o«e solution y = y0(t) on a maximal interval (<o_, a)+) such thaty0(t) e ii(C+) for co_ < / < <w+. In particular, when C+ has a compactclosure in E, then C0: y = y0(t) exists on (—00, oo) and C0 U A(C0) VJft(C0) <= H(C+).

An orbit C0:y = y0(0» <w_ < ^ < f>+, which is contained in some£2(C+), + " Q> is called an (co-) /im// orftiV. If, in addition, y = y0(t) isperiodic, y0(t + p) = y0(/) for all f and some/> > 0, the orbit C0:y = y0(0is called an (co-) limit cycle. (The condition C+ <t C0 assures that notevery periodic C0:y = y0(/) is a limit cycle; cf. the case of a family ofclosed orbits.)

Proof. Let 10 < tt < .. . and tn -> oo, yn~+y0 as n -» oo, wherey« = y+('«). Then yj(t) = y+(t + fj is a solution of

It follows therefore from Theorem II 3.2, where fn(t, y) =f(y) for « =1,2, . . . , that (1.2) has a solution y0(f) on a maximal interval (o>_, co_,_)and that there exists a sequence of positive integers «(1) < n(2) < ...

)

such that

holds uniformly on compact intervals of <w_ < / < o>+. It is clear thaty0(t) E Q(C+) for o>_ < / < o>+. This proves the first part of the theorem.

The second part concerning existence on (— oo, oo) follows at once fromTheorem II 3.1 which implies that the right maximal interval [0, <o+) fory = ^o(0 is either [0, oo) or y0(r) tends to BE as / -* o>+ < oo.

The last part concerning A(C0) and Q(C0) follows from (1.4) and thefact that Q(C+) is closed.

Remark 2. If solutions of all initial value problems associated with (1.1)are unique, then "selection" in the proof of the theorem is unnecessary;thus yn — y+(tn) -> y0 as n —> oo implies that

holds uniformly on every closed, bounded interval in (co_, o>+).Corollary 1.1. If Q(C+) consists of a single point y0 e E, then y0 is a

stationary point and y+(t) -> y0 as t —*• oo.

2. Uralaufsatz

In the plane (d = 2), where the Jordan curve theorem is available, thenotions of the last section can be carried much further to give the Poin-car£-Bendixson theory. This will be done in §§ 4-6. In order to avoid aninterruption of the proofs, the idea of the index of a plane stationary pointwill first be discussed.

Recall that a Jordan curve J is defined as a topological image of a circle;in other words, / is a y-set of points y = y(t], a t b, where y(i) iscontinuous, y(a) = y(b), and y(s) ^ y(t) for a s < / < b. The Jordancurve theorem will be stated here for reference. For a proof, the reader isreferred, e.g., to Newman [1, p. 115].

Jordan Curve Theorem. If J is a plane Jordan curve, then, its comple-ment in the plane is the union of two disjoint connected open sets, £x and E2,each having J as its boundary, dEl = dE2 = /.

One of the sets El or £2 is bounded and is called the interior of /;furthermore the interior of/ is simply connected.

Consider a continuous arc J:y = y(t), a t b, in the y = (y1, y2)plane. Let 77 = rj(t) ^ 0, a < t b, be a continuous 2-dimensionalvector attached to the point y(t), i.e., 77 jt. 0 is a vector field on J. Consideran angle y = <p(i) from the positive ^-direction (1,0) to rj(t), so thatcos<p = 77VNI, nnV-iflM, where II^H2 = W2 + W2- Theseformulae determine <p(t) up to an integral multiple of 2ir but if <p(t) is fixed

||

at some point, say t = a, then <p(t} is uniquely determined as a continuousfunction. By <p(t) below is always meant such a continuous determination.Define/„(/) by

For example, if rfo) is continuously differentiate,

If y = jj + /2 in the sense that J:y = y(r), a^t ^b, and a <c <b,J\'-y — y(t)> a^t-^c and 72:y == y(t), c t b, then

Figure 1.

Actually, if j?(f) is given,y,,(./) has nothing to do with J but, in applications,rj(t) will be a "vector beginning at the point y =• y(f)" of/.

The main interest below will be in the case that J is a Jordan curve, inwhich case it will always be assumed that J is positively oriented and rj j£ 0is a continuous vector on J [so that y(d) = y(b) and 17(0) = »?(£)]. [OnlyJordan curves J:y = y(t) which are piecewise of class C1 will occur, sothat the positive orientation means that the normal vector (—dy2jdt,dylldt) T£ 0, defined except at corners of J, points into the interior of /.]It is clear thaty"(-^) is an integer. It is called the index oft] with respect to J.

Theorem 2.1 (Umlaufsatz). Let J : y — y(i), 0 Is / 1, be a positivelyoriented Jordan curve of class Cl andr\(t) = dy\dt (5* 0) the tangent vectorfield on J. Thenjn(T)= 1.

Proof. On the triangle A : 0 <j * f 1, define r)(s, r)=[KO- 2/(s)]/tl rt/) - 9(5>| if 5 * r or (j, 0 * (0,1),»?(/, r) - y'(t)lly'(t)\\, and

97(0,1) = —77(0,0). It is clear that YI(S, i) is continuous and r)(s, /) 0 on A.Note that ??(0, /) and r)(t, 1) are oppositely oriented vectors; see Figure 1.

Suppose that the point y = y(Q) on J is chosen so that the tangent linethrough y(0) is parallel to the t^-axis and no part of J lies below thistangent line. Since A is simply connected, it is possible to define (uniquely)a continuous function y(s, t) such that g?(0, 0) = 0 and <p(s, t) is an anglefrom the positive -direction to TJ(S, t). Then 2-nj^J) = <p(l, 1) — 97(0, 0),as can be seen by considering <p(t, t).

Figure 2.

The position of J implies that 0 g?(0, ?) TT and that <p(Q, 1) is an oddmultiple of TT, hence g?(0,1) = TT. Similarly, a consideration of <p(s, 1) —??(0, 1) = y(s, 1) — TT for 0 s ^ 1 shows that 93(1, 1) — ir = IT. Con-sequently, 9?(1, 1) = ITT. Since 2irjn(J) = <p(l, 1) — 9?(0, 0) = <p(l, 1), thetheorem is proved.

The "rounding off" of corners in the case of a / which is piecewise ofclass C1 gives the following:

Corollary 2.1. Let J : y = y(t), 0 t < 1, be a positively orientedJordan curve which is piecewise of class C1 with corners at the t-values(0 < ) * ! < • • • < tn « 1) and»?(/) = dyjdt for t * tk. Let Jk:y = y(t),*k-i ^ *^ t» for * — 1, . . . , n + 1 with /„ = 0 and tn+l = 1. Then

n+l n2™ 2 M^k) + E(<Pk~ ^ — 2w, where <pk is the exterior angle 0 q>k

*=i *=i2?r a/ y(tk).

Note that ^ — IT is an angle, — IT ^ 9?,. — TT TT, from »;(/fc — 0) torj(tk + 0); see Figure 2.

r


The essential idea in the proof of Theorem 2.1 is contained in:Lemma 2.1. Let J: y = y(t\ a t :Ss b, be a Jordan curve, £(0 and

rj(t) two vector fields on J which can be deformed into one another withoutvanishing. Then jf(/) = /„(/).

The possibility of a deformation without vanishing means the existenceof a continuous vector 77 = r)(t, s) for a / ^ b, 0 s 5: 1, such that77(/, 0) = |(0, rj(t, 1) = »?(/), 77(0,5) = r,(b, s), and that r,(t, s) * 0. Forexample, »?(r, j) = (1 — j)f(0 + J»y(/) is such a deformation if £(f),»?(/)are not in opposite directions for any t.

Proof. Lety'(*) be the index of»/(/, j) for a fixed s. It is clear thaty'(-0 isa continuous function of s. Since y'(-s) is an integer, it is constant. Inparticular, y'(0) = y'0)-

3. Index of a Stationary Point

In what follows,/(y) = f(y1, y2) is continuous on an open plane set E.As before, a point where / = 0 is called a stationary point and a pointwhere/^ 0 is a regular point.

Let J:y — y(t), a is / ^ b, be an arc in E on which/(y) j& 0. Definey}(J) to bej^J), where »?(/) =/(y(0)- F°r example, if/(y), y(/) are of classC1, thenyyC/) is given by the line integral

When / is a positively oriented Jordan curve in E on which/5^ 0, theintegery/J) is called the index of f with respect to J.

Lemma 3.1. Let J0 and J^ be two Jordan curves in E which can bedeformed into one another in E without passing through a stationary point.Thenjf(J^=jf(Jl).

The assumption here means the existence of a continuous y(t, s),a ^ t ^ b, O ^ s ^ 1, such that (i) for a fixed s, J(s):y = y(t, s) is aJordan curve in £; (ii) 7(0) = yo, /(I) = /x; and (iii)/(#(/, s)) ^ 0. Theproof is the same as that of Lemma 2.1.

Corollary 3.1. Let J be a positively oriented Jordan curve in E such thatthe interior of J is in E and thatf(y) 7* 0 on and inside J. Thenjf(J) = 0.

Proof. Since the interior of a Jordan curve is simply connected, J can bedeformed (in its interior) to a small circle J^ around a point y0 of its interior.Since/(y0) 5* 0, it is clear that if the circle Jt is sufficiently small, the changeof the angle between f(y) and the ^-direction around Jl is small. Since

yX/j) is an integer, y/^) = 0. By Lemma 3.1,y//) = 0.Let y0 e E. Lemma 3.1 shows that the integery//) is independent of the

Jordan curve/ in the class of curves y in £ with interiors in E containing

)

)

no stationary point except possibly y0. This integer jf(J) is called the indexMVo) ofVo w»th respect to/. By Corollary 3.1,jf(y0) = 0 if y0 is a regularpoint. For this reason, only the indices of isolated stationary points y0 areconsidered.

Corollary 3.2. Let J be a positively oriented Jordan curve in E on whichf(y) ^ 0 and let the interior ofJ be in E and contain only a finite number ojstationary points ylt..., yn. Thenj,(J) =jM + ••• +jf(yn).

Figure 3.

For J can be deformed into a path consisting of circles around eachstationary point and "cuts" between the circles traced in both directions;see Figure 3.

Exercise 3.1. Show that according as ad — be > 0 or ad— be < 0,the index of the origin with respect to f0(y) = (ay1 + by*, cy1 4- dy*) is+ 1 or -1.

Exercise 3.2 (Continuation). Let f0(y) be as in the last exercise andfi(y) a continuous function defined for small \\y\\ such that^(y)/j|y|| -*• 0as y-»-0. Show that if/(y) =/0(y) +/i(y), then y « 0 is an isolatedstationary point and the index y/0) = ± 1 according as ad — be ^ 0.

Exercise 3.3. Let/(y) be of class C1 on an open set E with a Jacobiandeterminant det (dffdy) = dC/1,/2)/^1, y8) different from 0 wherever/ = 0. Let J be a positively oriented Jordan curve in E with interior / in Eand/(y) yt 0 on /. Show that there are at most a finite number of station-ary points y l5..., yk in / and that/X-O =* n+ ~~ n-> where n+ or n_ is thenumber of these points at which det (df/dy) > 0 or det (9//3y) < 0.

Theorem 3.1. Letf(y) be continuous on an open set E and let y = y9(t)be a solution ofy' ~/(y) of period p, yp(t + p) — yv(t)for — oo < t < oo.Let y a= yv(t), 0 t p, be a Jordan curve with an interior / contained inE andf(yv(i)) ^ 0. Then I contains a stationary point.

))

)

Proof. Let J be the Jordan curve, y = yv(t), 0 t < p, with a positiveorientation. By Theorem 2.\,jf(J) =1^0 . Thus the theorem followsfrom Corollary 3.1.

A return to tRe study of stationary points and their indices will be madein § 6 below.

4. The Poincare-Bendixson Theorem

The discussion of the differential equation y' =f(y) in § 1 will now becontinued for the plane case (d = 2) with the aid of the Jordan curvetheorem. The main result is the following theorem of Poincare-Bendixson.

Figure 4.

Theorem 4.1. Letf(y) =f(yl, y2) be continuous on an open plane set Eand let C+:y = y+(t) be a solution of

for t ^ 0 with a compact closure in-E. In addition, suppose that y+(tt) 9*^+(^2) f°r 0 = 'i < '2 < °° and tnat Q(C+) contains no stationary points.Then Q(C+) is the set of points y on aperiodic solution Cv\y = yv(f) of{4.1).Furthermore, ifp > 0 is the smallest period of y v(t\ then yv(ti) ^ yp(t^for0 ti < tz < p\ i.e., J\y — yv(t\ 0 = t p, is a Jordan curve.

In the case that initial value problems associated with (4.1) have uniquesolutions, either y+(/i) 5^ y+(tz) for 0 ^ < r2 < °° or y^(/) is periodic[i.e., y+(t + p) = y+(t) for all t for some fixed positive number/?]. In thelatter case (excluded in Theorem 4.1), Q(C+) coincides with the set ofpoints on C+.

The proof of Theorem 4.1 will show that C+ is a spiral which tends tothe closed curve Q(C+):y = y9(t) either from the exterior or from theinterior; see Figure 4. It will have the following consequence:

y

Corollary 4.1. Assume the conditions of Theorem 4.1 and let p > 0 be aperiod ofy = yv(t). Then there exists a sequence (0 ) t± < /2 < ... suchthat

uniformly for 0 / 51 /? OJM/

Proof of Theorem 4.1. A closed, bounded line segment L in E is calledtransversal to (4.1) if/(y) ,4 0 for y e L and the direction of/(y) at pointsy e L are not parallel to L. All crossings of L by a solution y — y(t) ofy' =/are in the same direction with increasing t.

Figure 5.

The proof will be divided into steps (a)-(e).

(a) Let y0 e E, /(y0) 5** 0, L a transversal through y0. Then Peano'sexistence theorem implies that there is a small neighborhood E0 of y0and an e > 0 such that any solution y — y°(t) of the initial value problemy' = /, y(G) = y° for y° e E0 exists for \t\ «• and crosses L exactly oncefor |/| e. In fact, if d > 0 is arbitrary, E0 and e can be chosen so thaty°(/) exists and differs from y° + tf(y0) by at most d \t\ for |/| 51 «=. Thusif £0 is sufficiently small, y = y°(/) crosses L at least once, but can cross Lat most once for \t\ ^ e since crossings of L are in the same direction.

In particular, it follows that if y = y°(t) is a solution of y' =f on aclosed bounded interval, then y = y°(t) has at most a finite number ofcrossings of L.

(b) Let Z, be a transversal which, without loss of generality, can besupposed to be on the y2-axis, where y = (y\ y2). Suppose that y — y+(i)crosses L at /-values /x < /2 < ..., then y+

2(/n) is strictly monotone in n.In order to see this, suppose without loss of generality that crossings of

L occur with increasing y1 (i.e., y1 changes from negative to positive valuesat crossings). Consider the case that y+

2(/i) < y+z(t2); see Figure 5.

The set consisting of the arc y = y+(t), tt ^ / ^ /2, and the line segment2/+2('i) = yz = y+z(t^) on the ya-axis forms a Jordan curve /. For allt > tz, y = y+(i) is in the exterior ofJ or in the interior of J, by the assump-tion on y+(t) and the fact that crossings of L occur only in one direction.This makes it clear that y+\t^ < y+2(/3) and the argument can be repeated.

(c) It will now be verified that if L is a transversal, Q(C+) contains atmost one point on L. For if y0 e L n Q(C+), part (a) implies that y —y+(t) crosses L infinitely many times [in fact, whenever y+(t) comes neary0]. With increasing t, the intersections of y = y+(t) and L tend monoto-nously along L to y0, by part (b). Thus L n O(C+) cannot contain anyother point y = y0.

(</) Since C+ is bounded, Q(C+) is not empty. Let y0 e (C+). By

Theorem 1.2, y' = /, y(Q) = y0 has a solution C0 :y = y0(0, — co < / < oo,contained in O(C+); thus Q(C0) <= Q(C+).

Q(C0) is not empty. Let y° e Q(C0), so that y° is a regular point sinceQ(C+) contains no stationary points. Thus there is a transversal L°through y° and y = y0(t) has infinitely many crossings of L° near y°, but y°and every such crossing is a point of H(C+). By (c), these points coincide.In particular, there exist points t± < t2 such that y° — y0(^i) = ^0(^2)- Itfollows that (4.1) has a periodic solution y = yv(t) of period p = tz — such that yp(0 = y0(f) for /x / < /2. Since y0(r) is not constant on any^-interval, it can be supposed that y^t0) ^ yv(t0) for 0 t0< t° <p.

(e) It must be shown that H(C+) coincides with its subset CP:y = yp(0,— oo < t < oo. If not, Q(C+) — Cp is not empty. Then Cv contains apoint y± which is a cluster point of Q(C+) — Cp, since Q(C+) is connectedby Theorem 1.1. Let Lj be a transversal through yv Any small sphereabout yl contains points y2 ^ (C+) — Cv. For any such yz, y' =/has asolution y = yz(t), — oo < / < oo, such that y2(0) = y2 and y2(t) iscontained in Q(C+), by Theorem 1.2. If yz is sufficiently close to yl5 theny2(/) crosses the transversal L!. The crossing is necessarily at the point ylby part (c).

Since i/2 £ C,,, this is impossible when solutions of initial value problemsbelonging to (4.1) are unique. That it is impossible in the general case canbe seen as follows: Let y2(fp) e CP, while yz(t) $ Cv for t between 0 and tp.Since yz(tp) is a regular point, there is a transversal Lv through y2(tp)- Thena small translation of Lv in a suitable direction is a transversal LVO whichmeets Cv and y = yz(t) in two distinct points; see Figure 6. This contra-dicts part (c) and proves the theorem.

Remark. For subsequent use, note that the argument in part (e) showsthat whenever C+:y = y+(t\ t^.Q, has the property that y+(t^ ^ y+(t^)for /! j^ /2

and ^o6 &(C+) n £ is a regular point, then there is a neighbor-hood EQ of j[0 such that the solution of y' =/, y(0) = y0 in H(C+) n EQ is

)

unique. If, in addition, ii(C+) is connected and there is a periodic orbitCj>:y = yv(t) in Q(C+) consisting only of regular points, then ii(C+) = Cp.

Proof of Corollary 4.1. Let y° = yP(G) and let L° be a transversalthrough y°. Let the successive crossings of L° by y = y+(t) occur at (0 )/i < /8 < • • • • Then y+(/n) tends monotonically along Z,° to y°. Sincey = yj>(0 is the unique solution of y' =/, y(0) = y° in n(C+), an analogueof Remark 2 following Theorem 1.2 shows that (4.2) holds uniformly forbounded /-intervals, in particular for 0 / p.

Figure 6.

Note that y+(tn +/»)-»• y9(p) = y°, n -* oo. Thus if t > 0 and n islarge, y+(t) crosses L° in the interval [tn + p — c,tn + p + e]. Hence'n+i^'n + /> + *• Also lly+fo +/) - y,(OII is sma11 for larSe »»0 ' < e ^ / < / ? — f <p, which implies that there is a ^ > 0 such thatlly+(ffi + /) — y°j| ^ <5 for 0 < e / ^ /? — e. In particular, there is nocrossing of L° for «• ^ / p — e. Hence /n+1 tn+ p — e for large n.This proves the corollary.

Theorem 4.2. />f/, C+ 6e «5 in Theorem 4.1 except that Q(C+) containsa finite number n of stationary points of (4.1). If n = 0, Theorem 4.1applies. Ifn — \ and £1(C+) is a point, Corollary 1.1 applies. If I ^ « < ooa/w/£i(C+) is nctf a point, then li(C+) consists of stationary points ylt yz,...,yn and a finite or infinite sequence of orbits C0:y = y0(t), — oo a_ < t <a+ oo, w/H'cA fito not pass through a stationary pointbut^0(

a±) = ^m y(0»aj / —>• a±, exw/ a»^ are among the set y l 9 . . . , yn.

It is possible that y0(a+) — yo(a-)- It is not claimed that (a_, a+) is the

maximal interval of existence of y = y0(0- But when initial conditionsuniquely determine solutions of (4.1), so that the only solution of (4.1)through a stationary point yk is y(f) = yk, then it follows that <x_ = — ooand a+ = oo; cf. Lemma II 3.1.

Proof. Consider the case that n 1 and Q(C+) is not a point. SinceQ(C+) is connected by Theorem 1.1, it contains regular points y0. For any

such point y0, there exist solutions C+'.y = y*(t\ ^oo < t < oo, ofy' =/, y(0) = y0 in Q(C+). Let C* denote any such solution.

Consider only C#+:y = y^(t) for 0. The treatment of / 0 is

similar. There are two cases: (i) there is a first positive / = a+, where^(a^) is a stationary point (hence one of the points ylt... ,-yJ or (ii)y+{t) is not a stationary point for any finite f > 0.

Consider case (ii). If fi(C#+) contains a regular point y°, then, by part

(d) in the proof of Theorem 4.1, C*+ contains a periodic solution pathCv:y =* yj(t). But then Cv contains a stationary point yt; otherwise,- bypart (e) and the Remark at the end of the proof of Theorem 4.1, Q(C+) =CP. This is impossible since y*(t) & yt for t 0. Hence in case (ii),£}(C*+) can contain only stationary points and, since it is connected, onlyone stationary point y+.

Thus, in case (ii), (i(C,,,+) is a stationary point y+ and y+(i)^y+ ast -*• oo by Corollary 1.1. Thus any regular point y0 e O(C+) is on an arcC9:y = y0(t), a_ < / < a+, in Q(C+) of the "type specified.

It remains to show that the set of such arcs C0 is at most denumerable.Note that if y0 e Q(C+) is a regular point, the solution C* :y — y+(i)above is unique on a sufficiently small interval |/| < e; cf. part (e) and theRemark following the proof of Theorem 4.1. Thus, no two of the arcsC0 can meet.

Since y+(ti) 7* y+(tz) for tz > tlt it can be supposed that y+(0 is not oneof the stationary points yl9..., yn for / 0. Otherwise, y+(t), t 0, isreplaced by y+(t), t t0, for a suitable t0 > 0. Also if y0 e Q(C+) is aregular point, then y+(t) ^ y0 for / J2: 0 for the sequence of intersections of2/ ~ ^+(0 witn a transversal through y0 tends to y0 in a strictly monotonesense; cf. part (b) of the proof of Theorem 4.1. Thus C+ and C0 have nopoints in common.

Suppose, if possible, that there exists a nondenumerable set of C0 whichcan then be assumed to join the same, not necessarily distinct, pair ofstationary points. Any one or two of these joined together forms a Jordancurve J. Since there is at most a denumerable set of/ with pairwise disjointinteriors, there exist three such distinct /, say Ji,Jz,Ja, such that /3 iscontained in the closure of the interior /„ of Jn for n = 1,2. This isimpossible, for since C+ does not meet Jn, C+ is either between Jt and J2 orJz and J3.

In the next theorem, the assumption "y+(O ^ y+(ty) for fx ^ tz" isomitted.

Theorem 4.3. Letf(y) =/(y1, y2) be continuous on an open plane set Eand C+:y = y+(t) a solution of (4.1) for ttZQ with compact closure in E.Then ii(C+) contains a closed (periodic) orbit Cv:y = yf(t) of (4.1) whichcan reduce to a stationary point yv(t) = y0.

Proof. Suppose that O(C+) does not contain a stationary point. Lety0eQ(C+) and C0

+:y = «/0(0> ° = t < oo, be a solution supplied byTheorem 1.2, so that C0+ O(C+). Since Q(C+) is closed, fl(C0

+) <=Q(C+). If t/o(fi) ?* yo('a) for 0 < /! < f2 < oo, then Q(C0+) is a closedorbit y = ^(r) by Theorem 4.1. If y0('i) = Vo(tz) for certain f l5 ra with0 fj < ?2 < °°> then the orbit C0

+ contains the periodic solution pathy = y»(0 of period p — t2 — which coincides with y0(t) for rx / fa-in either case Q(C+) contains a closed orbit y = yj(t).

Theorem 4.4. Letf(y) be continuous on an open, simply connected planeset E where f(y) ^ 0 and y = y(t\ a solution of (4.1) on its maximalinterval of existence (o>_, eo+). Then y = y(t) does not remain in anycompact subset E0 of E as t —> co+ [or t -*• w_].

Proof. If, e.g., C+ :y = y(t), t0 t < to+, is in a compact set £0 in Efor some f0, then co+ = oo by Theorem II 3.1 and Q(C+) contains aperiodic solution y = yv(t) by the last theorem. Since/-^ 0 on E, yv(t)does not reduce to a constant on any /-interval. Let ?x be the first / > r0,where yv(t^ — y,(/0)- Then y = yp(t), t0 t tlt is a Jordan curve /.Thus (4.1) has a periodic solution y = y0(t) of period fx — t0 such thaty0(f) = yv(i) for ?o = ' = ^i- Since E is simply connected, the interior ofJ:y •= y0(t), t0 t /19 is contained in £. By Theorem 3.1, this interiorcontains a stationary point. This contradicts / 5^ 0 and proves thetheorem.

5. Stability of Periodic Solutions

Return to the situation in Theorem 4.1.Theorem 5.1. Letf, C+, Cv be as in Theorem 4.1 Then there exists an

c > 0 such that ify0 is within a distance e of£l(C+) = Cv:y — yv(t) and onthe same side (interior or exterior) of Cv as C+, then

has a solution C0:y = y0(0 for t *£ 0 such that y^) ^ y0(tj for ^ tzanda(CQ+) = Cv.

Proof For sake of definiteness, consider the case that C+ is exterior toC,. Let y° = yv(Q), L° a transversal through y°, the successive crossingsof L° byy+(t) occur at (0 ) t± < tz < Thus y+(tn) tends monoton-ously along L° to y°.

Let Jn be the Jordan curve consisting of the arc y = y+(t\ tn t tn+1, and the open segment /„ of L° joining y+(tn), y+(tn+i). Let Dn be theinterior of Jn and £„ = /)„— 5n+1. Note that £B is a simply connectedopen set in £ since /n+1, except for the point y+(/B+1), is interior to /„;see Figure 7.

It is clear that if e > 0 is sufficiently small, then (jEn contains all pointsy0 £ C+ U Cp within a distance e of Cv on the same side as C+. It is alsoclear from Corollary 4.1 that the union U£"n, n^.N, for largeN is within a distance of Cv. Since/(y) ^Qforye Cp, it can be supposedthat (jEn contains no stationary points; otherwise, tlt / 2 , . . . is replaced

ty *jv> 'iv+i»

Figure 7. Shaded area is Am-

In the proof, it is sufficient to consider only those y0 $ C+ U CP, so thaty0 E En for some n. Let y = y0(t) be a solution of (5.1) for t near 0. ByTheorem 4.4, a continuation of this solution for increasing t meets theboundary dEn of E at a finite f-value. Let f0 be the first /0 > 0 at whichy0(/0) £ #£"„, where 9fn

c C+ U /„ U 7n+1. It can be supposed thaty^t1) j£ y0(t2) for 0 f1 < t2 /0. Because of the direction of crossingsof solutions on 7n, y0(f0) £ 7n so that y0(^o) e C+ U 7n+1. If y0(/0) e C+,saX yo(fo) = y+('°). tnen yo(0 can be defined for t ^ /0 to be y0(r) =y+(f + t° - /0). If y0(^o) e 4+i. then y0(/) exists for t(> /0) near /„ and is*Wi-

Continuing this procedure, we obtain y0(t) defined for t 5: 0 so thateither y0(t) = y+(t + a) for some a and large t or the solution y = y0(t)with increasing / successively passes through En, En+l,.... This completesthe proof.

Let Cv:y = yp(t) be a periodic solution of y' =/of period/? > 0 suchthat/:y = yp(i), 0 t /?, is a Jordan curve. Cj, is called orbitally stablefrom the exterior as t -*• + oo if, for every e > 0, there exists a <5 = 5 > 0

e

with property that if y0 is exterior to but within a distance <5 of/, then allsolutions C0

+:y = y0(t) of (5.1) exist and remain within a distance e of /for t zzQ. Cv is called asymptotically orbitally stable from the exterior ast -*• oo, if there exists a «5 > 0 such that if y0 is exterior to but within adistance d of J, then all solutions CQ+:y = y0(t) of (5.1) exist for t 0 andQ(C0+) = /. Similar definitions hold with "exterior" replaced by "in-terior" and/or 'V -> oo" by "/ -* - oo".

Theorem 5.2. Let f(y) be continuous on an open plane set E with theproperty that initial values determine unique solutions of y' =/. LetCp'.y — yp(t) be a periodic solution ofy' *=fwith a least positive period p.(i) Then Cp is asymptotically orbitally stable from the exterior as t ->• oo ifand only if the orbit Cp is Q(C0

+)/or some solution C0+:y = y0(0» t = 0.

on the exterior of Cp. (ii) Then Cv is orbitally stable from the exterior ast -*• oo if and only if either the orbit CP is Q(C0+) for some solution C0+exterior to Cp or, for every e > 0, there is a periodic solution of (1.1)exterior to and within a distance e of Cp.

Proof. In (i), "only if" is trivial and "if" follows from Theorem 5.1.In (ii), "if" is clear from Theorem 5.1. In order to prove the converse,suppose that Cv is orbitally stable from the exterior. Since solutions ofinitial value problems (5.1) are unique,/(y) *£ 0 on Cv and, hence, in some 0-vicinity of Cv. Let C0+:y = y»(t\ t 0, be a solution of y' = /,exterior to but within a distance e < e0 of Cp. Then C0+ has a compactclosure in E and Q(C0+) contains no stationary points, so that Q(C0

+) is aperiodic solution path ofy' = /by Theorem 4.1. Thus either £2(C0

+) = CPor Q(C0+) is periodic solution path exterior to and within a distance e of Cp.

6. Rotation Points

In §§ 6-9, the behavior of solutions of

near an isolated stationary point will be considered. Let/(y) be definedfor small ||y||, say, on an open set containing \\y\\ ^ b, and let

Note that if (6^.1) has a unique solution for all y0, then Theorems 4.1 and4.2 imply that if 0 < j|y0|| < b and C0

+:y = y0(r) is the solution of (6.1) onthe maximal interval [0, co+), then only the following (not mutuallyexclusive) cases can occur: (i) there is a least /0, 0 < /0 < to+, such thatllyo('o)l — b\ (ii) w+ = oo and the solution path C0

+ is a Jordan curvewith y = 0 in its interior; (iii) co+ = oo and C0+ is a spiral approachingsuch a closed orbit; (iv) <w+ = oo and y0(t)-+Q as /-*• oo; and (v)a>+ = oo, C0

+ is a spiral around y = 0 and Q(C0+) consists of a finite or

infinite sequence of orbits y(t), — oo < / < oo, such that y(t) -* 0 as/-» ±00.

By a spiral y — y0(0» 0 = t < co+, around y — 0 is meant an arcyQ(t) j& 0 such that a continuous determination of arc tan y<?(t)ly0

l(t)tends to either oo or — oo as / -»• oo+,

If every neighborhood of y — 0 contains closed orbits surroundingy — 0, the stationary point y = 0 is called a rotation point.

When y = 0 is a rotation point and solutions of arbitrary initial valueproblems (6.1) are unique, the set of solutions of y' =/in a neighborhoodof y = 0 can be described as follows: there is a neighborhood E0 of y — 0such that the solution C0:y = y0(t) of (6.1) for every y0 E E0 is either aclosed orbit surrounding y — 0 or is spiral such that ft(C0), A(C0) areclosed orbits surrounding y — 0.

This is illustrated by the following: Consider the differential equations

where u, v are continuous, real-valued functions of r = \\y\\ for smallr ^ 0. In polar coordinates, these equations become

Example 1. Suppose that u(r) = 0, v(r) = f$ ^ 0. Then (6.3) is

and (6.4) is r' = 0, 6' =0, so that all orbits (y & 0) are circles.Example 2. Suppose that u(r) = r sin (1/r), v(r) = 1. Then (6.4) is

r' = r2 sin (1/r), 6' = 1. Thus, besides the trivial solution y = 0, there areclosed orbits r = 1/mr, w = 1, 2 , . . . . Between the orbits, r = \\rnr andr = \j(n + \)TT, (— 1)V > 0 and so the corresponding orbits are spiralswhich tend to the circles r = I Inn, r = l/(n + l)?r as t -* oo, t -*• — oo or/ —»> oo, f ->• — oo, depending on the parity of n.

Example 3. In the last example, u(r) can be redefined to be 0, say,between r = \\rnt and r — !/(« + !)TT for a finite or infinite sequence of/j-values 1, 2 , . . . . Correspondingly, the spiral orbits between r = I//ITTand r — lj(n + l)ir are replaced by circular orbits.

Exercise 6.1. Let C be a closed set on 0 r ^ 1. Show that thereexist functions w(r), v(r) which are uniformly Lipschitz continuous on0 r < 2, «2(r) + v\r) 7* 0 for r 0, and the solution of (6.3) withinitial condition t/(0) = y0 is a closed orbit if 0 < ||y0|| e C and is a spiral

if 11*11 *C,0< I I S f o H ^ l .A rotation point y = 0 such that all orbits, except y = 0, in a vicinity of

y = 0 are closed curves is called a center. The simplest illustration of acenter is the linear system (6.5) in Example 1.


7. Foci, Nodes, and Saddle Points

Assume that the only solution of y' = f(y), y(Q) = 0 is y = 0 [so thatno solution y(t) ^ 0 can tend to 0 as / tends to a finite value].

The simplest nonrotation points are called "attractors." The isolatedstationary point y — 0 is called an attractor for / = oo [or / = — oo] if allsolutions y — y0(0 °f (6.1) for small ||y0ll exist for t 0 [or t 0] andy0(t) -*> 0 as t -> oo [or t ->- — oo]. If, in addition, all orbits y0(t) ^ 0 are

Figure 8

)

spirals, then the attractor y = 0 is called a focus. If all orbits, y0(f) ^ 0have a tangent at y — 0; i.e., if a continuous determination of 0(f) ==arc tan y<?(t)lyj(t) tends to a limit 00, — oo < 00 < oo, then the attractory = 0 is called a node. A node is called a proper node if for every 00 mod2rr, there is a unique solution y = y0(/) such that 6(t) -»• 00; otherwise it iscalled an improper node.

Illustration of these cases of attractors are given by real linear equations;see Figure 8. The system

is an attractor for / = oo [or / = — oo] if a < 0 [or a > 0]. It is a focusif a, 0 5^ 0 and a proper node if a ^ 0, $ = 0. In the case,

y = 0 is an improper node if AXA2 > 0, Ax 5^ A2. The case

where A ?* 0 is also an improper node.There are nonrotation points which are not attractors and attractors

which are neither foci nor nodes. The simplest example of a stationarypoint which is not an attractor is a saddle point. This is a stationary pointy = 0 with the property that only a finite number of orbits tend to 0 as/ — » + o o o r f - > — oo. This is illustrated by (7.2) where Alf A2 are real andA! A2 < 0.

Exercise 7.1. Verify the statements just made about (7.1)-(7.3).Exercise 7.2. Consider the linear system y' = Ay, where A is a real

constant 2 x 2 matrix with det A y* 0, so that y = 0 is the only stationarypoint. Let Als A2 be the characteristic values for A. Show that y = 0 is anattractor for t = oo [or t = — oo] if and only if Re Afc < 0 [or > 0] fork — 1,2; y = 0 i sa center if and only if Re At = Re A2 = 0; y = 0 is afocus if and only if Aj, A2 are complex conjugates, but not real or purelyimaginary; y = 0 is a proper node if Ax = A2 and the elementary divisorsare simple; y = 0 is an improper node if Als A2 > 0 but either A! 5^ A2 orAj = A2 and the elementary divisor is multiple.

8. Sectors

The general nonrotation point will now be considered. It will beconvenient to have the following terminology: A solution y = y(t) & 0°f y' =/defined on an interval [0, CD) (or an interval (—<u, 0]) for 0 <CD ^ oo is called a positive [or negative] null solution if y(0 -»• 0 as t -*> co[or —o>]. When the solution of y' =/, y(0) = 0 is unique, then necessarilyft) = 00.

Lemma 8.1. Letf(y) be continuous for small ||y||,/(0) = 0, andf(y) 5*0/or y y* 0. Suppose that y = 0 /s not a rotation point. Then there existsat least one null solution.

Proof. Suppose that e > 0 is so small that there is no closed orbit in||y|| ^ e surrounding y = 0 and suppose, if possible, that there is no nullsolution in ||y|| ^ e.

Then there is no solution C0:y = y0(0 0 of y' =/ defined andsatisfying ||y0(f)ll ^ « for t 0. For otherwise y0(O ?* Vofrt) when'i '2 (since there are no closed orbits in ||y|| ^ e) and Theorem 4.2implies that there exists at least one positive null curve. Similarly, nosolution y = y0(t) & 0 if y' — f exists and satisfies ||y0(/)|| ^ e for t 2s 0.

Thus, if ||y0|| < c and y0(t) is a solution of (6.1), there is a boundedinterval -s^t^O such that y0(Q) = y0, ||y0(f)ll < « for -5 < r < 0and ||yQ(—5)|| = e. Correspondingly, the solution y = y0(/ — s) is definedfor 0 f < 5, ||yc(/ - 5)|| < e for 0 < / ^ 5, ||y0(f - 5)|| = e for t = 0and y0(f — s) = y0 for / = s.

By considering a sequence of points y0 = ylt yz,... tending to theorigin, we obtain a sequence of solutions y = yn(t), 0 t 5M, such thatl!f«(0)| = e, lly«(0il < e for 0 < r ^ sn, \\yn(sn)\\ -0 as «-> oo. Aftera selection of a subsequence and renumbering, it can be supposed thaty0 = lim yn(Q) and co = lim sn, n -> oo, exist, where ||y0|| = « and 0 «> ^ oo. Also, if a> > 0, it can be supposed that y0(t) = lim yn(t), n -> oo,exists uniformly on every closed bounded interval of [0, <o) and is asolution of y' = f.

It is clear that at > 0. For otherwise, for large n, y = yn(t) is in a smallvicinity of y0 for 0 / ^ 5n; thus yn(sn) ->• 0, n -> oo, is impossible. If0 < to < oo, then Peano's existence theorem shows that, for large n,yn(t) can be defined on an interval containing 0 t eo, and y0(0 —lim yn(0 exists uniformly for 0 51 r o>; thus, y0(co) = lim yn(5n) = 0.Finally, if a> = oo, then y0(/) is defined and ||#0(0!l = « f°r ^ = 0- This isa contradiction and proves the lemma.

Hypothesis. In what follows, assume that solutions of arbitrary initialvalue problems y' = f(y), y(0) = y0

are unique.Let C be a positively oriented Jordan curve surrounding y = 0. A

solution y = y(t) of y' =/(#) is called a positive or negative base solutionfor C if y(t) is defined for either / 0 or t 0, y(0) 6 C, y(f) is interior toC for / ?«£ 0, and y(/) is a null solution.

Let y = 3^(0, y&) be base solutions for C. The open subset 5 of theinterior of C with boundary consisting of y = 0, the arcs y = y^t), ya(0and the (oriented closed) subarc C12 from ^(0) to yz(Q) will be called thesector of C [determined by the ordered pair y — y^(t\ y = y«(f)]• It ^s not

excluded that ^(0) = y2(0) so that C12 can be C or reduce to a point.

)

Consider the case that there exists a solution y = y0(t), — oo < / < oo,of y' =/ which is interior or on C for all t and y0(t + /x) m yt(f) for* 0. y<>(' + '*) = y»(0 for / 0 for some tlt t2(^ /O; see Figure 9.The point y = 0 and the arc y = y0(r), — oo < t < oo, form a Jordan

Figure 9. Elliptic sectors.

curve J with interior /. If S contains /, then it is called an elliptic sector.When tv = /2 [so that yx(0) = ya(0) = y0('i)]»

and C12 reduces to the pointy0(ti\ then S is elliptic and coincides with /. When f t 5* fa» «S can containpoints not in /. By considering the possibilities (i)-<v) mentioned after(6.2), it is seen that if y = y(0 is a solution of (6.1) with y(0) e 7, then y(f)exists for — oo < / < oo and y(/) -»> 0 as t -> db oo.

Figure 10. (o) Hyperbolic sector, (b) Parabolic sector.

A sector S with the properties that it is not an elliptic sector and thatS U C12 contains no base solution is called a hyperbolic sector; seeFigure 10(0). Part (a) of the proof of Theorem 9.1 below implies that oneof the boundary arcs y = yi(t), y2(r) of a hyperbolic sector is a positiveand the other a negative base solution.

A sector S with the properties that both boundary arcs yi(t\ y2(0 are

positive [or negative] base solutions and that the closure of S contains no

negative [or positive] base solution is called a positive [or negative]parabolic sector; see Figure 10(£).

Note that any type of sector S (elliptic, hyperbolic, or parabolic) cancontain solutions y = y(t), — <x> < / < oo, such that y(t) -»• 0 as / ->± oo.Such an arc and the point y = 0 constitute a Jordan curve. There can evenbe an infinite sequence of such orbits in S with the interiors of the corre-sponding Jordan curves pairwise disjoint. Denote by Se, the elliptic portionofS, the sets of points of S on such orbits and the point y — 0. Then Se isclosed. In the case of hyperbolic and parabolic sectors S, Se and C12 aredisjoint.

Hyperbolic sectors S do and parabolic sectors S can contain opensolution arcs y = y(t) having both endpoints on C12. The closure of theset of points y on such arcs will be called Sh, the hyperbolic part of S. Itwill be clear from parts (a) and (b) of the proof of Theorem 9.1 thaty — 0 E Sh or y — 0 £ Sh according as S is hyperbolic or parabolic.

Lemma 8.2. Let f(y) be continuous on a simply connected open set Econtaining y = 0 such that /(O) = 0, f(y) 5* 0 if y ^ 0, and that thesolutions of initial value problems y' = f(y), y(Q) — y0 are unique. Let C bea Jordan curve in E surrounding y = 0. Then there is at most a finitenumber of elliptic and hyperbolic sectors in C.

Proof. If the lemma is false, then there is a point yQ e C and a sequenceof points y(i)(0), y(2)(0),... of C tending monotonously to y0 along Csuch that y(2w)(0), y(2n+i)(0) is the initial point of a positive, negative basesolution, y(2n)(/), 2/<2«+i)(0» respectively. Then y = yM(t) is in a sectorwith boundary arcs y = y{n-l}(t\ y{n+u(t) for n 1. Clearly, yw(t) =lim y,2B)(0» y(1)(0 = lim 2/<2n+i)(0» »-*•<», exist uniformly on boundedintervals of t 0, t 0, respectively, and are solutions of y' = f. But aspoint sets, the two arcs y — y(Q}(t\ y(l\t) are identical. This is impossibleas can be seen by considering yw(t) for small / ^ 0 and y(1)(f) for small—/ ^ 0. Thus the lemma is correct.

Lemma 8.3. Letf(y), C be as in Lemma 8.2. If the closures of all of thehyperbolic and elliptic sectors are deleted from the interior of C, then theresidual set is either empty, the interior ofC, or the union of a finite numberof pairwise disjoint parabolic sectors.

Proof. It is sufficient to consider the case when there exist hyperbolicand/or elliptic sectors and that the residual set is the union of a finitenumber of disjoint sectors. It has to be shown that these sectors areparabolic.

Let S be a sector not containing any hyperbolic or elliptic sectors.Suppose first that of the two boundary base solutions y = (0, y*(i) of 5,one is positive and the other negative. It will be shown that this leads to acontradiction.

For the sake of definiteness, let yx(f) be a positive, y2(f) a negative basesolution; see Figure 11. Moving on the boundary arc C12 from y^O) toy2(0), there is a last point yx* [possibly y^O)] such that the solutiony = yx*(f) of y' =/, y(0) == yx* exists and is in S for t 0 (and hence is apositive null solution). Then yx* y2(0), since S contains no ellipticsectors. Moving on C12 from y2(0) toward yf, there is a last point y2*such that the solution y = y2*(0 °f y' =/» #(0) = V** exists and is in Sfor f ^ 0. Then y2* T^y-f since 5" contains no elliptic sector.

Let C*2 be the subarc of C12 joining yt* and y2*. The solution y —y\*(t) [°r yz*(t)] with f increasing [or decreasing] from 0 has a last point yu

[or y22] on C, where yu [or yZ2] can coincide with * [or y2*] and is on thearc (or point) of C from ^(0) to yt* [or yz* to y8]. Let Cu [or C22] be thesubarc on C from yn to yx* [or ya* to y22] and C12 = Cu u C*2 U C22.Let y = ynf/) [or y22(/)] be the solution of y' = /, y(0) = yn [or yM] fort ^ 0 [or f < 0]. Then there is a sector 5* with boundary consisting ofy*= 0, the base solutions y = yu(f)» y*z(t), and the arc C12.

Since 5* is subset of 5, it is not an elliptic sector. No solution y = y0(t)of (6.1) with y0 an interior point of C12 is in 5* U C12 for / 0 or / ^ 0.This is clear if y0 6 Cf2 by the definition of the endpoints of yt*, y2* of C^.It is also clear if y0 ci Cn U C22, for the solution arc beginning at such apoint y0 either leaves S or is a part of the solution arcs y = yi*(0» ^2*(')which for large 1/| are not in S* (but on the boundary of 5*).

Thus 5* is a hyperbolic sector in S. This contradiction shows that theboundary base solutions y = yx(0, y2(0 cannot be of opposite type (i.e.,positive and negative).

Figure 11.

Consider the case that both boundary base solutions y = yi(t), y2(t) ofS are positive [or negative]. Then if S is not parabolic, it contains anegative [or positive] base solution y = ya(t). But then y^(t), y3(t) andyz(t), y3(t) define sectors of the type just discussed. This is impossible,hence S is parabolic and the lemma is proved.

In order to avoid a consideration of special cases, the following con-vention will be adopted: If y = 0 is a nonrotation point, then, by Lemma8.1, there exists at least one base solution y = y^(t) if C is in a sufficientlysmall neighborhood of y = 0. If there are no hyperbolic or elliptic sectors,the arcs y^t) and y2(0 = y\(f) define a parabolic sector with C12 = C.Thus, for small C around a nonrotation point, there is always a decom-position of the interior of C into a finite number of elliptic, hyperbolic,and parabolic sectors.

Exercise 8.1. Let/, C be as in Lemma 8.2. Suppose that the closure ofthe interior I of C contains no periodic solutions. Show that there is apoint yQ e C such that the solution C0:y — y0(t) of (6.1) exists and is in /for either / 0 or t 0. Thus y0(f) is either a null solution or is a spiralaround y = 0 such that Q(C0) or A(C0) contains a solution y = y(/),— oo < / < co, which is both a negative and a positive null solution.

9. The General Stationary Point

The object of this section is to prove the following theorem:Theorem 9.1. Let f(y) be continuous on a simply connected open set E

containing y — 0 such that /(O) = 0 and f(y) j& 0 for y y& 0 and thatsolutions of initial value problems (6.1) are unique. Let C be a Jordan curvein a sufficiently small neighborhood of y = 0 surrounding y — 0, ne thenumber of elliptic and nh the number of hyperbolic sectors in C. Thenthe index j/Q) of the point y =* 0 is given by

It is clear that for a rotation point y = 0, ne = nh = 0 andy/0) = 1, sothat (9.1) holds and such points need not be considered. Thus it can besupposed that there is a decomposition of the interior of C into elliptic,hyperbolic and parabolic sectors.

Proof. C will be replaced by a piecewise C1 Jordan curve C', aroundy = 0, made up of solution arcs and orthogonal trajectories, as shown inFigure 12, where E, H, P represent elliptic, hyperbolic, and parabolicsectors. If -rj is the tangent vector to C' and y is a subarc of C', withendpoints which are not corners, then, in this proof 2ir/n(J) will representthe contribution of/to 2irjn(C') = 2*r, i.e., the variation of the turning of9? along J taking into account discontinuities of rj in accordance withCorollary 2.1. Furthermorey/0) =y/C').

)

The PoincareVBendixson Theory 167

If L is an arc joining two points A and B, the notation [AB], (AB), etc.will be used to denote the corresponding closed arc, open arc, etc. Forany sector with boundary base solutions y = y\(t\ y2(f)> l

et At = y,(±e0)for a fixed e0 > 0, where ± is chosen according as y£f) is a positive ornegative base solution.

(a) Hyperbolic Sectors. Let S be a hyperbolic sector determined byy = y\(t), ya(0- 1° order to fix ideas, suppose that y = y±(t) is a positivebase solution. Thus Al — yi(+e0), where 0 > 0. It is clear that A^ $ Se,the elliptic part of S. Since Se is closed, points y of S near Al are not in Se.

Figure 12.

Consider the differential equation

for the orthogonal trajectories of solutions of y' =/. Let Lt = M.JJJ,i = 1, 2, be a solution arc of (9.2) with initial point Ai such that (/4.-5J isin S — Se; see Figure 13. It will be shown that if Lx = [A^] is suffi-ciently short and y9e(A1Bl]t then the solution of y = y0(0 of y' = /,y( o) = ^0 exists and y0(/) e 5 on an interval t0 r /*, y0(f*) »s on andis the only point of y = y0(/) on L2 = (AZB2], and y0(^*) ->• 2 as y0 -> y<lt

Let T > 0, > 0. Then, by Theorem V 2.1, there exists a d == <5(c, F) >0 such that ||y0 — A^l < d implies that y0(f) exists and satisfies \\y0(t) —y\(t)\\ < for e0 < r r. In particular, yn(t) E S for e0 / Tif Tand1/e are sufficiently large. Let t1 = tl(y0) be the least t > e0 such thaty0(tl) e C12, so that t1 > T. The fact that S is hyperbolic and that y0 £ 5.implies the existence of t1. It is clear that tl(y0) -> oo as y0 -*• /4t.

Choose y01, ^ 0 2 , - - . such that y0n e (A&], At = Hm y0n, y° = Urn y0n(tl)exists on C12 as «-*• oo, where y0n(t) is the solution belonging to2/0 = yon and t1 = ^G/on)- Tnus, the solution y_n(0 = y0n(

f + t1) exists

and is in S U C12 for — t1 r 0 and y_n(0) -»• y°, n -* oo. By Theorem

]

)

)

V2.1, y-ooC/) = lim y_B(0 exists uniformly on every bounded interval-T <; t < 0, is the solution of y' -/, y(0) » y°(0) e C12, and y.J/) e 5for / 0. Since S is a hyperbolic sector, y_a,(0 is on the boundary of 5"for large —/; i.e., y.^t — *0) = y2(0 for t < 0 and y-^—t^ is the lastpoint of y_oo(0 e C12 when / decreases from 0.

It follows that according as /„ > 0 or t0 = 0, y0(tl — t0) or yQ(tl) tendsto y2(0) as y0 -+• A^. In particular, if y0 is sufficiently near to Alf thereexists a least /* > 0, r * = /*(y0)» such that y0(t*) e L2 = O^z]- Also

y0(r*) -> AI as y0 -* ^Let Lx = f^!^] be so small that /*(y0) exists for all yQ e Lx. Choose B2

to be y0(f*) f°r 2/o = ^iJ see Figure 13. Let Ct be an interior point of

[y4t-J?J and J the arc C{BVBZCZ consisting a piece of the orthogonal trajec-tory LI, the solution arc y = y0(t) joining Bl and Bz, and the piece [C2-ff2]of Lz. It is easy to see that

for the angle from r\ to/is \TT on [Cj^). It changes to 0 on going throughBI and i^O on (B^B^. It jumps to — 7r/2 on passing through Bz and is—7T/2 on (.SaCaJ. This gives 2tr[jf(J) — jn(J)] = —ir\2 — -n\2 = —IT, i.e.,(9.3). If y^(t) is a negative and y2(f) a positive base solution, the formula(9.3) is still valid.

(b) Parabolic Sectors. Let S be a parabolic sector. In order to fixideas, let S be a. positive parabolic sector, so that Ai is a point y,(e0) where*o>0.

Figure 13.

Note that y = 0 £ Sh, the hyperbolic part of 5". For otherwise, thearguments of part (a) show that there exists a negative base solution in5 U C12. Also, if y 0 is a point of Se, then y Sh by the definitions ofSe, Sh. Hence Se n Sh is empty, so that dist (Se, Sk) > 0. It is also clearthat Se n C12 is empty. Furthermore, yt(t) $ Se for / = 1,2 and t 0,for otherwise yt(t) E Se for / = 1 or 2 and all t 0 and, in particular,y,(0) e C12 n Se, which is impossible. Similarly, yt(t) $ Sh for / = 1, 2 and/ > 0, otherwise y,(/) e 5A for / = 1 or 2 and all t 0 and, in particular,the limit point y = 0 e Sh.

If Sh is not empty, there are points y0E S for which a solution y =y0(r) of (6.1) on some interval <x_ < / a+ is such that y0(t) e 5" for <x_ <? < a+ and y0(

a±) e Qz- The arc y = y0(t), a_ ^ / ^ a+ and the corre-sponding subarc of C12 joining y0(

a±) form a Jordan curve. There is afinite or infinite sequence of such maximal Jordan curves Jlt /2, ... in thesense that the interiors of Jlt Jz,... are pairwise disjoint and the union of/!, J2,... and their interiors contains S n Sh; see Figure 14. The set ofpoints on C12 not on any Jn together with the points on the closures of thearcs /„ n 5 form a Jordan arc C(z in S U C12 joining (0), y2(0). Let S'be the interior of the Jordan curve consisting of €{%, y = 0, and the arcsy = y<(0 for i = l , 2 and t 0.

If Sh is empty, let S = S' and C12 = Ci2. Then, whether or not Sh isempty, Se c 5', and so 5e n C/2 is empty. Hence there is a polygonalpath P; y = p(s), Q^s^ 1, which joins /41=/?(0), /42 =/>(!) andX*) eS' - SeforQ<s < 1.

Figure 14.

If 0 < / < 1, the solution y — y,(t) of y' =/, y(Q) = p(s) is such thatyt(t) 6 S for / 0 and y,(~O e CM for some '« > 0- Through each pointp(s) eP, draw an open orthogonal trajectory arc L\ i.e., a solution arc of(9.2) through y =p(s) such that the closure *of L is in S' — St. In thecases XO) =s Ai and/>(l) = At, let the corresponding L be half-closed andhave At as an endpoint instead of an interior point.

Figure 15.

The orthogonal trajectories L can be taken so short that they lie inS' — Se. It follows that the solution y = y0(0 of y' = f, y(G) — y9eLexists and is in S for t 0 and meets C for some t < 0.

The set of orthogonal trajectory arcs L can be considered to form an"open covering" of the j-set 0 s 1 in the sense that s is "contained"in an L if an arc y = ys(i), 0 f /* or — /* t 0 is in S and containsa point of L. If an SQ is "in L," then s near SQ is "in L" Thus, by thetheorem of Heine-Borel, there is a finite set Lx = [Alt B1), La = [/42, 2?2),La = (.43, J?3),... of these arcs such that every solution y = ys(t) meets atleast one of the L^ L2 , . . . at some /* (^ 0); in which case ys(t) E S forf£f.

Let Al = y(l}, y(2),. .., y(n) = Az be the endpoints of the arcs LltLz,.... The solution y(t)(0 of y' =/(y), y(0) = y(k} exists and is in S for/ ^ 0. Also, since y(k} e S' — Se, there is a least f = /* > 0 such thatPk = y{k)(—Q e C. Thus y(k}(t — tk), / ^ 0, is1 a positive base solution.After a suitable change of enumeration, it can be supposed that P^ =yi(0)» A. • • • . Pn = ^(0) are ordered on C and that /»< ^ P, for i 7^7;see Figure 15.

Each solution pair y = ylk}(t — tk), y — y<fc+i)(/ — tk+i) defines a sectorSk which is a subsector of S. Let s0 be the largest s-value, 0 :5s s 1, such

th&tp(s) meets the arc y — y(k}(t — tk) and jj the least j-value sl > s0 suchthat/>($!) is on the arc y = y(k+l}(t — f^). Now y = yt(t), for.j fixed inJo < s < *i> is in $k and meets at least one of the selected Llt L2,. L ..Since the arcs Lx, L2 , . . . have no endpoints in 5*, there is at least one ofthe arcs Llf L2,... which meets every solution arc y = ys(t) for s0 <j < jj. Thus, there are closed orthogonal trajectpry_arcs Ll,..., Ly~1

each of which is a subarc of one of the closed arcs Llt L2 , . . . , such that Vjoins y — y(k)(t — tk) and y = y<fc+i)(f — +1). It can be supposed thatLl, LN~l begin and terminate, respectively, at Alt A^\ say L1 = [A-^B^Z*-1 = [BZAZ].

Let Q, C2 be interior points of L1, L^"1 and / the arc joining Q, C2

consisting successively of subarcs of L1, y = ylZ)(t — fa), L2, y =y<3)(' ~ ^)» • • • » y = y(N-i>(* ~ ^-i)» ^-1- It will be verified that

Note that the tangent vector r\ to J at a point y e L* is in the direction±g(y), where (y) occurs in (9.2) and ± is independent of A:. (That the ±is independent of k can be seen as follows: Suppose that g(y) points into51! at AH then it clearly points into S2 at B^ By continuity it pointsinto S2 along the solution arc y = y^(t) and, hence, at the end-point ofLz on y — !/(2)(0- Similarly, it points into 5S at the endpoint of L2 ony = y3(0- This argument can be continued and shows that ± does notdepend on k.)

Thus the sign of the sine of the angle from rj to/is 1 and, consequently,the angle is of the form 2mr ^f \TT, where n is an integer and is independ-ent of k.. On L1 = [Qflj], the angle from y to/is J7r(sothaton L2,...,L^~1,itisoftheform2wr + £TT). On the part of J consisting of the solutionarc y = y (2)(t — /2), the angle from rj to/becomes 0 or rr. Hence on L2, it isJTT. Continuing this argument, it is seen that the angle from r\ to/is \tr onevery V. Thus 2ir[j,(J) -;„(/)] = fa - \tr = 0; i.e., (9.4) holds.

It is readily verified that if S is a negative parabolic sector, a similarconstruction also leads to (9.4).

(c) Elliptic Sectors. Let 5 be an elliptic sector with boundary solutionsy — yi(0. (0 which are subarcs of some y = y0(0» — oo < / < oo, in S.Suppose that the constructions just described have been made on all of thehyperbolic and parabolic sectors. Since S is adjacent to such sectors, thereis a solution arc [A^A^ on the arc y = y0(t) [containing the points yx(0),y2(0) in its interior] and two orthogonal trajectory arcs [C^j), (A^C^ inthe interiors of the adjacent sectors, respectively; see Figures 16 and 17.If / is the arc [C2CJ consisting of the orthogonal trajectory [CtA& thesolution arc M^J, and X^iQ]» then

)

For, on [C2A2) in Figure 16 [or in Figure 17], the angle from r\ to / is— \tt [or \rt\\ on (A^At), it is 0 [or rr]; and finally, on (A^i], it becomes\IT [or 37T/2J. Thus 2ir[jf(J) -jn(J)] is |TT - (-\ir) - IT [or 37r/2 -\TI = IT]; so that (9.5) holds.

(</) Completion of the Proof. In the constructions in parts (a) and (A),the same number e0 > 0 has been used for all of the hyperbolic andparabolic sectors. Thus, if a base solution y = y(t) is on the boundary of

two such adjacent regions, there is an orthogonal trajectory arc [CgCJcutting y = y(t) at a point A. For the arc / = [CjCJ, it is clear that

Thus if the relations (9.3), (9.4), (9.5), and (9.6) are added for all arcs /,we obtain 2njf(G) = 2ir — imh + imet where the 2rr on the right is ITTSy,(7) by the (Umlaufsatz) Theorem 2.1 and its Corollary 2.1. Thisproves (9.2).

Remark. For the purpose of the next exercise, note that the assumptionin Theorem 9.1 that "the solution of y' ==/(«/), y(0) = y0 is unique" canbe relaxed to the assumption that "the solution of y' = /(y), y(Q) = y0 isunique when y0 y& 0." (This involves an obvious modification of thedefinitions of "base solution," "elliptic sector," "hyperbolic sector," and"parabolic sector.") For y' = f(y) can be replaced by y' = h(y), whereh(y) = \\y\\ f(y)- The two indicesyXOX/AW are obviously equal. It is clearthat an arc y = y(t\ where y(t) ^ 0, is a solution of y' =f(y) if and onlyif it becomes a solution of dy\ds = h(y) after the change of parameters/ -*• s where ds = dtl\\y(t}\\. Thus "ne, nh" are the same for both systemsy' ~f(y\ y' = h(y). Finally, since y' — h implies that \\y'\\ ^ \\y\\ forsmall ||2/||, it follows that y = 0 is the only solution of y' = h(y), y(0) = 0.

)

Exercise 9.1. Let U(y) be a real-valued function of class C1 for \\y\\ < bsuch that (7(0) = 0 and the gradient g(y) = (dUjdy1, dUfdy2) vanishesonly for y = 0. Thus if/(y) = (dU/dy*, -dU/dyl), then f/ is constant onsolutions of y' = /. Show that either (i) there is an > 0 such that U(y) ^0 in 0 < \\y\\ < or (ii) the set of arcs in 0 < ||y|| 0; furthermore, jf(0) = I — n. [Note: The initial valueproblems y' = /(y), y(0) = y0 ^ 0 have unique solutions. (Why?)Case (i) occurs only if y = 0 is a rotation point, in which case it is a center.This happens only if U has a strict local maximum or minimum at y = 0.Case (ii) occurs if y = 0 is a nonrotation point, in which case there are noelliptic or parabolic sectors for any Jordan curve C surrounding y = 0.]

Theorem 9.2. Letf(y) be continuous on a simply connected open set Esuch that solutions of initial value problems (6.1) are unique. Let C be apositively oriented Jordan curve of class C1 in E with the property thatf(y) ^ 0 on C and thatf(y) is tangent to C at only a finite number of pointsylt. .., yn of C. Let ne, nh be the number of these points y, where thesolution arc y = y(t) ofy' = /, y(0) = ytfor small \t\is internally, externallytangent to C at yt (so that ne+ nh <ri). Then 2y,(O = 2 + ne — nh.

The solution arc y(t) of y' = /, y(0) = yt e C is said to be internally[or externally] tangent to C at yt if there exists an e > 0 such that y(f) isinterior [or exterior] to C for 0 < |/| ^ e.

Proof. Let 77 denote the positively oriented tangent vector on C. If/ie = /i* = 0, then it is clear that the angle from r\ to/along C does notpass through a value 0 mod IT. Hence the two integers jf(C),j^(C) areequal. Since jn(C) = 1 by Theorem 2.1, it follows that 2jf(C) = 2 ifne = nh = 0.

Hence it can be supposed that not both ne and nh are zero. A point y0of C will be called an elliptic [or hyperbolic] point if the solution arcthrough y0 is internally [or externally] tangent to C at y0. Thus ne [or «*]is the number of elliptic [or hyperbolic] points.

Let J — [AB] be a subarc of C such that A, B are elliptic or hyperbolicpoints but no interior point of/ is elliptic or hyperbolic. Then at a pointy0 interior to 7, the solution arc through y0 crosses C in a direction (frominterior to exterior or from exterior to interior of C) independent of y0.Thus by considering the angle from rj to/, it is seen that 2irjf(J) — 27r/fl(J)is — rr, 0, or IT.

If one of the points A, B is elliptic and the other hyperbolic, then/(y)and the tangent vector v\ have the same orientation at both A, B or havethe opposite orientation at both A, B; see Figure 18o. In this case,2nj,(J) - 2-njn(J) = 0.

If both points A, B are elliptic or both are hyperbolic, then/(y) and the

)

))


tangent vector r\ have the same direction at one of the points A, B and havethe opposite orientation at the other point. In this case, 2irj£J) —2irjn(J) =*±TT. It will be left to the reader to verify that Inj^J} — 2irjn(J} is tr or — TTaccording as both A, B are elliptic or both are hyperbolic; cf. Figure 186.

Thus, if «'(/) or n\J) denotes the number (0,1, or 2) of the endpoints ofJ which are elliptic or hyperbolic, then

fbj

fa)

Figure 18.

holds in all cases. Summing this relation for all subarcs /gives the desiredresult since

10. A Second Order Equation

An application of the theorem of Poincare-Bendixson will be given inthis section in Corollary 10.1. It concerns the second order equation

for a real-valued function u or the equivalent autonomous first ordersystem

If, in (10.1), g(u)u > 0, then the term g(u) is a "restoring force" (as for theharmonic oscillator u" + u = 0). If h > 0, then the "frictional" term hu'tends to decrease the speed \u'\ (as in the equation u" + hu' = 0 withh > 0 a constant). The next theorem can be interpreted as saying that ifthe restoring force and frictional term are not too small, then the solutionsof (10.1) are bounded.

Theorem 10.1. Let g(u), h(u, v) be real-valued, continuous functions forall u, v with the properties: (i) solutions o/*(10.2) are uniquely determinedby initial conditions; (ii) there exists a number a > 0 such that

(iii) there exists a number m > 0 such that

(iv) ifh0(u) = inf/i(w, y)/or — oo < v < oo, then

Then there exists a Jordan curve C bounding a domain E, containing theorigin (u, v) — (0,0), such that no point of C is an egress point for E andthat ifu(t), v(t) is a solution o/(10.2) starting, say, at t = 0, then u(t), v(t)exists for t^Q and (u(t), v(t)) E Efor large t.

A sufficient condition for (iv) is that there exists a number M > 0 suchthat h(u, v) M > 0 for |w| j> a, — oo < v < oo. Actually, the proofbelow does not use the full force of (iv) but only that h(u, v) > 0 for\u\ a, that h°(u) > 0 for u a if h°(u) - inf h(u, v) for v S -t?0, and

r«that h\u) ^0 for u a and A1^) rfr -* oo as M -»• oo if h\u) =

Joinf h(u, v) for y > t?0 > 0 for a fixed y0.

Proof. Along a solution arc u(t), v(t) of (10.2),

Below M', i?' refer to the derivatives in (10.2) and dvjdu refers to (1 .Let |#(M)| b for |w| ^ a. Then, by (10.5) and (10.8),

Let <p(«) be a positive, continuous increasing function of u > a such

x(u)Then v -<p(u) < - f-^4 implies that v < 0, h(u, v) ^ h0(u) >

g(u) #. t ^(M)

— 2—-; so thatv

Consider the arcs

for a large constant a > 0. These arcs are symmetric with respect to thew-axis. A portion of these arcs and a segment of the line u — a [or u = —a]

Figure 19.

form a Jordan curve bounding a domain .E+(a) [or £!_(a)]; see Figure 19.Along a solution «(/), KO of (10.2), the quantity y(0 — |z>2(0 + G(u(t))has a derivative y>' = v(—hv — g) + gv = —h(u, v)vz < 0 if |w(r)| 2z a,v(t) 5^ 0. Thus y»(r) is decreasing and so the arc u = u(t), v = v(t) entersE± (a) with increasing t as soon as it meets the curved boundary Ca of E±(<x.)and remains in £"±(a) as long as |w(f)| ^ a.

For convenience, the construction of C will refer to Figure 20. A letteron this diagram denotes either a point or one of its coordinates; e.g., vzdenotes either the point (a, v^ or the ordinate vz.

Choose a > a so large that

Choose a so that Ca passes through the point A — (a, —<p(a)); i.e.,a = \<p*(a) + G(a). In particular, Ca passes through the point (M, v) =(a, qp(cr)). Let T > 0 denote the point where Ca meets the w-axis.

Let w°(0, y°(0 denote the solution of (10.2) determined by the initialcondition (u0(0), u0(0)) = (T, 0). As t decreases from 0, «°(f) decreases and

p°(/) increases until w°(f) takes the value a at some point t = tz < 0. Thisis clear from v' = — hv — g ^ — g < 0, hence u' = v > 0, as long asu a. Let C° denote the arc (u, v) = (w°(0, y°0))- In view of the remarksabove, the part of C° for /2 / ^ 0 has only the point T in common withCa. In particular, if y°(r) = vl when w°(/) = a and if y0 = <p(a), then

Figure 20.

T! > v0. Also, by (10.8), dv\du <; -&(«, i>) ^ -h0(u) on C° for a < u <a. Hence, if vz = v°(tz), then

Put y = v0 + £#(ff) < v2 — \H(a). Thus the slopes of the line seg-ments y to Vi and — y to — y0 are at least \H(o)j2a and thus exceed 2m,by (10.1.3). Define £ by j8 = ^y2 + G(d), so the arcs C^ pass through thepoints ±y.

Let C denote the Jordan curve consisting of the arc C° from T to i?2, theline segment from v2 to y, the arc C^ from y to —7, the line segment from— y to —1>0, the horizontal line segment from —1>0 to A, and Ca from .4 toT. Let £" denote the interior of C.

It will first be verified that the points of C, except for the points on C°,are strict ingress points of E. To this end, it is sufficient to verify that the

solution arcs of (10.2) reaching the lines segments on C cross Cas indicatedin Figure 20. This is clear along the horizontal segment from —1?0 to A,for u' — v < 0 and dvfdu < 0 by (10.11) and the monotony of <p(u) foru a. The slope of the segment from — y to — v0 is H(cf)f4a > 1m while,along this segment, u' = v < 0, dv\du ^ 2m by (10.9) since v0 = <p(a) ^b\m by (10.13). Similarly, the slope of the segment from y to vt exceeds2m and on this segment v 2: y > v9 A//w so that i/ — p > 0 anddv/du ^ 2m.

It remains to show that every solution w(/), v(t) of (10.2) starting at/ = 0 exists for t 0 and that (u(t), v(tj) E E for large /. For this purpose,rename a to a0 and the Jordan curve C to C(a0). Then, for each a *z cr0,the above construction leads to a Jordan curve C(a) and its interior E(a).The sets E(a) are increasing with a and the union U = UC(<r) for a > a0is the exterior of C(<r0). Let M(/), t>(0 be a solution of (10.2) starting at apoint of U for / = 0. Then, as long as «(/), v(t) remains in U, there is aunique <r = a(t) such that (w(0> 0(0) e C(cr(0) and #(/) is a nonincreasingfunction of /. This implies that (u(t), v(t}) exists for t s£ 0; cf. Corollary.II 3.2.

Suppose, if possible, that u(t), r(f) does not enter £"((10) eventually, sothat a(t) ^ a0 for all / 0. Let = lim a(t) as t -+ oo. Thus (M(/), o(r))is arbitrarily near to C(a^ for large /-values. It is clear in this case,however, that (w(f), y(r)) e E(oJ for arbitrarily large /. But then <r(r) < o-jfor large t. This is a contradiction and proves the theorem.

Corollary 10.1 In addition to the assumptions of Theorem 10.1, supposethat g(u) 5^ 0/or u Q [so that g(u)u > 0/or u 0 am/ //re origin is theonly stationary point for (10.2)] and suppose that the origin is not an (o-limitpoint for every solution of (10.2). Then (10.2) has a periodic solution(tt0(/), ^o(O) 5^ 0 which is asymptotically orbitally stable from the exterior ast -> oo and the interior of the Jordan curve u = w0(0, v = v0(t) contains theorigin.

Proof. If some solution (u(t), v(t)) does not have the origin as aneo-limit point, then the theorem of Poincare-Bendixson (Theorem 4.1)implies that its set of co-limit points is a periodic solution (u0(/), v0(0)>since (u(t), v(t)) e E for large t.

The Jordan curve u = u0(f), v — v0(t) surrounds the origin by Theorem3.1. It also follows that if there are two periodic solutions, then one of the.corresponding Jordan curves is contained in the interior of the other.Since all periodic orbits are contained in the compact set E U C, it followsthat there exists a unique periodic solution i/0(/), v0(t) such that the Jordancurve C0:w = u0(t), v = v0(t) contains all other periodic orbits in itsinterior.

The periodic solution w0(/), v0(t) is asymptotically orbitally stable from

a

the exterior as / —>• oo. In order to see this, consider a solution u(t), v(t)starting for t = 0 at a point exterior to C0. Since the origin is not anco-limit point for M(/), v(t), its set of ey-limit points is a periodic orbit(Theorem 4.1) which is necessarily C0. Thus the assertion follows fromTheorem 5.2.

Exercise 10.1. (a) Let g(u),h(u,v) be continuous for all u, v and lete(t) be continuous for all /. Suppose that solutions of

are uniquely determined by initial conditions. Let there exist positiveconstants a, e0, m, M such that (i) h(u, v) M for |w| ^ a, \v\ a;(ii) h(u, v) > — m for all w, v; (iii) \e(t)\ < e0 for all /; and (iv) ug(u) > 0for large \u\ and lim inf | («)| > ma + e0 as |w| -*• oo. Then there exists aJordan curve C in the (u, u)-plane such that if «(f) is a solution of (10.15)starting at / = r0, then u(t) exists for / ^ /0, (w, v) = («(0, «'(0) is in tne

interior £ of C for large / and the (M(/), w'(0) cannot leave E with increasing/. (/>) If, in addition, e(f) is periodic of period p > 0, then (10.15) has asolution of period p. See Opial [7].

Exercise 10.2. Show that Theorem 10.1 remains correct if condition(iii) is strengthened to h(u, v) j> m > 0 for |w| ^ a, — oo < t; < oo, and(iv) is relaxed to /I(M, y) g: 0 for \u\ ^ a.

A particular case of (10.1) is the equation

If g(u) = u, (10.16) is called Lienard's equation. An elegant simpleargument shows, under certain conditions, the existence of a periodicsolution which is unique (up to translations of the independent variable /).The equation (10.16) will be treated as the first order system

where

Theorem 10.2. Let h(u), g(u) be continuous for all u with the properties:(i) that solutions o/"(10.17) are uniquely determined by initial conditions;(ii) that h(u) = h(-u) is even, H(u) <OforO<u<a, H(u) > 0 and isincreasing for u > a, H(u) —* oo as u -*• oo; finally, (iii) that g(u) =—g(—u) is odd and ug(u) > 0 for u 0. JAe/i (10.17) ACJ exactly one

periodic solution (u0(t), v0(tj) ^ 0, up to translations of t, and this solutionis asymptotically, orbitally stable (from the exterior and interior) as t -> oo.

Of course, («(/), 0(0) >* a [periodic] solution of (10.17) if and only if u(t)is a [periodic] solution of (10.16).

C

Proof. The functions on the right of (10.17) are odd functions of(u, v)i so that if w(0> v(t) is a solution, so is — u(t), — v(t). The tangent to asolution arc u(t\ v(t) is horizontal at a point (M, v) if and only if u = 0 andis vertical if and only if v = ff(u).

The arguments to follow refer to Figure 21. Along a solution startingat yx > 0, u increases and v decreases until v = H(u), say, at the point y.Then u decreases, v continues to decrease and the solution arc remains

Figure 21.

below the curve v = H(u) until the solution meets the y-axis, say, at v = v2.Otherwise the solution would have a horizontal tangent at a point whereM 5»* 0 or v would tend to — oo as u tends to a finite value, which is im-possible by

By symmetry, it is clear that a continuation of the solution is closed ifand only if v2 = — vv It is clear that we can start at any point y, i.e.,(y, H(y)\ and determine the points vlt v2 by moving along the correspond-ing solution arc with decreasing, increasing /. Denote |(ya

2 — v-f) by9>(y). Thus if y(w, v) = \vz + G(u), then

where the integral denotes a line integral along the solution arc from v1 toy to v2.

If 0 < y a, then 97(7) > 0, for along a solution arc, dyldt = VD' +g(u)u' = —g(u)H(u) > 0. If y > a, let a and $ denote the points where the

solution arc meets the line u — a; see Figure 21. Write

Along the solution arcs contributing to <plt dy\du — v(dvldu) + g(u) =-g(u)H(u)l(v - //(«)), by (10.19). Thus dy\du > 0, du > 0 on i^a anddy\du < 0, du < 0 on |fty2, so that <pi(y) > 0. As y increases, the arc t^ais raised, so thatg(w) \H(u)\[(v — //(«)) decreases; and fiv^ is lowered, sothatg(w) \H(u)\l\v — H(u)\ decreases. Hence <pi(y) decreases as y increases.

On the solution arc <xy0, dv < 0 and dyjdv = v + g(u) dufdv = H(u) >0. Thus

If ay/3 has a parametric representation « = u(v) = u(v, y), it is clear thatu(v, y), hence H(u(v, y)) is an increasing function of y. Thus dv < 0implies that <pa(y) decreases as y increases. If u a + <5 > a, then/^(M) e > 0 for some constant «. This assures that <pa(y) -»• — oo asy-> oo.

Thus, <p(y) > 0 for 0 < y a, <p(y) = ^(y) + <p2(y) is a decreasingfunction of y > a, and <p(y) ->• — oo as y —* oo [since 9>2(y) -> — oo and<pi(y) is decreasing]. Hence, there exists a unique y0 such that <p(y0) = 0.The corresponding solution of (10.17) is a nontrivial periodic solution,which is unique up to translations of the parameter t.

The fact that <jp(y) < 0 for y > y0 My) > 0 for 0 < y < y0] makes itclear that this periodic solution is asymptotically orbitally stable from theexterior [interior] as / -*• oo.

Exercise 10.3. (a) Verify that Theorem 10.2 applies to van der Pol'sequation

where /* > 0 is a constant, (b) If (10.17) is the system corresponding to(10.20), what is the nature of the stationary point (u, v) = (0, 0)?

Exercise 10.4. Show that if the condition "H(«) > 0 and is increasingfor u a" is omitted in Theorem 10.2, then the conclusions of Theorem10.2 are valid with "exactly one" replaced by "at least one" and "exteriorand interior" replaced by either "exterior" or "interior."

Exercise 10.5. Let h,g satisfy the conditions of Theorem 10.2; inaddition, let g(u) be increasing for M > 0; and let /t > 0 be a constant.Let r(ft) denote the limit cycle in the (M, i?)-plane belonging to

Figure 22.

i.e., the unique nontrivial periodic solution arc of

Show that, as ^ -^ oo, F(//) tends to the closed curve consisting of twohorizontal line segments and two arcs on v = H(u) as in Figure 22.Compare Lefschetz [1, pp. 342-346].

APPENDIX: POINCARF.-BENDIXSON THEORY ON2-MANIFOLDS

Although the proof of the Poincare-Bendixson Theorem 4.1 for planeautonomous systems depended on the validity of the Jordan curvetheorem, it turns out that analogous results hold under suitable smoothnessassumptions on arbitrary 2-dimensional differentiable manifolds. Theseanalogous results will be the main object of this appendix.

11. Preliminaries

The objects of study of this appendix will be flows on 2-manifolds.Definition. 2-manifold of class Ck. Let M be a connected, Hausdorff

topological space for which (i) there is given an open covering M = UC/a,where a e A and A is some index set, and for each a, a continuous, one-to-one map gx of t/a onto an open (plane) square such that (ii) if Ux n Uft isnot empty, then^fe-1) is a map of &,(£/, n Uft) onto &,(£/« n Up) whichis of class C* (so that, if k ^ 1, this map has a nonvanishing Jacobian).Then (A/; t/a;gj is called a 2-manifold of class Ck.

If m denotes a point of M, then yx = gx(m) is a function of m e Ua, thevalues of which are binary (real) vectors ya — (y^, ya

2). As m varies over(7a, ya varies over a square; e.g., \y*\, \y^\ < 1. t/a is called a coordinateneighborhood of any point m e t/a and ya the local coordinates of m —g*lM- Condition (ii) concerns the map yft = gfa*l(yj). (The paren-thetical part concerning the Jacobian d(yft

l, y^)jd(yai\ ya2) is redundant)

since the inverse maps g^(g^1) and gjigjf1) are assumed of class Ck,k°Z 1.)

It is important to emphasize that the 2-manifold {M; Ux; ga} consists ofthe space M, the given covering U C/a, and the given set of homeomorphismsga. Usually, Ua and ga are fixed and {M; (7a; ga} will be abbreviated to M.

Definition. Let M be a 2-manifold of class Ck, k ^ 0. Let U, g be an.open set of M and y = g(m) a homeomorphism of U onto an open square.Then U is called an admissible coordinate neighborhood on M and y =g(ni) (admissible) local coordinates of m e U provided that {M; Ua and17; gz and g} satisfies conditions (i) and (ii) of the last definition.

Definition. Let M be a 2-manifold of class Ck, k^.0. By a flow

of class Ck is meant a function /*(/, ni) defined for — oo < / < oo, m e Mwith values in M such that (i) for a fixed t, T*: M -> M is a homeomorphismof M onto Af; (ii) T* is a group of maps

in particular, /*(0, w) = /«; (iii) /*(f, m) is a continuous function of (f, m);finally, (iv) if k ^ 1, then p(t, m) is of class Ck as a function of (f, m).

The last two conditions have the following meaning: Consider anygiven (f0, ro0), let Ua be a coordinate neighborhood containing w0, ya «^a(m) local coordinates on t/a, and let p(t0, m0) E (/^. Then (iii) meansthat jj,(t, ni) E Up for (/, ni) near (/0, w0) and (iv) means that

is of class Ck as a function of (t, ya) on the open (t, y^-set on which theright side is defined.

Lemma 11.1. Let k ^ 1. Then, for a fixed t, the Jacobian of the map5^:ya -*• yft given by (11.3) does not vanish (wherever S^ is defined).

Proof. For, by (ii), 5^ has the inverse

which by assumption is also of class Ck. This implies the lemma.A "flow" is a generalization of the concept of "general solution of

autonomous differential equations on M"; cf., e.g., § 14 or § IX 2.For any given point m0, the subset C(m0) = (m0* = ft(t, m0), — oo <

/ < 00} of M is called the orbit through m0. Since each point m0 E Muniquely determines its orbit, it is clear from the group property (ii) thattwo orbits are either identical or have no points in common.

When the orbit C(m0) reduces to the point m0 [i.e., /*('» wo) = wo f°r

— co < / < oo], then m0 is called a stationary point.

1)

Lemma 11.2. Let k ^ 1 and m0 E Ua. Then w0 is a stationary point ifand only ify — gj(^(tt AWO)) has the derivative 0 at t = 0.

Exercise 11.1. Verify this lemma.A subset Ml of M is called an invariant set if TiMl = Ml for every /,

— oo < / < oo. Thus Mx is an invariant set if and only if T*Mi ^ M1

for — oo < / < oo or, equivalently, if and only if C(m^ c Ml wheneverml E MI. In particular, if Mj is a closed invariant set and ml E Mlt thenC(mj), the closure of the orbit through mlt is contained in Mv

A subset N of M is called a minimal set if (i) # is a closed invariant setand (ii) N contains no proper subset which is closed and invariant.

For example, if m0 is a stationary point, then the set consisting of m0 is aminimal set. More generally, if ji(t, /w0) is periodic [i.e., if there exists anumber p > 0 such that ju(t + p, m0) = n(t, /w0) for — oo < r < oo], thenC(/w0) is a minimal set.

Exercise 11.2. Let M be a 2-manifold and T* a flow on M. (a) IfMlt M2 are invariant sets, then Ml U A/2, Afj n Afa and Afj — Af2 areinvariant sets. (6) If Afx is an invariant set and dMlt Mj° = Ml — dMtare its boundary and set of interior points, then dM-^ and Mj° are in-variant sets, (c) If TV is a minimal set, then either N = M or N is a nowheredense set.

Let w0 E M and let C+(w0) denote the semi-orbit C+(w0) = {//(/, w0) for/ ^ 0} starting at w0. A point /w e M is called an co-limit point of C+(w0)if there exists a sequence of /-values 0 < t^ < /2 < ... such that tn -> ooand fj,(tn, m0) ->• m as n -> oo. The set of co-limit points of C+(w0) will bedenoted by O(w0). The analogues of Theorems 1.1 and 1.2 are valid.

Exercise 11.3. Let M be a 2-manifold and T* a flow on M of class C°.Let m0 e M. (a) Let O'(m0) = {m = p(t, w0) for / /} = C+(^(y, m0)).Then n(w0) = ^^/Wo) n £2(/w0) n ... and Q(/w0) is closed, (b) If j^ EQ(/MO), then C^j) c ii(w0); i.e., Q(/w0) is an invariant set. (c) If, inaddition, C+(m0) has a compact closure, then Q(/w0) is connected.

Exercise 11.4. Let A/, J1' be as in Exercise 11.3, TV a minimal set, andm0 E N. (a) Then £l(w0)

c Ctyw0) = A . (6) If, in addition, n(w0) is notempty (e.g., if N is compact), then Q(/HO) = M

Let M, T* be of class Ck, k ^ 1. Let f/a be a coordinate neighborhoodon M and # = g^m) local coordinates of m E Ua; thus £a: (7a -»• 5 is ahomeomorphism of f/a onto a square 5, say, .Srly1!, \y*\ < 1. A closed[or open] arc Y:y = y0(,y) for |.y| < fl [or |j| < a] of class C* in 5 or itsimage T*:m0(s) = g^l(y0(sy) in £/a is called a transversal of the flow I7'if noorbit is tangent to T, i.e., if the differentiable arc y = £a(//[/, WoC*)])*which is defined for small |/| and starts at y0(s) when t = 0, is not tangentto F for any s on |s| a [or |j| < a].

Lemma 11.3. Let k "^ 1, w0 6 t/a a nonstationary point of the flow T*

)

and y = g^m) local coordinates on Ua. Then there exists a Ck transversalarcY;y = y0(s) orF:m0(s) = g^(yQ(s))for \s\ a0satisfying w0(0) = m0.Furthermore, for any such transversal and sufficiently small a > 0, the setU = {m = n(t, mQ(s)), where \t\ < a, \s\ < a} and y = g(m) = (s, t) forme U are an admissible coordinate neighborhood and local coordinates onU, respectively.

Exercise \ 1.5. Verify this lemma.

12. Analogue of the Poincare-Bendixson Theorem

The main object of this section is to. prove:Theorem 12.1 (A. J. Schwartz). Let M be a 2-manifold of class Cz and

T* a flow on M of class C2. Let N be a nonempty compact minimal set.Then N is either (i) a stationary point m0; or (ii) a periodic orbit (which ishomeomorphic to a circle); or (iii) N — M.

In case (iii), M = Nis compact and has a flow T* on it without stationarypoints. It follows from the Euler-Poincare formula (relating genus and thesum of indices of singular points of a vector field on M) that the genus ofM is 1; thus M is homeomorphic to a torus or a Klein bottle. Actually,a flow on a Klein bottle without stationary points necessarily has aperiodic orbit. Thus, in case (iii), M is a torus. For a discussion of theseremarks, see H. Kneser [2].

The assumptions concerning the C2-property of T* cannot be reducedto C1 even if M is of class C°°; see Exercise 14.3.

Proof of Theorem 12.1. In view of Exercise 11.2(c), N = M or TV is anowhere dense set. Suppose that the theorem is false. Thus N j& M is acompact, nowhere dense, minimal set in M which does not contain astationary point or a periodic orbit. It will be shown that this is impossible.

Let /H0 e N and let 0 and y = g(m), where y = (s, /), be an admissibleneighborhood and local coordinates as furnished by Lemma 11.3. Thus,ifm =£-%) =g-l(s,t),v/heTe\s\,\t\ < a,thenw0 = g-^O, 0),f :m0(s) =g~l(st 0) is a transversal arc of class C2 and, for a fixed s, g~*(s, t) =ft(t, w0(j)) is part of the orbit C(w0($)).

A point m and its local coordinates (s, t) will be identified; e.g., m0

will be referred to as the point (0,0). Let F represent the open linesegment \s\ < a, t = 0, which is the transversal arc T:y = g(m0(sy) =(s, 0). The point (s, 0) of F will be referred to simply as s.

Since AT contains no interior points, it is clear that K = N n F containsno j-interval, so that AT is a nonempty, nowhere dense set, closed relative toF. Let

so that W is an open j-set and has a decomposition W = U(a,., /?7) intopairwise disjoint, open ^-intervals (o^, &), (a2, /32),

Let t/ be the set of j-values on \s\ < a such that the semi-orbit C+(m0(s))starting for / = 0 at the point m0(s) e F meets f for some / = t(s) anddoes not pass through an endpoint of f for 0 < t < tfs). It isclear that Uis an open j-set on \s\ < a. For s e £/, let t(s) be the leastpositive /-value [in fact, t(s) ^ a] such that /J,(t, m0(s)) e f and let f(s)denote the ^-coordinate of p(t(s), w0(j)); i.e.,

We can obtain 7(5) and/fa) in another way. Let SQ E U and define thereal-valued functions a, T of (s, t) by

for small |j •— j0|, |f — /(JD)|. Thus, on the orbit starting at m0(s), (<r, T)are coordinates of the point corresponding to the time t. In particular

a(s, t(s))i=f(s) and r(s, t(s*)} = 0. Then a, r are of class C2 with a non-vanishing Jacobian d(a, r)/d(s, t) by Lemma 11.1.

Since T is a transversal, dr(s, r)/5r 3? Oat-r =/(j), hence t(s) isof class C2 by the Implicit Function Theorem. Hence

Note that cr(s, t), T(S, T) satisfy

for small (5 — s0\, \t\. Since 3((T, r)/d(s, t) ^ 0, it follows that the Jacobianof (a(s, t + t(s)), t) with respect to (s, t), which is the product of theJacobians d(a, T)/<?(J, /) and d(s, t + t($y)/d(s, /), does not vanish. Hencethe partial derivative of a(s, t -f- t(s)) with respect to s is not 0. Sincef(s) = a(s, t + t(s)) for / = 0, it follows that

Since A^is a compact minimal set, m^eNimplies that C(m^ = N; cf.Exercise 11.4. This shows that if sl e K = N n F, then s1 e U, i.e., thatAT ci £/. Let F be an open subset of U such that

In particular, (12.3) and (12.4) imply that there exist constants R and Csuch that

)

It is clear that

is a one-to-one map and has an inverse/"1^). Below/*(.y), k = 0, ± 1,...,denote the iterates of/and/-1; e.g., f°(s) = 5 and if s e U &ndf(s) e U,then fz(s) =/(/(.s)). In particular, if k > 0, then fk(s) is defined onU C\f~\U) n • • • r\f-(k-v(U)\ a similar remark applies to k < 0.

In addition to the properties (12.6), (12.7), the function/(j) has theproperties that

since N is invariant; that

since N contains no periodic orbits; and finally, that

(12.11) K is a nonempty minimal set with respect to/

in the sense that K contains no proper subsets K0 such that KQ is closedwith respect to K and that/(#0) = K0 [or equivalently,/±1(/:0) c K0].

The remainder of the proof consists in proving the assertion:Lemma 12.1. If K is a closed set and K, U are open sets of \s\ < a

satisfying (12.5), then there cannot exist a function f(s) defined on Usatisfying (12.3)-(12.11).

Proof, (a) Assume that this lemma is false and that/(.$•) exists. Put

Note that if (a,, /?,) is an interval of W = (—a, a) — Kt then

since a3, ft e /C

(6) It is easy to see that

are the endpoints a,, /?. e A". For s0 e K — K^ if and only if 50 is a limitpoint of both K n (—a, s0) and K n (s0, a). Thus (12.14) follows from(12.4) and (12.9).

(c) It will be shown that there exists an integer / such that a = at,j8 = ft satisfy

Let Q be the finite set of endpoints a,, ft of intervals (a,, ft) of W suchthat ft — o^ > c. In view of (12.10), there is an integer n such that

)

f*M ^Qfork^n. Then, by (12.14),/"(ai) = a,. or/'X) = & for somei and |/*(&) -/*(oti)l <eforn^k. Thus (12.15) follows from (12.13).

(d) Let » 0 be an integer and [/>, q] a closed j-interval such thatfk(lp, q])c V for 0 A: < w. Then

for 0 rSji £ « and/? r, s <s ^. In order to verify this, note that/*+1(^) =f(fk(s)). Thus if Df°fk(s) denotes the derivative of/evaluated at/*(?),then we have

(12.17) Dfk+\s) = [D/°/*(s)]Z)/fc(s), hence Dfk+\s) =n/)/o/^).>=o

Consequently,

and, by the mean value theorem of differential calculus,

for suitable 0, between f*(r) and/'(j). Thus (12.16) follows from (12.6),(12.7).

(e) Let a, ft be as in (12.15) in (c). Then

In order to see this, note first that Df°(s) = 1, so d 1. There is a Ok c(a, /S) such that

Since the intervals with endpoints/*(a),/"*(/?) are contained in (—a, a) andare pairwise disjoint for k ^ 0 by (12.10), it follows that

In addition, by (12.16),

This proves (12.18).(/) Let d denote the number

)

hold for * = 0, 1, In view of/*(a) e K and the definition (12.12) of , it is seen that (12.20*) is a consequence of (12.21*); cf. (12.13).

The relations (12.21*), (12.22*) are trivial for * = 0. Assume that(12.21*), (12.22*) hold for * = 0, . . . ,« . Then, by (12.16) in (d},

By the mean value theorem and (12.22/), fory = 0 , . . . , n,

where 0, is between a and s. Thus (12.18) implies that

But since 3C/?rf<5 < tja ^ 1 and e < 3, the inequality (12.22 n + 1)holds. This inequality shows that

where |a - 0| < |* - a|. Since C> 1 and /? > 1 imply that 3dd < e,the inequality (12.21 n + 1) holds. This proves (12.20*)-(12.22*) for* = 0, 1,

(#) In view of (12.18) and (12.22*),

By the minimality property of K, the closure of the sequence of points/J'+n(oc) for n = 0, 1,... (and j fixed) is the set K. Hence there exists alarge value of * (>0) such that

Thus |/fc(a db d) — a| < d, and so/*(.s) — 5 has opposite signs at 5 = a ± d.Hence there is an 5-value s0 such that fk(s0) = 50 and |.s0 — a| < d. Inaddition, fnk(s0) = s0 for n = 1, 2,. . . and, by (12.23), /nfc(a) -* J0 asn -*• oo. Since A^is closed and a £ A implies that/n*(a) E £, we have s0 e A.This contradicts (12.10) and proves Lemma 12.1 and Theorem 12.1.

))

Lemma 12.2. Let M be a 2-manifold and Tl a flow on M of class C°.Let Ml be a nonempty, compact, invariant subset. Then Ml contains atleast one nonempty, minimal set.

Since the intersection of invariant sets is an invariant set, this lemma isan immediate consequence of Zorn's lemma. (For the statement andproof of the latter, see Kelley [1, p. 33]). Theorem 12.1 and Lemma 12.1give an analogue of the Poincar^-Bendixson theorem:

Theorem 12.2. Let M be an orientable 2-manifold of class C2, Tl aflow on M of class C2, and m0 e M. Suppose that Q(m0) -^ M and thatQ(m0) is a nonempty compact set which contains no stationary points. ThenD(m0) is a Jordan curve and C+(w0) spirals toward O(w0).

When N is a Jordan curve on M containing no stationary points, thenC+(w0) is said to spiral toward N if, for every m e N, there is a transversalarc F = F(m) through m such that successive intersections of C+(/w0) ={m0* = /t(t, w0), / =£ 0} and F tend monotonically to m.

The concept of the "orientability" of M is used here to mean thefollowing: Let jV be a Jordan curve on M of class C2 . Then there existan open set V containing Wanda C^-diffeomorphism of the cylinder {(s,V1, y*)'-\s\ < 1, fr1)2 + (y2)2 = 1} onto V such that the circle s = Oismapped onto N. Thus either (s,yl) or (s,y2) are local coordinates.

Proof. By Lemma 12.2, O(AW°) contains a nonempty, compact minimalset Wand, by Theorem 12.1, N is a Jordan curve (containing no stationarypoints). Let V be a neighborhood of N described above. By consideringthe image of the flow on the cylindrical image of V, it is clear that thearguments in parts (b) and (c) of the proof of the Poincare-BendixsonTheorem 4.1 are valid. Hence C+(m0) spirals toward N. This implies, inparticular, that N = n(w0) and completes the proof of the theorem.

13. Flow on a Closed Curve

In view of the remarks following Theorem 12.1, it is seen that the torusis an exceptional 2-manifold M in that it can admit flows for which M is aminimal set. It seems therefore of interest to examine flows on a torus.This section is a preparation for such an examination and concerns a"flow" on a Jordan curve F or, equivalently, a topological map of F ontoitself.

Let F be a Jordan curve and S: F -» F an orientation preservinghomeomorphism of F onto itself. The discrete group of homeomorphismsSn, n = 0, ± 1, . . . , of F onto itself will be called a flow on F. As in § 11,we can define the orbit C(y):{Sny:n = 0, ±1,...} through a point y e F,semi-orbit C+(y) = {Sny :n = 0,1,...}, the set Q(y) of w-limit points ofC+(y), invariant sets, minimal sets, etc.

The points of the curve F can be considered to be parametrized, sayy = y(y), where 0 y :£ 1, and the points y E F corresponding toy = 0 and y = 1 are identical. Or, more conveniently, F can be consideredas a line, — oo < y < oo, on which we identify any two points yl9 yz forwhich yl — y8 is an integer. Then yx = Sy can be represented as real-valued function yr = /(y) satisfying

(i)/(y) is continuous and strictly increasing,(")/(y + 1) =/(y) + 1, so that/(y) - y has the period 1.

The fact that/is nondecreasing merely reflects the fact that S is orientationpreserving. The condition that / is strictly increasing and satisfies (ii) isa consequence of the fact that S is a homeomorphism. Conversely, anyfunction /(y) satisfying (i) and (ii) induces an orientation preservinghomeomorphism S of F onto itself.

Let /°(y) = y and f~\y) be the function inverse to /(y). Let f*(y) =/(/&)), /%) =/(/%)),... and similarly /-'(y) =/-1(/-%))./~%) =f ~l(f-\y)),..., so that/n(y) corresponds to S" for « = 0, ±1,... andeach/n(y) satisfies (i) and (ii).

Lemma 13.1. Let /(y) be a continuous function for — oo < y < oosatisfying (i) and (ii), then there exists a number a such that

in fact, ifdn, €n are defined by

ybr — oo < y < oo, 50 f/iaf

//re«

The number a is called the rotation number of the map S or of the flowSn on F.

Proof, (a) The continuous function

has the period 1, by the analogue of (ii), and satisfies

For suppose that (13.4) does not hold. Then ^(y^ — 9>m(y2) = 1 forsome points ylt y2. Since <pm is periodic, it can be supposed that ya <Vl < y2 + 1. Then, by (13.3), /"(yj -/ro(y2) - 1 + ft - y, > 1; sothat /""(yO > 1 +/CT(y2) =/m(l + ya)- But this contradicts the in-creasing character of/"1.

(£) Let k ^ 1 be an integer. Then (13.3) shows that

Hence

in particular, for k = 1,

Then if « > 0 and n — kd + r, where 1 r < k, a sum of these relationsfory = 1,. . . , d and / = kd + 1 , . . . , kd + r gives

By the definition of /Sn, /?B, and 9?n(y), this implies that

Consequently

But since /Sfc — pk -»- 0, A: -* ex), by (13.4), it follows that these upper andlower limits are equal, say, to a. Thus <pn(y)ln —*• a as « -> oo uniformlyin y. This gives the part of (13.1) concerning n —*• + oo.

By (13.4) and (13.5), kfik ^ yk(y) ^ kfik and k0t < A:a ^ kp. Hence ifdk = fca - A: and jf c = A;/S* - fca, then (13.2) holds for n = 0, 1

Note that/-%) - y = z —fn(z) if 2 =f~n(y) and so (13.2) holds .with#_n = n and e_n = 5n for « = 1, 2 , . . . . This completes the proof.

Lemma 13.2. The rotation number a of S is a topological invariant, i.e.,is independent of the parametrization of F.

Proof. Let S determine/(y). A change of parameters z = g(y) on F,preserving orientation, is given by a function g(y) satisfying the analogueof (i) and (ii). If R:z = g(y), then, in the z-parametrization, S becomesRSR~l:zl = h(z), where h(z) = g(f(g~l(z))) satisfies (i) and (ii). Since(RSR~l)n = RSnR~l:zn = hn(z) = g(fn(g~\z^ it follows that hn(z) =g(g-^) + ** + rn) where -dn <rn^€n by (13.2). Letv(z) = g& - *,so that y(z) has the period 1. Then, as n -> oo,

This proves the lemma.Theorem 13.1. Let Sn be a flow on a Jordan curve F. Then the rotation

number a of S is rational if and only if Sk has a fixed point y0» Sky0 = y0,for some integer k > 0.

Proof. Let 5 belong to/(y). Suppose that S*}>o = Xo f°r some > 0,i.e., that there is a number y0 and an integer r such that/*(y0) = y0 + r.

kk

) ()

))))

Hence

is rational.Conversely, suppose that a = r/k is rational, where k > 0. Then/%) —

y — r attains non-negative and non-positive values by (13.2). Hence thereis a y0 such that/%0) — yQ — r — 0; i.e., SkyQ = y0 for the point y0

e Fcorresponding to y0. This proves the lemma.

Lemma 13.3. Let S have an irrational rotation number a. Let y0 e F befixed, yn ==• Sny0,j and k fixed integers, and F0 either of the closed arcs onF bounded by y,, yfc. Then there exists an integer n > 0 such that F c: Fn,where

/fewce, ify £ F, f/zere w an i = /(y), 0 i n, such that Sili~k\y e F0-Proof. For the sake of definiteness, let F0 be the oriented subarc of

F from yfc to y, and let F(n) be the union of arcs

Thus F" = F(n) or F" = Sn(i~k}r(n) according asy > k or; < k and so,F c F" if and only if F c F(n). Thus it suffices to show that if n > 0 islarge, then F c: r(n). The wth and (m + l)st arc on the right of (13.6)abut at a common endpoint5-m0-fc)yfc. Hence if F <£ F (n)forn = 1,2,.. . ,the endpoints S-n(i~*}yk tend monotonously to a point y° on F as n -*• oo.But then y° = lim S-n(i~k)yk = lim S-(n~l)(j-k}yk = S'-*y° whereas, byTheorem 13.1, S7'~fc has no fixed point. This contradiction shows thatF c r(B) for large n.

Thus, if y e F, then y e S-mlj-fc|F0 or Sm|j-fcly e F0 for some m = m(y),0 m n. This proves the lemma.

If 5*y0 = y0 for some k > 0, the orbit C(y0) consists of the points7o> -tyo* • • • , Sky0. If no Sk, k ^ 0, has a fixed point, the situation is asfollows:

Theorem 13.2. Let Sn be a flow on a Jordan curve F having an irrationalrotation number a. 'For any y e F, let Q(y) be the set of (o-limit points ofthe semi-orbit C+(y). Then N0 = Q(y) is independent of y and hence is aunique minimal set of Sn. Furthermore N0 = F or NQ is a perfect nowheredense set on F.

When JV0 = F, the flow 5n is said to be ergodic.Remark. Since N0 is a unique minimal set of the flow Sn, as well as

S~n, it is clear that Sn is ergodic if and only if 5~B is ergodic. Thus the setAT0 is the set of limit points of any semi-orbit C+(y) = (SBy, n = 0,1,...}or of any semi-orbit C~(y) = (S~ny, n = 0,1,...} or of any orbitC(y) = {S"y,n = 0, ±1,...}.

Proof. Consider the semi-orbit C+(y0) and its set O(y0) of co-limitpoints. Let y° e Q(y0) and y any point of F. Since there are pointsy,, yk e C+(y0) arbitrarily near to y° and the smaller of the arcs F0boundedby them contains a point S'y for some i 0, it follows that y° e Q(y);i.e., Q(y0)

c &(y)« Interchanging the roles of y and y0 shows thatTVo = Q(y) is independent of y.

The set #<> is closed and invariant. If y E NQ, then y is a limit point ofC+(y) c= N0t hence a limit point of N0. Thus N0 is perfect. Clearly AT0

is minimal. Hence N0= T or N0 is nowhere dense; cf. Exercise lf,2(c).This proves Theorem 13.2.

Lemma 13.4. Let S have an irrational rotation number a. For a giveny0, the function g(yn + k) = na + k, where yn —fn(y0) andn,k = Q,±l,... 75 an increasing function on the sequence of numbers {yn + k}.

Proof. It has to be shown that

Applying/-"1 to the first inequality in (13.7) gives

Since fn^m(y) — y cannot be an integer, max (fn~m(y) — y) <j — k andso (13.2) implies that -(n — w)oc <y — k. This proves (13.7) and thelemma.

For the next theorem, the following simple lemma will be needed.Lemma 13.5 (Kronecker). Let a be an irrational number and Sn the

flow on a Jordan curve F such that the correspondingly) isf(y) = y + a.Then Sn is ergodic (i.e., N0 =• T).

In view of the Remark following Theorem 13.2, this is equivalent tothe statement that the set of points {y = HOC + k, where k, n = 0, ± 1,...}is dense on — oo < y < oo. In other words, if [y] denotes the largestinteger not exceeding y and yn = «a — [net.] is the fractional part of not,then the lemma is equivalent to the statement that the set {y = yB, wheren = 0, ± 1, ...} is dense on 0 y 1.

Exercise 13.1. Prove Lemma 13.5.Theorem 13.3. Let Sn be a flow on a Jordan curve F having an irrational

rotation number a. Then Sn is ergodic (i.e., N0 = F) if and only if S istopologically equivalent to a "rotation" i.e., if and only if there exists anorientation preserving change of parameters R:z — g(y) such that

Proof. Let there exist an R satisfying (13.8). The minimal set of themap RSR~l is the set N0 of limit points of a, 2a,... on F by Theorem

)

13.2. Since a is irrational, this minimal set N0 is F. Hence the minimalset for S is F.

Conversely, let N0 = F be the minimal set for S. Let y0 = 0 and yn =/"(O) in Lemma 13.4. Then the increasing function z = g(y) is defined onthe sequence {yn + k, n, k = 0, ± 1,.. .} which is dense on — oo < y < <x>.It is continuous since {Hex. + k, for n, k = 0, ± 1,...} is dense on — oo <z < oo and, hence has a unique continuous extension to an increasingfunction for — oo < y < oo. Furthermore, g(y + 1) = g(y) + 1. ThusR'z = g(y) defines a change of parameters on F.

The relation g(/n(0) + k) = no. + k implies that if y =/n(0) + k,then g(f(y)) = g(fn+1(ty + k) = (n + l)a + k = g(y) + a. Since theset of the points {/"(O) + &} is dense on — oo < y < oo, it follows thatgtfW) = gto + « for all y. That is, HSR*:^ = g(f(f\z)) = z + a.This proves the theorem.

Exercise 13.2. Let F be a circle and -/V0 any perfect, nowhere dense seton F. There exists an orientation preserving homeomorphism S of Fonto itself having (an irrational rotation number a and having) NQ as itsunique minimal set. See Denjoy [1],

Lemma 12.1 occurring in the proof of Theorem 12.1 has the followingconsequence.

Theorem 13.4. Let Sn be a flow on F having an irrational rotationnumber a. Let S belong tof(y) and suppose thatf(y\f~l(y) are of class C2.Then S is ergodic (i.e., NQ = F).

This assertion can be improved slightly as follows:Exercise 13.3. (a) Let the condition that "f(y),f-l(y) are of class C2"

be relaxed to "f(y),f~l(y) are of class C1 and dffdy is of bounded variationfor 0 :5s y rfs 1." Then S is ergodic. Denjoy; see van Kampen [1].(b) This assertion is false if the condition that "df/dy is of boundedvariation for 0 y < 1" is omitted. See Denjoy [1].

14. Flow on a Torus

A torus M will be viewed as a square O^x^l, Q<y^l in the(x, t/)-plane in which the points (a;, 0), (x, 1) or (0, y), (1, y) on oppositesides of the square are identified or, even more conveniently, as the entire(x, y)-plane in which pairs of points (xlt y^, (xzt y2) are consideredidentical if and only if xt — x2, yl — y2 are integers.

Consider a continuous flow

on the torus M, so that f, r\ are continuous for — oo < /, x, y < oo andT* is a group of homeomorphisms of the (x, y)-plane onto itself. Further-more, since (x, y), (x + 1, y), (x, y + 1) are considered identical points of

z))

(y)

(y)

y


the torus M

[The first line means, e.g., that if the point (z, y) is translated to (a; + 1, y),then its orbit is merely translated a distance 1 in the ^-direction.] Thegroup property r*+t = T*T* is equivalent to

Suppose that £, 77 have continuous derivatives with respect to /. ThenFI(*> y) = EW, y)/^lf-e» (*» y) = I^» *, y)/3/]t=0 satisfy

and x — (/, x0, y0), y = 77 , a:0, yfl) is a solution of the initial valueproblem

by (14.3).If, e.g., £ and 77 are of class C1, then it follows from Theorem III 7.1

that (14.5), (14.6) has the unique solution x = g(t, x0, y0), y = rj(t, x0, y0).Lemma 14.1. Let M be a torus and (14.1) a flow on M of class Ck,

k ^ 1 [or a continuous flow such that £, r\ have continuous partials withrespect to /]. Suppose that T* has no stationary point [or that F^ + F£ j* 0].Then there exists a Jordan curve F of class Ck [or of class C1] on M which istransversal to the flow. Furthermore, such a transversal curve F cannot becontracted to a point on M, so that F does not bound a 2-cell (i.e., does notbound a subset of M homeomorphic to a disc).

For example, suppose that F^(x, y) does not vanish and let F(x, y) =FtjFi, so that (14.5) has the same solution paths as

Here every circle x = const, on M is a transversal curve F.Proof. Consider the differential equations for the orthogonal tra-

jectories

Let x0(t), y0(r) be a solution of (14.8), so that (z0(/), y0(0) const, since

Fj* -f F22 0. Also (z0(f), y0(0) is of class Ck [or of class C1] since

(F!, F2) is of class C*-1 [or class C°].Suppose first that x = x0(t), y = y0(t) is a closed curve F on Af, i.e.,

let there exist a least number/? > 0 and integers r0, s0 such that x0(t + p) =^(0 + ro» y<>(' + p) = y0(0 + J0- It is c'ear that F is a transversal curve.

If a?0(f), y0(f) is not a closed curve on M, then the half-trajectory #„(/),y0(t) for r 0, viewed as a path on M, has at least one w-limit point, say,(^i. 2/i)- The point (afj, 3^) is contained in arbitrarily small curvilinearrectangles R:ABCD on M, in which the arcs AB, CD are solution arcs of(14.5) and BC, AD are solution arcs of (14.8). The point «0(0» &>(') isinside /? for some large t = t0 and leaves at some point /\ on CD [or /4J?]at a first time f t > /„ and then meets AB [or CD] at a point P2 f°r somefirst /2 > /j. It is clear that if /? is small enough that there exists an arc/V*! in R which together with the arc x = ar0(/), y = y0(f) for /j ^ ? /2

constitutes a transversal curve T of class C*.Suppose, if possible, that a transversal curve F on M can be contracted

to a point. Then it has an image F in the (x, y)-plane which is a Jordancurve of class C1 bounding an open set Q°. Clearly the points of F are allegress or all ingress points of Q° for (14.5). Then the index of F relative to(14.5) is +1 or —1; cf. §2. Hence H° contains at least one stationarypoint by Corollary 3.1. This is a contradiction and completes the proof ofLemma 14.1.

In what follows, it is supposed that(Hj) Mis a torus, T* is a flow on M of class Ck, k ^ 1, without stationary

points.Let F be a transversal (Jordan) curve on M. After a suitable Ck homeo-

morphism of the plane, it can be supposed that(H2) The circle F :x = 0 w a transversal curve. In particular, Fjfn, y) 9*

0 if « = 0, ±1,. . .. Without loss of generality it can be supposed that

for otherwise / is replaced by — t.Let m0 be a point of F on M and suppose that the semi-orbit C+(/w0)

through m0 meets F again for some tr > 0. Let mt be the first such point.In other words, if m0 = (0, y0), then there is a unique tl > 0 such that£('i,0,y0) = l [cf. (14.9)], so that mx = (1, l5 0, y0)). Put /(y0) »*?('n 0» 2/0)- Tne set U of points y0 where U is defined is open and of period1 (in the sense that y0 e U if and only if y0 + 1 6 U). It will be supposedthat

(H3) Every orbit meets F andyl =f(y) is defined for all y, — oo < y < oo.The hypothesis (H3) holds, e.g., if F^x, y) does not vanish [so that (14.5)

is "equivalent" to (14.6)] or if there are no closed orbits.


Exercise 14.1. Verify the assertions of the last paragraph.The function/(y) has the following properties: (i)/(y) is strictly in-

creasing; (ii)/(y + 1) =/(y) + 1, so that/(y) — y has the period 1;(iii) f(y) is of class C*. The property (i) follows from the fact that twoorbits are either identical or have no points in common. The property(ii) follows from the last line of (14.2); for if x — 0 and / = tlt then£('i> 0, y} = 1 and f(y) = rj(tlt 0, y). The proof of property (iii) issimilar to (but simpler than) the proof of (12.3) in § 12.

The results of § 13 can .now be transcribed to results about certainflows on a torus. Let the number a of Lemma 13.1 for the function/(y)be called the rotation number of the flow Ti. Thus Lemma 13.2 impliesthe assertion:

Theorem 14.1. Let T* be a flow of class C1 on a torus M satisfying (Hj),(H2), and (H3). Then there exists a periodic (closed) orbit if and only if therotation number a. of T* is rational.

Exercise 14.2. Let Tf be as in Theorem 14.1 and let the rotation numbera be rational. Show that every semi-orbit C+(w0) on M is either aJordan curve or that Q(/w0) is a Jordan curve and C+(/w0) spirals toward0(m0).

Theorems 13.2 and 13.4 giveTheorem 14.2. Let T* be a flow of class C2 on a torus M satisfying (Hx),

(H2), and (H3) and let its rotation number a be irrational Then M is aminimal set (and every semi-orbit C+(/w0) is dense on M).

When M is a minimal set, the flow T* is called ergodic.Remark. This theorem is false if the condition "T* is of class C2" is

relaxed to "7"' is of class C1." The situation becomes even worse if it isonly assumed that "71' is continuous, £ and y have continuous partialswith respect to /, and /V + F2

2 ^ 0," although the other results abovehave analogues in this case. This is indicated by the next exercise which isan extension of the results of Exercise 13.3.

Exercise 14.3 Let points (x, y), (x + 1, y), and (x, y -j- 1) be identifiedso that the plane becomes a torus M and the line x = 0 a circle F. (a)Let NQ be an arbitrary, perfect, nowhere dense set on F. There exists acontinuous function F(x, y) = F(x + 1, y) = F(x, y -f 1) such that initialvalue problems belonging to (14.7) have unique solutions and such that\fTt:xtsst + x0, y* = rj(t, x0, y0) is the flow induced by (14.7) on M andS is the corresponding homeomorphism on F, then (the rotation numbera is irrational and) N0 is the unique, nonempty, minimal set of S. (b)There exist functions F(x, y) = F(x + 1, y) = F(x, y + 1) of class C1 suchthat the corresponding flow T* of class C1 has an irrational rotationnumber a but 7* is not ergodic (i.e., M is not a minimal set). See Denjoy[1]-

The simplest flow on M is given by

arising from the differential equations

where a is a constant (in fact, the rotation number of the flow). Considera flow of the form

arising, e.g., from a system of differential equations of the form x' = 1,y' = F(x, y). The function/(y) belonging to (14.11) is

In examining the orbits of (14.11), it is sufficient to consider r)(t, 0, y),i.e., the orbits beginning for t — 0 at a point (0, y) of the y-axis. For theorbit starting at (x, y) meets the z-axis when t = —x, and so

since J* = Tt+xT~x.It will be verified that

for — oo < r, y < oo. To this end, note that /w(y) = rj(n, 0, y), so that

by Lemma 13.1. If t = n + 01? where 0 0! < 1, then rj(t, 0, y) =rj(n, 0, »7(0!, 0, y)) satisfies

Also, T^(/, 0, y + 1) = 1 + ??(/, 0, y) shows that if y =j + 02» where0 02 < 1, then

The last two relations give (14.14). The assertion (14.14) can be greatlyimproved in the ergodic case. This is the essence of the following result.

Theorem 14.3 (Bohl). Let T1 be a continuous flow on the torus M of theform (14.11) such that T1 is ergodic and has the (irrational) rotation numbera. Then there exist continuous functions y(y), G(t, z) such that

and that


Note that for a fixed y [ or t], ?y(/, 0, y) — cut — y is an almost periodicfunction of t [or periodic function of y].

Proof. Let z = g(y) be the function supplied by Theorem 13.3 and lety(y), y»0(y) be the functions of period 1 defined by

Make the change of variables /?:(#, 2) = (x, g(y)). Then T' becomes

where

By the analogue of (14.2)

and, by the group property of RT'R-1 [cf. (14.3)],

Introduce the abbreviations

Note that, since f(y) = 77(1, y), Theorem 13.3 gives £0» 2) = z + a.Hence (14.20)-( 14.22) imply that

Thus the continuous function

has the period 1 with respect to / [or 2] for fixed 2 [or /]. Write the lastrelation as

In view of (14.19), r](t, y) * ^»(C[^ (y)D, so that, by (14.17),

Consequently (14.23) shows that

if (7(/, 2) is defined by

It is clear that G satisfies the last part of (14.15). Finally, (14.24) and thefirst part of (14.17) give (14.16). This proves the theorem.


Notes

Most of §§ 1-9 of this chapter is contained in (or related to) the four-part memoir ofPoincare [3]; see also Bendixson [2], L. E. J. Brouwer [1] considers similar problemswithout analyticity assumptions and even without the assumption that there is onlyone solution through a given point.

SECTION 1. The terminology "a- and co-limit points" is due to G. D. Birkhoff [3];"limit cycle" to Poincar6 [3].

SECTION 2. The Umlaufsatz (Theorem 2.1) goes back to Riemann [1, pp. 106-107).It was first given a formal statement and proof by G. N. Watson [1]. The proof in thetext is that of H. Hopf [1]; cf. also van Kampen [4].

SECTION 3. The definition of "index" was given by Poincar6 [3,1].SECTION 4. See PoincarS [3] (in particular, part II) and Bendixson [2].SECTION 5. See Poincar6 [3].SECTIONS 6-7. See Poincare [3].SECTIONS 8-9. For Lemma 8.1, cf. Bendixson [2, p. 26]. In the analytic case, Theorems

9.1 and 9.2 are due to Poincar6 [3] (in particular, part I) and to Bendixson [2]. Theexposition jn the text of Theorem 9.1 is an adaptation and simplification of Brouwer'streatment [1, II] (which is complicated by the fact that initial conditions do not determinea unique solution). For Exercise 9.1, see Schilt [1].

SECTION 10. Problems of the type considered here go back to van der Pol [1] and toLienard [1]. Theorem 10.1 is a variant of a result of Levinson and Smith [1]. Ananalogous result concerning (10.15) was given by Levinson [1], improved by Langenhop[1], and further improved by Opial [7] as in Exercise 10.1. Theorem 10.2 is a general-ization of a result of Lienard [1J. For Exercise 10.5, see, e.g., Lefschetz [1, pp. 342-346]and references to Flanders and Stoker [1], LaSalle [2], and Stoker [1]. For fullertreatments of related problems and for references to the work of Cartwright, Littlewood,Duff, Massera, Reuter, Sansone, Schimizu, etc.; see Andronow and Chaikin [1],Bogolyubov and Mitropol'ski [1], Bogolyubov and Krylov [1], Lefschetz [1], Minorsky[1], Conti and Sansone [1], and Stoker [1].

APPENDIX. This type of investigation was initiated by PoincarS [2], [3, III] for thecase of differential equations on a torus; see notes on §§ 13-14.

SECTION 11. The ideas in this section are due to Poincare.SECTION 12. The results and methods of this section are those of A. J. Schwartz [1J;

for a generalization, see Sacksteder [1]. The analogue of Theorem 12.1 was knownearlier for the torus; Denjoy [1]. See notes on §§ 13-14.

SECTIONS 13-14. Except for Theorem 13.4 and Exercises 13.1, 13.2 in §13 andTheorems 14.2,14.3, and Exercise 14.3 in § 14, the results and methods of these sectionsare essentially in Poincare [3, III]. The presentation in the text follows the completedarguments and simplifications introduced by Denjoy [1] and van Kampen [1]; cf.also Siegel [1]. (After Poincare, proofs for the main part of Lemma 13.1 have beengiven by E. E. Levi, Bohl, H. Kneser, Nielsen, and Denjoy; see Bohl [1] and vanKampen [1] for references.) Theorem 13.4 and its consequence, Theorem 14.2, wereconjectured by Poincare [3, III] for the analytic case. They were first proved by Denjoy[1] in a slightly stronger form (cf. Exercise 13.3). A very simple proof of Denjoy'sresult has been given by van Kampen [1]. A similar proof which avoids, however, theuse of the rotation number has been given by Siegel [1]. Denjoy [1] has given examples(cf. Exercises 13.2 and 14.3) showing that the results are false if the smoothnessassumptions are lightened. Theorem 14.3 is a result of Bohl [1].

Chapter VIII

Plane Stationary Points

This chapter continues the discussion of the behavior of solutions of planeautonomous systems. The basic existence theorems will be proved in thefirst section for autonomous systems of arbitrary dimension.

1. Existence Theorems

In this section, there will be considered an autonomous system

for a real (/-dimensional vector z = (z1,..., 2**). By a half-trajectory of(1.1) will be meant a solution arc in the 2-space: C+: z = z(t), 0 / <a>+ (^oo), or C~: z = «(/), 0 / > <o_ (^ — oo), defined on a right orleft maximal interval of existence. Correspondingly, F will be called oftype C+ or C~. The point z(0) will be called the endpoint of F.

Theorem 1.1. Let f(z) be continuous on an open z-set Q,. Let Q0 be anopen subset of H such that the part of its boundary in ii (i.e., 9£10 n ii)is the union of two disjoint sets L U R, where R is compact, the points ofLare egress points, and dQ0 n Q = L U R is not compact. Then QQ UL U R contains at least one half-trajectory F of (1.1) w/'/A endpoint

in L or R.For the definition of egress point, cf. § III 8. It is not assumed in this

theorem that L exhausts the set of egress points of iV Since the boundarydQ0 is closed relative to O, the condition that L U R is not compact impliesthat either L U R is unbounded or has a limit point £«, which is not in O.

It will be clear from the proof that either there exists a half-trajectory Fof type C~ in Q0 U L with endpoint on L or there exists a half-trajectory oftype C+ in Q0 U L VJ R with endpoint on R. Sufficient conditions forthe latter case are given by the next theorem. The conclusion of thistheorem does not specify, however, whether F is of type C+ or C~.

Theorem 1.2. In Theorem 1.1, assume that solutions o/(l.l) are uniquelydetermined by initial conditions, that R is not empty, and that L U R is

202

Plane Stationary Points 203

connected, then there exists a half-trajectory in Q0 U L U R, endpointon R.

The condition that "L U R is connected" can be relaxed to "L0 U R isconnected, where L0 cr £ and L0 U /? is either unbounded or has a limitpoint not in L U R (i.e., not in Q)."

Remark. In Theorem 1.2, the condition that solutions of (1.1) areuniquely determined by initial conditions can be omitted if there exist

Figure 1.

smooth functions /n(z), « = 1, 2 , . . . , on Q which approximate /(«)uniformly on compact subsets of Q and if Theorem 1.2 is applicable toz =/n(z). This will be illustrated in Corollary 1.1 below.

Proof of Theorem 1.1. For brevity, we write o>_(z0), <o+(z0), r(2o)» • • • >

although solutions are not determined by the initial point z0, so that o>_,o>+, T, ... depend on z0 and the selected solution. Let z(t, z0) be anysolution of (1.1) determined by z(0) = z0

and 'et its maximal interval ofexistence be «_(z0) < f < eo+(z0). If z0 e L, then z(r, z0) e Q0 for small—/ > 0. If there exists a z0 e L having a half-trajectory z(t, «0) in Q0 U Lon its left maximal interval eo_(z0) < f ^ 0, then the conclusion ofTheorem 1.1 holds.

Suppose therefore that, for every z0 e L and solution z(/, z0), thereexists a T = T(ZO), w_(z0) < T(ZO) < 0, such that z(t, z0) e Q»U L, r(z) <r < 0, but Z(T, «0) ^ HJJ I" Then Z(T, z0) e R; so that, -R is not empty.

Since /? is compact and L U R is not compact, there exists a sequenceof points Zj, z2 , . . . of L such that if rm = ^(z^), then, as m ->• oo, z° =lim Z(TTO, zm) E /? exists and either ||zw|| -> oo or zw = lim 2ro exists andz^ ^Q; see Figure 1.

Put zm(t) = z(t, z(rm, zj), so that 2JB(0) = z(rm, zj € R, zm(-rj =2m £ L, zm(t) e Q0 U L, 0 < t < -rm. Since zm(0) -+z° as m-+ao,Theorem II 3.2 implies that if the sequence zlt 22 , . . . is suitably chosen, itcan be supposed that there exists a solution z°(/) of (1.1) satisfying 2(0) = 2°,having a maximal interval of existence (co0, a>°), and such that

on any /-interval [r#, /*] in (ey0, o>°). In particular, (o_(zm) < ^ < /* <w+(2m) f^' large m.

Figure 2. 2°(/) e Q0 U fl for 0 / < (t>+(z°).

Sujppose, if possible, that z\t) $ Q0^ L u R,Q < / < o>°. Then z°(0 H0 n O = Q0 U L u /? for 0 / < o>° and, hence there is a t — tlt0 < /! < w° such that z\tj # H0. Let /t < /2 < co°. It follows that, forlarge w, zm(/) is defined for 0 f ^ /2 and that (1.2) holds for 0 / /2.But then zm(tj) <£ D0 for large m. Consequently, 0 < —rm<tl for largew. In this case, z°(—TW) — zm(—rm)-*0, w-^oo, by (1.2). Since2m(—T

m) = 2™ and either j|«m|| -> oo or zm -» z^ as t -> oo, it follows thateither ||2°(—TTO)|| ->- oo or z°(—TTO) -> z^ ^ Q as /w -> oo. This is impossibleas — rm < fj < w°. Hence Theorem 1.1 is proved.

Proof of Theorem 1.2. In view of the proof of Theorem 1.1, it sufficesto consider the case that there exist points z0 e L such that (the unique)z(t, 20) is in Q0 U L, hence in Q0 u L U R, for o>_(20) < r < 0. Let Lvbe the (nonempty) subset of L of such points. Clearly Lx is closed relativeto L, since L consists of egress points; cf. Theorem II 3.2.

If there is a point 2° e R which is a limit point of Llt then C~ : z(t, 2°)e Q0.UX U /?, w_(z°) < / 0. Suppose therefore that Lj has no limitpoint in R. Thus Lt is closed relative to L U /?.

Consequently L — Lt has a limit point 2^ in Lx. Otherwise L U R

has a decomposition into nonempty, disjoint sets (L — Lj) U R and Ll5

which are closed relative to L U R. But L U R is connected.Thus the constructions of the last proof can be repeated with the

modifications that zm E L — Lx and zm^zaoeLl. If z°(f) e &„ UZ, U /?,0 51 f < co+(z0), the proof is complete; cf. Figure 2. If not, it becomesnecessary to examine the conclusions at the end of the proof to the effectthat — rm < ti < <t>° and z°(—rm) -*• «„ as m -*• oo.

Hence there exists a / = TOO( 0) which is a (finite) limit point of thesequence rlt r2 , . . . . Then z°(—r^) = 2W. It is clear that TW < 0 for

Figure 3. z°(/) eOoV R for o>_(z°) < (0 <; 0.

2°(0) e /?, ZOQ 6 L! (and L, R are disjoint). Since z^ E Lx and «(/, «„.,) isuniquely determined by z^, it follows that z(t, 200)£Q(ru L U /?, w_(2oo) <f < 0; see Figure 3. [This is the critical point where the assumption thatsolutions of (1.1) are uniquely determined by initial conditions is used.]

In particular, the half-trajectory z°(t — T^) — z(t, ZoJeQoUL U R,((^(ZOQ) < t < TQO . Since the endpoint of this trajectory is Z(TOO, z^) =z° e R, the theorem is proved: z°(0 e Q0

u L U /? for w_(z°) < f ^ 0.Corollary 1.1. Let dim z 2, /(z) 6e continuous on an open set Q

containing the closure of the spherical sector

wAere d, -r\ are positive numbers and ||z*|| = 1, and let

Le/ L, /? be the lateral and spherical parts of the boundary ofO^,

respectively. Suppose that every point of L is an egress point for Q0. Then(1.1) has at least one half-trajectory F in the set &0 with endpoint in R(and defined on a half-line t > 0 or t 0); see Figure 4.

Proof. Suppose first that the solutions of (1.1) are uniquely determinedby initial conditions, then Theorem 1.2 is applicable if the open set £1 ofCorollary 1.1 is replaced by the open set obtained by deleting 2 = 0 fromQ, so that 9O0 n Q = L U R. Then there exists a half-trajectory, e.g.,C+ : z(0, 0 <; / < CD+ (^ oo), in Q0 UZ, U R, 2(0) e /?. If co+ < oo, then

w '»'

Figure 4. (a) T of type C+. (b) T of type C~.

2(0 ->• 0 as t -> o>+; cf. Lemma II 3.1. But, in this case, the definition ofz(t) can be extended to 0 t < oo by defining z(t) = 0 for / > w+.[Actually, this situation cannot arise when the solutions of (1.1) areuniquely determined by initial conditions, since z(/0) = 0 for some/0impliesthat z(t) ^ 0. It can arise, however, in the general case to be considerednow.]

It has to be shown that Corollary 1.1 is valid if it is not assumed that thesolutions of (1.1) are uniquely determined by initial conditions. Sincez e L is an egress point, if follows that the trajectory derivative of «(z) =II2/II2II — 2*ll2 — *?2 is non-negative at 2 e L; i.e.,

fOfO < ||2|| £ ($,||2/||2|| - 2*11 - 17; Cf.§1118. ThuS,if/e(2) =/(2) - €2*,

then

which exceeds a positive constant ecn > 0 when

for large n.

<

z)

Let/!(z),/2(z),... be a sequence of smooth functions such that fm(z) -»>/(z) as m -*• oo uniformly on an open set Q.1 ~=> &0. Let the integer n > 0be so large that l/« < 6. Then if/m(z) is replaced by/TO(«) — z*fn, ifnecessary, it can be supposed that there is an m = m(ri) such that

when (1.9) holds, and that m(ri) -> oo as n -> oo.Let £1B be the open set obtained by deleting the sphere ||z|| ^ l/« from

Q1. Let

and let Ln, R be the lateral and spherical parts of the boundary of QOB inQn. By (1.10), z e Ln is an egress point of with respect to the differentialequation

Thus, by Theorem 1.2, (1.11) has a half-trajectory TTO in Q0n U£BU /?,endpoint on /?. For the sake of definiteness, let Fro = Cm

+ : z = zm(t\Q^t<rm(^ oo), where zm(0) 6 R.

Here 0 / < rm is the right maximal interval of existence if (1.11) isconsidered only on Qn. Let the right maximal interval of existence ofzm(t) be 0 < / < a>m (^ oo) if (1.11) is considered on Q1. Thus rm ^ ojmand rm < com implies that \\zm(rm)\\ = l/«.

By choosing a subsequence, if necessary, it can be supposed that z0 =lim zm(0) e R exists as m = m(n) —*• oo. By Theorem II 3.2, it can also besupposed that (1.1) has a solution z0(f) satisfying z0(0) = z0 and thatzm(t) —> z0(f) as m — m(n) —> oo uniformly on every compact interval of theright maximal interval [0, o>0) of existence of z0(0 relative to Q.1.

Suppose that the half-trajectory F : z0(r), 0 / < o>0, is not in theset QO.. Thus zo('i) ^ o for some tlt 0 < rt < w0. If e > 0, then 0 <

TTO < ?i + € anOasm(«)-^ oo. Thus if r^ is a limit point ofrlf TZ, . . ., then ZQ^^) = 0. Here, it is possible to change the definitionof z0(0 as follows: if t0 > 0 is the least /-value where z0(/) = 0, putz0(r) = 0 for t^ t0. Now T : z0(r) is defined for 0 t < oo and zm(r) *>z0(/), m(/i) -* oo, uniformly for 0 t /0 < oo. Repeating the argumentjust concluded, it follows that z^t) is in the set &0 for t 0. Thiscompletes the proof.

)

Corollary 1.2. Let dim 2 = 2. In addition to the assumptions of Corol-lary 1.1, suppose that

Then the half-trajectory F : z(t) in Corollary 1.1 w defined for / > 0 or/ 0 am/ satisfies z(t) —> 0 flj / -> oo or / -*• — oo.

Proof. If this corollary is false, then F : z(t) for / ^ 0 or / ^ 0 remainsat a positive distance from 2 = 0; cf. Lemma II 3.1. In view of (1.12)this is impossible by the general theory of plane autonomous systems;cf. Theorem VII 4.4.

Figure 5.

Corollary 1.3. Let the assumptions of Corollary 1.2 hold, (i) If, inaddition, z -f(z) < Ofor z e R, then any half-trajectory F in the set fi0

with endpoint on R is defined for t 0. (ii) If z -f(z) > 0 for z € R andz(t) is any solution of (1.1) with 2(0) interior to R [i.e., ||z(0)|| = <5, ||z(0)/|| 2(0) || - 2*|| < »?], then 2(/) can be defined for t <; 0 so that z(t) e &0

and 2(0 -> 0 as t -»• — oo.Exercise 1.1. Prove Corollary 1.3.A different situation is considered in the next result.Corollary 1.4. Let dim z = 2, so that L in (1.5) is the union of two

disjoint open line segments Lj, L2. Let the conditions of Corollary 1.2 holdbut, instead of assuming that every point z of L is an egress point for ii0,assume that every z e Ll is an egress point for Q0 and that every z E L2 is aningress point for Q0. Then there exists no, or at least two, half-trajectoriesin D.0 with endpoint on L U R.

Proof. Assume that there exists a half-trajectory F° in Q0 with end-point 2° e L U R. Suppose that F° is of type C+ (otherwise, replace / by—t and interchange LI and L2)- Then 2° e Lz U R.

Suppose that 2° e Lz. Then every point 20 of the segment of L2 from0 to z° is the endpoint of a half-trajectory F° of type C+ in Q0 U L2;cf. Figure 5(a).

Suppose that z° is an interior point of R and that no point zx of L2 is anendpoint of a half-trajectory in Q0 <J L2. Then a solution arc of (1.1)through *! does not meet F° for increasing t, but does meet R', cf. Figure5(b). The arguments used in the proof of Theorem 1.1 show that thereexists a half-trajectory F0 of type C~ in D0 U R with endpoint 20 e R.This proves Corollary 1.4.

Exercise 1.2. In Corollary 1.4, assume that z-/(z)^0 for z e R.Show that there exists no, or infinitely many, half-trajectories in Q0 withendpoints on L U R.

For arbitrary d = dim z, Corollary 1.3 has the following analogue.Corollary 1.5. Let dtZ.2. Let the assumptions of Corollary 1.1 hold and,

in addition, z •/(«) 0 /or 2 e Q0 U R. Then any half-trajectory F :z(0 inthe set HO with endpoint in R is defined on a half-line t ^ 0 or t ^ 0and tends to z = 0 as t-*- oo or / -> — oo according as z -/(z) < 0 orz •/(«)> 0.

This is clear because ||z|| is decreasing or increasing with t according asz -/(z) < 0 or z •/(«) > 0.

2. Characteristic Directions

In this section, dim z = 2. Write z = (x, y) and (1.1) as

It will be supposed that X, Y are continuous for small \x\, \y\ and that

Introducing polar coordinates z = r cos 0, y = r sin 0 transforms (2.1)into

A direction 6 = 60 at the origin is called characteristic for (2.1) ifthere exists a sequence (rls 0J, (r2, 02)» . . . such that 0 < rn —> 0 and0M -> 00 as n -> oo; (JTn, yj = (JT, 7) at (*, y) = (r, cos 0n, rB sin 0W) isnot (0, 0), and

The condition (2.4) means that the angle (modTr) between the vectors(Xn, Yn) and (cos 00, sin 00) tends to 0 as n -*• cx>.

Lemma 2.1. Let X(x, y), Y(x, y) be continuous for small \x\, \y\ and

X2 + Y* 0 according as xz + yz ^ 0.(*(0. y(t))for 0 t < <o (^ oo) swc/r /fort

Le/ (2.1) possess a solution

Lef r(0 = (z2(0 + y2(0)'* > 0 and 0(0 a continuous determination ofarc tan y(OMO- Let 0 — 00 be a noncharacteristic direction. Then either0'(0 >Qor 0'(0 < 0/or all t near <ofor which 0(0 = 00 mod 2ir.

Figure 6.

Proof. Clearly 0'(0 ?* 0 for all t near w for which 0(0 = 00 mod 27r.Otherwise there exists a sequence ^ < tz < ... such that fn -*• to, 0(tn) ~00 (mod 27r) and 0'(/M) = 0. But then (2.4) holds with (rn, 0n) = (r(/n), 00)because the expression in (2.4) is zero by (2.3). This is impossible since0 = 00 is noncharacteristic.

Suppose if possible that the lemma is false. Then there exists /j < tz <. . . such that tn -* to, rfo) >! •&)>••• , (-I)n0'(>«) > 0 and 0(fn) = 00

(mod 2n); see Figure 6. Let ©(/•, 0) denote the right side of the secondequation in (2.3), so that (— I)n0(r(fn), 00) > 0. By the continuity of0(r, 00) with respect to r > 0, it follows that there exists an rn such thatKO > rn > K'»+i) and 0(rn, 00) = 0. Since JT2 + 72 0 if *2 + y2 0,it follows that (2.4) holds with (rn, 0J = (rn, 00); i.e., 00 is characteristic.This is a contradiction and proves the lemma.

Theorem 2.1. Let X(x, y), Y(x, y) and (x(t), y(t)) be as in Lemma 2.1.Suppose that every 0-interval, a < 0 < /?, contains a noncharacteristicdirection. Then either

or (#(/), y(t)) is a spiral; i.e.,

In the case (2.6), 6 = 00 is a characteristic direction.Proof. Suppose, if possible, that lim 0(/), as / —»• co, does not exist

either as a finite or infinite value. Then there exist numbers a, ft such that,as / -> co,

By assumption, there is a noncharacteristic 00 satisfying a < 00 < fi.Since 0(/) is continuous, there exist /-values arbitrarily close to co where0(0 = a < 00 and /-values close to co where 0(0 = /3 > 00. It followsthat there are /-values arbitrarily close to co where 0(/) = 00, 0'(/) 0 andother /-values where 0(/) = 00, 0'(/) 0. This is impossible by the lastlemma. Hence lim 0(/), / —»• co, exists as a finite or infinite value. Sincethe last assertion of Theorem 2.1 is clear from the definition of "character-istic direction," the proof is complete.

Exercise 2.1. Let X, Y be continuous for small |ar|, \y\ and satisfyX'— Y = 0 at (0,0). Let y(r) be a positive continuous function for smallr > 0 such that v<+0) = 0. Suppose that the limits

exist uniformly for 0 near 00 and that/?2(0) + qz(Q) ¥* 0. Show that 0 = 00

is a characteristic direction if and only if ^(0) cos 0 — p(6) sin 0 = 0 at0 = 0o-

Theorem 2.2. Let X(x, y), Y(x, y) be continuous on the triangle T:0 < x < 0, \y\ »/a; a/w/ 5McA /Ac/ A" 5^ 0. £e/ co(x, u) be a non-negative,continuous function for 0 < # ^ a , O ^ M ^ 2??a; w//A //re properties thatCD(X, 0) = 0 and that the only solution u(x) of

for small x > 0 satisfying

w M = 0. Le/ the function

satisfy

and (x, yj), (x, y^ E T. Then, up to replacement of the parameter t by t +const., (2.1) has at most one solution (x(t), y(t)) which for large t [or —t]satisfies

as t [or —t]-+ oo.Exercise 2.2. Prove Theorem 2.2. Note that (2.1) is equivalent to

dy{dx=U(x,y).Exercise 2.3. Show that the conclusion of Theorem 2.2 is valid if

(2.11) is replaced by U(x, y^ — U(x, y^ ^ (yt — yjfx for — yx <^ yl <2/2 = n* and 0 < x = a- In particular, this is the case if X > 0 andX, Y have continuous partial derivatives Xv, Yv with respect to y satisfyingXYV- XyY< X*fxon T.

Exercise 2.4. Replace T in Theorem 2.2 by /^rO < x < a, \y\ b;also, let a>(x, u) be continuous on 0 < x a, \u\ 2b. Show that ananalogue of Theorem 2.2 is valid if "u(x)jx -» 0" and "KOMO -* 0" aredeleted from (2.9) and (2.12), respectively.

3. Perturbed Linear Systems

The results of §§ 1 and 2 will be applied to a 2-dimensional systemobtained by perturbing a linear system

in which z = (x, y) is a real 2-dimensional vector and £ is a constantmatrix with real entries. Unless otherwise specified, it will be assumedthat

Let A!, A2 De the eigenvalues of £; so that A1} A2 are real or are complex

conjugates since £ is real. If A = Ax or A = is a simple eigenvalue or if^ = AJ = A2 and £ has a double elementary divisor, then, up to factors±1, there is only one (real) unit eigenvector z* satisfying £z* = Az*.

Recall from Exercises VII 7.1 and 7.2 that if A^ A2 = ±ift are purelyimaginary (with /? j& 0), then z = 0 is a center; that if Ax, Aj = a ± ifi arecomplex conjugates but not real or imaginary (a, ft real, ^ 0), then z = 0is a focus (at r = ±00 according as a ^ 0); that if A1? A2 are real and

det E — A^ > 0, then z = 0 is an attractor, in fact, a node for /= ± ooaccording as Als A2 ^ 0 (and a proper node only if Aj = A2 <*0 and E hassimple elementary divisors); finally, if A15 A2 are real and A^ = det E < 0,then z = 0 is a saddle point.

After a real linear change of variables, it can be supposed, when con-venient, that E is in one of the normal forms

(simple elementary divisors) (double elementary divisor)

The system to be considered in this section is of the form

where F(z) is continuous for small \\z\\ and satisfies

Theorem 3.1. Assume (3.2) and that the continuous F(z) satisfies (3.5).Let z — 0 be an attractor for (3.1) at t — oo, so that <x.k = Re Afc < 0/ork= 1,2.

(i) In this case, z = 0 is an attractor for (3.4) at t = oo. More generally,*f<*k< —c< Qfor k = 1,2, then there exists a constant M = M(c) suchthat if \\ZQ\\ ?£ 0 is sufficiently small, every solution of (3A) satisfying theinitial condition z(0) = z0 exists for t ^ 0 and satisfies

w/itfre a = ax or a = a2.(ii) 7f at! < az < 0, so //ifl/ Ax = alf A2 = a2 are real, and J/(3.7) Aoto,

Me«

exists and is an eigenvector of E belonging to a (=At or A^. In particular, ifE is in the normal form (3.3-3), then

according as a = At or a = A2.(iii) Le/ ax < a2 < 0. 7/"z° w either of the two real unit eigenvectors of E

belonging to A = a,, then (3.4) /raj at least one solution z(t) satisfying (3.8)and (3.7) w///r a = at. Ifz°is either of the two real unit eigenvectors of Ebelonging to A = a20m///||20|| 5^ Oaw/||z0/||z0|| — 2°|| are sufficiently small,then any solution of (3.4) determined by z(0) = z0 ex«/j ybr t 0 arm/satisfies (3.8) a«*/ (3.7) vwY/i a = a2.

Proof of (i). After a real linear change of variables, it can be supposedthat E is in one of the real normal forms (3.3). In (3.3-5), it can also besupposed that e > 0 is so small that Re X + |c < — c. It is then readilyverified that if r = ||z|| and r 0 is small, then r' — cr. This impliesthat along a solution z(t), r(t) ^ r(Q)e~ct for small t > 0. Consequentlyif r(0) > 0 is sufficiently small, then z(t) exist for t ^ 0 and satisfies (3.6)with M .= \.

When E is not in a normal form and L is a nonsingular matrix such thatL~1EL is in the form just used, then (3.6) holds with M = 1 if 2 is replacedby 17 = Lz. In this case, (3.6) holds if M = ||L|| • UZr1!!.

When a.t = a2 < 0 and £" is in a normal form (3.3-2), (3.3-4), or (3.3-5),it is easy to see that 0 z(t) -> 0 as t -* oo implies (3.7). This completesthe proof of (i) for this case. [The cases ax ^ a2 will be considered in theproofs of (ii) and (iii).]

Proof of (ii) and (iii). Assume that E is in the normal form (3.3-3), sothat the unit eigenvectors of E are (±1,0) for A = Ax and (0, ±1) forA = A2. On introducing polar coordinates z — (r cos 0, r sin 0), (3.4)takes the form, as r —> 0,

It follows that the only characteristic directions (mod 2-n) are 0 = 0,7T/2, TT, 3rr/2; cf. Exercise 2.1. Hence, by Theorem 2.1, if 0 2(r) -* 0 as/ -> oo, then z = z(f) is a spiral or (3.9) holds.

Let ao be a wedge Q0:0 < ||z|| < 6, \\zl\\z\\ - z°|| < r), where 2° =(0, ±1) and 6, r) are small. It is seen that if a solution z(r) starts in Q0 orenters in Q0, it remains in Q0, for the boundary points are strict ingresspoints for D0; cf. Corollary 1.3 (ii) with t replaced by — t. Thus suchsolutions satisfy z(/) -> 0 as t -> oo and the second part of (3.9), i.e., (3.8).

The first equation of (3.10) implies (3.7) with a = A2. In particular, nosolution z = z(i) ^ 0 tending to 0 as t -»• oo is a spiral.

If, in the definition of the wedge Q0, z° is taken to be (±1,0), theexistence of solutions z(r) 0 satisfying (3.8) follows from Corollary1.3(i). As above, such solutions satisfy (3.7) with a = . This provesTheorem 3.1.

Theorem 3.2. Let the eigenvalues of E be a ± ifi, where a 5J 0and ft T£ 0 are real; let (3.5) hold; let z(i) be a solution of (3.4) suchthat 0 < ||z(/) || < OQ for all t and 0(t) a continuous determination ofarc tan y(t)lx(t). Let 0 < e < \ft\. Then there exists a df > 0 such that ifd0 ^ de, then

In particular, 0(/) ->• ± oo as r —»• oo according as ft ^ 0. //", /'« addition,a < 0, f/ie/i

T%MJ, if z = Q is a center for (3.1), // w a center or focus for (3.4) fl«f/ */z = 0 w a focus for (3.1), /Aew /f w a focus for (3.4).

Proof. It is readily verified that neither the assumptions nor conclusionsare affected if z is subjected to a real linear transformation. Hence, it canbe supposed that E is in the normal form (3.3-2), with a < 0. Let F(z) =(F1, F2) and write (3.4) as

Introducing polar coordinates gives

where

so that R(r, 0), S(r, 0) -* 0 as r -»• +0. Thus, there exists a 6e > 0 suchthat |5(r, 0)|< if 0 < r (5e. Hence |0' - /8|< if 0 < ||z(0|| ^ <56,and (3.11) holds. If a < 0, then, by Theorem 3.1, r(t) ->• 0 as t ->• oo, andso S(r(r), 0(0)-^0 as /-»> oo. In this case, (3.12) follows. This provesTheorem 3.2.

As can be expected, the property that z = 0 is a center (i.e., that asolution starting at any point z0 y& 0 at t = 0 returns to exactly the pointz0 at some positive t) is very sensitive to perturbations. This is illustratedby the following exercise which shows that no condition of smallness onF(z) & 0 at z = 0 can assure that if z = 0 is a center for (3.1), then it is acenter for (3.4).

Exercise 3.1. Let h(r) be a continuous function for 0 _^ r 1 suchthat h(r) -* 0 as r ->• 0. Consider a system (3.4) of the form

where r = (a;2 + 2/2)>x*, the function .F(2) = (xh(r), yh(r)) is continuous for\\z\\ = 1, and Hf{z)||/N = |A(r)| -* 0 as r -> 0. If h(r) = 0, i.e., F = 0,then 2 = 0 is a center. Show that if h(r) < 0 for 0 < r <: 1, then 2 = 0is a focus (at t = oo) for (3.16).

It might also be guessed that the other cases, ^ = A2, determined byequalities (rather than by inequalities) are sensitive to perturbations.This turns out to be the case. For example, the next two exercises showthat if 2 = 0 is a node for (3.1) with Aj = A2 < 0, then, even if (3.5) holds,2 = 0 can be a focus for (3.4), whether or not E has simple or doubleelementary divisors. However, as will be shown in Theorems 3.5 and 3.6,suitable conditions of smallness on Fat z — 0, more stringent than (3.5),preserve the character of this type of stationary point.

Exercise 3.2. Let E = diag [A, A], A < 0. Show that there existcontinuous functions F(z) for ||z|| < <5 satisfying (3.5) and such that (a)2 = 0 is a focus for (3.4); and (b) the equation (3.4) has a solution z(f) -> 0as t -> oo satisfying any one of the seven possibilities compatible with

Exercise 3.3. Let E be as in (3.3-5) with A < 0 and e = 1, so that Ehas a double elementary divisor and is in a Jordan normal form. Showthat there exist continuous F(z) for ||2|| <s <5 satisfying (3.5) such that (a)all, (b) some but not all, (c) no solutions z(t) of (3.4) which tend to 0 as/ -»• oo are spirals (i.e., |0(f)| —»• oo as t —*- oo). [Case (b) cannot occur ifthe solutions of (3.4) are uniquely determined by initial conditions]. SeeTheorem 3.3.

Theorem 3.3. Let E be as in (3.3-5) with A < 0 and e = 1. Let F(z) becontinuous for small \\z\\ and satisfy (3.5) and let z(t) y& 0 be a solution of(3.4) for large t satisfying z(t) ->• 0 as t -*• oo. Then either z = z(t) —(*(0, KO) '•* « J/ww/ (/.«., |0(OI -*• °° as t -+• oo) or 0(f) -* 0 (mod TT) as*-» oo.

Exercise 3.4. Prove Theorem 3.3.Conditions which assure that all or that no solutions in Theorem 3.3

are spirals will be considered subsequently; cf. Exercise 4.5.Theorem 3.4. Let E = diag (H19 A2), where ^ < min (0, A2); let F(z)

be continuous and satisfy (3.5). 7^2° = (1,0) or (— 1,0), then (3.4) has atleast one solution z(t], t 0, satisfying (3.8) and (3.7) with a = Ax;

furthermore, if A2 > 0 0/k/ 2(/) w a solution of (3.4) ybr /arge r MC/I f/wtf2(/) -»> 0 <w / -» oo, then (3.7) /ro/</5 w/7/i a = A! andy(t)jx(t) -> 0 flj / -»• oo.

)

Exercise 3.5. Deduce Theorem 3.4 from Corollary 1.3 (i).Exercise 3.6. Let E = diag (Als A.J), At < min (0, Aa), and let F(z) be

continuous and satisfy (3.5) and

(with «! 5^ z2). Then, up to reparametrizations (i.e., replacements of / byt + const.), (3.4) has unique* pair of solutions z±(f) for large t such that0 z±(t) -* 0 as / -* oo. These solutions satisfy z±(0/l|z±(OII -»• (± 1, 0)as t -> oo, and, hence (3.7) with a = AJ.

Exercise 3.7. Let £ = diag (Ax, A2) with < min (0, A^ and let F(z)be continuous for small \\z\\ and satisfy (3.5). Use Theorem 2.2 and/orExercise 2.3 to find conditions, more general than (3.17), to assure that(3.1) has at most one solution (up to changes of the parameter) satisfyingz(0 -* 0 and z(0/lKOII -» (1,0) as / -* oo.

This completes the discussion of (3.4) under the assumption (3.5).Except in the case of a center, assumptions slightly stronger then (3.5)suffice to preserve the character of the stationary point z = 0 in passingfrom the linear system (3.1) to the perturbed system (3.4). Results ofthis type are consequences of general theorems in Chapter X (inparticular, in § X 16). Some will be stated here for the sake ofcompleteness. The deduction of these theorems from results in Chapter Xwill be given as Exercises 3.7-3.11; cf. also Theorem X 13.1 and itscorollaries in §X 16.

The first condition to be imposed on F(z) will involve the function

[so that <p0(r) is a continuous, nondecreasing function for small r 0and <p0(Q) = 0] and the condition

This last condition is satisfied if, e.g.,

for some e > 0, since (3.20) implies that 9>0(r)/rl+ ->• 0 as r -*• 0. Conditions

of the type (3.18), (3.19), or (3.20) are invariant under linear changes of thevariables, z—*Lz where L is a constant matrix, so that, in the theorems tofollow, the assumption that the matrix £ is in a normal form is no loss ofgenerality. This does not apply, e.g., if (3.18)'is replaced by y0(r) =max ||F(2)|| for ||z|| = r and it is not assumed that <p0(r) is monotone.

Theorem 3.5. Let F(z) be continuous for small \\z\\ and satisfy (3.18)-(3.19) and let z(t) be a solution of (3.4) satisfying 0 z(t) -> 0 as t -*• oo.

(i) Let E be as in (3.3-2), where a < 0, /? j* 0. Then there exist constantsc > 0, 00 swc/i //wf

ast-+ oo; conversely ifc > 0, 00 are given constants, there exists a solutionz(t)of (3A) satisfying (3.21).

(ii) Le/ £ = diag [Al5 A2] w/f/j Ax < A2 < 0. Then there exists a constantcl^Qora constant c2 7* 0 such that either

or

conversely ifclj^0 and cz 5^ 0, r^ere exist solutions z(t) o/(3.4) satisfying(3.22) a«f/ (3.23), respectively.

(iii) Le/ £" = diag [Ax, A2] vv/'/A Ax < 0 < ^2. 7%e« there exists a constantc1 T£ 0 JMC/I fAa? (3.22) holds; conversely, ifc^ ^ 0, there is a solution z(t) of(3.4) satisfying (3.22).

(iv) Let E — diag [A, A] >v/Y/j A < 0. 77ie/i f/zere exist constants cl5 ca,«o/ both 0, swc/z //»«?

conversely, if clt c2 are given constants, not both 0, then (3.4) Aarj a solutionsatisfying (3.24).

Exercise 3.8. Denote by Theorem 3.5* the analogue of Theorem 3.5 inwhich the hypothesis (3.18)-(3.19) is replaced by the slightly heaviercondition:

(a) Deduce parts (i), (ii) of Theorem 3.5* from Theorem X 1.1 (i.e., fromvariants of Corollary X 1.2). (b) Deduce parts (iii), (iv) of Theorem 3.5*from Lemma X 4.3 (i.e., from Corollary X 4.2).

Exercise 3.9. Using an analogue of the Remark 2 following LemmaX 4.3 and the result of Exercise 3.8, prove Theorem 3.5.

Theorem 3.6. Let the condition (3.18), (3.19) of Theorem 3.5 be replacedby the assumption that there exists a non-negative, non-decreasing, continuous

function <p(r)for small r 0 such that

and that (3.26) /zo/<fa. Then the constants c, 00 in (3.21), f/ie constant c± in(3.22) HI 6of/i (ii) and (iii), am/ f/ie constants cx, ca /'» (3.24) uniquely deter-mine the solution z(i). (In particular, in case (iv), z = 0 w a proper node.)

Exercise 3.10. (a) Deduce the assertions concerning (3.21) and (3.24)from Theorem X 1.1 (i.e., from variants of Corollary X 1.2). (b) Deducethe assertion concerning (3.22) in both parts (ii), (iii) from Exercise 3.6.

In the case of a multiple elementary divisor for E, the condition (3.19)of Theorem 3.5 has to be strengthened to

This condition is also satisfied if (3.20) holds.Theorem 3.7. Let F(z) be continuous for small \\z\\ and satisfy (3.18),

(3.28). Let E be as in (3.3-5) with A < 0 am/ = 1. Let z(t) be a solutionof (3.4) with small |z(0)|| j£ 0. Then z(t) exists for t 0 and either thereexists a constant cly^Q such that

or there exists a constant c27* Q such that

conversely, if cl 0 and czj£Q are given, then there exist solutions z(t)satisfying (3.29) and (3.30), respectively.

Exercise 3.11. Deduce Theorem 3.7 from Corollary X 4.1. In order toobtain the assertions concerning (3.29) [or (3.30)], make the change ofdependent variables 2 = (x, y) -»• (u, v) defined by x — euu,y = eut(u + v)[or x = eMu]t, y = eu(u -f v)] and the change of independent variable/ = es. This deduction is more straightforward if the condition (3.18),(3.28) is strengthened to (3.25),

Otherwise, the necessary arguments involve the analogue of Remark 2following Lemma X 4.2; see also Corollary X 16.3 and Exercise X 16.2.

Theorem 3.8. Let the condition (3.18), (3.28) of Theorem 3.7 be replacedby the assumption that there exists a non-negative, non-decreasing con-tinuous function <p(r) for small r ^ 0 satisfying (3.27), (3.31). Then theconstant cz in (3.30) uniquely determines the solution z(t).

Exercise 3.12. Deduce Theorem 3.8 from the changes of variables inExercise 3.11 and from Theorem X 8.2.


4. More General Stationary Point

A discussion similar to that of the last section will be given for a planeautonomous system of the form

where P, Q are homogeneous polynomials of degree m 1 and

In terms of polar coordinates, x = r cos 6 and y = r sin 6, define

thus /?, 5" are homogeneous polynomials of sin 6, cos 6 of degree m + 1.In terms of polar coordinates, (4.1) can be written as

where

tend to 0 as r —*• 0 uniformly in 0.If 5(0) 0 and R(d) & 0, then a necessary condition for 0 = 00 to

be characteristic is that S(60) = 0 and a sufficient condition is that5(00) = 0, R(60) 7* 0; cf., e.g., Exercise 2.1. If 5(0) =£ 0, it has only afinite number of zeros (mod 1-n). Theorem 2.1 implies the following:

Theorem 4.1. Assume (4.2), (4.3) and S(0) & 0. If (x(t), y(t)) is asolution of (4.1) for large t > 0 [or — t > 0] satisfying

then a continuous determination of 0(0 = arc tan y(t)jx(t) satisfies either

and 5"(00) = 0 or

The question to be considered first is the following: If 5(00) = 0, dothere exist solutions of (4.1) satisfying (4.8), (4.9)? After a rotation ofthe (x, y)-plane, it can be supposed that 00 = 0. Suppose 0 = 0 is a zeroof degree k > 0 for 5(0),

In order to state the next result, introduce the sector

Theorem 4.2. Assume (4.2), (4.3), (4.11) and that k is an odd integer.Then if d, r\ > 0 are sufficiently small, (4.1) has a half-trajectory F in£i0(<5, rj) with endpoint on r — 6. For any such half-trajectory, (4.8) and(4.9) hold with 00 = 0. If, in addition, R(G) 9* 0, then F is defined for larget > 0 or — t > 0 according as R(Q) $ 0.

Exercise 4.1. (a) Using Theorem 2.2 (and Exercise 2.3), obtainsufficient conditions to assure that F in Theorem 4.2 is unique (up to thereplacement of t by t + const.), (b) In addition to the conditions ofTheorem 4.2, assume that jR(0) 0. Using Exercise 2.4, applied to (4.6)rather than (4.1), deduce sufficient conditions for the uniqueness of F.For example, show that if

satisfies

and small r > 0, where y>(r) > 0 is continuous and satisfies

Proof. If 0 < r d and 0 = ±??, where d, YJ > 0 are sufficientlysmall, then, by (4.11), c00' ^ 0. Thus, if c0> 0, then the lateral boundaries0 = ±»7 of Q0 are strict egress points and Corollary 1.1 is applicable. IfCQ < 0, this corollary becomes applicable if t is replaced by —t. Also,(4.3) implies that Corollary 1.2 can be used. This gives the existence ofT : («(0»y(0) satisfying (4.8). Also, (4.9) follows from Theorem 4.1since r) > 0 can be taken so small that 0 = 0 is the only characteristicdirection in |0| :£ 751. This gives the first part of Theorem 4.2. The secondpart of Theorem 4.2 is much simpler since Corollary L3 (or even Cor-ollary 1.5) can be used in place of Corollary 1.1.

In order to obtain refinements of Theorem 4.2 and to deal with the casethat k > 0 is even, suppose that

Theorem 4.3. Assume (4.2), (4.3), (4.11) with k > 0 an even integerand R(Q) ^ 0 [i.e., (4.13) with j = OJ. Then, if d, r, > 0 are sufficientlysmall, (4.1) has no or infinitely many half-trajectories F in Q0(<5,»?). Forany such half-trajectory, (4.8) and (4.9) hold with 00 = 0.

The first part of this assertion follows from Exercise 1.2; the secondpart from Theorem 4.1. See Theorem 4.5 and Exercises 4.6, 4.7 forcriteria for the alternatives in Theorem 4.3.

8, then R is unique

Wheny = 0 [so that R(0) = d0 7* 0], k is odd, and

then (4.1) has infinitely many half-trajectories F satisfying (4.8) and (4.9)with 00 = 0; cf. Corollary 1.3(ii). In the next theorem, there will be noassumption on the parity ofy", k; instead, it will be supposed that

[It can be mentioned that if bothy, k are even, then condition (4.14) in-volves no loss of generality. For the substitution 0 -»• — 6 (i.e., y -»• — y)changes the sign of c0 if k is even (cf. (4.6)) but not that of d0 if y is even.]Whenever (4.14) holds, it can also be supposed that

otherwise t is replaced by —t.Exercise 4.2. Show that there are examples of P, Q with c^ < 0,

k > 0 even,y > 0 odd, k > j + 1 such that no condition of smallness onp, q assures that (4.1) has a half-trajectory (z(0, y(0) satisfying (4.8).

Theorem 4.4. Assume (4.2), (4.3), (4.11), (4.13), (4.15), (4.16). Thenthere exists a positive 0 = e0(c0, d0,j, k) such that if

holds for small r > 0 [e.g., ifp, q = o(rm+e) as r -*• -fO/or some € > 0],then (4.1) possesses infinitely many half-trajectories defined for t^Qsatisfying (4.8) and (4.9) with 00 = 0.

In order to prove this, introduce the following notation: Let d, r) > 0;ex and dl are positive constants satisfying

Let y>i(r), v»2(r) be positive continuous functions for 0 < r d such thatthe functions (4.7) satisfy

and that

It follows from (4.6) that if r\ > 0 is small and <5 = d(rf) > 0 is sufficientlysmall, then

)

Let QI be the set

where dz is any fixed constant, 0 < dz < d^\ see Figure 7. Then (4.6)implies that, on Ol5

Thus, along a solution of (4.1) in Ql5 / < 0 and it is permissible to

Figure 7.

introduce r as as independent variable, so that

In addition, by (4.21) and (4.23),

Theorem 4.4 will be deduced from the following:Lemma 4.1. If there exists a continuously differentiable function

6 = 00(r), 0 < r ^ <5, satisfying the differential inequality

and dzBj s£ y>2(r), then (4.1) /KM infinitely many half-trajectories definedfort^O satisfying (4.8) and (4.9) w/7/i 00 = 0.

Proof of the Lemma. If the new variable

is introduced, (4.28) becomes

where

The relation (4.30) implies that 00(r) -> 0 as r -> 0. For 00(r) 0 isincreasing and if 00, hence v, has a positive limit as r -*• +0, then, by(4.30), r dvjdr ^ const. > 0 for small r > 0. Since this leads to the con-tradiction v(f) < v(rj) — const, log (rjfr) -*• — oo as r -> +0, it followsthat 00(r) -* 0 as r -* +0.

Let r0 > 0 be so small that r0 < <5, 00(r0) < r). Let 0 = 0(r) be asolution of (4.25) satisfying an initial condition r\ > 0(r0) > 00(r0); seeFigure 7. By (4.27), 0(r) < ?y on any interval [rlt rc), ra > 0, on which0(r) exists. Since (4.26) holds as long as (#, y) = (r cos 0(r), r sin 0(r)) is innx and since 00(r) satisfies (4.28), Theorem III 4.1 implies that 00(r) <0(r) < r} on any interval [rlt r0), r± > 0, on which 0(r) exists and thecorresponding point (x, y) e Qj. Since dzd0

j(r) ^ y>8(r), the solution 0(r)can be defined on (0, r0] and the corresponding point (x, y) e Qj. This.implies the lemma.

Proof of Theorem 4.4. In view of (4.17) and (4.19), it is possible tochoose

A

For a constant « > 0 to be specified, put

so that

The inequality (4.28) or (4.30) is equivalent to

Since X > 1, it is clear that if 0 > 0 is sufficiently small, it is possible tochoose an c > 0 satisfying this inequality. Finally, the condition

«WW ^ ViM becomes, by (4.29),

But if </2 > 0 and e > 0, this holds for small r since A; > j + 1 > j. ThusTheorem 4.4 follows from Lemma 4.1.

When j — 0 in Theorem 4.4, the result can be sharpened somewhat.Theorem 4.5. Assume (4.2), (4.3), (4.11) with c0 < 0, /?(0) = d0 < 0,

fl/w/ A: > 0 ere«. Put

(i) Le?/ 0 < 0 < e*, <5 > 0, vj > 0. Suppose that a(r, 0) in (4.7) satisfies

Then (4.1) tow infinitely many half-trajectories defined for t^Q satisfying(4.8) and (4.9) w/rA 00 = 0- (») ief c* < «°, <5 > 0, »y > 0. Suppose that

7Ae« wo half-trajectory satisfies (4.8) #«</ (4.9) w/7A 00 = 0.The proof of (i) is similar to that of Theorem 4.4. Let 0 < — c0 < clt

0 < dl < — d0, yj(r) be defined as in (4.19) and

It is clear from (4.25) that if «5, ?/ > 0 are sufficiently small, then (x, y) E Qx

implies that

and (4.27). The required analogue of Lemma 4.1 is the following:Lemma 4.2. Let c = cjdlt y(r) = ViWM- #" re ^^w/j a co«-

tinuously differentiate function 00(r) > 0, 0 < r ^ ?; satisfying

//ten /Ae conclusion of(i) in Theorem 4.5 Ao/cfr.Exercise 4.3. (a) Prove Lemma.4.2. (6) Deduce Theorem 4.5(i) from

it.For the proof of Theorem 4.5 (ii), let 0 < cx < —c0, —d0 < </x, and

Then if d, r) are sufficiently small,

Lemma 4.3. Let c = cjd^ y<r) = yVWi- If, for every small rj > 0and r0 > 0, the solution of

satisfies 6(r^ — —r\ at some r^ 0 < rt < r0, //ze/j //ie conclusion of Theorem4.5(ii) Ao/rfs.

Exercise 4.4. (0) Prove Lemma 4.3. (£) Deduce Theorem 4.5(ii) fromit.

Exercise 4.5. Apply Theorem 4.5 to the case m = 1, P(x,y) = A#,(?(#, y) = a; + Ay, and A < 0 [so that (4.1) is the system considered inTheorem 3.3].

Exercise 4.6. The proof of Theorem 4.5(i) makes it clear that ify(r) 0 is any continuous function for 0 r < 77, y(0) = 0, and if

c > 0, has a solution 00(r) > 0 for 0 < r »?, then we can obtainanalogues of Theorem 4.5(i) by replacing (4.33) by — a(r, 6) ^ ey>(r) for asuitable > 0. This exercise deals with conditions on \p j> 0 to assurethat (4.39) has positive solutions for 0 < r ??. Introduce the newindependent variable t defined by r = e~t/c, so that dr/r = — dt/c andr = 0 corresponds to t = oo. Writing g?(f) = if>(e~t{c)lc and A = &transforms (4.39) into

^ > 1, and q?(/) 0 is continuous for large /. The problem is to findconditions on continuous <p(t) ^ 0 to assure that (4.40) has a positivesolution for large f; cf. § XI 7 for the case A = 2. For brevity, a function<p(t) ^ 0 continuous for large / for which (4.40) has a positive solutionfor large t will be called of class N^.

fco(a) Show that if y(/) is of class N^, then 9?(f) dt < oo. (b) Show

that <p(t) e NI if and only if there exists a continuously differentiablepositive function 6 — 00(t) for large t such that

hence if 9?0(/) e N* and 0 q>(t) ^ (p0(t), then <p(0 e VA. (c) If // =A/(A - 1), e* = max (u - MA)/(A - l)*^-" for w > 0, and 0 g>(/) <

e*//*, then ??(/) 6 N^. (d) If <p(0 dt < oo and

is max [M - (A - !)«*] for u > 0, then <p(t}eN^. (e) If f "V'^Vfr)</f < oo, then 93(0 e N

Exercise 4.7. Formulate analogues of Theorem 4.5 (i) using parts (d)and (e) of the last exercise.

Notes

SECTION 1. Theorems 1.1 and 1.2 may be new and are suggested by the result ofHartman and Wintner [1] refining a paper of Perron [3]. The results of this sectionhave the advantage of permitting the treatment, e.g., of some cases of (4.6) whenR(6), S(6) have a common zero.

SECTION 2. The main result (Theorem 2.1) of this section goes back to Bendixson [2]under conditions of analyticity. The treatment in the text follows that of Nemytskiiand Stepanov [1]; cf. Hartman and Wintner [11] and Kowalski [1]. For resultsrelated to Theorem 2.2, see Hoheisel [1], Hartman and Wintner [1], Hartman [1], andKeil[l].

SECTION 3. For early references on the subject of this section, see the encyclopediaarticles of Painlev6 [1] and Liebmann [1]; see also Dulac [1], [2]. Investigations on thequestions considered here were begun by Briot and Bouquet [1] for equations of theform x dyfdx = ax + by + ... which, because of § 2, contain most of the cases of(3.4) when the eigenvalues of E are real. Poincare [1 ] initiated the discussion of solutionsof (3.4), under conditions of analyticity, when (3.S) holds. Perron [3], [5] was the firstto systematically investigate the questions of § 3 for nonanalytic differential equations.He obtained existence and uniqueness theorems of the type Theorems 3.5-3.8 but withmuch heavier conditions. Weyl [4] obtained existence and uniqueness theorems for thecases of real eigenvalues under conditions similar to those of Theorems 3.6, 3.8; cf.also Hoheisel [1]. Wintner was the first to omit a Lipschitz condition of the typeoccurring in Theorems 3.6 and 3.8 in considering questions of existence. Theorems3.5, 3.7 are due to Wintner [6], [11] (and are based on his papers [3], [8]). This type ofresult has been generalized for nonautonomous systems of arbitrary dimension byHartman and Wintner [19], see § X 13. Examples of the type occurring in Exercise 3.1and 3.2(a) were given by Perron [5,1] and were modified by Hartman and Wintner [11]to obtain all the assertions of Exercises 3.2 and 3.3.

SECTION 4. Many of the papers mentioned in connection with § 3 are relevant herein some cases of m = 1. Most papers in the literature on the problems of this sectioninvoke the hypothesis S(60) = 0, R(d0) ^ 0 (i.e., deal with the cases k > Q.y = 0); see,e.g., Frommer [1], Forster [1], Lonn [2], Grobman and Vinograd [1], and Nemytskiiand Stepanov [1]. The particular uniqueness criterion involving y(r) in Exercise4.1(6) is due to Lonn [1]; for other criteria, see Vinograd and Grobman [1]. Theorem4.4 may be new. Theorem 4.5 is a result of Lonn [2]; for the case k «= 2 correspondingto (4.1) of the type x' = fa + p, y' — x + Ay + q as in Exercise 4.4, the number e* in(4.32) is —</0*/4c0 and part (i), but a less sharp form of part (ii) was given earlier byLonn [1].

y

Chapter IX

Invariant Manifolds and Linearizations

This chapter concerns the behavior of solutions of an autonomous system(of arbitrary dimension) in the vicinity of a simple type of stationarypoint or of a periodic solution. Many results to be obtained will beextended to nonautonomous systems in the next chapter by very differentmethods; cf., e.g., §1X6 and §§X8, 11. The lemmas of this chapter,however, dealing with local maps from one Euclidean space to anotherhave intrinsic interest, give some insight which is not furnished by othermethods, and are applicable to the study of both stationary points andperiodic solutions.

1. Invariant Manifolds

For every (real) t, let T*: £ -> £t be a continuous mapping of a neighbor-hood Dt of f = 0 in a Euclidean £-space into a neighborhood of £ = 0in the same space, with T'(0) = 0. A set S is called invariant with re-spect to the family of maps {Tf} if T\Dtr\ S) c S for all t. A set S iscalled locally invariant with respect to {T*} if there exists an e > 0 suchthat I e S1 implies that r'«£ e S for all /„ for which ||r*||| < e on ther-interval joining 0 and t0.

The problem of the behavior of solutions of a smooth autonomoussystem near a stationary point can, in some cases, be viewed as the com-parison of the solutions of a linear system with constant coefficients

and solutions of a perturbed system

Unless the contrary is stated, it will be supposed that F(£) is of class C1

for small ||£|| and that

228

}

Invariant Manifolds and Linearizations 229

or, equivalently,

where dsF is the Jacobian matrix of F with respect to £.Let |t = rj(t, |0) be the solution of (1.2) satisfying the initial condition

»?(0, |0) = |0. For a fixed /, consider |t = »?(/, |0) as a map r':|0 -* |tof a neighborhood Z>4 of | = 0 in the |-space into a neighborhood ofI = 0 in the same space. The map 7** is defined on the set Dt of points |0for which the solution r)(t, |0) is defined on a ^-interval containing 0 and/. [The maps T* are the germ of a group; cf. (2.2).]

A set 5 in the |-space which is [locally] invariant with respect to thefamily of maps T* will be called [locally] invariant with respect to (1.2).S is invariant [or locally invariant] with respect to (1.2) if and only if ithas the property that |0 e 5 implies that rj(t, |0) e S for all / on the maximalinterval of existence of the solution rj(t, |0) [or for some e > 0, r)(t0, |0) e Swhenever \\rj(t, |0)|| < e on the ^-interval joining 0 and t0].

If 5 is an invariant set, then the intersection of S and a sphere ||||| < eis locally invariant. Conversely, if S is a locally invariant set, then 50 =\JT\S n Dt) is an invariant set. Thus the investigation of invariant setscan be reduced to the study of locally invariant sets and vice versa. This isconvenient by virtue of the following remark: If F(|) is altered outside of asmall sphere, ||£|| < e, and an invariant set SQ is determined for thenew differential equation, then the intersection of S0 and the sphere||III < e is a locally invariant set for the original differential equation (1.2).

Locally invariant sets are convenient for another reason. The condi-tions imposed on Fare of "local" nature and it is not reasonable to expectthat invariant sets, involving notions "in the large," should be simple sets.For example, suppose that dim 1 = 2 and that (1.2) has a pair of solutions£ = fi(0» £i(0 for - oo < / < c», such that £x(f), £2(f) -*• 0 as t -*• ± ooas in Figure 1. Then the set S0 consisting of I = 0 and the points I =fX0» — oo < f < oo and j = 1,2, is an invariant set 50. S0 is a curvewith a self-intersection. But each of the sets Si = (I = (I1, 0), II1] < e}or 5, = (I = (0, |2), ||2| < e}, for a sufficiently small e > 0, is a locallyinvariant set and is a C^-arc.

After a linear change of variables, with a constant matrix N,

the equation (1.2) becomes

Suppose that N is chosen so that

where P is a d x d and Q an e x e matrix with eigenvalues plt..., pdand qlt..., qe, respectively, where d > 0, e 0 and

Write £ = (y, z), where y is a ^-dimensional vector, z an e-dimensionalvector and (N^EN)!, = (.Py, £z). Thus, in ^-coordinates, the lineaiequation (1.1) becomes

The solution (y(t), z(tj) of (1.9) with an initial point (y(0), z(0)) = (y(Q), 0)satisfies z(i) = 0 and \\y(t)\\ ^ const. /^ const. e(<x+f)t for some integer

Figure 1.

j and for arbitrary e > 0 and large /; cf. § IV 5. In addition, if (y(t), z(t))is a solution of (1.9) for which \\(y(t), z(t))\\ ^ elft~tV for some e > 0 andlarge / §: 0, then z(t) = 0. Thus the J-dimensional flat z = 0 in the£ = (y, z)-space is invariant with respect to (1.9) and is made up of allsolutions (y(t), z(t)) satisfying \\(y(t), z(0)ll ^ e(fi~e)t for some c > 0 andlarge /.

The first question concerning (1.2) to be considered is whether or notan analogous situation holds for (1.2). More precisely, for the system(1.2) of (1.6) written as

where T7^ F2 are of class C1 for small ||y||, ||z||,

is there a af-dimensional locally invariant manifold 5 of the form S:z =^(y) defined for small ||y|| which is made up of all solutions (y(t), z(tj) of

(1.10) in a neighborhood of (y, z) = 0 for large / satisfying ||(3/(/), z(/))ll ê(P-fK for some «• > 0. It will be shown in § 6 that the answer is in theaffirmative.

2. The Maps T*

(i) Consider the unique solution £ = rj(t, |0) of the initial value problem

Since the solution ??(/, 0) = 0 for £0 = 0 exists for all /, T?(/, £0) exists on anarbitrarily large interval \t\ /0 if |fj is sufficiently small; cf. TheoremV2.1.

For a fixed t, consider £( = ??(/, £0) as a map r':£0-> £t from the £-space into itself. The set of maps J1* behaves like an Abelian group in thesense that if ||fj is so small that £t = rj(t, f0) is defined on a /-intervalcontaining t = 0, tlt tz, and tl + t2, then

for the given ^0, i.e., rjfa + t2, 0) = r)(tlt r)(tz, f0)) since the solution of(2.1) is unique.

(ii) Consider a change of variables R: £ = Z0(f) which together with itsinverse R*1:!; = X0(£) is of class C1. Then (2.1) becomes of the form

where G(0 = o(||{||) as C-^0 and N is the Jacobian matrix N =(3Ar

0/90{=o — ^cô(0)- In general, G(£) is not of class C1. Solutions£< = £(t, Co) °f (2-3) are, of course, unique since solutions of (2.1) areunique and the map R is one-to-one. The map Co -+ Co is given by

This can be seen by considering the action of RPR*1; thus R'1^ is apoint £0 = A^Co), F'/J-ô is the solution |t = ri(t, £0) of (2.1) for fixed£0 and (r'/r1^) is therefore the solution ^ = C(A Co) of (2.3) for fixedCo-

(iii) By Theorem V 3.1, rj(t, £0) is of class C1 and its Jacobian -H(t, ^0) =d^ with respect to f0 satisfies the linear initial value problem

In particular, if f0 = 0,

thus

Therefore, the expansion for rj(t, £0), for a fixed t, in terms of linear termsin £0 and higher order terms is of the form

where

3. Modification of f(g)

In order to avoid technical difficulties (as, e.g., the fact that the domainDt of the map r':^0 -> £t depends on t), it will be convenient to replaceF(£) in (2.1) by a function which is defined for all £, is identical with F(£)for small ||£||, say, for ||£|| < $s, and vanishes for ||£|| ^ J > 0. If thenew function is called .F(£) again, then the solution £ = rj(t, £0) of (2.1)is defined for all £. Thus, for every f, the domain of T* :£„-»• £t *s tne

entire £0-space and the set of maps T* is indeed a group.Lemma 3.1. Let F(£) be a vector function of class C1 for small ||£||

satisfying F(G) = 0, d^F(0) = 0. Let 0 > Q be arbitrary. Then thereexists a number s = s(0) > 0 (which tends to 0 with 0) and a function G(£)of class C1 defined for all £ satisfying G (£) = F(£)/or ||£|| & (?(£) = 0for |1 £|| > j, a«^ ||a{G|| < 0/or a// £.

In this lemma, F and £ need not be of the same dimension. Here andin the remainder of this chapter, the norm \\A\\ of a rectangular matrix Ais the norm of A as a linear operator from one Euclidean space intoanother, i.e., the least constant c such that \\Ay\\ ^ c \\y\\ for all y.

Proof. Let s >0 be so small that | F(£)|| ^ 0/8, in particularll^(£)ll ^ 0 II £11/8, for || £|| ^ 5. Let <p(0 be a smooth real-valued functionof / for t 0 such that <p(t) = 1 for t (^)2, 0 < <p(0 < 1 for (i$)2 <f < s2, <f(t) = 0 for / > sz and 0 < -dyjdt ^ 2/52 for all t 0. PutG(£) = /'(£M||£||2) or G(£) = 0 according as ||£|| or > s. ThendfG = Ofor ||£|| > s. For ||£|| j, d^G = (dFW)? + 2(F^) dyldt&ndso ||afG|| ^ (0/8) + 2(0 ||£||2/8)(2/j2) < 6, This proves the lemma.

Thus, in dealing with solutions of (2.1) only in a small neighborhoodof £ = 0, Lemma 3.1 shows that there is no loss of generality in supposingthat F(£) is of class C1 for all £,

for all £, and

where s = s(6).It will now be verified that there exist s0 = s0(s, 0) > 0, 00 = 00(s, 0)

such that s0, 00 -*• 0 as s, 0 -> 0 and that if the solution £ = r)(t, £0) of

(2.1) is written as (2.8), then

In order to see this, note that (3.1) implies that ||F(f)|| ^ 6 ||f ||, thus asolution of (2.1) satisfies ||f'|| ^ c0 ||f|| for c0 = ||£|| -f 0. Hence thesolution f = £(f) of (2.1) satisfies If (OH ^ II foil exp(-c00; cf. LemmaIV 4.1. Thus, if ||fol|^Jo» where J0-Jexpc0, then ||f(0|| ^ J for0 ^ / ^ 1. In this case, (2.1) reduces to f = Ef, f(0) = f0 for 0 / ^ 1and the solution f(/) is e£tf0; i.e., in (2.8), E(t, f0) = 0 for 0 < / 1 andll£oll ^ *o-

The relation S(f, f0) = ^(f, f0) - em^ implies that dfE(t, f0) =//(?, fo) ~ eEt or

aloS(f, f0) = em[K(t, f0) - /], where K(t, f0) = e-E<H(r, f0).

The matrix K(t, f0) has the derivative K' = e~m(H' - EH) or, by (2.5),

K'(t, f0) = e-£t dsF(r))emK(t, f0), X(0, f0) = /.Since \\e~Et dsF(ri)em\\ has the bound cx0 for 0 S r 1, where ^ =(el|£|1)2, it follows from Lemma IV 4.1 that \\K(t, f)|| has a bound ofthe form exp c$ for 0 <£ / < 1. Hence ||A"|| ^ (c^expc^, and so||A:(r, f0) - / || < fafl) exp c^ for 0 f ^ 1. Consequently,

i.e., (3.4) holds with 60 = e11*11^^) exp cj.

4. Normalizations

After a linear change of variables, f = #£> (2.1) can be written as(4.1) y' = Py + F&, z\ z' = Qz + F2(y, z) and y(0) = y0, z(G) = 20,

where N~%EN — diag [P, Q], It is supposed that the eigenvalues pft q*ofP, Q satisfy

The eigenvalues of the nonsingular matrices

are e*', ««*, respectively, where 0 < \e*'\ ^ e" < 1, |e«*| ^ t? > e". Thusif « > 0 is arbitrary, there exist real nonsingular matrices Nlt Nt such thatN^ANt = txp(N^lPNl) has a norm ^ e"+e and that N^1C~1N^ =exp (N%lC~lNi) has a norm < e~^+e. This follows by considering the"real" analogue of the Jordan normal forms in which the usual 1 on thesubdiagonal is replaced by an arbitrarily small e; cf. § IV 9.

Since diag [A, C] = exp diag [P, Q], it can be supposed that N is re-placed by the product N diag [Nly N2], so that

It will be supposed that e > 0 is so small that

satisfy

It will also be supposed that Flt Fz are of class C1 and

Correspondingly, the general solution of (4.1) defines, for fixed t, a mapT* from (y0,

zo) to the point (y, 2) = (yt, zt) such that J* is of the form

where

for all t,

and 0 / ^ 1. In (4.12)-(4.13), 00, s0 depend on (0, j) in such a way that0o> •sro~>0 as ^> J—»«0. Finally, the set of maps T1 form a group:j'<1+<2 — y^T'2.

5. Invariant Manifolds of a Map

One of the basic results to be proved concerns one map T: (y0, z0) -»•(^i. 2i) rather than a group of maps T1. In the application of this result,r= r1.

Lemma 5.1. Let A be a d x d matrix, C an e x e nonsingular matrixsuch that (4.5), (4.6) hold. Let T: (y0, ZQ)-* (yly zj be a map of the form

wAere 7, Z are of class Clfor small ||y0||, ||20|| and satisfy (4.11). Then there

vanish at for

for all and

for

and the Jacobian matrices vanish at

for all

and

exists an e-dimensional vector function z — g(y) of class C1 for small \\y\\such that

and that the maps

transform T into the form

where(5.5) U, V and their Jacobian matrices vanish at (MO, v0) = 0

and(5.6) F(Wo,0) = 0.

Condition (5.6) means that the set of points («0, y0) near the origin onthe flat y0 = 0 is invariant under the map (5.4); i.e., the manifold z = g(y)is locally invariant for (5.1). In applications of Lemma 5.1, the followingtwo remarks will often be used. Remark 2 will be used in §§ 8-9.

Remark 1. In view of (4.11) and Lemma 2.1, it can be supposed thatY, Z are of class Cl for all (y0, z0) and satisfy (4.12)-(4.13), where 00, s0 arearbitrarily small positive numbers. Let 00 satisfy

It will be shown, in this case, that g(y) can be defined for all y [so thatRTR~l is defined for all («„, i>0)J and (5.6) holds for all MO. Furthermore,there is a constant a = <r(00) such that

and a -*• 0 as 00 -> 0.Remark 2. If, in addition, it is assumed that c > 1, then g(y) —*• 0 as

llyll - oo.Lemma 5.1 will now be proved by the method of successive approxi-

mations* Another proof will be given in Exercises 5.3 and 5.4 at the endof this section.

Proof of Lemma 5.1 and Remark 2. Assume for a moment that R [i.e.,g(y)] is known. Then (5.1), (5.3) show that

*See also Appendix, page 271.

Hence, by (5.4)

and (5.6) holds if and only if

Hence, it must be shown that the functional equation (5.11) for g(ii) hasa solution of class C1 satisfying (5.2)-(5.8).

The equation (5.11) will be solved by successive approximations. Let

and, if £n_i(w) has been defined, put

Below let .! = (ifl, g°n_v = g^Au + Y°), where Y° = Y(u, (w)),Z° = Z(M, gn_i(«))- It is clear that g0, glt... are defined and of classC1 for all M. In addition, if dgn is the Jacobian matrix of gn, then

where, e.g., dyY° = dvY(y,z) at (y,2) = fag^u)).Define the number a by

by (5.7). It will be shown by induction that

It is clear that (5.16) holds for n — 0. Assume that (5.16) holds when n isreplaced by n — 1. Then, by (5.14) and a < 1,

Since ^[orCa + 300) + 00] = <*» (5.16) follows and the induction iscomplete.

It will now be verified that dg0, dglt... are equicontinuous. For anyfunction /=/(«) or/ = /(y,z), let A/=/(M + AM) -/(«) or A/ =/Of + Ay, z + Ac) -/(y, 2). Put

where dVit Y, Z means any of the four Jacobian matrices dyY, BgY, d^Z,d,Z of 7(y, z), Z(y, z). It will be shown by induction that

where

It is clear that (5.18) holds for n = 0. Assume its validity if n is replacedby n - 1. Note that, by (5.16),

hence

where the last two inequalities follow from (4.12) and (5.7). Using theanalogue of A[/i(w)/2(i/)] =f^u + AM) A/2 + (A/i)/g(«) and <r < 1, itfollows from (5.14) that if ||Aw|| ^ <5 < 1, then

where the right side is /j(<$) by (5.19).Next, it will be shown that the sequence g0, gi,... converges uniformly

on every bounded w-set. This is true if there exist constants M, r such that0 < r < 1 and for/i = 1,2,. . . ,

This inequality holds for n = 1, if M and r are chosen subject to Mr = a.Assume the validity of (5.22) when n is replaced by n — 1. By (5.13),c !(£„(") - £«-i(")ll is at most

The first term is majorized by

Hence c \\gn(u) - ^n_!(w)|| is not greater than

which is at most Mr11-1 ||u|| (fl + 400). Thus, if r =^ (a + 400)/c andM = <r/r, then (5.22) holds and r < 1 by (5.7).

Consequently, g(w) = lim£B(M) exists uniformly on every boundedM-set. In view of (5.13), this limit function g(u) satisfies the functionalequation (5.11). Finally, since the sequence dg0, dglt is uniformlybounded and equicontinuous, there is a subsequence which is uniformly

convergent on every bounded w-set. It follows that g(u) is of class C1.This completes the proof.

Proof of Remark 2. Let M = max || («)|| for \\u\\ <> s0. By (4.13) and(5.11),£(«) = C-*g(AU)X\\u\\ ^ s0andg(u) = C-»g(Anu)if \\A»-*ul\ > SQ.Thus if \\An-lu\\ ^ s9 but \\Anu\\ < s0, then || («)|| Mc~n. This impliesRemark 2 since c > 1 and, for large ||M|(, there exixts a unique integern = n(u) satisfying M"-1!*!! ^ s0 > \\Anu\\ and «(«) -»• oo as ||w|| -* oo.

Exercise 5.1. (a) This part of the exercise concerns variants of Lemma5.1 under various smoothness assumptions on Y, Z, Instead of the assump-tion that Y, Z is of class C1 and satisfies (4.11)-(4.12), suppose thatYy Z satisfies one of the following hypotheses: (i) Y = 0, Z = 0 at(y, 2) — 0 and Y, Z is uniformly Lipschitz continuous with an arbitrarilysmall Lipschitz constant for small ||y||, |J2|| [i.e., if c > 0 is arbitrary, then||A7|| + ||AZ|| ^ e(|Ay|| + ||A2||) for sufficiently small (\\y\\,<\\y + Ay||,||21|, 12 + A2||)]; (ii) F, Zis of class Cm, 1 m oo, and satisfies (4.11);(iii) Y, Z satisfies (ii) with 1 m < oo and its partial derivatives of orderm have a degree of continuity majorized by a (constant times a) monotone,non-negative function hm(8) -> 0, <5->- + 0; (iv) Y, Z are analytic andsatisfy (4.11). Then the analogue of Lemma 5.1 holds with a &(y)having the corresponding property (i), (ii), (iii) or (iv), instead of being ofclass C1.

(b) Verify that if Flt F% in (4.1) have the analogues of property (i), (ii),(iii) or (iv), then 7= Y(t,yQ,z0\ Z = Z(f,y0,20) in (4.10) have thecorresponding property (i), (ii), (iii) or (iv) with respect (y0, z0) uniformlyf o r O ^ / ^ 1.

Exercise 5.2. Show that the restriction a < 1 in Lemma 5.1 is notneeded. [Note that the condition a + 200 < 1 was used in the proofonly in connection with (5.20), (5.2l)i]

Corollary 5.1. Let T, g(y), 00 be as in Lemma 5.1 and Remark 1 followingit. For a given (y0,20), put (ylf 2j) = T(y0,20), (y2, 22) = T(ylt zj,Then, on the one handy z0 = g(yQ) implies that \\(yn, zn)\\ — O((a + 00)

n)as n -*• oo (in fact, if (4.2), (4.3) hold and y0 ^ 0, then yn T£ 0 for all n,\\zn\\l\\yn\\ ~+ 0 and lim sup n~* log ||(yn, 2B)|| < a as n -+ oo); and, on theother hand, z0 5* g(y0) implies that (c — 200)

B = O(\\(yn, 2n)||) as n -> oo.Remark 3. If c > 1 (so that a < 1 < c) then the manifold z — g(y)

in a neighborhood of (y, 2) = (0, 0) can be described as the set of points(y0, «o) such that (yn, zn) = Tn(y0,20) satisfy ||(yw, 2rt)|| ->0 exponentiallyas n -»• oo and/or ||(yn, 2B)|| -*• 0 as n -*• oo and/or (yn, zn) remains in aneighborhood of (0,0) for w = 0, 1, 2,.... In the case a < 1 < c, themanifold 2 — g(y) is called the stable manifold of (5.1) as n -> oo; the

corresponding manifold for n -»• — oo is called the unstable manifold of(5-1).

Proof of Corollary 5.1. Note that z0 = g(t/0) is equivalent to y0 = 0.In this case, VQ = vl — • • - = 0 by (5.4), (5.6). Correspondingly,«„ = Aun-i + tf(«_i, 0), so that ||ttj ^ (a + 00) IK-i!l and K|| (a + 0o)n II"oil -**0 as « -* oo. Thus, if e > 0 is arbitrary, there is anN = N( such that ||wj| ^ (a + e) Hw^J for w > JV and ||i/B+A-il ^(a + *)n IIM for n 0. Since yn = «„, zn = g(un) = O(||MB||) as n -^ oo,it is seen that \\(yn, 2B)|| < (1 + a) \\un\\ and the first assertion follows. Alsolim sup n-1 log \\(yn, ZB)|| ^ log a. Since a linear transformation of the^-variables can bring log a arbitrarily near to a, it follows that lim sup n~l

log||(yn,zn)||<a.By virtue of (5.10), we have the relationdvy(u, v) = d,Z(U, v + g(u» - 3g(Au + Y[u, v + g(u)}) dzY(u, v +g(u)),

so that \\dvV\\ <B0 + a60 ^ 200, and by (5.6), || V(u, v)\\ <> 200\\v\\- Thus,vn = OP-I + n"n-i, »«-i) implies that H»J ^ (c - 200) H^.JI or ||i;B|| (c - 200)« KB. Also ||(yB,2n)|| UK, O|| - \\g(un)\\ > (1 - a) \\(un,vn)\\which implies the last assertion.

Theorem 5.1. In the map T: |0 -+ £1,

/e/ S(^0) be of class Cl for small ||£0|| a«J jfl/i^ S(0) = 0, dl(E(0) = 0;F a constant, nonsingular matrix having d, e0, e eigenvalues of absolutevalue less than 1, equal to 1, greater than 1, respectively, where d, e0, e 0.Then there exists a map Rofa neighborhood o/£0 = 0 onto a neighborhoodof the origin in the Euclidean («0, y0, wQ)-space such that R is of class C1

with a nonvanishing Jacobian, and RTR~l is of the form

where A, B, C is a square d x d, eQ x e0, e X e matrix with eigenvalues ofabsolute value less than 1, equal to 1, greater than 1, respectively, U, V,W and their partial derivatives vanish at the origin', and

The condition (5.25) [or (5.26)] means that the plane v0 — 0, vv0 = 0 ofdimension a? [or w0 = 0, w>0 = 0 of dimension e] is a locally invariantmanifold. When F has no eigenvalues of absolute value 1, so that dimlo = d + e, then the variables w0, wl are absent in (5.24).

Proof. (Details will be left to the reader.) By Lemma 5.1, there is amap R0: £0-+(uQ, yo» ^o) of class C1 with nonvanishing Jacobian suchthat if RoTRo'1 is given by the right side of (5.24), then (5.25) holds. IfLemma 5.1 is applied to (RoTR^1)-1 = RoT^R^1, a new map Rt resultsand the desired map R in Theorem 5.1 is given by R = R^.

The results of this section are applicable to differential equations byvirtue of the arguments used to obtain the following corollary of Lemma5.1 and Corollary 5.1.

Corollary 5.2. Let (4.10) be a group of maps T* of class C1 for all( 0. 2o) satisfying (4.11) and (4.12) for 0 / 1, where P, Q are constantmatrices such that (4.2), (4.3), (4.5), and (4.6) hold, and 00 satisfies (5.7).Let g(y) be the function furnished by Lemma 5.1 and the Remark \ followingit when T = T1. Then RT'Rr1 is of the form

Furthermore, ifyQ^Q and ZQ = g(y^ then zt = g(yt)for all t, yt ^ Of orall t, \\zt\\l\\yt\\ -+ 0 and lim sup rl log ||yt|| ^ a <w t -+ oo; //z0 5* (y0),//*« (c - 200)' . 0(||(y,, 2<)||) of / -^ oo.

If c > 1 > a, a remark similar to that following Corollary 5.1 isapplicable here.

Proof. It will first be verified that if n < / « + 1, then there existpositive constants clt c2 such that

In order to see this, note that T* = T*~nTn. Thus (4.10) and (4.11) for0 / — n < 1 give

and a similar inequality for zt. These inequalities imply (5.29).Let z0 = ^o)> then the behavior of (yw, zn) for large n is described by

Corollary 5.1. The last part of (5.29) gives lim sup rl log \\(yt, zt)\\ ^ aas t -> ao. Suppose, if possible, that zt -^ g(yt) for some t, say / = /0,then (c - 260)« _ O(||(yn+to, zn+to)||) as « -^ oo by Corollary 5.1. But thisis a contradiction. Hence zt = (yt) for all f; i.e., vt = 0 for all f so that(5.28) holds.

Note that if yt = 0 for some /, then zt — g(yt) implies ||2(|| ^ a \\yt\\ =• 0for the same /. But then (yt, zt) = 0 for all / by the group property of T'.The remaining assertions of Corollary 5.2 follow from those of Corollary5.1 and from (5.29).

The following two exercises give another proof of Lemma 5.1 basedon the methods of §§ X 8-10 for nonautonomous differential equations?

*See Appendix, page 271, for simplification and extension of theseexercises.

)

The main part of the proof is Exercise 5.4(6), which leads to a com-paratively simple proof of Lemma 5.1 because it deals with maps of theform Tn = Sn ° 5n_! ° • • • ° Slt where Sk depends on k, rather than withTn = T° • • - ° T, cf. parts (</) and (e) of Exercise 5.4.

Exercise 5.3. (a) For n = 1, 2 , . . . , let £ -*• SJ; be a continuous mapof the | = (f1 , . . . , £*) space R = R* into itself and Tn = Sn ° Sn^Si. Let S be a compact and Klt K2,... closed f-sets such that 5n(R —#n_i) c R - #„, and A"n n rn(S) is not empty for n = 2, 3, Thenthere exists a point £0 e 5 such that rw£0 e #„ for « = 1, 2 , . . . . (6) Let£ be a nonsingular constant d x d matrix, F(£) a continuous vector-valued function for all f such that F(g) = 0 for large ||£||. Then £ -* ££ +F(£) maps the |-space onto itself (i.e., the equation £| + F(£) = r\ has atleast one solution | for every r\ e R*1).

Exercise 5.4. (a) Let y4, C be matrices as in Lemma 5.1. For n = 1,2,. . . , let Yn(y, z), Zn(y, 2) be continuous functions for all (y, z) whichwhich vanish for large ||y|| + ||z||. Let Sn denote the map

Tn the map Tn = Sn ° 5n_i ° • • • ° 5lf and P the projection P(y, 2) = 2 ofthe (y, 2)-space onto the 2-space. For a fixed y0, show that

is a map onto the 2-space. (b) Let 7n, Zn satisfy

where 0 < 4<5 < c — a. Let /T be the cone ||2|| < ||T/|| and, for a fixedy0, 5 the sphere S = {(y, 2): y = y0, ||z|| ^ ||y0||). Show that 5n(R - ^)c R — K and that there exists a z = 2 (n ) such that P ° Tn(y0, z(n)) = 0,hence (y0,2(n))e5 and /C n rw(5) is not empty. Consequently, thereexists a (y0, 20) e S such that (yn, 2n) = Tn(y0, z0) e for n = 0, 1,(c) Show that if (yn, zn) = rn(y0, 20) e K for « = 0, 1,...., then ||zj| ^||yn|| ^ (a + 2«5)» ||y0|| , where a + 26 < \(a + c); but if

for some n (hence for all large «), then, for large n,\\Vn\\ > llzJI ^ (const.)(c - 28)n > 0, where c - 2<5 > K« + c).

(</) In addition to the conditions of parts (a), (b), and (c), assume thatQn= Yn,Zn satisfy

for all y, 2, y*, 2*. In terms of a sequence (yw, ZB) = Tn(y0, z0), introducethe maps

where

so that if (2/B*,2n*)=rn(y0*,Zo*), then 5 * f y n * - yn, z* - zn) -(y*+i — 2/n+n 2*+i — 2M+1XShow that y0 in part (6) uniquely determinesz0; in fact, if (yn, zn), (yn*, zn*) e K for n = 0,1, . . . , then

(e) Assume the conditions of (a), (b), (c), (d) and, in addition, that Yn(y, 2),Zn(y, 2) are of class C1. Let z0 = g(y0) be the unique z0 in (b). Show that#0/o) i& °f class C1 tand that the partial derivatives of g vanish at y0 = 0if the partials of Yn, Zn vanish at (y, 2) = (0,0)]. In fact, let 7£* =•S1** ° s**i °'"° s**> where S** is the linear map

with the Jacobian matrices dVtSYn, Zn evaluated at (yn_!, 2n_1); let e, =(0,. . . , 0, 1, 0,. . . , 0) be the vector with the &th component ef = 0 or6? = 1 according as k 5*7* or k =/ Then («„, O = (dyn(y0)ldy0\dzn(y*)l<hjj\ n = 0. !»•••» exists and is the unique sequence satisfying(ii., iv) = T?(u» »«), «o = «„ and |»J ^ ||«J| for n - 0, !, . . . .(/)Deduce Lemma 5.1 from part (e) with the choice 5X = 5"2 = • • • =s r.

6. Existence of Invariant Manifolds

A consequence of Corollary 5.2 is the following:Theorem 6.1. In the differential equation

let F(£) be of class C1 and F(0) = 0, d(F(0) — 0. Let the constant matrixE possess d (>0) eigenvalues having negative real parts, say, dt eigenvalueswith real parts equal to di, where at.l < • • • < ar < Oanddt + - • • + dr = d,whereas the other eigenvalues, if any, have non-negative real parts. If0 < e < — ar, then (6.1) has solutions $ = £(/) 5* 0 satisfying

and any such solution satisfies

Furthermore, for sufficiently small > 0, the point £ = 0 and the set ofpoints £ on solutions £(t) satisfying Ifm t~l log ||£(0|| a» /"" a fixed1 [or limsupr~1log ||f(f)ll <Q] as t-+ ao constitute a locally invariantC1 manifold St [or Sr] of dimension dl + • • • + di [or dl + •' - + dr = d].

It will be clear that the proof has the following consequence.Corollary 6.1. Let F(£) be of class C1 for small ||{||, F(Q) = 0, and

3 (0) = 0. Then (6.1) has a solution £ = £(/) 0 satisfying (6.2) forsome e > 0 if and only if E has at least one eigenvalue with negative realpart.

For another proof of Theorem 6.1 and a generalization to nonautono-mous systems, see §§ X 8, 11.

Proof. If "lim" is replaced by "lim sup," the last part of Theorem 6.1follows from the normalizations of § 4 and from Corollary 5.2 witha = a,, /? = ai+1 [or a = ar, /? = 0]. This argument also shows that, as/-*oo,

lim inf r1' log ||f(0|| < a,-+1 implies that lim sup r1 log |||(/)|| at,

for / = 1,. . . , r with ar+1 interpreted as 0. Hence lim sup r1 log ||£(OII —<xf for some / implies that lim inf = lim sup.

Remark. We have similar results for the solutions |(f) 9* 0 satisfying||f(Oil e~~f'-> 0 as f-* — oo. This follows by replacing / by the newvariable — t, so that (6.1) becomes

and applying Theorem 6.1 to this equation.The arguments used to obtain Theorem 6.1 and Corollary 5.2 giveTheorem 6.2. Let E, F(|) be as in the last theorem. In addition, let E

have e (>0) eigenvalues with positive real parts. Let f, = £(t, £0) be thesolution of (6.1) satisfying f(0, fo) = fo ana T*tne corresponding map T*:£t = f(/, |0). Let e > 0. Then there exists a map Rofa neighborhood of£ = 0 in the £-space onto a neighborhood of the origin in the Euclidean(u, v, w)-space, where dim u — d, dim v — e, dim u + dim v + dim w =dim f, such that R is of class C1 with a nonvanishing Jacobian and RT'R"1

has the form

U, V, W and their partial derivatives with respect to HO, v0, w0 vanish at(«o» ^o» wo) = 0- Furthermore, V = 0, W = 0 // i?0 = 0, w0 = 0; andU = 0, W = 0 // MO = 0, H>O = 0; finally \\ep\\ < 1, ||e-°|| < 1 fl/i^/ //reeigenvalues ofep° are of absolute value 1.

It can be remarked that the change of variables R:£ -> (u, v^w) trans-forms (6.1) into differential equations of the form

where Flf F2, F3 are o(\\u\\ + H| + ||w||) as (u, v,w)^0 (but Flt F» F3need not be of class C1); Ft = 0, F3 = 0 if v — 0, w = 0; and F^ — 0,Fz = 0 if u = 0, w = 0.

The condition V = 0, JT = 0 when y0 = 0, n>0 = 0 [or U = 0, W = 0when i/0 = 0, u>0 = 0] means that the ^-dimensional plane % = 0, w0 = 0[or e-dimensional plane w0 = 0, vt>0 = 0] are locally invariant manifolds.When £ has no eigenvalues with real part 0, then the variables w, w0 areabsent; in this case, the manifold «0 = 0 [or t>0 = 0] that consists of thesolution arcs which tend to 0 as t —»• oo [or / -* — oo] is called the stable[or unstable] manifold of (6.1) through £ = 0.

7. Linearizations

In the differential equation

suppose that no eigenvalue of E has a vanishing real part. The remarksconcerning (6.5) suggest the question as to whether or not there is a C1

change of variables R: £ ->• £ with nonvanishing Jacobian in a neighborhoodof | = 0 which transforms (7.1) into the linear system

in a neighborhood of £ = 0. In general the answer is in the negative ifdim £ > 2; see Exercises 7.1 and 8.1-8.2. A discussion of this problemis given in the Appendix of this chapter.

Exercise 7.1. Let f, »?, £ be real variables and consider the system ofthree differential equations:

where a > y > 0 and e 5^ 0. Show that there is no map /?: (I,»?, £) ->(w, v, H') of class C1 with nonvanishing Jacobian from a neighborhood of(£, ??, £) = 0 onto a neighborhood of (w, y, w) = 0 transforming the givendifferential equations into the linear system

See Hartman [21].For a topological, rather than a C1, map /?, we have:Theorem 7.1. Suppose that no eigenvalue of E has a vanishing real part

and that f(f) is of class C1 for small ||£||, £(0) = 0,df£(0) « 0. LetTl:£t = r)(t, £0) andV:^ = <?*' £0 &? the general solution of (1.1) and (1.2),respectively. Then there exists a continuous one-to-one map of a neighbor-hood of £ = 0 onto a neighborhood of £ = 0 such that R^R'1 = V\ inparticular, R:g-* £ maps solutions of (7A) near 1 = 0 onto solutions of(7.2) preserving parametrizations.

Thus the topological structure of the set of solutions of (7.1) in aneighborhood of £ — 0 is identical with that of the solutions of (7.2) near£ = 0. This is no longer true if some eigenvalues of E have vanishingreal parts. For in this case, (7.2) has closed solutions paths arbitrarilynear £ = 0, but (7.1) need not have closed solution paths near £ = 0;cf. Exercise VIII 3.1. Theorem 7.1 will be proved in § 9.

8. Linearization of a Map

Instead of the problem of linearizing a group of maps T\ the correspond-ing question involving one map Twill be considered first.

Lemma 8.1. Let A, C be non-singular constant matrices, where A is ad x d matrix, C an e X e matrix, and

Let T:(y0, z0) -> (ylt zx) be a map of the form

where Y, Z are functions of class C1 for small \\y0\\, ||z0|| which vanishtogether with their Jacobian matrices at (y0, z0) = 0. Then there exists acontinuous, one-to-one map

of a neighborhood of(y, z) = 0 onto a neighborhood of(u, v) — 0 such thatR transforms T into the linear map

Remark. In view of Lemma 2.1, it can be supposed that Y, Z are ofclass C1 for all (y0, z0) and satisfy (4.12)-(4.13), where 00, s0 are arbitrarilysmall positive numbers. It will be shown in this case that if 00 > 0 issufficiently small, then R can be chosen so that it is a continuous, one-to-onemap of the (y, z)-space onto the (w, u)-space, and that (8.4) holds for all("o» *><>)•

The following exercises give positive and negative results concerningthe existence of linearizing maps R which are smoother than (8.3) in Lemma8.1; see also the Appendix.

Exercise 8.1. (a) Using the example J: a;1 = ax, yl = ac(y + cxz),z* = cz, where 0 < c < 1 < a, ac > 1, and e > 0, show that if R is anylinearizing map, then R and R~l are not of class C1. See Hartman [21].(h) Let dim y = dim u 2. Let the map T:yl = Ay + Y(y) be of classC2 for smalj ||y||; Y and its first order partials vanish at y = 0; A is a

constant matrix having no eigenvalue of absolute value 0 or 1. Show thatthere exists a map R:y -> u of class C1 of a neighborhood of y = 0 ontoa neighborhood of u — 0 with nonvanishing Jacobian such that RTR~l isthe linear map RTR'1:^ = Au. See Hartman [20, Part 3]. (c) Using theexample T: x1 — czz + y2, yl = cy, where x, y are real variables ande > 0 is fixed, show that R in part (b) cannot always be chosen of class C2

even if T is analytic and \\A\\ < 1. See Sternberg [3, p. 812].Exercise 8.2. Contractions, (a) Let dim y = dim u = d be arbitrary.

Consider a map T:yl = Ay + 7(y), where A is a nonsingular d x dmatrix such that \\A\\ < 1 and Y(y) is of class C2 for small \\y\\ with7(0) = 0, dv 7(0) = 0. Show that there exists a map R:y-+uofa neigh-borhood of y = 0 onto a neighborhood of u = 0 such that /?(0) = 0,R is of class C1 and has a nonvanishing Jacobian, and RTR~* is the linearmap RTR-1: ul = /Iw. See Hartman [20]. [Note, that by Exercise 8.1 (c),there may not exist an R of class C2 even if Y(y) is analytic in y.] (b) Inpart (a), let al5. . . , ad be the eigenvalues of A and suppose that (*)a, 5«£ a ™ » a ™ 2 . . . a™* for any set of non-negative integers (mlt.. ., md)satisfying 1 < 2mfc ^ n, where n is an integer such that n > log |a,|/log|aj for 1 y, A: < d. Let 7(y) be of class Cn. Then /? in part (d) can bechosen of class Cn. Also, if (*) holds for all sets of non-negative integers(/»!,..., /wd) satisfying Swk > 1 and 7(y) is of class C00 [or analytic],then R can be chosen of class C°° [or analytic]. See Sternberg [3] andAppendix to this chapter.

Proof of Lemma 8.1. In order to prove Lemma 8.1, we prove twosimple lemmas

Lemma 8.2. Let B be a nonsingular m * m matrix and letbj = ||5"7|l . Let S:x0-+Xj be a map of the form

where X(xQ) is defined for all XQ and satisfies a Lipschitz condition

where Qjbj < I. Then S is one-to-one and onto the Xj-space. If,in addition, \\X(xQ)\\ < c° for all XQ and

then lUWll ^ blc° foratt *!•Proof. Note that

In order to see that S is one-to-one, it suffices to show that

is one-to-one. But this is clear from 1 - bj 8^ > 0 and

In order to show that S is onto, it suffices to show that B'1 S isonto; i.e., that if x^ is given, then there exists an XQ satisfyingthe equation in (8.8). Note that (8.8 ) can be written as

We show the existence of a solution xn by using successivef\ U

approximations defined by x = 0 and

By (8.7), for n > 2,

A simple induction gives | \xn - xn~l \\ £ (b1 Q^'1 \\ x1 - x° \ \ for

n> 1. Since 0 < 6; < 1, the series Z (xn - x"'1) is convergent; i.e.,

the sequence x ,x ,... has a limit, say XQ, as n+ ». The lastpart of (8.10) shows that the limit x^ satisfies (8.9), hencethe equation in (8.8).

Finally, the last part of the lemma follows from X (Xj) =^(Bxff-xJ =- B^X^Q), This completes the proof.

In order to formulate the next lemma, we introduce somenotation and terminology. Since A in Lemma 8.1 is non-singular, it has an inverse. Let

Let bj, 87, 9 be constants satisfying

If C0 > 0 is a constant, let fl (6j,c0) denote the set of pairsof functions ( Y(yQ,zQ), Z(yQ,zQ)) defined and satisfying, forall f>0,V>


where A7 = Y(yQ + LyQ, ZQ + bzQ) - Y(y0,z0), etc.

It is clear that Lemma 8.1 is contained in the case Y, = 0, Z, = 0(so that U = L) of the following:

Lemma 8.3. Let A,C be as in Lemma 8.1, (Y,Z) and(YyZj) a pair of elements of n(Qj,c0) and

Then there exists a unique continuous map

defined for all (y,z) such that \ (0,0) = 0, Q(O.O) = 0, A, and 0are bounded,

Furthermore, RO is one-to-one and onto the (u,v)-space.Proof (a) It follows from Lemma 8.2 that T has an inverse

defined for all (yj.Zj), say,

and that

(b) The equation (8.19) is equivalent to the pair of equationsAy + Y +\ (Ay + Y, Cz + Z) = A(y +A ) + Yj(y +t^ ,z + Q),Cz + Z + Q(Ay + Y, Cz + Z) =C(z + $} + Zjfy +fi , z + 6 /

where the argument of y,Z,A,0 is (y,z). Using (8.20), the firstof these equations can be written

where A = A(y,z) and A( r ; )=A«r- ; with argument (y,z)Consequently, (8.19) is equivalent to the last two equations whichcan be written

(c) Existence of RQ. We prove the existence * of RQ byshowing the existence of a solution (A , 6 ) of the functionalequations (8.22),(8.23), using successive approximations. These aredefined by A ° = 0, 0 ° = 0 and

for n = 1,2,... . Thus A W ,0 W are defined and continuous for all(y,z). They are also bounded, for it is clear that A°, ©^ and A 7 ,Q1 are bounded and, if we put

where [||...||| = sup j|...|| , then (8.24),(8.25) give

On adding, we get

by the last part of (8.12). Hence, rn < rl Q*1'1 for n = 1,2,...,

so that the seriesZ(?An -A11"1), EC^ - 8P"1) have the convergentmajorant r, Z 6n*7, independent of (y,z). Consequently thelimits A = limAn, 0= lim 011 exist uniformly on the ^z^-space,as n-*- w. These Umits are continuous and bounded. Thedefinitions (8.24),(8.25) imply (8.22),(8.23) which are equivalentto (8.19).

(d) Uniqueness. This follows in the usual way.(e) RO is one-to-one and onto. Denote the unique RO

satisfying (8.19> by RTU, so that RTUT = URTU • If we inter-change the roies of T and U, it follows that there exists a uniqueRUT satisfying the prescribed conditions and RyjU = TRyf. Hence

* The existence of R0\ also follows from the Banach space analogue ofLemma 8.2.


By virtue of uniqueness , RfyRifp = Ryu = I and RffjR'pi/ =R-PP = I, where I is the identity map. Hence both RO = RJ.JJand RJJT are one-to-one and onto. This completes the proof.


In the proof of the theorem, it will be supposed that E has d eigenvalueswith negative real parts and e eigenvalues with positive real parts. (Thecase when d = 0 or e = 0 is easier and is contained in the general case bythe addition of dummy components to £.) After preliminary normaliza-tions, it can be supposed that T* is of the form (4.10) for all (t, y0>zo)>where (4.11) holds, (4.12) and (4.13) hold for 0 / 1, and A = ep,C = eQ satisfy the conditions in Lemma 8.1.

It will also be supposed that 00 is so small that Lemma 8.1 and theRemark following it are applicable to T = P. Denote by R0 the corre-sponding map supplied by Lemma 8.1, so that RQT1RQ1 = L, Put

Then

If s — t is introduced as a new integration variable, say s, the last integralbecomes

In the first of these integrals, the integrand can be written as

since L = R^Rg1. Thus (9.2) becomes

Thus, in order to complete the proof, it is sufficient to verifythat R = RQ. To this end, let U = L in (8.18) and RQ is

given by (8.19) in Lemma 8.3. The definition (9.1) of R impliesthat R has the form prescribed in Lemma 8.3. Hence R = RQ

follows from / = 1 in (9.3) and the uniqueness of RQ.


10. Periodic Solution

Lemmas 5.1, 8.1 will now be applied to the study of solutions in theneighborhood of a periodic solution of an autonomous system:

Lemma 10.1. Let /(£) be of class C1 on an open set containing 1=0.Let £ — ??(r, £„) be the solution o/(l0.1) satisfying ??(0, £0) = |0. Supposethat y(t) — rj(t, 0) is periodic of least period p > 0. [Thus y(t) ^ const.,/(y(/)) 7* 0 and r)(t, £0) exists on an open t-interval containing [0, p] if|| £ 01| is sufficiently small.] Let -n be the hyper plane -n: £ -/(O) = 0 orthogonalto the curve #: f = y(t) at £ = 0. Then there exists a unique (real-valued)function t = T(|O) of class C1 for small ||£0|| such that r(0) = p andq(t, £0) 6 TT when t = T(£O); />.,

Roughly speaking, if a solution starts at £0 near 0, then at a time/ = T(f0) near/?, the solution meets the hyperplane TT; see Figure 2.

Proof of Lemma 10.1. This is an immediate consequence of the implicitfunction theorem. The equation r)(t, £0) -/(O) = 0 is satisfied if t = p,f0 = 0. Also the derivative »?'(/, £0) -/(O) =/(r?(/, f0)) -/(O) at £0 = 0is/(y(0) '/(O). At / =p, it becomes |/(0)|2 0 since y(^) = y(0) = 0. Thisgives the lemma.

If we consider small |||0||, £0 e then

is a map from one neighborhood of f = 0 on TT into another. The meaningof T is clear; the solution f = i?(f, f0) starts for t — 0 at £0 6 ?r and|x = r(£0) is the first point (t > 0) where the solution £ = ??(/, £0)

agam

meets TT. The applicability and consequences of Lemmas 5.1, 8.1 will beconsidered. Roughly, we can expect the following type of results: On theone hand, if the Jacobian matrix of this map at £0 = 0 has a norm lessthan 1, then £ = y(t) is orbitally asymptotically stable (in the sense thatif f° is sufficiently near a point of the curve #: | = y(t), then the solution

£ = ri(t, £°) "tends" to # as / ->• oo, i.e., is in an arbitrarily small neighbor-hood of # for large 0- On the other hand, if dim £ = m and the Jacobianmatrix at £0 = 0 of this map has only </(< m — 1) eigenvalues with absolutevalue less than 1 and (m — 1) — d with absolute value greater than 1, thenthe set of £0 e -n for which £ = r)(t, £0) "tends" to #, as / -> oo, constitutesa ^-dimensional manifold £.

Figure 2.

In order to calculate the eigenvalues of the Jacobian matrix of the mapT:!;Q -*• £j at £„ = 0, consider £0 arbitrary for a moment (i.e., not subjectto £0 e TT). The matrix H(t, £0) = d$ji(t, £0) satisfies

In particular, for £0 = 0,

Note that H(t, 0) is a fundamental matrix of (10.5) and that the coefficientmatrix, dsf(y(t)), is periodic of period p. It follows from the Floquettheory in § IV 6 that H(t, 0) has a representation of the form

is a periodic matrix function and D is a constant matrix. In particular,T(0) = p and K(p) = AT(0) = / imply that

The characteristic roots (= eigenvalues) elt ez,..., em of the matrixff(p, 0) = e^ are called the characteristic roots of the periodic solution£ = y(t). Note that ek ^ 0 since /f(/?, 0) is nonsingular. Correspondingly,p-1 log tfj, ~1 log e& ... are called the characteristic exponents, so thatonly the real parts of the characteristic exponents of £ = y(t) are uniquely


defined. The characteristic exponents, modulo 2m, are eigenvalues of D(and D is not unique).

When £ is subjected to a linear change of coordinates, £ = Nt, it isreadily verified that //(/, 0) is replaced by NH(t, Q)N~\ so that the set ofcharacteristic roots are not changed.

Lemma 10.2. Let /(£) be as in Lemma 10.1, dim | = m, and T themaP £o -*• £1 in (10.3), where £0, £t E -n. Let elt..., em be the characteristicroots of£ = y(t). Then one of these, say, em is 1 and elt..., em_^ are theeigenvalues of the Jacobian matrix of the map T at £0 = 0 £ n. In fact, ifthe coordinates in the £-space are chosen so that

and IT: £m = 0, then the last column of H(p, 0) is (0,. . . , 0, 1) and the(m — 1) x (m — 1) matrix obtained by deleting the last row and columnfrom H (p, 0) = eDp is the Jacobian matrix of the map Tat £0 = 0.

Proof. It will first be verified that H(p, 0) satisfies

i.e., A = 1 is an eigenvalue of H(p, 0) and/(0) is a corresponding eigen-vector. Note that y'(t) =/(>;(0)- If tm's relation is differentiated withrespect to t, it is seen that f = /(/) is a solution of linear initial valueproblem

Since H(t, 0) is a fundamental matrix for this linear system reducing to /at / = 0, /(/) = H (t, 0)/(0); cf. § IV 1. For t = p, this relation becomes(10.9).

Thus, if (10.8) holds, then

The Jacobian matrix of the map (10.3), without the restriction £0 e n, is

At |0 = 0, this becomes

The first term on the right is the matrix (/l(0) dr(0)/d£00> so that thefirst m — 1 rows are 0 by virtue of (10.8). The last term is H(p, 0).Consequently, the lemma follows.

11. Limit Cycles

The remarks of the last section make it clear that, in a study of thesolutions of (10.1) near £ = y(f), the real parts of the (nontrivial)

characteristic exponents of f = y(t) play a role similar to that of the realparts of the eigenvalues of £ in a study of solutions of (1.2) near £ = 0.

Theorem 11.1 Let /(£) be of class C1 on an open set and let (10.1)possess aperiodic solution £ = y(t) of (least) period p > 0. Let dim £ = mand let the real parts of m — I characteristic exponents of | = y(t) benegative, say, less than a//>< O.Then there exists a <5 > 0 0m/ a constant Lwith the property that for each |° on f/ie open set dist (|°, #) < 6, where*%'• % = y(0» 0 = f =/>» ^^ "" an asymptotic phase t0 such that thesolution £ = »?(/, £°) of

•ttJf//5/?ey

/« particular, £ = y(t) is a limit cycle and is asymptotically, orbitally stable.Proof. As before, let y(Q) = 0. The assumptions make it clear that

after a linear change of the variables £, it can be supposed that the map(10.3) is of the form

where S(£0) is of class C1 for small ||£0|| and vanishes together with itsJacobian matrix at £0 = 0 and that a & \\A\\ < e*. Thus, if ||£0|| issufficiently small, say, ||£0|| < *, then ||£J ^ e" ||£0||. Hence fw = T»£0is defined for n = 1, 2 , . . . and ||£J| ^ ena ||£0|| -^ 0 as « -* oo.

Since the solutions of (11.1) are unique, it follows from Theorem V 2.1that if c > 0 is arbitrary, then there exists a d = <5 > 0 with the propertythat if dist ( , £°) < <5, then there exists a least positive value t = T°(|°)such that fi(t, £°) exists for 0 <s / r° and »?(TO, f°) 6 TT, || (T°, |°)|| < «.

Let fo = »?(^0» £°) e TT. Put r1 = T(|O) + r° and, for n = 2, 3, . . . ,rn = ,-(£„_!) + r"-1, so that fn = ^(T», f°). Clearly, r(0) = p impliesthat r(£n) ->/7 and rn/«p -*• 1 as n —> oo. Actually, J0 = lim (rn — «/?)exists as n -> oo and there is a constant Lj such that

In order to see this, note that

Since r(0) =/? and T(£) is of class C1, |T(£n_j) — p\ is majorized by£o llfn-ill = ^a(n-1) ll£oll for a suitable constant L0. Thus the existenceof a /„ satisfying (11.3) with Lt = L0/(l — e") follows.

Since 7y(/, ^°) is of class C\ \\r)(t + rn, ?) - y(t)\\ = \\r)(t, |n) -i)(t, 0) | <; L2 HIJI ^ Lze*n |||0|| for some constant £2 and 0 / </>.

Also, the boundedness of /(£) in a vicinity of # implies that \\r)(t +rn, f°) - if(/ + «/> + /«, f°)|| ^ VM Hloll by virtue of (11.3). Hence

\\n(t + np + t0, £°) - y(r)|| < (L2 + L3K" llf.ll for 0 < t < p.If t + np is replaced by /, it follows that

for np :5s ? (« 4- !)/> and / = 0, 1 , . . . . This proves the theorem.Theorem 11.2. Let dim £ = m,/(£) fee of class C1 on an open sef, a/w/

7ef (10.1)/Mmew aperiodic solution £ = y(f) of least period p > 0. (i) T/ie«there exists a solution £ = £(f) ?£ y(/) defined for large t and satisfying

/or some e > 0, vv/iere ^: £ = y(/), 0 / ^ /?, f/ a«</ on/y // f = y(/)/ia^ ar least one characteristic exponent with negative real part, (ii) If| = y(t) has exactly d (^ m — 1) characteristic exponents with negativereal parts, then the set of points £ near *& on solutions f = £(0 o/(10.1)satisfying (11.4) /or some e>0 constitutes a (d + Y)-dimensional C1manifold S. (iii) If at least one characteristic exponent has positive realpart, then £ = y(t) is not orbitally stable, (iv) IfO < d < m — I in (ii) and(m — 1) — rf characteristic exponents have positive real parts, then S canalso be described as the set of points £ near % on solutions £ = £(/)satisfying

awd/or

for some sufficiently small € > 0.It will be clear from the proof and from Exercise 5.1 that the C1 assump-

tion on/and assertion on S can be weakened somewhat or strengthenedeither to analyticity or to C™, 2 m oo. In case (iv), there is, ofcourse, an analogous (m — </)-dimensional manifold corresponding tot—>— oo which intersects S transversally along ^; S and the corre-sponding (m — d)-dimensional manifold are called the stable and unstablemanifolds of ft.

Proof. Only the proof of (ii), when 0 < d < m — 1, will be indicated.It can be supposed that y(0) = 0 and that (10.8) holds. After a linearchange of variables in -n [i.e., in the (I1, . . ., l^'^-subspace], it can besupposed that T in (10.3) is of the form (5.1), where Y, Z are of class C1

for small ||t/0||, ||z0||, and Y, Z and their Jacobian matrices vanish at(y0) 2o) = 0. Here dim y0 = d, (y0, z0) = (I1,.. ., £m-'), a = \\A\\ < 1and 1/c = HC-11| ^ 1 + 6, where 6 > 0 is arbitrarily small. Let y = g(z)

be the manifold supplied by Lemma 5.1. Then r%0,20) is defined forn = 1,2,... and \\Tn(yQ,z0)|| tends to 0 exponentially as n -> oo if andonlyifz0 = £(y0).

If || f0|| is small and £0 e n, there is a (y0,20) such that f0 = (y0, z0,0),where the last 0 is the real number 0. Consider the set of l-points givenby S: | = ??(/, £0), where £0 = (y0, £(y0), 0) e TT and 0 / T(|O).

It will be left to the reader to verify that the subset S of £ in a smallopen vicinity of satisfies the assertion of the theorem.

Remark. From Lemma 8.1, we can deduce the topological nature ofthe set of solutions of (10.1) near # when the real parts of the nontrivialm — I characteristic exponents of £ = y(t) are not 0.

Exercise 11.1. Under the conditions of Theorem 11.2(0, let £(0 be asolution of (10.1) satisfying (11.4) for some e > 0. Show the existence ofnumbers t0 and c > 0 such that ||£(f + /0) — y(t)\\ ect->0 as t-+ oo.

Theorem 11.3. Letf(g) be of class C1 on an open set and such that (10.1)possesses aperiodic solution $ = y(t) of (least) period p > 0. Put

If A > 0, then $ = y(t) is not orbitally stable. If dim 1 = 2 and A < 0,then £ = y(t) is exponentially, asymptotically, orbitally stable.

Proof. For arbitrary m = dim |, (10.5) implies that

cf. Theorem IV 1.2. Thus

so that e* is the product e^ ... em. When A > 0, then | t| > 1 for somek and when m — 2, it can be supposed that ez = 1 and = eA. ThusTheorem 11.3 follows from Theorems 11.1, 11.2.

Exercise 11.2. Let dim £ = 2,/(£) be of class C1 on a simply connectedopen set E, and tr d|/(£) = div/(£) 5* 0. Then (10.1) does not possess aperiodic solution (^ const.) nor a solution f = $(t), — oo < t < oo, suchthat |(± °°) = lim f(0 as t -* ± oo exist, are equal, and £(± oo) e £.

APPENDIX: SMOOTH EQUIVALENCE MAPS

12. Smooth Linearizations

As pointed out earlier, Theorem 7.1 and Lemma 8.1 become false if"the continuity of R and R~v' in the assertions is replaced by the assertionthai 1R, R~l are of class C1." This and the following two sections concern

the existence of smooth linearizing maps R, R~1 under additionalhypotheses.

Theorem 12.1. Let n > 0 be an integer [or n = oo]. Then there existsan integer N — N(ri) ^2 [or N = oo] with the following properties:If T is a real, constant, non-singular d X d matrix with eigenvalues ylt ..., ydsuch that

for all sets of non-negative integers mlt..., md and if, in the map

3(f) is of class CN for small ||f|; satisfying 3(0) = 0, 3{3(0) = 0, thenthere exists a map R: £ = Z0(£) of class Cn for small ||£|| MC/I ffotf

Note that (12.1) implies that |yfc| ^ 1 for £ = 1,..., d for, since F isreal, its eigenvalues are real-valued or occur in pairs of complex conjugatese-g-> if IXil = 1 and ya = y\ = l/y8, then y^2 = yj. When r* is the"group" of maps associated with the differential equation (7.1) and Tin(12.2) is T = T1, then F = eE\ so that if ex , . . . , «?„ are the eigenvaluesof E, then y, = exp e, and (12.1) is replaced by m^ + • • • + mded ^ ek.

Exercise 12.1. Formulate an analogous theorem concerning thelinearization of the differential equation (7.1) and prove it by usingTheorem 12.1 and the device involving (9.1) in obtaining Theorem 7.1 fromLemma 8.1.

Remark. The proof of Theorem 12.2 supplies a choice of N(n) or,equivalently, of X = N — n (probably, far from the best choice): Let0 < a < a < 1 be such that the eigenvalues of F satisfy 0 < a < min(lytl> l/IVkl) < a < 1 for k = 1,. . . , d. (In particular, in a suitablecoordinate system, the norms of F, F"1 are less than c = I/a; furthermoreF = A or F = diag [A, B], where the norms of A and B~l satisfy a < \A\,\B~l\ < a.) Then N can be chosen to be « + X if A is an integer such that

is the number of partial derivatives of order n of a function of d variables;cf. (14.29) and steps (/) and (m) in the proof of Theorem 12.2 in §^14.

Theorem 12.1 will be obtained from a more general result, Theorem12.2. The fact that Theorem 12.1 is contained in Theorem 12.2 is impliedby the following lemma which depends on a simple formal calculation.

Lemma 12.1. Let N ^.2 be an integer [or N = ao]. Let T be a real,constant, nonsingular matrix with eigenvalues satisfying (12.1) and letS(£) be of class CN for small ||£|| satisfying 5(0) = 0, dsE(0) = 0. Thenthere exists a map J?t: £ = Z^g) of class CN for small ||£|| satisfyingZx(0) = 0, 5^(0) = /; and Ex(£) in

where 7\ = /^TTJf1, has the property that all partial derivatives of Ej(£)of order < N vanish at £ = 0.

A generalization of Theorem 12.1 involves the "equivalence" of twomaps T, 7\ without the assumption that either is linear:

Theorem 12.2. Let (12.5) be a map of class CN for small ||£||, where2 N ^ oo and Hj(0) = 0, djS^O) = 0, am/ suppose that the eigenvaluesyk of F satisfy jy fcj^ 0, 1. Le/ « > 0 be an integer. Then there exists aninteger A = A(w) > 0 depending only on n and F with the following property:IfN ^ n + A(«) and Tin (12.2) is of class CN such that all partial derivativesof E(£) — Hj(£) of order ^ TV vanish at £ = 0, //^« /Aere e;c/$/s a w#/>/?: £ = Z0(£) o/c/flw Cn/or small ||£|| satisfying (12.3) a»^/

In the proof of this theorem, the definition of R will not depend on n,thus R is of class Cn for every applicable n. In particular, if N — oo, thenR E C°°. Also, the proof will show that for a given n > 0, the assumptionthat H, Hj are of class C^ can be relaxed to the assumption that S, Hjare of class Cn+1 and that each partial derivative of 3(£) — 5 1) oforder & « + 1 is majorized by const. H^H^0"* for a fixed N0 5: « 4- ^(w),in which case RE Cn is such that each partial derivative of R£ — £ oforder y « is majorized by const. ||f H^0""^.

Lemma 12.1 will be proved in the next section and Theorem 12.2 in§ 14. The proof of the following theorem which is the analogue ofTheorem 12.2 for differential equations depends on a simple modificationof the proof of Theorem 12.2 and will be left as an exercise; see Exercise14.1.

Theorem 12.3. In the differential equation

let ^(0 be of class CN for small ||£||, where 2 N oo and (0) = 0,0^(0) = 0, and suppose that no eigenvalue of E has a vanishing real part.Let n > 0 be an integer. Then there exists an integer A = A(«) > 0depending only on n and E with the following property: If N ^ n -f A(«)and F(£) in

is of class Cy such that all partial derivatives of F(£) — /\(£) of order<; N vanish at £ = 0, then there exists a map R: £ = Z0(£) of class Cn

for small ||£[| satisfying (12.3) and transforming (12.7) into (12.8).The remarks following Theorem 12.2 on the smoothness of 5, El have

analogues concerning the smoothness of F, Fv In particular, the analogueof the last part of the remark concerning R implies the following result onasymptotic integration:

Corollary 12.1. Under the conditions of Theorem 12.3, there is a one-to-one correspondence between solutions £(t) of (12.8) and solutions £(f) =/?£(/) o/(12.7) satisfying £(f), £(0~*0 os t -»> oo [or — oo]; furthermoreII«0 - £(011 ^ const. ||£(0ir « / - oo [or - oo].

Exercise 12.2. (a) Let T and 7\ in (12.2) and (12.5)'be maps of classC°° for small ||£|| such that the eigenvalues yt of F satisfy |yj 5* 0, 1 andthat 5(0) = 0, dsE(0) = 0, SÔ) = 0, aÊÔ) = 0. Let T0, T10 denotethe (not necessarily convergent) Taylor expansion for T£, rt| at the origin.Then there exists a map R: £ = Z0(£) of class C°° for small j|||| satisfying(12.3) and 7\ = RTR~l if and only if there exists a formal power seriesmap R0: £ = £ -j- • • • such that formally TwRo = R0T0. (This assertionis a consequence of Theorem 12.2 and Lemma 13.1) The question of theexistence of R0 depends on the solvability of certain linear equations;cf., e.g., the proof of Lemma 12.1. (b) Formulate the analogue of (a)in which the maps T, 7^ are replaced by the differential equations (12.7),(12.8).


Case I (2 N < oo). After a suitable linear change of variables, itcan be supposed that F is in a form similar to a Jordan normal formexcept that the subdiagonal elements are 0 or e > 0, where e will bespecified below; cf. § IV 9. Thus the transformation (12.2) can be writtenin terms of components as follows:

where j = 1,. . . ,d and e, is 0 or e. In (13.1) and below, (/) represents a</-tuple (/!,..., id) of non-negative integers; |/| = i\ + • • • + id; and!<*> is the product £(i> = (&)'*... (!*)<<.

The map /?x will be determined so that each component of R^ is apolynomial in (I1,..., £d), say,

The object is to determine Rl so that /^TTZf x£ = T| + o(||l|hv), or,equivalently,

Up to terms of order oGIIH^), they-th component of the left side of (13.3) is

while that of the right side is

Thus (13.3) is equivalent to

Comparing coefficients gives a linear system of equations for rfm).It is easy to see that in view of (12.1) these linear equations uniquely

determine the numbers /fOT) fory = 1,... , d and 2 \m\ ^ N if «! =• • • fd = 0. In fact, the main part of the factor of r[m) on the left (i.e., thepart of lowest order in £) is (y, — II y?")f<fw), so that we can successivelydetermine rj

(m) first for \m\ — 2, then \m\ = 3, etc. This shows that ife1 = • • • = ed = 0, then the determinant of the matrix of coefficients ofthe unknown r\m) is not zero.

It follows that if e > 0 is sufficiently small, where c, is 0 or e, then thematrix of coefficients is nonsingular. This proves Lemma 12.1 when2 N < oo.

Remark. For the purpose of treating the case N = <x>, note thefollowing corollary of the proof just completed: if 2 N < oo, thenthere exists a unique map RI of the form (13.2) satisfying the conclusionsof Lemma 12.1. Actually, the proof shows that Rl is unique after a certainlinear change of variables (which leaves the form (13.2) unchanged) andhence is unique before the linear change of variables.

Case 2(N — oo). Since T is of class C°°, it has a formal (not neces-sarily convergent) Taylor expansion at £ = 0;

where F = (y,.fc). The proof just completed and the remark following itshow that there is a unique formal power series map

such that RtT =* TRl formally.

In order to complete the proof, grant the following fact for a moment:Lemma 13.1. Corresponding to any formal power series

with real coefficients, there is a function r(f) of class C°° having (13.5) as itsformal Taylor development.

For if Lemma 13.1 holds, a desired C°° map R! is obtained as R^ =(r^f),. • • , r4^)) in which r'(£) is of class C°° and has the formal Taylorseries on the right of (13.4). Thus in order to complete the proof ofLemma 12.1, it only remains to verify Lemma 13.1.

Proof of Lemma 13.1. Let O(/) be a real-valued function of class C°°such that O(/) = 1 for |/| < 1/4, 0 < <&(/) < 1 for 1/4 < |/| < 3/4, andO(/) = 0 for \t| 3/4. It is easy to see that

is a uniformly convergent series for ||{|| < 1 and represents a C°° functionwith the desired property. The (w)-th term is zero unless

in which case the (m)-th term of (13.6) is majorized by

if \m\ ^ 2. Thus the series in (13.6) is uniformly convergent for ||£|| ^ 1and r(0) = r(0). Similarly, the series in (13.6) can be differentiated for-mally any number of times to give a uniformly convergent series anddlmlr(^fd(^)mi • • • WT* = r^m^. '. ... md\ at f = 0. This provesLemma 13.1.14. Proof of Theorem 12.2

In what follows it is supposed that T, Tl are of class CN and thatthere exists a constant CN such that

and M = 0, 1 , . . . , N, where D(m} = 8'wil/9(l1)mi - . . d(£d)m*< In thefollowing only \\£\\ ^ r0 < 1 is considered, so that ||£||J ^ UII* ify ^ A:.

In view of the normalizations of § 4 and of the results of Exercise 5.1, itis no loss of generality to suppose that J1, 7\ are defined on the entire£-space, are one-to-one, and reduce to the linear map Ff for large ||£||;that T = diag [A, B] and £ = (£_, £+); that Jis of the form

that 7(0, f+) = 0 and Z(f_, 0) = 0, i.e., that

are invariant manifolds; and that there exist constants

such that

where ||£± \\ — ||£|| ±. These inequalities will also be used in the equivalenform

Both sets of inequalities are illustrated in Figure 3 when dim £_ = dim f + = 1

Figure 3.

Let K+, K°, K- denote the £-sets

Thus #° is a conical hypersurface. Let Kj = T'K" for j = 0, ±1,.. .,so that T-*K*= K° and, in view of (14.4), Kf c K+ if j > 0 and Kj c K~ify < 0. Let Q° denote the |-set between K° and /^ including K° but notJP, i.e.,

In addition, let Qj — PQ° the corresponding set between Kj and Ki+l

including Kj but not Ki+\ so that

cf. Figure 4. It is clear that


Figure 4.

that the sets Qj are pairwise disjoint, that K+ — M+ is the union of Qi

forj ^ 0, and that K~ — M~ is the union of Q' forj < 0. For £ £ M+ UM~, let &(£) denote the unique integer k such that £ e g*.

For r > 0, let S(r) denote the sphere ||£|| r, #±(r) = K± n S(r),£°(r) = #° n S(r), and g'(r) = Q* ^ S(r). For a given r0 > 0, thereexists an TJ = r^/o) > 0 such that 0 < ^ < r0 and that if f e Qk(r^) f°r

A: > 0 [or A: < 0], then T-'£ e S(r0) for j = 0,1,..., k [or -j = 0,1,...,*].

The proof will be divided into steps (a)-(m). For brevity, most discus-sions will be given for £eAT+ ; the obvious analogous statements anddiscussions for f 6 K~ will not be given, but will be used.

(a) //*(!) £ 0, then, f o r j - 0,. . . , *(£),

This is clear from (14.4) and from the fact that T-'& e K+ fory = 0,. . . ,*(£).

(b)Letk(ft>0. Then

In order to verify this, let y = T~*(f)f, so that r\ 6 Q° <= ^T+ but l"-1^ ^AT- Then, by (14.4),

where k = A:(|); so that (14.8) follows from ||»?||_ ^ \\ij\\+ and ||r- ||_ >lir-^||+.

(c) TAere e^w/ constants c < 1 OT«/ K > 1 .SMC/I /Aaf //" /:(!) > K, then

[In fact, c can be chosen arbitrarily close to a if K is sufficiently large.]By (14.4) and (14.8),

Thus if K is so large that a~2a4" + a8 < 1, the result follows with c —(a-V« + a8)1 .

(rf) Let ||7^11 C H I I I /or ||||| r0 a/i^ Ar/ T^l, TfyeS(rj> forj = 0,1,. . . , h. Then

Put £(j) . r/| - Tin so that CO') - T&j- D for y - 1,..., A.Hence ||£{j)|| ^ C ||C(y- 1)|| and (14.10) follows.

(e) Let r0 > 0 be fixed, rx = /^(r,,) > 0 as before part (a). Let C 1be a constant such that ||r|||, \\T^\\ ^ C \\t\\ for ||£|| ^ r0. Let A, N beintegers such that

awrf /e/ //iere exw/ a constant CN such that (14.1) holds for ||f || r0. 7%e«f/iere ex/ste an r2 = r2(A), 0 < rz < r1} 5i/cA //ia/ (i) for j = 0, . . . , &(£),

and (ii) f/iere exwW a constant CN such that

Proof. Let v\ E K+(r^) and suppose that T'TJ, T^rj are defined for j =0, 1, . . . , h for some A ^ £(£). Put

Then ^0') = T'ltT'r^ - ^^1 + (^i - T)T^\ i.e.,

^-^(/-o + m-r)^-1^Since £(0) = 0, an easy induction gives

Hence, by (14.1),

Choose ?? = r-fc({)^, so that

By (14.7), for; = 0 , . . . , A,

Hence, if C f l A < l ,

so that

fory = 0, . . . , h.From the definition of £(/) and 77,

hence ||T/T- || ||£(;)l| 4- ||r^(^f||.

This inequality, (14.7), and (14.15) make assertion (/) clear. The definitionof £(;') and the choice j = Ar(£) in (14.15) give (14.13).

(/) Definition of R. Let R{0} be a map of the closure of Q° into the|-space such that R(0} is of class CN, reduces to the identity / on K° andto T^T-* on K1, and satisfies an inequality of the form

small ||£||, and some constant dN. Define a map R(k} on Qk, the closure ofQk, by putting

and k = 0, ±1, If £ £ 3 , put

The conditions Rw = T^r-1 on JF, Rw = I on K° imply that /? is con-tinuous at £ 6 jK1 and hence for all £ £ M*. It will also be supposed thatjR<0) is chosen so that

Such a choice of R(0} is clearly possible; it suffices to choose /?<0)£ = £for 0 £ near K°, and /Z<°'£ = T^T-^ for 0 5^ £ near .

A possible construction of Rw is as follows: Note that if 0 5* £ 6 JK1 =7*°, then a2 ||£||_/||£||+ ^ «2 and if 0 £ 6 0°, then a2 ||£||_/||£||+ 1. Let 0 < a2 < l < 2 < 1, so that the set {£: €l ||£|j+ ||£||_ e2||£||+,£ * 0} is interior to Q° while the set {£: a21|£||+ ||£||_ ||£||+) contains5°. Let 0>(0 be of class C°° for a2 / ^ 1 such that 0>(0 = 0 fora2 < / ^ t! and O(0 = 1 for e2 / < 1. For 0 5^ £ e 5<0), put

and R(0}£ = 0 if £ = 0. Note that Rw = / on /:°, ^<°> = T^-1 on AT1

and

from which (14.16) readily follows.(g) The map R satisfies, for £ £ M±,

(14.20) *r£ = 7\/?£

This is clear from (14.17) and (14.18), for if £ e Qk, then T£ 6 Qk+l and/?r£ = rf+l/?«»r-*£ = rXJR£.

(h) Assertion (e) remains valid //(14.12), (14.13) are replaced by

respectively. Note that

The right side is majorized by C' dN \\ r-*<*>£||^ in view of (14.16) and henceby 2NdNC* ||£||/V<*>Ar in view of (14.7). Assuming that C s> 1, it followsthat, for; = 0, ...,*(£),

In view of (e), this makes the validity of (h) clear.The object of the remainder of the proof is to obtain estimates for the

derivatives of /?£ — £ for £ <£ M±. To this end introduce the followingnotation:

To avoid confusion with other superscripts, the coordinate index of a vectorwill be written before, instead of after, the vector symbol; e.g.,/ = (*/,..., df)and £ - C£, . . . , *£)• For a function/^), let/} = d/ld^S). If (a) = («„ . . . , ad)is a rf-tuple of non-negative integers, write/,a) = D(a,/ = dMfld(l£)"i... d^)**.Let/<0[) ° r), ff o ^ denote the value of the corresponding derivative of/evaluatedat the point I = ?/. Similarly for a map, say T, the abbreviations jT,£ or r(a)£meana(rs

e)/ae£)or(r£)(a).

It will be shown that if X = A(n) in (e) is sufficiently large, then for asuitable constant CA(|<x|)

for £ e K+(rz) — M+ and |a| S n. The proof will be by induction on |a|.The relation (14.21) corresponds to the case |a| = 0.

For 0 7* £ e Qk, define

By (14.17),

hence

or

f

where

The chain rule of differentiation gives

Repeated differentiation leads to a formula of the type

where the second summation is over all ^/-tuples fi = (&,..., /Jd) with|)8| = |a|; ft"'? is a product of |a| factors of the form h(T~^; ¥*•«•* is apolynomial (independent of &) in the components of T^} ° R^T*1!-,(r-%,1 for |0| |a| and /lj*jo 7*-i£ for |y| < |a|. Note that the poly-nomial in T1'01'* does not depend on k; the dependence on k arises fromits arguments 7*1(W°tf^r-1! and /$}« r-1!.

If the second term on the right of (14.24) is written as TtT~1^ —T! IT~l£, where / is the identity operator, it follows that the analogue of(14.26) holds, i.e.,

where ''a is obtained by replacing Tm ° R(k>T-*£, Rfv) ° T~l£ by Tm oT-1 , 7(y) o r-i£, respectively, inT*'-"1*. Consequently, (14.24) implies that

where

(/) The sum

w^/cA w defined for 0 ?£ I 6 6t+1(r2) /KW o ^ownt/ L((X| independent ofk*=0,1,. . . . For, if C is a bound for the first order derivatives of 7\£,T-^ on Hil l ^ r0, then |Oa | C|w| and || Tlm ° /?<fc)r-^|| ^ C. Hence amajorant for the sum in (/) is

Remark. By considering sufficiently small r0, the number C, henceL,a,, can be made to depend only on the norms of F and F~*.

(j) There exists a number C0 such that

By (14.25), a differentiation of F£ gives

Repeated differentiations show that *F<0)£ is a polynomial in *(7\ — T)(y)°.T-*£ and m(77-1)(y) for |y| < |j8| and m = 1, . . . , d, and that each termcontains a factor of the first type. Since there exists a constant C0', suchthat

for 0 |y| ^V and small ||£||, the assertion (/) follows.(A:) 7%ere exists a number C0 such that

If C' is a bound for the second order derivatives of 7\£ for small ||£||, themean value theorem of differential calculus shows that

In view of (14.21), the assertion (A:) holds with C0 = C"C|a|</(S l)CN.(/) I^r 1 « JV — L Let the conditions of (e) hold and, in addition,

let K be so large that L,aA < 1 for v = 1, . . . , » , where Lv is given inpart (/). Then there exists a constant C^da)) such that (14.22) holds for£eK+(rJ-M+ifO^\ai\^n.

Proof. Introduce the abbreviation

The assertion to be proved can be cast in the form

The proof will be by induction on n. The case n = 0 is contained in (14.21)in assertion (h). Assume 0 < n N — A and the, desired result for0, l , . . . ,n- 1.

It will first be shown that there is a constant €NO(ri), independent of £,such that

and 0 9* £ e QkJr\rt). Note that the descriptions of ¥''•*•* and T'-1 follow-ing (14.26) imply that the expression on the left of (14.32) is majorized by

where "const." is independent of k. Since N > n, the argument used inpart (k) shows that there exists a constant K\(n), depending on n butindependent of k, such that the first sum does not exceed Kx(n) HT*-1^.Hence the existence of Cxo(n) in (14.32) follows from the induction hy-pothesis.

In view of (14.27) and (/), (/), (fc), (14.32), and (14.30),

It is clear from (14.16) that there exists a constant such that

It follows from (14.33) and a simple induction on k that similar inequalitieshold if S<0-n), 0° are replaced by S'*-"', Qk. Thus if K is given in step (c),then, for the finite set of ^-values, k = 0, 1, . . . K, there exists a constantCv(n) such that

Note that, by (c), Lncs~n ^ Lnc* < 1 if K is (fixed) sufficiently large.

An induction on k will now be used to show that (14.34) holds for allk = 0, 1, Assume that (14.34) holds for some k ^ K. Then, by(14.33) for 0 5* £ 6 C^fo),

But || T"-1!!) ^ c ||III by (c), so that the right side is majorized by[LBCA(«) + CyoOi) + 2C0]r

v-» U\\*-n. Since (14.35) implies that thefactor of ||£||A:~" is at most Cv<«), the inequality (14.34) holds.if & isreplaced by k + 1. This completes the induction on k and also on71.

(m) Let the conditions of(e) and (/) hold. Then R can be defined on M*so that R is of class C"-1 on \\£\\ < rt.

Proof. It has been shown that R, considered on A"+(r2) — M+, is ofclass C-v and that its partial derivatives of order ^ n are bounded ifN — n A. Without loss of generality, it can be supposed dim M+ <d — 1 (e.g., by increasing d, by adding dummy coordinates to l-+). Thuspoints of K+(rt) — M+ which are near can be joined, by short rectifiablepaths in K+(r%) — M+. It follows that R and its partial derivatives of

order ^ n — 1 are uniformly continuous in AT+(r2) — M+. Hence R hasan extension to AT+(r2) of class C""1. This proves (m).

In view of (g) and the continuity of R, (14.20) holds on ||£|| ^ r2.Since R is close to the identity for small ||£||, R has an inverse of classC""1. This proves Theorem 12.2. [We should remark that A dependsonly on F; cf. (c) and the Remark following (/).]

Exercise 14.1. Prove Theorem 12.3. To this end, treat the "group"of maps T\ TV instead of the differential equations and the problem offinding a suitable R satisfying RT1 = TV R. For £ £ M±

t define /(£) sothat r-((l)£e#°. Put R£ « r{<«r-((l>|. The restriction of this * to5° gives an Rlo} as in (/). It is only necessary to verify (14.16); but thiscan be deduced from (e).

APPENDIX: SMOOTHNESS OF STABLE MANIFOLDS*

In this Appendix, we give another proof of Lemma 5.1 which hasthe advantages that (1) it is valid for infinite dimensional Banachspaces and (2) it yields easy proofs for additional smoothness as inExercise 5.1(a),(ii). Additional differentiability is deduced from"continuity" if one allows parameters; cf. Chapter V§§3-4. Thesimplification of the proof of, say, differentiability results by dealingwith a more general situation: first, the iterates Tn of T arereplaced by products Tn = Tn o ... o TI of maps Tj,T2,...and, second, the introduction of terms Un(w) as in (1.2) below.The first generalization permits "changes of dependent variables"and the second permits the analogue of the "interchange ofparameters and initial conditions" as in Chapter V § 1.

"This Appendix is reprinted with permission of Academic Pressfrom Hartman [S3] in J. Differential Equations 9(1971)361-372.For other treatments, see Hirsch.Pugh and Shub [SI] and referencesthere.

271A Ordinary Differential Equations

1. NOTATION AND HYPOTHESES

Let W, X, F, Z = X x Y = XY, E =W X X x Y = WXY beBanach spaces. Norms for w e IV, x e X, y e Y, z — (x, y) e Z, e = (w, x,y)&Ewill be denoted by

Finally, let Xr(xQ) = {x e X : \\ x — x01| < r}.

DEFINITION STABILITY SET. Let X° C X be open; Tn': X° -> X a mapfor n — 1,2,...; @(Tn' ° ••• ° T^) the (possibly empty) open set whereTV o ••• o TV is defined. The (possibly empty) subset 2 = ®({Tn'}, X°) =C\@(Tn' o ••• o 7\') of X° will be called the stability set of the sequence(TV), with respect to X°.

We shall be interested in stability sets of sequences of maps Tn' : Z^(G) —> Zof the form

where AneL(X, X), BneL(Y,Y) are bounded linear operators, Bn isinvertible, || An \\ < 1, ((B^11| < 1; Fn' and Gn' are maps from Zj(0) to J%T andF, respectively, such that Fn'(Q) = 0, Gw'(0) = 0; \\Fn'(xj-Fn'(zJM,|| Gn'(%) - Gn'(z2)\\ ^t\\Zl-zz ||; and || xn ||, || y || < c; where e > 0 is afixed small number. In order to reduce (1.0) to the case xn = 0, yn = 0,we can replace Z, Tn' by R1Z, T^, where

It is clear that the stable set @({Tn'}, Zj(0)) will be determined if®({T"n}, (R1Z\(QD is known; in fact,

This reduction to the case tn — 0, xn = 0, yn = 0 has been accomplishedat the cost of replacing the linear map x H>- Anx, having a norm a < 1, bya map (£, x) i-> (f, y4njc), having norm 1. It will turn out that, in view of theform of the latter, nothing is lost and, indeed, there are other advantages.

For these "methodical" reasons, we shall not consider maps (1.0) below,but maps Tn : Er(Q) -» E = W x X x Y = WXY of the form

Invariant Manifolds and Linearizations 27IB

where r is fixed (0 < r < oo); AneL(X, X) and BneL(Y, Y) are linearbounded operators, Bn is invertible,

Un : WT(Q) -+W;Fn,Gn: Er(Q) -> X, Y; and

The stability set of {Tn} with respect to {Er(Q)} is

The first result (Section 2) will give, under suitable assumptions on Un,Fn and Gn, the existence of a continuous function y0 : (WX)r(Q) -> Yr(Qi)such that

In such a case, we shall speak of the stable manifold @ \y — y0(w, x) of thesequence {Tn}, with respect to Er(Qt). Actually, the assumptions on Un,Fn,and Gn will imply that y0 e C1, but this and other smoothness properties willbe deduced from the existence proof for a continuous y0(w, x) = y0(w, x, a)carried out for the case that Tn depends on parameters a.

HYPOTHESIS (H). Let £ be a metric space and o e H. Letrne C°(Er(Q) x 27, E). For each aeS, assume that Tn satisfies (1.2)-(l-7)>where a, 6, S are independent of a. Let Tn have a Frechet derivativeDerneC°(£r(0)x2:).

HYPOTHESIS (H0). Let H = A x U x T, where A, U are metric spacesand T = [0, e] is a real f-interval. We say that T:, T2,... satisfies (H0) if (H)holds, Tn0 = (Tn)t=0 is independent of u e 17, and Tw , Z)eTn -> Tn0, DeTn0

uniformly on U, for fixed (e, A), as f -> +0.By (1.4), we have

271C Ordinary Differential Equations

also, if

then a < 1 — 28 <1— 8 — 82 implies that

Generally, we shall not exhibit the dependence of functions on parameters,unless convenient to do so.

2. MAIN LEMMA

In this section, we prove

LEMMA 2.1 (a). Assume hypothesis (H). Then there exists y0(w, x) —ya(w, x, a) e C°((WX)r(Q) X E, YT(ty) such that, for a fixed a, the stability set3> = ®({Tn}, Er(0)) is given by (1.9); also if e0 = («;„ , x0 , y0(w0 , *0)) andek = Sk(eQ) = (wk , xk , yk), then, for k ^ 0, ek = 0 if (WQ , x0) = 0,

and the manifold y = y0(w, x) is "invariant" that is,

(b) If, in addition, (H0) holds and a = (A, u, t), then yu(w, x, A, u, 0) isindependent of u and y0(tv, x, A, u, t) -> y0(w, x, A, u, 0) uniformly on U, forfixed (w, x, A), as t -> -f ()•

Note that (2.1), (2.2) imply

where w = w0 , x — x0 . But if y ^ yQ(^, x) and eQ = (w, x, y) e 2n forsome n > 0, then ek — Sk(e0) — (wk , xk , yk) satisfies

Invariant Manifolds and Linearizations 27ID

where (2.18) holds; cf. the proof of Proposition 2.5. In particular, if

<*" I I yow(wt *) — y II > r> then (w> *• y) $ &N •PROPOSITION 2.1. Let F be a complete metric space, and 27 as in (H);

C = C(y, a) : F x 27 -> T continuous and, for fixed a, C0 = C(-, a) : T -v Ta contraction

w>#A 0 independent of a. Then, for each a e 27, the map C0 : F —> F has a uniquefixed point y = y(a) and y(a) e C°(27, F). If 2 = A x U x T as in (H0),C(y, A, w, 0) w independent of u e (/, ant/ C(y, A, «, /) -> C(y, A, «, 0), fltft —»> -f-0, uniformly on U for fixed (y, A), ?^en y(A, M, 0) w independent of u andy(A, M, <) —>• y(A, M, 0) uniformly on £7, /or i.m/ A, a^ / —*• -f 0.

For, by the proof of the contraction principle, if y0 e F, then y(a) =lim C/(y0), as » -> oo, and dist(y(a), Ca"(y0)) < 6* dist(C0(y0), y0)/(l - 0).

PROPOSITION 2.2. Let n > 0; e0,e°E@n = @(Sn}\ and ek = Sk(e0),* = S&)fork = Qt...tH.

(a) The inequality

0 < m < n (2.4)

(/or example, wm — wnt, *m = xm), implies that, for k = m + l,...,n,

where a. > 1; t/. (1.10).

(b) The inequality

(/or example, yn — yn) and (1.11), (1.12) twpfy that, for k = 0,..., n,

Proof (a). Since || e || = max(|| w ||, || * ||, || y ||), (2.4) implies that

27IE Ordinary Differential Equations

Hence, by (1.6) and (1.7), || wm+l — zvm+11| < || wm - wm \\ and

so that, by (2.4),

Also, (2.9) and (1.6), (1.7) give

The last two inequalities and (1.10) give (2.5) for k — m -f 1, and an induc-tion gives it for k = TW + I,---. «•

Proof (b). The inequalities (2.7) follow from part (a). In order to obtain(2.8), note that (2.7) implies that

Hence, by (1.7),

An induction gives (2.8).

PROPOSITION 2.3. Write Sn(e) = (Pn(e), Qn(e), ^n(e)), where ee@n andPn eW,Qne X, Rn e Y. For a fixed a and n 1, there exists a function

w/cA fAaf (w, x, yQn(w, x)) e n ; y0n — 0 if (w, x) — 0; and

If, in addition (H0) holds, then yQn is independent of u when t = 0 andyon(w> x> A> «• 0 ~* on(w» *» A» «. 0). uniformly on U for fixed (w, x, A), <w/-> +0.

Proof. We shall give the proof for a fixed a, but with some detail, so thatthe continuity and uniformity assertions will be clear from Proposition 2.1.

Let w° = w0 = w, X° = XQ = x and e0 — (a?0 , XQ , y0), e° = (w°, x°, y°) e @n .Thus (2.4) holds for m — 0, so that by Proposition 2.2(a),

Invariant Manifolds and Linearizations 27IF

Thus, for fixed (w, x), the equation

has at most one solution y. Define the set

Jn — {(w> #) '• (2-13) has a solution y, (w, x, y) e @n}.

In particular (tv, x) = 0 e /„ ^ 0.We shall show, by the implicit function theorem, that /„ is open. Let

(w0 , *0) e /„ and Rn(wQ , x0 , y0) = 0. By (2.12), DyRn(w, x, y) is invertible*if(x,y)e@n , also \\[DyRn]~l \\ < I/a". Put D0 - DvRn(wQ , x0,y0) and write(2.13) as y — DQIRH(W, x,y) — y, so that a solution y of (2.13) is a fixedpoint of the map y t-*- y — DQIRn(w, x, y) — Cw x(y\ depending on param-eters w, x. Since DyCWtX = 0 at (w0 , x0 , j0), || £>„<?„,,„ || < 6 < 1 for (w, x, y)near (WQ , XQ , j0). It follows that there are positive numbers e, s such that,for fixed (w, x) e Ws(w0) X Xs(xQ), CWiX is a contraction mapping of Yf(y0)into ye(.yo)» with - contracting factor 6. Hence, Ws(zvQ) x ^s(^o) e/« » ant^/n is open.

Let (w, x)ejn, so that «OB(w, x) = (w, x, y0n(w, x)) e 2n and put ekn(wt x) =•S"«(^onK x)) for k = 0, 1,..., n. Thus, ?„„(«;, Jf) = Rn(e0n(w, x)) = 0; sothat if (w°, ^°), (w0 , x0) e /w , Proposition 2.2(b) gives

By the uniform continuity (2.14) of y0n(w, x) on /„ , yQn(w, x) and eQ(w, x) —(w, x, J0n(zf, x)) have unique continuous extensions to Jnr\(WX)r(Q)satisfying (2.14)-(2.17). In particular, e0(w, x)eS>n and (2.11) holds on/„ n (WX)r(Q). Thus, /„ is both open and closed relative to (WX)r(Q) andso, /„ = (WX)r(Q). This completes the proof of Proposition 2.3.

PROPOSITION 2.4. Let y0n(w, x) be given by Proposition 2.3. Then

*When dim Y =<x>, @n in the definition of Jn should be replaced byits connected component containing e =• 0.

271G Ordinary Differential Equations

on (WX)r(ty x Z. Also, the stability set (1.8) satisfies

a*/ (2.1), (2.2) hold.

Proof. Use the notation of the last proof, and let 1 < k < n. Since«?„„(«>, #) = wQk(w, x) = w and xQn(w, x) = xok(w, x) = x, Proposition 2.2(a),with m — 0 in (2.4), and ykk = 0 give

This proves the statement concerning (2.18), since a > 1.Keeping A: fixed and letting n -> oo in (2.14), (2.15) gives (2.1), (2.2),

hence \\yk || < (1 - 4S2)r < r for A > 0, and || xk [ (< (« + 2S)r < r for^ 1. This implies (2.19).

PROPOSITION 2.5. The relations (1.9) <WK/ (2.3) fo/tf.

Proof. On (1.9). It only remains to prove the reverse inclusion to (2.19).Suppose that (w, x,y)e& and y y0(wt x). Let ekn(w, x) = 'Sk(w, x, yQn(w, x))for k = 0,..., n and ek(to, x) = Sk(w, x, y) for k = 0, 1,.... Since w0n = w0 = w,*o« = *o = x, Proposition 2.2(a) and ynn = 0 give \\yn \\ = \\y - yn \\ «n j| yQn(w, x) — y || ~ oin || yQ(w, x) — y \\ -> oo, as n —* oo. This contradicts(a>, #, _y) e 2.

On (2.3). In view of what has been proved, for every (w, x) E(WX)r(G),there is one and only one y such that e = (to, x, y) is in the stability set<&({Tn}, Er(Q)). The relation (2.3) follows by applying this uniquenessstatement to the sequence 7^ , Tk+1,....

3. SMOOTHNESS OF STABLE MANIFOLDS

Let eQ(w0 , XQ) = (w0 , XQ , y»(wQ , XQ)) e 2\ en(w& , x0) = Sn(eQ) forn = 1, 2,...; and

Banach spaces of bounded linear operators. Define a sequence of linear mapsof Q3H into itself by

Invariant Manifolds and Linearizations 271H

where n = 1, 2,... and

so that rn0 depends on parameters («>0 , #0 , a). By Lemma 2.1, there exists afunction

in C°(&s x (WX)r(Q) X ), such that for a fixed (w0, *0, a) and P > 0,the restriction y | (^S)p(0) gives the stable manifold ({T^}, (i3S//)p(0)).

THEOREM 3.1. Under assumption (H), ffe stability set @({Tn}, Er(Q)) is amanifold y — yn(w, x) — jV0(w, x, a), where yQ satisfies the conclusions ofLemma 2.1 (a) and has a Fre"chet derivative D(w X)y0 , continuous in (iv, x, a),given by

D

This will be deduced from Lemma 2.1 in the next section. In the case (1.0)or equivalently (1.1), we can obtain the existence of derivatives of y0 withrespect to some of the parameters on which it depends, and also higherorder differentiability. In the next theorem, we assume that the parameterspace E is of the form

where V° is an open ball in a Banach space V and A is a metric space. Weshall also assume that

THEOREM 3.2t. Assume (H), (3.6), (3.7), and (3.8). Suppose that the mapTn in (1.2) is of class Ck, k ^ 1, as function of(w, x,y, v), that the derivativesof An , Bn ,Fn,Gn are continuous functions of (w, x, y, v, A), that the firstorder derivatives are bounded on {small v-balls} x ES(Q) x A x {n — 1, 2,...}for all s, 0 < s < ry and that all derivatives of order ^ k are bounded on{small (v, w, x, y)-balls} X A x {n = 1, 2,...}. ThenyQ(w, x, v, A) is of class C"with respect to (w, x, v) and its derivatives are continuous functions of(w, xt v, A).

Furthermore, tfe0 — (w, x, y0(w, x, v, A)) e E and en = Sn(e0)for n = 1,2,...,£fon ffo ^zrsf order derivatives D(w>XtV)en are bounded on

{small v-balls} x Es(0) x A x {« = 0, 1,...}, 0 < * < r,

and all derivatives of order ^ k are bounded on

{small (w, x, v)-balls} x A X {» = 0, 1,...}.

4. PROOF OF THEOREM 3.1

Let e > 0 be arbitrarily small and s = r — e. We shall prove Theorem 3.1on Es(0). From now on, assume that (w0, x0) e(WX)a(G) and (w, x) e(WX)1(Q),so that (WQ + tw, XQ -{- tx) 6 (WX)r(0) if t e T = [0, e]. Using the notations ofSection 2, let <?0 = («J0 , xQ,y0(w0 , ^0)) and *„ == 5n(e0) for » = 1, 2,....For 0 < t s^ e, put

By (2.1), (2.2) in Lemma 2.1,

We can write

where Tw' is the linear map

in which

and the argument of the integrands is

Although t > 0 is required in (4.1), (4.2), we allow t = 0 in (4.5); so thatthe operators there do not depend on (w, x) at t = 0 and tend, as t -*• +0, to

Invariant Manifolds and Linearizations 271J

their values (3.3) at t = 0 uniformly for (w, x) e (WX)i(Q), with (WQ ,xQ,a)fixed.

By Lemma 2.1, there is a function

defined on

such that, for a fixed set of parameters,

for all p > 0.One can associate with (4.4) a linear map of QSH into itself,

Again, by Lemma 2.1, there is a function 07 = y(w, £), defined on ^5" anddepending continuously on parameters (ZVQ , x0 , a, w, x, t), such that thestable manifold @({Tn}, (Q3H)P(QJ) is 77 = y \ (Q3)P(Q) for any P > 0. Att = 0, (4.6) reduces to (3.2), (3.3) and is independent of (w, *).-By Lemma 2.1,

as an H = L(W!AT, F)-valued function, is continuous in all of its variablesand, as t —>• +0,

uniformly for (w, x) e (WX^^Q), for fixed (w, |, w0 , x0 , a). In particular,if (w, £) = (idw, idx) is fixed, then

as f -> +0, uniformly for (w, #), (w, Jc) 6 (WX)j(G), and fixed («J0 , XQ , a).On the one hand, it is clear that

On the other hand, (4.2), (4.3) imply that e0E&({Tn'}, £p(0)} for p > 1;so that, by(4.10),

27 IK Ordinary Differential Equations

Thus, (4.7) and (4.8) give, as t -+ +0,

uniformly for (a>, #) 6 (WX^G), for fixed (a»0 , xti , a). This proves Theorem3.1.

5. PROOF OF THEOREM 3.2j

Since the assertions to be proved (that is, the existence and continuityof D(wxv) y0 near a point (a?1, x1, v1)) are local, we can restrict our attentionto v near w1. It will also be clear that the only e —• (a>, x, y) which comeinto consideration are e E £s(0), where s (<r) is chosen so that Es(0) containsall of the iterates en = en(w, x), n — 1, 2,..., for (a>, x, v) near (a?1, x1, vl).Thus we can suppose that there is a constant C such that

on Ef X F° X A. By replacing z; by a new variable a1 + ew, we can supposethat C is arbitrarily small, that V° is centered at 0. We can also supposethat V° = F.(0).

In what follows, we consider the space W to be replaced by the spaceVW, and deal with a sequence of maps Tn , T2l,..., depending on a param-eter A, from (VE),(0) into F£ = VWXY,

where the linear maps Kn , Anl = An(Q, A), and Bnl — Bn(Q, A) depend onlyon A, and

Note that the linear map (v, w) —»• («;, Knw) has a norm 1, and that a Lipschitzconstant for Fnl, Gnl with respect to (v, e) is S2 -}- 2C -f- Cs S'2. Since Cis arbitrarily small, we can suppose that the inequalities (1.4) hold if 8 isreplaced by 8' > 0.

It follows from Theorem 3.1 that the set ®({Tnl}, (F£)s(0)) is a manifoldy = y<>(w' x> *>) = y<fc°> x> ». A)» (w> x> v) £(VWX)S(Q), and that D(w^v)y0exists and is a continuous function of (a>, x, v, A). By the analogues of (2.1),(2.2), we also have that if e0 — (w, x, y0(w, x, v)) and en — Tn(en_^), so that(v, en) = Tnl(v, €„_,), then || D(w^v)en || < 1 on (VE)S(0) X A. This provesTheorem 3.22.

Invariant Manifolds and Linearizations 271L

6. PROOF OF THEOREM 3.2fc, k > 1Let k > 1 and Theorem 3.2fc_1 hold. For a moment, assume that no

parameters v occur in Tn. Consider the map (3.2), (3.3), where KnQ = Kn

depends only on a — A. The map rn0 depends on parameters («;0, x0, A)and, on any given sphere (Q3H)P(Q), satisfies the conditions of Theorem3.2fc_!, in which (zv0, XQ) plays the role of the parameter v. Thus, byTheorem 3.2k_i, the function (3.4) is of class C*1"1 with respect to(to, £, «>0 , x0) with derivatives continuous in (w, £, w0 , x0 , A) and thespecified boundedness properties. Theorem 3.2^ then follows, for the caseunder consideration, from (3.5).

In the case that parameters v do occur in Tn , apply the same argumentsto the maps belonging to (5.2) in the same way that (3.2)-(3.3) belongsto (1.2). This completes the proof.Notes

The idea of reducing the study of the behavior of solutions of ordinary differentialequations to the study of maps is due to Poincare; cf. e.g., VII Appendix. In particular,in connection with periodic solutions of autonomous systems, see Poincare [3, IV] or[5, III, chap. XXVIII. In the latter context, he introduced the concept of invariantmanifolds of a map (actually, "invariant curves" in his case). Poincare's fundamentalidea of studying maps in the theory of differential equations was further exploited byG. D. Birkhoff and has led to a large body of research associated with the name of"dynamical systems" or "topological dynamics."

SECTIONS 5-6. In the case where y0, z0 are 1-dimensional and the map T is analytic,Lemma 5.1 is due to Poincare [3, IV, pp. 202-204] or [5, III, chap. XXVII]. Thecorresponding result where 7 is C1 is due to Hadamard [2] (who, however, did not showthat his invariant curve is of class C1) and when 7 is C1 (or as in Excerise S.I), it is dueto Sternberg [1]. D. C. Lewis [1] extended Hadamard's method to the case of arbitrarydimension when A, C have simple elementary divisors, carrying out the details in theanalytic case. Lemma 5.1 was stated by Sternberg in [3] where he indicated a ratherincomplete sketch of the proof based on the successive approximations (5.13). Theproof in the text, using these approximations, is taken from Hartman [20] (cf. [28]).(The proof contained in Exercises 5.3, 5.4 may be new. These exercises give a simpleproof of the assertion in Exercise 5.4(6) due to Coffman [2]; a weaker existenceassertion and the uniqueness statement in Exercise 5.4(^0 are contained in Perron [13];Exercise 5.3(6) is in Coffman [2].). The analogous result for differential equations(i.e., Theorem 6.1) is due to Lyapunov [2, p. 291], under conditions of analyticity,and to Coddington and Levinson [2, p. 333] in the nonanalytic (and, more general,nonautonomous) case; cf. § X 8 and the reference there to Petrovsky [1J.

SECTION 7. Theorem 7.1 with F, R, R~l analytic was proved by Poincare in 1879(see [1, pp. xcix-cx]) under the assumptions that the elementary divisors of fare simpleand that the eigenvalues Aj Ad of E lie in an open half-plane, Re ei6A > 0, of thecomplex A-plane and (*) A, 9* m^i + • • • + m<h for all sets (mlf..., md) of non-negative integers satisfying m^ + • • • + md > 1. For an analogous result for smooth,but nonanalytic F, R, R~l, when Re A, < 0, see Sternberg [3]; cf. Exercise 8.2(6).For Re A, < 0, Fe C2 and R, R~l E C1, but without Diophantine conditions of type(*), see Hartman [20]; cf. Exercise 8.1 and 8.2(a). When it is not assumed thatA , , . . . , Aj lie in an open half-plane Re eldA > 0, but Re A, ?* 0, the problem for analytic

F, R, R~l has been considered by C. L. Siegel [2] under conditions stronger than (*).For smooth, but nonanalytic, F, R, R-1, see Sternberg [4], Ise and Nagumo [1], andChen [1]; cf. Appendix. Theorem 7.1 is due to Grobman [1J, [2] and to Hartman [28],[21] with different proofs.

SECTIONS 8-9. The 1-dimensional case of Lemma 8.1 (i.e., y0 is 1-dimensional and20 is absent) with T, R, R-1 analytic goes back to Abel [1, II, pp. 36-39]; Schroder [1],[2]; and Koenigs [1], [2]; see Picard [3, chap. IV]. For a similar treatment of the1-dimensional, smooth (but nonanalytic) case, see Sternberg [2]. The case of a con-traction (i.e., y0 of arbitrary dimension d and z0 missing) with 7", R, R'1 analytic wastreated by Leau [1] under the assumption that the eigenvalues otj,. .., ad of A satisfya, ^ a™ia™z . . . a™d for all sets of non-negative integers ( / M I , . . . , md) subject tomi + • • • + md > 1. Lemma 8.1? as stated, is due to Hartman [21], [28].

The papers of Sternberg [3], [4] and Hartman [20], [21], [28] mentioned in connectionwith § 7 are relevant here and, in fact, deal principally with maps. The device involving(9.1) which permits the deduction of Theorem 7.1 on differential equations from Lemma8.1 on maps is due to Sternberg [3] and is used in these related papers. Sternberg [3]also considers the question of normal forms for RTR~* different from the linear oneswhen there are relations a> = <x™ia™s . . . a™<» and generalizes results of Lattes [1],[2] on 2-dimensional analytic maps. For related papers, see also Sternberg [5], Moser[1], C. L. Siegel [3], and Chen [1]; cf. Appendix. A recent important paper of J.Moser [3] is related to the critical case not considered here when some eigenvalues ofthe linear part of T have absolute value 1 and concerns the problem of the existence ofclosed invariant curves.

SECTIONS 10-11. As mentioned earlier, the principal results of these sections are dueto Poincare in the analytic (3-dimensional) case. Their validity in more general casesdepends on the extensions of Poincare's results on maps given in the earlier sections ofthe chapter.

APPENDIX (SECTIONS 12-14). Theorem 12.1 is due to Sternberg [4] and its general-ization, Theorem 12.2, to Chen [1]. (Particular normal forms other than linear oneshad also been considered by Lattes [1], [2] and by Sternberg [3].) The proof in the textfor Theorem 12.2 is based on Chen [1 ] which, in turn, is a modification and simplificationof Sternberg [4]. Chen's improvement consists essentially of noting that Sternberg'sprocedure is valid without first using the result in Sternberg [3] to obtain a linearizationon the invariant manifolds. An admissible choice of A = N — n in Theorem 12.1 isgiven by Sternberg [3] for the case of contraction maps; for n = 1 and the general caseof differential equations, see Ise and Nagumo [1]. Exercise 12^2 is due to Chen [1],

*The proof in the text is a simplification of that of Hartman [21], [28]due to Pugh [SI].

Chapter X

Perturbed Linear Systems

This chapter is concerned with methods for the asymptotic integrations ofdifferential equations f' = ££ 4- F(t, £) which can be considered asperturbations of linear systems with constant coefficients £' = ££.

The first section of the chapter concerns the simple but important caseE = 0. Since a very easy argument, which has wide applications, gives thedesired result in this case, it seems worth isolating it.

One of the most important methods to be used for an arbitrary E isbased on a simple topological principle, discussed in §§ 2-3. This principlehas wide applications beyond the scope of this chapter. A very differentmethod for obtaining results analogous to those of §§ 13 and 16 is discussedin Part III of Chapter XII.

In this chapter, for convenience and generality, we shall allow thecomponents of f to be complex-valued, so that a linear change ofcoordinates permits the assumption that £ is in a suitable normal form;cf. § IV 9. Correspondingly, if f1} £2 are two vectors, then |j • £2 denotesthe scalar product ]£ ^ikizk-

k

1. The Case E = 0

This section concerns the equation

where Fis "small" in a suitable sense. The main results are the following:Theorem 1.1. Let F(t, |) be continuous for t 0, ||£|| < <5(= °°) and

satisfy

where y>(f) is a continuous function for t 0 such that

273

If ||£0|| is sufficiently small, say

then a solution £(/) of (LI) satisfying £(0) = £0 exists for t 0. Further-more, //£(/) w fl solution of (LI) for large t, say t f0> ^*ew

Ufuft fl/if/ loo 5^ 0 wrt/m £(/) = 0.In other words, the solutions of (1.1) for large / behave like the solutions

of £' = 0, namely, like constants. Theorem 1.1 has the following extension:Theorem 1.2. Let F(t, £) be as in Theorem 1.1, and let g^bean arbitrary

vector such that

Then (1.1) has at least one solution £(t) for t^Q satisfying (1.5). If, inaddition, F(t, £) satisfies the following type of Lipschitz condition

then for a given £«,, there is at most one solution £(t) of (LI) which existsfor large t and satisfies (1.5).

The last part of Theorem 1.2 states that condition (1.7) establishes aone-to-one correspondence between solutions £ = £,„= const, ofI' = 0 and solutions of (1.1), with the understanding that H l ^ H is suffi-ciently small when 6 < oo.

Proof of Theorem 1.1. Multiply (1.1) scalarly by £, so that (1.1), (1.2)imply that

Since d || |(OII2/dt = 2 Re £ • £', a quadrature gives

if £(/) exists on a /-interval containing t and f0» where 11 /0. In particular,if /0 = 0 and £(0) = £0 satisfies (1.4), then £(/) exists for f ^ 0.

More generally, if £(t) exists for t /0, then it is bounded,

Hence, (1.1), (1.2) show that ||£'(OII ^ V<0 llf('o)ll M'o)- Consequently,

Perturbed Linear Systems 275

I £(t)dt is absolutely convergent and so the limit (1.5) exists. In fact

Note that the inequality (1.9) shows that £(/) = 0 if and only if £(/)vanishes at some point tQ. When £(/) & 0> the first inequality in (1.9)implies that £ „ 5^ 0. This proves Theorem 1.1.

Proof of Theorem 1.2. Consider first the existence assertion. For agiven /0 0, let £ = £(/, /0) be a fixed solution (which is not necessarilyunique) of the initial value problem

Since (1.9) holds for any t at which £(f, /„) exists, it follows from (1.6) and£(>o) = £oo> that £(/, f0) exists for / £ 0. Also, ||£(r, /0)|| ^ ||£J| A/(0) < (5,where M(0) is defined in (1.10). Hence (1.2) shows that ||£'(/, /0)|| V<0 II£00II ^(0) for all / 0; thus, for 0 / < /0,

In particular, the family of functions £(/, f0) are uniformly bounded andequicontinuous on every bounded /-interval. Hence there exists a sequence'i < '2 < • • • of /0-values such that /„ -> oo as n -* oo, and

exists uniformly on every bounded /-interval. Furthermore £ = £(/)is a solution of (1.1). Putting /0 — tn in (1.13) and letting n -*• oo, with /fixed, gives

This implies (1.5) and completes the existence proof.Uniqueness will now be proved under the assumption (1.7). Let

£ = £i(0» £2(0 be two solutions of (1.1) for large /, say / T, satisfying(1.5). Let £(/) = &(/) - £2(/). Then (1.1) and (1.7) give (1.8), hence (1.9)for t0 > / ^ T. If / is fixed and t0 -» oo in (1.9), it follows that £(/) = 0since £(/„) ->• 0 as /0 -> oo. This completes the proof of Theorem 1.2.

The majorant y(/) ||£|| in (1.2) involving a factor ||£|| is convenient inTheorems 1.1 and 1.2 only to assure that certain solutions exist for / 0.A simpler result involving existence for "large /" is given in the followingexercise.

Exercise 1.1. Let F(t, £) be continuous on a product set {/ 0} x D,where D is a bounded open £-set. Let F satisfy ||F(/, £)J < y(/), / ^ 0

and £e D, for some continuous function y>(/) satisfying (1.3). (a) Let£0 e D. Then there exists a number T, depending only on dist (£0, dD)and the function y>(0, such that if tQ T, then a solution £(f) of (1.1)satisfying £(*„) = £0 exists for t^T. Furthermore, any solution £(f) of(1.1) for large t has a limit £«, as /->• oo. (6) Let £«, e D. Then thereexists a number T, depending only on dist (£„<,, 3D) and the functiony>(0> such that (1.1) has a solution £(/) for / ^ T satisfying (1.5).

Exercise 1.2. Show that the solutions of (1.1) exist for/ 5:0 andare bounded if jF(f, £) is continuous for / 0 and all £ and if (1.2) isreplaced by

where y(r) is as in Theorem 1.1 and <p(r) is continuous for r 0 andsatisfies

Theorem 1.1 and 1.2 have corollaries for the case that (1.1) is re-placed by

where A(t) is a continuous d x d matrix. Here, solutions of (1.15) shouldbe compared with

Let Z(0 be a fundamental matrix for (1.16), so that the change of variables


Thus an application of Theorems 1.1 and 1.2 to (1.18) givesCorollary 1.1. Let A(t) be continuous for t 0 and Z(f) a fundamental

matrix for (1.16). Let G(t, 0 &e continuous for t 2t 0 o/w/ a// ^ am/ satisfy

wAere y(0 /J as w Theorem 1.1. Lef $(/) 6« a solution of (1.15) on somet-interval Then t,(i) exists for t 0,

exists and £«, 5^ 0 wn/ew ^(0 = 0; conversely, for a given !«,, f^ere is aw&iffoi ^(0 o/ (1.15) satisfying (1.20).

When Z(r) is bounded for / 2: 0, we can formulate a correspondingresult when G(t, £) is only defined for t 0, ||£|| < 6 < oo. In addition,we can obtain an analogue of the uniqueness assertion of Theorem 1.2.When A(i) = A is a constant matrix, Corollary 1.1 takes the followingform:

Corollary 1.2. Let G(t, £) be continuous for t ^ 0 and all £ and satisfy

where y(/) is as in Theorem 1.1. Let £(0 be a solution of

OH jome t-interval. Then £(f) exists for t 0,

existe anrf £„„ ,<£ 0 un/m £(0 = 0; furthermore, if £„ is given, there is asolution of (1.22) for t i> 0 satisfying (1.22*).

Exercise 1.3. Formulate theorems related to Corollaries 1.1 and 1.2as Exercises 1.1 and 1.2 are related to Theorems 1.1 and 1.2.

Generally, a result of the type given by Corollary 1.2 is only convenientwhen e±At are bounded for / 0. For example, suppose that d = 2 andA = diag [1, -1], so that eAt = diag [e\ <?-']. Then, if Corollary 1.2is applicable, (1.22) has a solution of the form £ = e\\ + o(l), 0(1)) as/ ->• oo, but not necessarily a solution of the form £ = e~*(o(l), 1 + o(l))as / -> oo. Furthermore the hypothesis (1.21) can be very severe for thetype of conclusion stated in Corollary 1.2. The results obtained in theremainder of this chapter are much better, under less stringent conditions,for the situation just described.

Exercise 1.4. Suppose that (1.1) is a linear homogeneous system, say

where G(t) is a continuous matrix for t 0. The system (1.23) will besaid to be of class (*) if (i) every solution |(f) of (1.23) has a limit £«, ast -*• oo, and (ii) for every constant vector £«,, there is a solution f(f) of(1.23) such that f(f) -> £«, as / ->- oo. (a) Show that (1.23) is of class (*)if and only if, for one and/or every fundamental matrix Y(t) of (1.23),Y^ = lim Y(t) exists as t -*• oo and is nonsingular (and that this is true if

/•OO

and only if Yx = lim Y(t) exists as t -» oo and | tr G(s) ds converges,

possibly conditionally), (b) The system (1.23) is of class (*) if and only ifthe adjoint system £' ?= —G*(t)£ is class (*); cf. § IV 7. (c) The system

(1.23) is of class (*) if f *||G(r)|| dt < oo [or, equivalently, if G(t) =/•« J

(&*(*)) and \gjlc(t)\ dt < oo for j, k = 1 , . . . , d\. This is merely a

consequence of Theorems 1.1, 1.2. (</) Show that (c) has the followingcorollary [which is a refinement of (c)]: The system (1.23) is of class (*)

f°°if G0(/) = 1 G(s)ds converges (possibly just conditionally) and either

JV(')Co(OII dt < oo or jVoWCWII A < oo.

2. A Topological Principle

Let y,/ be ^/-dimensional vectors with real- or complex-valued com-ponents and/(f, y) a continuous function defined on an open (t, y)-set Q.Let Q° be an open subset of Q, 9Q° the boundary and Q° the closure ofQ°. Recall, from § III 8, that a point (/„, y0) e Q n dQ° is called an egresspoint of Q°, with respect to the system

if for every solution y = y(t) of (2.1) satisfying the initial condition

there is an e > 0 such that (t, y(t)) e Q° for /0 — e r. < /„. An egresspoint (/0, y0) of Q° is called a strict egress point of £1° if (f, y(f)) £ &° forr0 < / r0 + e for a small e > 0. The set of egress points of £1° will bedenoted by Qe° and the set of strict egress points by Q£.

If U is a topological space and V a subset of U, a continuous mappingIT: U-+ V defined on all of U is called a retraction of J7 onto V if therestriction TT | V of IT to F is the identity; i.e., TT(M) e V for all « e U and7r(y) =s y for all v e F. When there exists a retraction of U onto F, K iscalled a retract of U. This notion can be illustrated by the followingexamples, which will have applications.

Example 1. Let U be a ^/-dimensional ball ||y|| ^ r in the Euclideany-space and V its boundary sphere ||y|| = r. Then V is not a retract of U.For if there exists a retraction TT: U-+ V, then there exists a map of Uinto itself, y-*- —ir(y), without fixed points, which is impossible by theclassical fixed point theorem of Brouwer; for the latter, see Hureiwiczand Wallman [1, pp. 40-41].

Example 2. Let C be the "cylinder" which is the product space ofa Euclidean sphere ||y|| = r and a Euclidean w-space, so that C ={(y, u): \\y\\ — r,u arbitrary}. Let S be a section of C, say, S ={(y, «o): llyll ^ r, u0fixed}; see Figure 1. Then S n C = {(y, «0): ||y|| = r,

s

«0 fixed) is a retract of C [as can be seen by choosing the retractionir(y, w) = (y, "o)l> but S n C is not a retract of S by Example 1.

Theorem 2.1. Letf(t, y) be continuous on an open (/, y)-set Q with theproperty that initial values determine unique solutions of (2.1). Let Q° be

Figure 1.

an open subset o/Q satisfying Qe° = Q°e; i.e., all egress points 0/"Q° arestrict egress points. Let Shea nonempty subset o/Q° U Qe° such that S niie° is not a retract of S but is a retract of Qe°. Then there exists at leastone point (f0, y0) e S n Q° such that the solution arc (t, y(t)) of (2.1), (2.2)is contained in Q° on its right maximal interval of existence.

Figure 2.

As an illustration, consider (2.1) where y is a real variable and/(f, y)is continuous on Q: (/, y) arbitrary. Let Q° be a strip \y\ < b, — oo <t < oo; see Figure 2. Thus the part of the boundary of &° in Q, i.e.,d£}° n Q, consists of the two lines y = ±b. Suppose that/(f, b) > 0 and/(*, -6) < 0, so that Qe° = Q«e = dQ° n Q. Let S be the line segmentS = {(*, y): / = 0, \y\ b}. Then 5 n Qe° is the set of two points (0, ±b)and is a retract of Qe° but not of S. Thus it follows from Theorem 2.1,that there exists at least one point (0, y0), |y0| < b, such that a solution of(2.1) determined by y(0) = y0 exists and satisfies \y(t)\ 0.

Proof of Theorem 2.1. Suppose that the theorem is false. Then for('o» 2/o) ^ S — Qe°, there exists a fx = /x(/0, y0) such that ^ > r0 and the

solution y(/) of (2.1), (2.2) exists on /„ / tlt (t, y(/)) e fl° for /„ / < /! and (/!, y('i)) e Q«° for t — tt. Define a map TTO : S -»• Qe° as follows:TO('O»yo) = (/i,K'I)) if (f0,yo)eS-{V and TTO(/O,y0) == (/0,y0) if(fo» y<>) e S n £W Since the solutions of (2.1) depend continuously oninitial conditions (Theorem V2.1) and Qe° = Q°e, it follows that TTO iscontinuous. In order to see this, let y(t) = »;(/, /°, y°) be the solution of(2.1) such that rj(t°, t°, y°) = y°, so that r)(t, t°, y°) is continuous. Supposethat (f0, y0) e S n Q°, and (f°, y°) is near (/0, y0), then ??(/, f °, y°) exists onthe interval [/°, /^/o, y0) + e] for some e and (/, rj(t, r°, y°))e O° on'* î(^ ^0) - « and (/, iy(/, /•, )) $ n° if t = rô, y0) + c. Thus,l^°, ^°) - 'i(4» ô)l < c, and so (tlt rfc^t9, y°), t°, y°)) is a continuousfunction of (t°, y°); i.e., rr0 is continuous at (f0, y0). A similar argumentholds if (foêSna.0.

Let TT: Qe° ->- 5 n Qe° be a retraction of Qe° onto S n Qe°. Then thecomposite map TTTTQ is a retraction of 5 onto S n Qe°. The existence ofsuch a retraction gives a contradiction and proves the theorem.

Exercise 2.1. Let U be a topological space; Vlt Vt subsets of U. Theset Kj is called a quasi-isotopic deformation retract of Vz in f/ if thereexists a continuous map TT : V2 x {0 * 1} -*• U such that (i) TT(VZ, 0) =r2 for vt E K2; (ii) w^, j) = yx for e Vl and 0 ^ 5 ^ 1; (iii) ir(v2, 1) eF! for y2 6 F2; and (iv) for fixed j on 0 5 < 1, TT(VZ, s) is a homeomor-phism of Vz onto its image. Let/, Q, Q° be as in Theorem 2.1; S^ asubset of De°; 5" a nonempty subset of Q° U 5t such that St is not aquasi-isotopic deformation retract of S U 5j in Q° U 5j. Then thereexists at least one point (/„, y0) e S n Q such that the solution arc (/, y(r))of (2.1), (2.2) is either in D° on its right maximal interval of existence orfirst meets dQ.° at a point of S — S^

3. A Theorem of Wazewski

The usefulness of Theorem 2.1 depends on suitable choices of Q°.One of the difficulties in the application is the determination of the set ofegress points. In some cases to be described, this difficulty can be overcome.

Recall from § III 8 that a real-valued function u(t, y) defined on an opensubset of ii is said to possess a trajectory derivative u(t, y) at the point(>o» y0) along the solution y(t) of (2.1)-(2.2) if u(t, y(t)} has a derivative att = t0; in this case,

If y (hence/) has real-valued components and «(/, y) is of class C1, thistrajectory derivative exists and is

where the last term is the scalar product of/ and the gradient of u withrespect to y.

When y has complex-valued components, a function u(t, y) is said to beof class C1 if it has continuous partial derivatives with respect to / and thereal and imaginary parts of the components of y. Write the kih componenty* of y as yk = <r* + IT*, where cr*, T* are real, so that a* = (y* + y*)/2,T* s=s (y* — yk)f2i. This suggests the standard notation, dujdy* =ftdulda* - i a«/ar*] and duldy* = |[a«/a<r* + i dufd-r*]. Thus if gradv M =(dufdy1,..., aw/a/) and grad^ u = (aii/Sy1,..., dujdjj*), then (3.2) shouldbe replaced by

as can be seen by writing (2.1) as a system of Id differential equations for(<r, r) = (a\ . . . X, r\ . . . , rd).

An open subset fl° of D will be called a (u, v)-subset of Q with respectto (2.1) if there exists an (arbitrary) number of real-valued continuousfunctions, u^t, y),..., ut(t, y), v^t, y),..., vn(t, y), on Q such that

and if f/a, F^ are the sets

Vm = {( , y): «a(f, y} = 0 and M,(f, y) 0, ufc(f, y) 0 for all;,'k},

then the trajectory derivatives wa, y^ exist on C/a, F^ and satisfy

respectively, along all solutions through (t, y). In this definition, either /or m can be zero.

Lemma 3.1. Let f(t, y) be continuous on an open (/, y)-set Q and Q° a(it, v)-subset of£l with respect to (2.1). Then

Proof. It is clear that dQ° n Q c ((J £/a) u (y j/p. In addition, Oe° nK^ is empty, for if (f0, y0) e V^ and y(f) is a solution of (2.1), (2.2), then(3.6) shows that vft(t, y(t) > 0 for rc — e < / < t0 for small c > 0, so that(r,y(/))^Q°. Thus

Let (/0, y0) e \J ua - U *V Then H/*O, y0) 0 and vk(t9, y0) < 0 forall y, A:. By (3.5), there is an > 0 such that wa(/, y(f)) < 0 or > 0according as /„ - € / < /„ or /„ < / <; t0 + e if (/0, y0) e l/a, «,(>> y(0) <0 for t0 - € < t < /0 + f if ('o, 2/o) £ ty, and i?fc(f, y(0) < 0 for /0 - e r f0 + « for all A:. Hence (;„, y0) e QJe; i.e., (J £/a - U c ^Se- Inview of (3.8), this proves the lemma.

Theorem 3.1. Letf(t, y) be continuous on an open (t, y)-set ti with theproperty that solutions of (2.1) are uniquely determined by initial conditions.Let £2° be a (u, v)-subset of£l with respect to (2.1). Let S be a nonemptysubset 0/"Q° U £le° satisfying S C\ Qe° is not a retract of S but is a retractof ne°. Then there exists at least one point (t9, y0) e S n O° such that asolution arc (t, y(t)) of (2.1), (2.2) is contained in Q? on its right maximalinterval of existence.

This is a corollary of Theorem 2.1 and Lemma 3.1. Sometimes, therequirement of the uniqueness of the solution of (2.1), (2.2) can beomitted:

Corollary 3.1. Letf, Q, £1°, S be as in Theorem 3.1. except that it is notrequired that solutions of (2.1) be uniquely determined by initial conditions.But, in addition, let S be compact and let Uj(t, y), vk(t, y) be of class C1

(with respect to t and the real and imaginary parts of the components ofy).Then the conclusion of Theorem 3.1 is valid.

Proof. Letyi(f, y),fz(t, y),... be a sequence of functions of class C1 onii which tend to/(f, y) uniformly, as n ->• oo, on compact subsets of Q.Let Qj, Q2 , . . . be a sequence of open subsets of ft, such that S <=: Qj,fln has a compact closure &n <= On+1, and ii = U ftB.

By replacing/j,/2,... by a subsequence, if necessary, it can be supposedthat

Thus if iitt° = ft0 n Qn, then ftw° is a (u, y)-subset of iin with respect tothe system

The set of (strict) egress points Qj^ of Qn° is Q.° n 1 ,. Hence iije n 5 =a.b n 5 is not a retract of S, but Q^ n 5 is a retract of Q^ c Qe«.

Thus by Theorem 2.1 there is a point (/„, yj e 5, such that the solutiony = yn(t) of (3.9) satisfying yn(/J = yn is in QB° on its right maximalinterval of existence [tn, TW) relative to Qn. If (3.9) is considered on Q,instead of Qn, let the right maximal interval of existence of yn(t) be[/„, o>w), so that rn o)n ^ oo and rn < con implies that (rn, yn(rn)) c8DM n £1

Since S is compact, there is a point (f0, y0) on 5 which is a clusterpoint of the sequence of points (fj, yj), (f2, y2), By Theorem II 3.2,there exists a solution y(r) of (2.1), (2.2) having a right maximal intervalof existence [t0, ca) and a sequence of integers n(l) < n(2) < • • • such thatyn(k)(t) -*• 2/(0 uniformly as, k -»• oo, on any interval [f0, f *] c [f0, o>).

It follows that (/, y(0) <= £° n Q for /„ / < w. For suppose thatthere is a /-value t°, tQ<t° < w, such that (f°, y(r°)) £ H°. Then, forn = «(£) and large k, (t°, yn(f °)) $ &°, so that (/°, yB(f °)) £ Qn°. Hencern < t° < o)n for n = n(k) and large A:. By choosing a subsequence, ifnecessary, it can be supposed that T = lim rn(k} exists as k -> oo, so that>o T < /° and (TB, yn(rn)) -* (T, y(r)) as n = n(k) -+ oo. But this givesa contradiction for (TB, yw(rn)) e 3Qn n Q, where /i =« «(A:), cannot havea limit point (T, y(r)) e (1

Since (f0, y0) e S c Q° n ne° and Qe° ~ Q , it is seen that (/„, y0) e Q°,otherwise (t, y(t)) $ ii° for r0 < f ^ /„ -f « for some e > 0. By the sameargument, (/, y(r)) cz Q° for /„ / < w. This proves the corollary.

4. Preliminary Lemmas

The theorems of § 3 will be illustrated by using them to obtain resultsabout the asymptotic integrations of

where £ is a constant matrix and F(t, f) is "small", say,

and y(/) is "small" for large r In this section, we state the basic Lemmas4.1, 4.2, 4.3. Their proofs are given in §§ 5-7 using the results of §3.Theorems on the asymptotic integration of (4.1) are stated in §§ 8,11, 13and 16 and are deduced, respectively, from Lemma 4.1 in §§ 9-10, fromLemmas 4.1-4.2 in § 12, and from Lemmas 4.1-4.3 in §§ 14-15.

If E has at least two eigenvalues with distinct real parts, we can suppose,after a linear change of variables with constant coefficients, that E =diag [P, Q], I = (y, 2), ££ = (Py, Qz), where dim y + dim z = dim &the real parts of the eigenvalues pi,p* • •., q\, q* . . . of P, Q satisfy

for some number ft. We can also assume that P, Q are in a suitablenormal form (cf. § IV 9), so that for an arbitrarily fixed e > 0 and some c,


we have the inequalities

Correspondingly, write (4.1) in the form

where F = (Flt F%). The initial conditions will be of the form

When (4.2) holds, (4.5) and (4.6) give

Sometimes, it will be convenient to suppose that E = diag (Alt AZ) Aa),£ = (*, y, 2), E£ — (Ai*, A$, A&), where the eigenvalues a,l5 a,2,... ofAJ satisfy

where (4.4) holds. Correspondingly, it will be supposed that

The initial value problem to be considered is

where F(ty |) = (Fl9 F2, F3), and

When (4.2) holds, (4.10) and (4.11) imply that

In what follows, x, y, z are (real or complex) Euclidean vectors;£ = (y,z) or £ = (*,y,z) and F=(FltF2) or F=(F1,F2,F3) areEuclidean vectors in the corresponding product space. The first lemmarefers to (4.6) and (4.7); the last two, to (4.11) and (4.12).

Lemma 4.1. Let p, e, c be constants andP, Q constant matrices satisfying(4.4H4.5). Let F(t, I) = (J^, FJ be continuous for t~£ 0 and \\y\\,

|| 21| < 6 (_ oo) and satisfy (4.2), where v>(0 > 0 is continuous for t _ 0,and

converges, so that there exists a T 0 JMC/Z f/rc/

Lef r0 > TandQ < ||y0|| < 6, Then there exists at least one z0, \\z0\\ < 6,such that (4.6)-(4.7) has a solution y(t), z(t) satisfying

on its right maximal interval t0 / < G> (^ oo). In particular, if the rightside of (4.11) is less than 6 for t _ f0> then to = oo.

The last assertion is a consequence of Corollary II32. The other partsof Lemma 4.1 will be proved in § 5.

Lemma 4.2. Let ft, e, c be constants and A^ Az, A3 constant matricessatisfying (4.4) and (4.10). Let F(t, f) = (Flf Fz, F3) be continuous fort ^ 0 and \\x\\, \\y\\, \\z\\ < 6 (^ oo) arid satisfy (4.2). Let y(0 > 0 becontinuous for t > 0,

converges, and let there exist a T 0 such that

Ler r0 > T, ||*0|| < 7ff(/0) ||y0||, 0 < ||y0|| < (5. Then there exists at leastone z0, \\z0\\ < d, such that (4.11)-(4.12) has a solution x(t), y(t\ z(t)satisfying

on its right maximal interval of existence t0^ t <. a>(< oo). In particular,if the right side of (4.21) is less than dfor t ^ fc, then 00=00.

In applications of Lemmas 4.1 and 4.2, it is convenient to know when

This is the case if

or, more generally, if, as / -*• oo,

Holder's inequality shows that a sufficient condition for (4.24) [hence, for(4.22)] is that

(4.25)

Actually, the next exercise states that if y > 0 and c — c > 0, then (4.24)is necessary and sufficient for (4.22) to hold.

Exercise 4.1. Let y(/) j> 0 be continuous for / 0 and c — « > 0.(a) Show that (4.24) implies (4.22). In fact, if <$(/) denotes the "sup" in(4.24), then

(b) Conversely, show that if either a(t) -»• 0 or r(t) -> 0 as / ->• c», then(4.24) holds.

Exercise 4.2. Let y>, c — e be as in Exercise 4.1 and let (4.25) hold for

some/?, 1 < p < 2. (a) Show that

Conclude that

Exercise 4.3. Show that Lemma 4.2 remains valid if c = (/) , c = c(t),H = ^u(;) are continuous functions of f for / 0 satisfying (4.4), (4.10)and if (4.18) is replaced by

where it is assumed that the last integral converges and (4.19) holds forsome T.

Lemma 4.3. In addition to the assumptions of Lemma 4.2, assume thaty(0 satisfies (4.24) [so that (4.22) holds]; that p = 0, e = 0 in (4.13);/Aaf an equality of the form

holds, where y>0(/) is a continuous function for t ^ 0 satisfying

finally, that

T/te» co = oo in the assertion of Lemma 4.2,

ejcis/s, a«</ y^ ^ 0. Furthermore there exist d1 > 0, T > 0 and, for everyt0> T, a positive constant (5a(/0), SMC/I f/iaf // y^ ^ 0, #0 are g/ue/i vectorsand ||y J < <$!, ||g0|| < 52 lly.ll, /Ae« there exist y0

flw^ 2o such that(4.11H4.12) /ia5 a solution for t /0 satisfying (4.20) am/ (4.29). [H^e/id = oo, <$! can fe ?aA:e« ?o 6e oo.]

Remark 1. The proof of this lemma will show that there exists aconstant C depending only on the integral of y0(f) over /„ / < oo suchthat the solutions mentioned satisfy

where / 10 and y+ can be either y0 or y^.Remark 2. In the proof of the first part of Lemma 4.3, the inequalities

(4.13) with IJL = = 0 and (4.26) need not hold for all ||£|| < 6. For inview of (4.21), the proof will involve only y satisfying cx ||y0|| < ||y|| <c2 ||y0ll» hence

by (4.20), where

Correspondingly, (4.13) and (4.26) need only be assumed when (4.30)holds. In the second part of Lemma 4.3, the same remains true if (4.30)is replaced by

These assertions permit the replacement of the assumptions (4.2) and(4.26) in the derivation of (4.13) by another type of hypothesis: For a

pair of numbers r, R satisfying 0 < r < R (^ oo), let there exist a con-tinuous function <prR(t) > 0 for / 0 such that

Then (4.33) implies that

which is the analogue of (4.2) with

Notice that with this choice of v>(0 and y>0(0 — v(0» (4-31) shows thatCj -> 1 as T-> oo. Hence if r < \\y0\\ < R/3 [or r < ||yj| < J?/3], thenthe first part [or last part] of Lemma 4.3 remains valid.

The case u, 7* 0 can be reduced to fj, = 0 by the change of variables£ = e^t, [when u > 0, it is necessary to assume that F(t, £) is defined for/^Oanda/ /£] :

Corollary 4.1. In addition to the assumptions of Lemma 4.2, assumethat v(0 satisfies (4.24) [so that (4.22) /io/<fo] a/if/ that an inequality of theform

holds, where y)Q(t) is a continuous function for t 0 satisfying (4.27). /jfp > 0, assume that d = oo (50 f/ia/ F w defined for t 0 #«</ a//1). T/ren/Ae assertions of Lemma 4.3 remain valid if (4.29) is replaced by

Exercise 4.4. Verify Corollary 4.1.The condition ^42 = pi can be replaced by the assumption that A2 is

a diagonal matrix (or has simple elementary divisors) and that all of itseigenvalues have the same real part p:

Corollary 4.2. Let the assumptions of Corollary 4.1 hold except thatAz = diag \ji -f 'Vi, /* + 'Va, • • • L vv/zere yl9 y2 , . . . are real numbers, and(4.37) is replaced by

Then the assertions of Lemma 4.3 remain valid if (4.29) is replaced by

Note that the last part of (4.40) means that the kth component y*(/) ofy(t) satisfies e~(tt+iY" V(0 -* oo* as / -> oo.

Exercise 4.5. Reduce Corollary 4.2 to Lemma 4.3 by the change ofvariables (x, y, z) —> (w, v, w) given by x = e^u, y = eA**v, z — e^lw.

Exercise 4.6. Let £ be a constant matrix with eigenvalues A x , . . . , Ad

such that A j , . . . , Afc are simple eigenvalues with Re A3 = Ofory = 1 , . . . , kfor some k, 1 k ^ d. Of the eigenvalues Afc+1,. . ., Ad, let m havepositive real parts, n negative real parts, where 05j /n , n d — k andm + n = d — k. Let G(t) be a continuous matrix for t 0 such thatG(t) -*• 0 as / -»• oo and the elements gw(0 are °f bounded variation for

/ •GO

f > 0 (i.e., I4fi3(0l < oo). For example, if G(t) is continuously differen-J TOO

tiable, let G(t) -* 0 as f -* oo and ||(/'(OII * < °°- For large '»the matrix

£ + G(0 has k simple continuous eigenvalues ^(f), • • • > Afc(r) such thatA,-(/) -> Aj, as / -*• oo; cf. Exercise IV 9.1. (a) Show that the linear system£' = [E + </(03£ nas w linearly independent exponentially small solutionsas / -^ oo. (b) If Re A,(/) < 0 for 7 = 1, .. ., k, then £' = [E + (/(/)]£has « + k bounded solutions as t —> oo. (c) If k = 1, then there exists avector c 7&Q such that £' = [E + G(0]£ nas a solution of the form

| = (c + o(l)) exp ^(s) ds as r ->• oo.. (d) If Re X^s) ds is bounded

fory = !,...,£, then there exist linearly independent vectors c1}.. ., cfc

such that £' = [£ + ^(0]^ nas solutions of the form

For applications of the corollaries of Lemma 4.3, see the exercises in§ VIII 3. Further applications and extensions of Lemma 4.3 and itscorollaries are given in §§ 13-16

Exercise IX 5.4 gives an analogue of Lemma 4.1 for difference equations.Exercises 4.8 and 4.9 to follow give analogues of Lemmas 4.2 and 4.3.

Exercise 4.7. Let R = Rd be the £ = (£ J , . . . , ^)-space. For n = 1,2 , . . . , let Sn be a map of R into itself and Tn = Sn ° £„_! ° • • • ° S^Let 5 be a compact and K0, Klt K2. . ., K00, K10t K20,. . . closed sets of Rsuch that S c K0 n #00, Sn(R - Kn_J c R _ KW S^K^ n ^w_1>0) cKn0, and Kn f~\ Tn(S) is not empty for n = 1 ,2 , . . . . Then thereexists a point £0 e S such that 7"n£0 c Kn n ATn0 for « = 1, 2,....

Exercise 4.8. Let .4, 5, C be square matrices satisfying

where ^ > 0 and 0 < < c. For n = 1, 2 , . . . , let Arn, Yn, Zn be

continuous, vector-valued functions defined for all (#, y, 2) which vanish

for large ||*|| + ||y|| + \\z\\ . Let Tn = Sn « S^ ° • - • « 5lf where Sn isthe map

(a) Let 0 < 0 < 1 and ||x0|| 6 ||y0||. Show that if 0 (5 (c - e)0/6and

then there exists a z0 such that (xn, yn, zw) = Tn(x0, y0, z0) satisfies

(£) Show that if 0 < // < 1 and

as («, a;, y, z) -»(oo, 0, 0,0), then (4.44) implies that ||a5j/||yn|| -» 0 andKII/llyJ|-^Oas«-^oo.

Exercise 4.9. Let A, Cbe matrices satisfying

Let (9, xTn, Fn, Zw be as in Exercise 4.8, with d < (c - 1)0/6 in(4.43). Let Tn = Sn ° Sn_t ° • • • ° Slt where 5n is given by (4.42) withB = I. Let V"o(0 + Vo(2) + ' ' ' be convergent, and

(a) Let (z0, y0, z0) be such that (an, yn, zj = rn(xffl, y0, z0) satisfies (4.44).Show that

exists. (6) In addition, assume that <5 < (1 — a)0/3 and that 0 3y»0(

w) < 1- Let £0»y« be given and satisfy ||a;0|| 0 IIVooll- Show that thereexists a (y0, «o) such that (xw, yB, zn) = rB(a?0, yc, z0) satisfies (4.44) and(4.45). (c) Formulate analogues of parts (a) and (b) when the matrix Bin (4.42) is /*/,/* 0 (instead of /).


In order to apply Corollary 3.1, let

It will be verified that Q° is a (u, y)-subset of Q determined by the onefunction u = ||z||2 - 25r2(0 ||y|l2. Let

Since u = 2(Re z • z' - 25r2 Re y • y' - 25rr' ||y||2), it follows from (4.8)that on £/, where 5r(0 < 1, ||z|j = 5r(0 ||y|| ||y||, and ||£|| < 2 ||y||, wehave

The last factor is positive since r satisfies the differential equation

and v > 0. Thus Q° is a («, y)-subset of Q and £/ = Qe° = Qje.Note that, by the definition of Q, the point (y, 2) = (0,0) is not in Q;

hence (y, z) e Qe° implies that y ^ 0.Let S = {(t0, y0, z): ||z|| < 5r(/0) ||y0||}. Thus S n Qe« = {(/„, y0, z):

INI - 5r(/0) ||y0||}. S is a ball, ||z|| 5r(/0) ||y0||, and S n Qe» is its bound-ary and is not a retract of S. Since U = Qe°, the map IT: iie° -*- 5" n Qe°given by TT(/, y, z) = (f0, #0, M'o) HyollMO 112/11) « continuous [sincey ?£ 0 on Q,,0 and r(t) > 0] and hence is a retraction of Q«° onto S n Q,,0.The existence of z0 and a solution y(f), z(r) of (4.6)-(4.7) satisfying (4.16)follows from Corollary 3.1.

Since (4.15), (4.16) imply that ||z(OII ^ \\y(t)\\, hence ||£(/)|| < 2 ||y(0!lthe inequality (4.17) is a consequence of (4.8). This proves Lemma 4.1.


This proof is similar. It depends on the choices

where

Define the sets

It is readily verified that (4.13), (4.19), and (4.20) imply that ||£|| < 3 ||y||and

Hence Q° is a (u, i>)-subset of Q and Oe° = C/ - K = {(r, *, y, z) 6 ft:w = 0, v < 0}.

Choose S to be the set {(t0, x0, y0, z): \\z\\ <^ 7r(/0) ||y0||}. As above, it isseen that S n £lf is not a retract of 5 but is a retract of £le°. Thus Lemma4.2 follows from Corollary 3.1.


Let (z0, y0, z0) and £(0 = (*(0> y(0» *(0) be as in Lemma 4.2. By (4.19)and (4.20), ||£(0» ^ 3 ||y(/)l|. Hence (4.26) gives ||y'|| < 3Vo(0 ||y||. Itfollows from (4.19), (4.20), and (4.28) that |(r) exists for / /0. The firstpart of (4.29) follows from (4.20) and (4.22). The inequality ||y'|| ^3y0(0 llyll implies the existence of the limit y^ and y^ ^ 0 as in the proofof Theorem 1.1.

The last part of Lemma 4.3 will not be deduced from Lemma 4.2 butwill be obtained from another application of Corollary 3.1. Let

where uit wa, v are defined by

and /0 is a positive constant to be specified. Let (7a be the subset of iiwhere wa = 0 and uf < 0, v 0, and F the subset of Q where v = 0 andMJ, M2 = 0. Then, as in the last section, M2 > 0 on C/j, i) < 0 on K. When«!, «2, y 0, then Hill ^ 3||y|| and

Since

a simple calculation shows that on t/1}

Let J be so large that

Thus, if /0 > T, it follows that u± > 0 on Uv In this case, Q° is a («, y)-subset of Q and Qe° = Q£ is the subset of O where ult wa ^ Q, y < 0 andeither wx = 0 or «2 = 0.

Choose <52(/0) to be

Thus, by (7.3), IKH < dz(t0) OyJ implies that ||*0|| < "('o) \\V\\ for y e &°.

Let S = {(/0, *„, y, 2): ||z|| ^ 7r(O ||y||, ||y - y^\\ << 1 \\y^\\ f \<fc}, soJtothat 5 cr QO u Oe°. Topologically, 5 is a ball in the (y, z)-space. (If y, z

Figure 3.

are 1-dimensional, then S appears as the shaded area in Figure 3.) It isclear that S C\ £le° is the subset of S on which u: = 0 or «2 = 0, sothat, topologically, 5 n Qe° is the boundary of 5 and is not a retract of S.On the other hand, S n De° is a retract of £le° for a retraction IT : Qe° ->51 n Heo is given by 7t(t, x, y, z) = (/„, a?0, y«, 2T(O ||y°||/T(/) ||y||), whereyo _ yO(^ y) js chosen so that y° — y00 = oc(y — y^) and

(That S n Qe° is a retract of Qe° is geometrically easy to see, becausethe projection (f, x, y, z) -> (/, «0, y, 2) of the (t, x, yt z)-space into the('»y-> z)-space carries Qe° into a set which is topologically the boundaryof a "cylinder" with S n Qe° corresponding to a section t = /„; cf.Example 2 of § 2.)

Thus by Corollary 3.1, there is a point (f0> #o> ^o> zQ) E S C\ Q° such that asolution arc (/, a^f), y(/),«(/)) belonging to (4.11)-(4.12) remains in Q° onits maximal right interval of existence [t0, o>). As in the argument at thebeginning of the proof of this lemma, to = oo if d — <x> or if \y^\ issufficiently small; in this case, (4.20), (4.29) hold.

8. Asymptotic Integrations. Logarithmic Scale

Consider again a system of the form

in which

holds. In this section it will be supposed that £ = (y, z), F = (Flt F2), andE = diag [P, Q], so that initial value problems associated with (8.1) takethe form

The eigenvalues pi,p&... and qlt q& ... of P and Q will be assumed tosatisfy

for some number u,.Theorem 8.1. Let (8.1) be equivalent to (8.3) H^er* the eigenvalues of

P, Q satisfy (8.5); F(t, £) & continuous and satisfies (8.2) for t 0 a/K/||y||, ||21| < (5 (^ ex)); c/i</ y(/) > 0 w continuous for t 0 am/ satisfies

W^en /* 0, assume that d — ao. Then there exist T Qand 6l > Osuchthat for every t0 7* a«</ y0 satisfying \\y0\\ < <5lt /A^re w c «0 with theproperty that the initial value problem (8.3)-(8.4) has a solution for t 3£ /0

satisfying either (y(t), z(t)) =Qor y(t) j± Of or t > t0 and

If fji in (8.7) is replaced by /u 4- «= > /«, this follows at once from Lemma4.1 [with ^(/Q) = oo if d = oo]. Since a linear transformation of they-variables with constant coefficients does not affect (8.6) but permits anarbitrary choice of > 0, Theorem 8.1 follows. Assertions (8.6), (8.7)will be improved in § 11 below.

Remark 1. This proof of Theorem 8.1 shows that if the y-variablesand z-variables are each subjected to a linear transformation with constantcoefficients and y(0 is replaced by const, y^r) for a suitable constant, thenit can be supposed that (4.5) and (4.8) hold. With these choices of co-ordinates and y, the inequalities (4.16)-(4.17) in Lemma 4.1 hold for anysolution (y(t), z(t)) & 0 of (8.3) satisfying (8.6H8.7).

Theorem 8.2. In addition to the conditions of Theorem 8.1, assume thatF satisfies the Lipschitz condition

that t0 is sufficiently large, and that ||yj is sufficiently small. Then z0

and (y(t), z(0) are unique andz0 = g(t0, y0) is a continuous function (in fact,uniformly Lipschitz continuous on compact subsets of its domain ofdefinition).

If, in addition, F is assumed to be smooth (say, of class Cm, m^ 1, oranalytic), then z0 = g(t0, y0) is of the same smoothness. Here, a functionof a vector with complex-valued components is said to be of class Cm ifit has continuous, mth order partial derivatives with respect to the realand imaginary parts of its variables. In this terminology, the result forFeCHs

Theorem 8.3. Let the conditions of Theorems 8.1,8.2 hold and let F(t, £)have continuous, first order partial derivatives with respect to the real andimaginary parts of the components of |. Suppose also that (A < 0. Thenz0 = g(t0, y0) is of class C1. If, in addition, the partial derivatives of F withrespect to the real and imaginary parts of the components of f vanish at| = Qfor all t, then the partial derivatives ofg with respect to the real andimaginary parts of the components ofy0 vanish at y0 = 0/or all fc.

The proofs in §§9 and 10 will show that Theorems 8.2 and 8.3 arecorollaries of Theorem 8.1, which is, in turn, an immediate consequence ofLemma 4.1. For applications, note that the proofs of Theorems 8.1-8.3imply the following remark.

Remark 2. Let e > 0 be fixed so small that / * - j - e < O i f ; u < 0 andthat Re^fc > fj. + e in (8.5). Then there exists a number />e > 0 with theproperty that if the condition on y(f) is relaxed to

then Theorems 8.1-8.3 remain valid if (8.6), (8.7) are replaced by thesingle condition

Notice that the "smallness condition" (8.2) does not seem appropriateif (8.1) is considered only for small £, e.g., if F(t, f) does not depend on t.In this case, more natural conditions are

and, of course, p < 0.

Corollary 8.1 Let the assumptions of Theorem 8.1 hold except that (8.11)replaces (8.2); also assume that d < oo and p. < 0. Then the conclusions ofTheorem 8.1 remain valid. If, in addition,

w/»e/i £x •?£ £2, then the conclusions of Theorem 8.2 hold in the followingsense: there exists a small «50 > 0 H>///I the property that if t0 is sufficientlylarge and \\y0\\ is sufficiently small, then there exists a unique z0 = g(t0, y0)such that the solution £(/) = (y(t), z(0) of (8.3)-(8.4) exists and satisfies111(011 ^ o^for t t0 a«</ f/ze conclusions of Theorem 8.1; furthermore,g(tQ, y0) is uniformly Lipschitz continuous. Also, if F satisfies the smooth-ness assumptions of Theorem 8.3, then the conclusions of Theorem 8.3 arevalid.

This generalizes the last part of Theorem IX 6.1 on the existence ofinvariant manifolds. The other part will be generalized later in §11.

Corollary 8.1 follows from the Remark 2 by virtue of the fact that(8.11) implies that, for every p > 0, there exist T 0 and <50 > 0 such that

and correspondingly, (8.12) gives

furthermore, if e > 0 is sufficiently small, then p + e < 0 and (8.10),(8.11) imply (8.6), (8.7).

For another deduction of the first part of Corollary 8.1 from Theorem8.1, make the change of variables

where 0 < a < —//. Then (8.1) becomes

and fji is replaced by /* + a < 0. For the applicability of Theorem 8.1,it is sufficient to verify the existence of a y(t) such that y(0 -> 0 as t -> ooand

Note that a > 0 and (8.11) imply that such a y»(r) is given by

Exercise 8.1. This exercise involves a proof of the conclusions ofTheorems 8.1 and 8.2 by the method of successive approximations rather

than by the use of Corollary 3.1 (via Lemma 4.1). In view of the changeof variables (8.14) with a suitable a, there is no loss of generality inassuming that ft < 0. If £ = (y(t\ z(tj) is a solution of (8.3) satisfying(8.7), then it is easy to see that

where it can be supposed that P, Q are such that

Conversely, if £ = (y(0»z(0) is a solution of (8.15) satisfying (8.7), thenit is a solution of (8.3). Show, by the method of successive approximations,that under the assumptions (8.2), (8.8), where y(0-*-° as t-+ oo, (8.15)has a solution (for sufficiently large t0, small ||yj if d < co) satisfyingy(t*) — Vo and (8.6)-(8.7). Let the Oth approximations be y0(t) =ep(t'l^y^ z0(t) = 0, and the nth approximation be obtained by writingMS), <*)) = (y*-i(s), z,-^)) on the right of (8.15) and (y(OXO =(?/„(/), 2n(0) on the left. See Coddington and Levinson [2, Chapter 13].This gives the existence Theorem 8.1 under the additional condition (8.8)and the condition y>(f) -> 0. Theorems 8.2 and 8.3 can also be proved bythe considerations of the successive approximations, but note thatTheorems 8.2, 8.3 are deduced in §§ 9 and 10 essentially from Theorem 8.1.(Despite the disadvantages of the method of successive approximationsin the present situation, this method has important applications in relatedproblems.)


It can be assumed that (4.4), (4.5), and (4.8) hold; cf. Remark 1 followingTheorem 8.1. In terms of the function a(t) in (4.18), define

It is readily verified (cf. § 5) that if (4.8) holds, T 0 is sufficiently large,and / ^ T, then

Uniqueness. Suppose that (8.1) has two solutions $XO — 0/XO> 3j(0)»where j = 1, 2, satisfying yfa) = y0 and (8.7), but 2^0) zz(t0). Putf(0 = WO - li(0 - (KO, *(0). Then (8.8) implies (4.8) hence, by (9.2),

du(t, y(0, z(0)A// > 0 if w(/, y(t), «(/)) = 0. Since y(t0) = 0, ; (/„, y('o),2('o)) > 0 and» consequently, w(/, y(t), z(0) cannot vanish for t t0. Thus

It follows from (4.8) and cr(/) -> 0 as / -> oo (cf. Exercise 4.1) that

But this contradicts ||z(0» < || 1(011 ^ HWOII + HWOII, since both£ = £15 £2 satisfy (8.7).

Continuity of z0 = g(to,y0). Let f0 > TQ, HyJ < ^^/o), Zj = ^(/0, yt) andfi(0 be the corresponding solution of (8.1). Introduce new variables into(8.1) defined by

so that (8.1) becomes

and, by (8.8),

It follows from the part of Theorem 8.2 already proved that if \\yz — yjis sufficiently small, then (9.6) has a unique solution £(0 which satisfies£(0 = 0 or lim sup r1 log || £(011 /* + * as t -» oo, £(/„) = (y2 - y lf...),and

for t^ /0 'f WO = f(0 + WO = (y2(0»22(/)). (The inequality (9.8) isthe analogue of (4.16)in Lemma 4.1.) It follows that £ = WO is a solutionof (8.1) and that z2 = g(t0> y2). Thus / = t0 in (9.8) gives

Let I = |(r, r0, ^0) = (y(^» 'o. ô)» 2(^» >o> Vo)) be the unique solution of(8.1) supplied by Theorem 8.1 and the first part of Theorem 8.2. Thus

The uniqueness of this solution implies that for /j ^ /0,

In order to examine the continuity of g(t0, y0) with respect to f0, considerf(', '<» 2/o) - ^('» î» yo) for /! r0 and small ||yj. In view of (9.11), thisdifference can be written as £(/, t\, y(t\, /o» ô)) ~" ^('» 'i» ô)- The analogueof (9.8) holds and at / = ti gives


Since £ = £(/, 'o> ô) remains in a compact |-set for /„ t /0 + 1, and||y0|l small, it follows from (8.1) that ||f'('> 'o.ô)! ^ Af, if M is a boundfor f on this set. Hence lUfo, /„, y0) - £(/0, /„, t/0|| ^ M(tv - t0), so that

Hence, for /0 /! r0 + 1 and M = M(/0, y0),

The inequalities (9.9), (9.12) complete the proof of Theorem 8.2.


It will be shown that £(t, t0, y0) is of class Cl; in particular, |(r0, /0, y0) =(y0, (/0, y0)) »s °f câss C"1. The proof will be given as if all variables andfunctions are real-valued. This is justified since a real system is obtainedby separating real and imaginary parts of (8.1); cf. the interpretationdw/3y* = $(duldak - i du^T*) if y* = a" + irk mentioned after (3.2).

Let e be a unit vector in the y-space, h 0 a small real number. By theLemma V 3.1, the difference

satisfies a linear differential equation of the form

where, in view of the continuity of £(/, 'o» ô)>

uniformly on bounded /-sets, and d^F denotes the Jacobian matrix ofF with respect to |. By the analogue of (9.8), the function (10.1) is boundedby

The derivation of (9.8) and the analogue of the inequality (4.17) in Lemma4.1 show that this is at most

0

Hence, for fixed (t0, y0), the family of functions (10.1) is uniformlybounded and equicontinuous in / on bounded /-intervals of / /0- Thusthere exist sequences hlt h2,... such that hn -> 0 and the correspondingfunctions (10.1) tend to a limit £(/) = ^(/, /„, y0) uniformly for bounded


t (= '(>)• This limit satisfies the linear system

an initial condition of the form £(/„) = (e, z*) for some z*, and ||£(OII ^r(r). The last inequality implies that £(?) = 0 or

By (8.8), || d(F(t, f)|| ^ y(t). Then Theorems 8.1, 8.2 imply that if/„ is sufficiently large, there is a unique z* such that (10.4) has a solutionsatisfying £(/„) = (e, z*) and (10.5). Consequently, the selection of thesequence hlt hz,... is unnecessary and

exists uniformly on bounded /-intervals and is the unique solution of (10.4)satisfying (10.5) and £(?„) = (e, z*) for a unique z*.

Hence £(/, /0, yQ) has partial derivatives with respect to the componentsof y0. The continuity of these derivatives as functions of (/, t0, y0) followsfrom (10.4) and arguments similar to those just used to prove (10.6).The existence and continuity of 3£(/, f0, y0)/d'o follows by the argumentsin the proof of formula (V 3.4) in Theorem V 3.1.

Note that if df(t, 0) = 0, then (10.4) reduces for y0 == 0 to £' = ££.The only solutions £ = (y(f)» 2(0) °f this linear system satisfying (10.5)have z(0 = 0; cf. § IV 5. Thus dVQz(t, t0, 0) = 0 and, at f = /„, this givesdVog(t0, 0) = 0 and proves Theorem 8.3.

11. Logarithmic Scale (Continued)

The object of this section is to obtain improvements of the assertionsof Theorem 8.1 without adding additional assumptions on F. To this end,let | = (x, y, z), E — diag [Al9 A2, A3], and F(t, x, y, z) = (Flf F2, F8), sothat initial value problems associated with (8.1) take the form

It will be assumed that the eigenvalues an, a,2,... of A^ satisfy

for some number //.Theorem 11.1. Let (8.1) be equivalent to (11.1), where A^ A& A3 are

matrices satisfying (11.3), and let F = (Flt Fz, F3), y(0» &> Qnd [A be as inTheorem 8.1. Then there exist dl>Q, T 0 and, for every t0^T, a

constant a(t0) > 0 such that if \\x0\\ < 7<r(/0) ||?J and 0 < \\y9\\ < 6lt thenthere is a z0 with the properties that (11.1)-(!!.2) has a solution for t f0

satisfying y(t) 7* 0 and

This theorem, which concerns certain solutions of (8.1), follows at oncefrom Lemma 4.2 (with <$! = CD when d = oo). Note that if // is the least[or greatest] real part of the eigenvalues of E (so that there are no x [or z]variables), a corresponding statement holds. In fact, this case is containedin Theorem 11.1 since dummy XOTZ variables can be added to the system(8.1), with suitable choices of Al or A^ and Ft s 0 or F, = 0. The nexttheorem concerns all solutions of (8.1).

Theorem 11.2. Assume the hypotheses of Theorem 11.1 on F(t,g). If6 = oo, let £0(f) 0 be any solution o/(8.1); and if d < oo, let £0(f) & 0£e a solution of (9.1) for large t satisfying

TTien (Ae limit (11.5) existe and is the real part p of an eigenvalue ofE. If, inaddition, coordinates in the £-space are chosen so that (8.1) is of the form(11.1), where (11.3) holds, then £(/) = (x(t), y(t), z(tj) satisfies (11.4).

It is clear that the first part of Corollary 8.1 has a similar improvement:Corollary 11.1. Let the assumptions of Theorem 11.1 [or Theorem 11.2]

hold except that (8.2) is replaced by (8.11), and let d < oo, < 0. Then theconclusions of Theorem 11.1 [or Theorem 11.2] remain valid.

Exercise 11.1. (a). Consider the case of a linear system of differentialequations

where G(f) is a continuous matrix for / 0 such that ||G(/)|| ^ y(0,where y»(0 is continuous and satisfies (4.24). Let E — diag [A l s . . . , AJ,and let the real parts /* t , . . . , /^ of A j , . . . , Ad be distinct. Then, for anyy,l ^ y ' ^ r f , (11.7) has a solution £(f) = (f»(0, • • • > ^(0) such that^(03*0 for large f, ||*(OI = o(|l'(0!) as ?-^cx) for A: ^y, and

/•oo

r1 log ||;(OI -»y", as r-»co. (£>) Show that if y>(0 <ft < oo, then

£'(f) = [c + o(l)] exp A,/, as / -> oo, for some constant c 0.Exercise 11.2. Let £ = diag [^j, ^2, v43], where >4; is a square matrix

with eigenvalues an, a,2, • . . satisfying (11.3). Let G(f) be a continuousmatrix for f ^ 0 and identify (11.1) with (11.7), where £ = (a;, y, z) andF(t, £) = G(r)^. Suppose that \\G(t)\\ < y(0, where y<f) is continuous

poo

and satisfies I ly^OI" dt < oo for some/?, 1 p ^ 2. Let AI be a 1 x 1»/

matrix, consisting of the constant A, Re A = /*; thus y is 1-dimensional.Let £(/) = (*(/), y(0, z(0) be a solution of (11.7) satisfying (11.4), (11.5).Show that there is a constant c 0 such that

where g(t) is the diagonal element of G(t) which is the coefficient of y inthe second equation of (11.1). Note that this equation is the form y' =Ay + 2<7,(/K + g(t}y + 2>*(/)z*, where (qlt qz,...,g, rlt r2,...) is a rowof <?(/).

Exercise 11.3. Let/(/, y) be continuous and have continuous partialderivatives with respect to the components of y on a (/, y)-domain and beperiodic of period p in t,f(t + p, y) =f(t, y)- Let

have a periodic solution y = y(t) of period p. Discuss the behavior ofsolutions of (11.8) and y(/0) = y0, where (/„, 2/o) 's near the curve (/, y(/)),0 / p, on the basis of the following suggestions: Introduce the newvariables

Thus (11.8) becomes

which can be written as

where

P(r) is a matrix function of period /> and //(/, 0 's continuous and hascontinuous partial derivatives with respect to the components of £, andH(t, 0) = 0, d{//(f, 0) = 0. The linear matrix initial value problem

has a solution which, by the Floquet theory in § IV 6, is of the form

where K(t 4- p) — K(t) and £"is a constant matrix. The change of variables

Consider the application of the theorems of § 8 and of this section to (11.15)to obtain generalizations of the results of §§ IX 10,11. (Note that e?need not have A = 1 as an eigenvalue in the situation here.)


It will be shown that it is sufficient to consider the case of linear equa-tions. Note that (8.2) implies that if the solution £ = £0(f) of (8.1)vanishes at one /-value, then it vanishes for all /. Hence £0(f) 0 forlarge t, say t j>: r0. Define a matrix G(t) = (gi]e(t)) as follows: if f =(F»,/*...). put

for t ^ /„. Since £ = £0(f) is a solution of (8.1), it follows that it is asolution of the linear system

Note that (8.2) and (12.1) imply that

Hence Theorem 11.2 is contained in the following:Lemma 12.1 Let G(t) be a continuous matrix for t ^ 0 such that

where y(t) > 0 is a continuous function satisfying (4.24). Let £ = £„(/) y* 0be a solution of (12.2). Then the conclusion of Theorem 11.2 holds.

Proof of Lemma 12.1. Let p\ < u,i < • " < i*f denote the differentreal parts of the eigenvalues of E. After a change of coordinates, it canbe supposed that E = diag [/flf B2,..., J?/]f where the eigenvaluespik of 5, satisfy Re &fc = pt. Correspondingly, let £ = (y l s . . . , y,),£| = (5^!,..., /y/), and let (12.2) be written as

where Grt(0 is a rectangular matrix and 11 (011 v*(0-

304 Ordinary Differential Eqonsationss

If 1 q /, /0 is sufficiently large, and y^ j& 0, then Theorem 11.1implies that (12.4) has a solution £ = (^(0, • • •»#/(0) satisfying

This solution, say f = £a(f, /0, yg0), is unique by Theorem 8.2. In fact, itis unique even if (12.6), (12.7) are replaced by

(cf. Remark 2 following Theorem 8.3). This uniqueness implies that£«('. 'o» y*)) is linear in yqn (for fixed /, /„, ?).

With the understanding that gg(t, /„, 0) s 0, it follows that there existunique yw,..., yf0 such that the given solution £0(f) is of the form

In fact, y10, . . . ,y / 0 are defined recursively as follows: if £„(/) =(yi(0, • • • , yf(0)» ^t 2/io = ^(^o); then let y20 = y2(/0) - y12(/0), wljereli(', ^o, 2/10) = (yu(0, y«(0, • • • , MO); etc.

Let ^ be the largest/-value such that yi0 ^ 0 in (12.9). It is clear that|0(f) = (^(f),..., y//)) satisfies (12.6), (12.7). This proves the lemma.

13. Asymptotic Integration

The object of this section is to study the asymptotic behavior of solu-tions £(f) of a perturbed linear system

rather than the behavior of |!(f)ll as m § *Suppose that £ is in a Jordan normal form E = diag [/(I),..., /(g)],

where J(j) is an /»(;) x h(j) matrix [as in (IV5.15HIV5.16)]. ThusJ(j) = h(j)Ih(i) + A^,), where 7A is the unit h x h matrix and Kh is 0 ifh = 1 or is the h x h matrix with ones on the subdiagonal and otherelements zero if h > 1. According as h = I or h > 1,

(13.2) J0)y,-Ay, or J(j)y, - (Ay/, V + y/ f . . . , Ay/+ y}"1),

where A = A(/), y, = (y/,..., y/), A = A(y)-Correspondingly, it is supposed that £ = (ylt..., yff),

and (13.1) is of the form

Let /j, denote one of the numbers Re A ( l ) , . . . , Re A(g). An index jwill be denoted by p, q, or r according as Re A(j) < /*, Re A(j) = ,a orReA(j)>//. Put

Lety"0 be an integer and /? a number satisfying

and %), fc(^) integers, if any, such that

The next theorem concerns sufficient conditions for (13.3) to have asolution with the following asymptotic properties as t —>• oo,

where ctf* are constants,

and 2]r = 0 if %), A:(^) do not exist.Note that if the o-terms are replaced by 0, then, since 1 < / < & in

2i» (13.7) becomes a solution of the linear system

The choice of the range of summation l(q) ^ / min (k, k(q)) is dictatedby several consideration^. On the one hand, results permitting / > k caneasily (but will not) be obtained as a consequence of Theorem 13.1; alsothe first term in the first line of (13.7) is not significant unless i ft, hencethe choice i min (k, k(q)) S min (k, 0) since k ^ h(q). On the otherhand, the condition / l(q) means that the degree of the polynomial£1 <V>*~V(& •" 0' d°es not exceed the given j0.

Theorem 13.1. In the system (13.3), let J(j) be a Jordan block; cf.(13.2). Let p — Re h(j)for some j. Let an index j = 1 , . . . , g be denotedby p, q or r according as Re A(y') < p, Re A(y) = fj,, or Re A(y') > (i, and

define h+ by (13.4). Let j0 be an integer and 0 a number satisfying

Let /(?), k(q) be integers (if any) satisfying (\1.6). Let F(t, £) = (Flt..., Fa)be continuous for t 0 and all £, and satisfy

where y'i(f) > 0 w a continuous function such that

Let m = v h(p) + 2, [%) — &(f)]. /or a/iy «?/ o/ constants cqk, l(q) ^

A: £(^), wo/ #// 0, there exists an m parameter family of solutions £(/) of(13.3) defined for large t and satisfying the asymptotic relations (13.7) ast-> oo.

The part of the assertion concerning "m parameter family of solutions"means essentially that it is possible to specify a partial set of m "initialconditions," as well as the asymptotic behavior (13.7) for f(f); cf. thestatement following (14.15) in the proof of Theorem 13.1.

Remark 1. Consider a system of differential equations

where E° is a constant matrix and F°(t, rj) is continuous for / 2: 0 and allrj. Let L be a nonsingular constant matrix such that L-1£°L is a matrixE = diag [/(I),..., J(g)] in a Jordan normal form. Then the changeof variables r) — Lf reduces (13.13) to (13.1) [i.e., to (13.3)], whereF(t, |) = L~lF°(t. L£). The applicability of Theorem 13.1, or at leastthe condition (13.11), can sometimes be verified without the knowledgeof L or the explicit reduction of (13.13) to (13.1). For it is clear that\\F°(t,r,)\\ £ ^(0 \\ri\\ implies that ||F(/, £)|| < (0 ||||| if, e.g., c-ll*-1! ' ll^ll-

Remark 2. The derivation of Theorem 13.1 from Lemma 4.3 willshow that the theorem remains valid if F(t, £) is defined only for / 0,III|| < 6 < oo if fj, < 0 (or p = 0, h+ = 1, and the constants \cq

k\ aresufficiently small).

Theorem 13.1 has a partial "converse" dealing with all (rather thancertain) solutions £(/) of (13.1) satisfying

(cf. Theorem 11.2):Theorem 13.2. Let E = diag [/(I),..., /(#)] and F(t, £) be as in

Theorem 13.1 except that (13.12) is replaced by

(and h0 is not necessarily an integer). Let £(/) 7* 0 be a solution o/(13.3)satisfying (13.14). Then there exists constants cQ

k, & = ! , . . . , A(^), no/all 0, swcA f/ifl/ ;/y'0 is defined by

P = h0 — y*0, andl(q\ k(q) are the least, greatest integers (if any) satisfying(13.6), then £(t) satisfies the asymptotic relations (13.7) as t -> oo.

Consequences and refinements of Theorem 13.1, 13.2 will be given in§16; see also § XII 9.


Change of Variables. In order to apply Lemma 4.3, make the linearchange of variables

(14.1) f - C W C

given in terms of £ = (ylt..., ya) and £ = (zlt..., za) by the formulae

where 0 < e < 1, 2i 's the sum over the /-range l(q) / min (A:, k(q))as in (13.8), and 2n is the sum over the other indices / on the range1 ^ i < A:, so that

A solution ^(/) of (13.3) satisfies (13.7) if the corresponding vector £(/),defined by (14.1), satisfies

To clarify the meaning of (14.1) and to calculate the resulting differentialequation for £, the map (14.1) will be given a decomposition of the form

to be described. This factorization is suggested by the fact that if tk~l> inthe first formula of (14.2) is replaced by tk~i (and written behind the signIn), then this formula becomes yq = e7^^.


The change of variables £ = Q0(t)w, w = (wlt..., wa), is given by

Thus (13.3) becomes

where h — h(j). Finally, let D(t) be the diagonal matrix such that w =/>(/)£ is given by

If the resulting differential equation for £ is written as

then the linear part £' = £0£ is given by

where 7 (y) is the matrix obtained by replacing the ones on the subdiagonalof/(j) by c; cf. § IV 9. The last part of (14.9) is easy to see if the trans-formation Wj-t-zj is made in two steps w*-*z^/e*-1 -»• r1- */ -1.

Finally, replace the independent variable / by j, where

Thus (14.8) becomes

where the linear part of this equation is

Preliminary Existence Result. Suppose that there is a continuousfunction y>0(/) for large / such that

/*00

The last condition is equivalent to I /v*o(0 ds < <x> since <fc = </f//.

Then if e > 0 is sufficiently small, Lemma 4.3 is applicable to (14.11) if xis a vector with components zp

k, and zak, k > k(q)\ y is the vector with

components zQk, l(q) ^ k ^ £(9); and z is the vector with components

zr* and zqk, k < l(q). Note that (14.12) shows that there is a constant

c > 0 such that Re (zk dzflds) ^ — c |zff*|2 or ^ c |z,fc|2 according ask>k(q)^$ or A: < %) < 0; also Re (z, • </z'/<fe) ^ -rf IM2 or^ cr ||z,||2 according as/ = p (i.e., Re A(y') < /i) ory = r (i.e., Re A(y') >fjt) if e > 0 is small and / > 0 is large.

Thus, by Lemma 4.3, (14.13)-(14.14) imply that if ca*, l(q) ^k< %),are given constants, not all 0, then there exists a solution £(0 of (14.11)such that, as t -*• oo,

In fact, we can also specify a set of m initial conditions for £: zp*(r) =zJ0 and z0*(r) = zJ0 for fc(^) < Jt < A(^) if T is sufficiently large and|z*0|, |z*0| are sufficiently small numbers.

The Norms \\Q\\, HC"1!!- In order to complete the proof, it remains toshow that the assumptions (13.11)-(13.12) imply (14.13)-(14.14) and that asolution £(0 of (14.8) satisfying (14.15) also satisfies (14.3). To this end,it will first be verified that there exist positive constants c, c' such that forlarge /,

From (14.2), the norm of Q(t) is easily seen to be O(el4tY), wherey = max [h+ — /8, h(q) — l(q}\ and the max refers to the set q. From(13.6), %) - l(q) < y0 and, from (13.10), A# - ft <> y0; hence \\Q(t)\\ =0(6*1**) as / -»• oo. It is similarly seen that e^t^ = O(^Q(t)\\) as t-»• oo.This gives the first part of (14.16).

The factorization Q —Q0D of Q into nonsingular matrices for / > T)shows that Q~l exists and is g"1 = D~*Qol- The inverse map


is easily seen, from £ = Q0w, w = D£ in (14.5), (14.7), to be

Thus, for large t, \\ Q^l(t)\\ is bounded from above and below by a positiveconstant times e~ftity, where y = max [ft — I , k(q) — 1]. Since k(q) — 1 fsft - 1, by (13.6), the last part of (14.16) follows.

Completion of the Proof. In view of (13.11),

Hence, by (14.16),

Thus (13.12) implies that (14.13),(14.14) hold if y>0(f) = c^+'o-i (0, andso (14.8) has a solution £(f) satisfying (14.15).

In view of the first part (14.9), the corresponding equations in (14.8) are

zj' = (gfc)th component of Q~*F(t, Q£),

so that, by (14.18),

where Fj is the (^/)th component of F. Hence,

by (13.11). In view of (14.16) and the boundedness of £(/) as r -* oo,

Consequently, k ^ ft shows that

This gives the first part of (14.3) and completes the proof of Theorem 13.1.


This theorem can be reduced to the case of linear equations by thedevice used at the beginning of § 12. Hence we can be suppose thatF(t, |) = (/(/)£, where G(t) is a matrix satisfying ||(7(011 Vi(0 and 03.1)is replaced by

Let q0 denote a fixed value of q and k0 an integer on the range 1 k0 %0). Then the equation (15.1) has a solution ffcotfo(0 satisfying, as t -»> oo,

where 'y = ^0 — A(^0) + k0 1. This follows from Theorem 13.1 withy0, p replaced by %0) — kd,y = hQ— [%„) — fc0] and the choice cf = 1or cf = 0 according as (qk) = (qjc0) or (qk) ^ (^o)-

The set of solutions lrt(0 is a set of Zh(q] linearly independent solutions.Also if n = ^h(p), then Theorem 8.2 implies that there are exactly nlinearly independent solutions £i(t),..., £„(/) satisfying

and n + Tih(q) linearly independent solutions satisfying

Hence if ^(0 5^ 0 is a solution of (15.1) satisfying (13.14), then there existconstants clt..., cn and cf such that

and that not all cf are 0. It will be left to the reader to verify that thisimplies Theorem 13.2.

16. Corollaries and Refinements

When the matrix E in Theorem 13.1 has simple elementary divisors(e.g., when the eigenvalues of E are distinct) or even if h+ = 1, thenh^ = 1,/0 = 0, /8 1, and condition (13.12) reduces to

cf. Corollary 4.2. Here, the asymptotic formulae (13.7) reduce to

For a fixed y'0, the smallest admissible value of ft in Theorem 13.1 is

/•ooP = ht—jQ in which case (13.12) becomes f**~1yl(0<#< oo. A

larger choice of /3 has the role of possibly increasing the number of signi-ficant terms in the asymptotic formulae (13.7) and of improving the errorterms. When (13.12) is strengthened to

the maximal number of significant terms is possible. In this case, we haveCorollary 16.1. Let E = diag [/(I),..., J(g)], ft, h+ be as in Theorem

13.1 and let F(t, £) be continuous for t ^ 0 and all £ and satisfy (13.11),where ipi(f) > 0 is a continuousfunction satisfying (16.1). Lett; = £0(f) 5^ 0be a solution of the linear system £' = ££ such that /~1log |||(/)|| ->/*as t -» oo. Then (13.1) has a solution £(0 satisfying ||f(f) — £0(OII e~^ -» 0,/-* oo.

In this corollary, E is not required to be in a Jordan normal form (cf.Remark 1 following Theorem 13.1). If it is, we can, in addition, assign apartial set of 2/j(/>) initial conditions, y „(/<>) = y^ for sufficiently large f0.Also, £(/) satisfies the asymptotic relations (13.7), where I(q) — 1, k(q) =h(a), cQ

k are suitable constants determined by I0(0.7o *s defined by (13.16),and ft = 2h+ — /„. This improves the asymptotic relation claimed in thecorollary.

The deduction of Theorem 13.1 from Lemma 4.3 shows that assumptions(13.11), (13.12) can be weakened somewhat.

Corollary 16.2. Let assumptions (13.11), (13.12) of Theorem 13.1 berelaxed to

or, more generally, to

where £ = (?(/)£ is given by (14.2) and y>(f), yo(0 ar^ positive continuousfunctions for t > 0 JMC/I ?/wz/

Then the conclusions of Theorem 13.1 remain valid.Exercise 16.1. By referring to Remark 1 following Lemma 4.3 and to

the proof of Theorem 13.1, find sharper estimates for the o-terms in (13.7)under the conditions (16.3)-(16.6) of Corollary 16.2.

Remark 2 following Theorem 13.1 and Corollary 16.2 have importantconsequences. For example, suppose that Fin (13.1) does not depend on

f, so that (13.1) can be written as

where /X£) is defined for ||||| < d < oo and satisfies

or, more generally,

or even

where 9?(/>) is a nondecreasing function for 0 p < <5 such that

Then (14.16) and (16.10) imply that if < 0 and ||g|| ^ 1, then

for large /. Thus the analogue of (16.2) holds with

If p = c^t'o is introduced as a new integration variable in the integral in(16.11) and it is noted that dpjp ~ // di and log p ~ fit as t -> oo, then itis seen that (16.6) is a consequence of (16.11).

Corollary 16.3. In (16.7), let E = diag [7(1),..., J(g)] be as in Theorem13.1, let\F(£) be continuous for ||£|| < <5(<oo) and satisfy (16.10), w/iere9?(p) w a nondecreasing function of p satisfying (16.11). Letft<Q. Thenthe conclusions of Theorem 13.1, with (13.3) replaced by (16.7), remainvalid.

Exercise 16.2. By involving the Remark 2 following Lemma 4.3, showthat conditions (16.10), (16.11) in Corollary 16.3 can be replaced by

where <p0(p) is a nondecreasing function of p, 0 (p) = <p0(p)lp, the monotony of <p0 does not imply that of (p.]

Analogously, we obtain the following consequence of the proofs ofTheorems 13.1 and 13.2.

Corollary 16.4. Let E, p, F, <p be as in Corollary 16.3 except that (16.11)is replaced by

(and h0 need not be an integer). Then the conclusions of Theorem 13.2, with(13.1) replaced by (16.7), are valid.

17. Linear Higher Order Equations

The results of §§ 4, 11, 13, 16 will be applied in this section to a lineardifferential equation of order d > 1,

for a real- or complex-valued function u. This will be viewed as a pertur-bation of the equation

with constant coefficients. The characteristic equation for (17.2) is

Equation (17.1) can be written as a linear system

for the ^/-dimensional vector | = (u{*~1},..., «(1), «<0)), where u = w(0)

and /?, (7(0 are the matrices

Note first that if at = • • • = ad = 0, then R is in the Jordan normal formand consists of one Jordan block with A = 0 on its main diagonal. If,the coefficients p$t),... ,pd(t) are small, then (17.1) can be consideredto be a perturbation of uw = 0 which has the linearly independentsolutions w = 1, r, . . . , t*~l. It will be verified that Corollary 16.2 hasthe following consequence.

Theorem 17.1 In (17.1), let ar = • • • = ad = 0 andletpt(t),... ,pd(i)be continuous complex-valued functions for t*Z.Q satisfying

and k = ! , . . . ,</. Then, for any j, 0 5jy d — 1, (17.1) has a solutionsatisfying u(t) = (tjlj$(\ + °(f~a)) as t-+ ao, and this relation can be"differentiated" d — 1 times, i.e.,

It will be clear from the proof that, for a given j (rather than for any j)on the range 0 j ^ d — 1, a sufficient condition for the existence of asolution satisfying (17.7) is that

Proof. Since R in (17.5) is in a Jordan normal form, (17.4) can beidentified with (13.3) if F(t, f) = G(f)l, where £ = (I1, . . . ,£") and£* — u

(d~k}. In order to verify the conditions of Corollary 16.2, note thatthe sets of/? and r are vacuous and that there is only one q. Correspond-ingly, h(q) = 0 and h(q) = d. Lety0 = j be the index j in (17.7), 0 = d —y'+ a, and l(q) — k(q) — d — j. Thus ]£i 'n (13.8) contains no terms ifk < d — j or exactly one term / = d — j if d —j ^ k ^ d. Also, letcj = 1 or c0' = 0 according as / = d — j or / 5^ d — j, so that the desiredasymptotic relation (17.7) is identical with (the first part of) (13.7).

Consider F(t, g(f)0 — G(t)Q(t)£. Since only the first row of G(t) con-tains nonzero elements, this can be written as F(t, g£) = (F1, 0 , . . . , 0),where, by (14.1)-(14.2) and (17.5),

^ = (z1 , . . . , 2**), and n is tne sum over the set of /-values, \ ^ i kand iy* d — j. Consequently, || Q~l(t)\\ ^ ctf-1 implies that

for a suitable constant c0 and large /. Since the coefficient of ||£|| is a/»oo

function y0(r) satisfying I y>0(f) dt < oo by (17.6), Theorem 17.1 followsfrom Corollary 16.2. ^

When all the roots of (17.3) are the same, say A, this can be reduced tothe situation of Theorem 17.1 by replacing u by the new dependentvariable v — ue~u. In the other extreme case, when A is a simple root of(17.3), we have

Theorem 17.2 Let (17.3) have a simple root, say A, and suppose that ifA0 is any other root, then Re X j£ Re A0. Let pi(t),.. ., pd(t) be continuousfunctions for t j^ 0 satisfying

and k — 1 , . . . , d. Then (17.1) has a solution u(t) satisfying

Proof. This is the simplest case of Corollary 16.2 when h(q) = 1. LetjQ = 0, ft = 1 + a. Let (17.4) be identified with (13.13) in Remark 1following Theorem 13.1. Then

for some constant c. Thus (13.11) holds with ^(f) = cS \pk(t)\ and thetheorem follows from Corollary 16.2.

Consider the general case where (17.3) has a root, say A = 0, of multi-plicity h, I ^h^d.

Theorem 17.3. LetA = Qbea root of(11.3) of multiplicity h,l^h^d;i.e., let aa_h+1 = - • • = ad = 0 and ad_h ^ 0; and suppose that if A0 j£ 0is fl«y o//zer /-oof, //ze« Re A0 0. Let p^(t),.. ., /^(f) />e continuous func-tions for t 0 5MC/J Ma/

for some a ^ 0. Then, for any j, Q <j ^ h — 1, (17.1) has a solutionu(t) satisfying, as t -> oo,

Exercise 17.1. Prove Theorem 17.3.Exercise 17.2. Restate Theorem 17.3 when A = 0 is replaced by an

arbitrary A.Theorems 17.1-17.3 depend on §§ 13, 16; we can also apply the results

of §11:.Theorem 17.4. Let A be a simple root of (17.3) and suppose that //"A0 is

any other root, then Re A0 5^ Re A. Let pi(f),..., pd(t) be continuous func-tions for t 0 satisfying

or, /wore generally,

ybr A; = ! ' , . . . ,</. 77re/i (17.1) possesses a solution u(t) ^ 0 for large tsuch that

Proof. It is sufficient to prove this theorem in the case that A = 0,otherwise ue~M is introduced as a new dependent variable in (17.1). Thusad = 0. Write (17.1) as the system (17.4), (17.5). Let Y be a constantnonsingular matrix such that Y~1RY = E — diag [/(I),..., J(g)] is in aJordan normal form. The first column of Y can be taken to be (0,..., 0,1),since this is an eigenvector of R belonging to the simple eigenvalue A = 0.Thus /(I) is the 1 x 1 zero matrix and the diagonal elements A(y') of /(/)are such that Re A(/) 0 for j = 2, . . . ,< / . The change of variables£ = Yr) reduces (17.4) to

If ?? = (r)1,..., rf), it follows from Theorem 11.1 that (17.16) has asolution r)(t) ^ 0 such that *?*(/) = 0(1 (01) as t -> oo for k =• 2,. .., d.The corresponding solution £(/) = Yr)(t) of (17.4), where f = (I1,...,£*), satisfies £*(f) = o(|f(OI) as r -» oo for k = I,..., d - 1. SinceM(r) = f(f) and w(d-*> = £* for k = 1,..., d - 1, the relations (17.15)follow.

It cannot be expected that condition (17.14) in Theorem 17.4 can beimproved. This is shown by the following exercise.

Exercise 17.3. (a) In the second order equation,

let ^(0 be continuous for / 0 and Re A ^ 0. Show that a necessarycondition for (17.17) to possess a solution u(t) which does not vanish for

large t and satisfies u'/u -> A as t —*• oo is that

(o) Prove that the necessary condition (17.18) in (a) is sufficient if A is apositive number and q(t) is real-valued; see Hartman [5]. For a relatedresult, see Exercise XI 7.5.

Theorem 17.5. Assume the conditions of Theorem VIA with (17.14)strengthened to

for k = 1, ...yd. Then a solution «(/) & 0 of (17.1) satisfying (17.15)also satisfies

where c 0 is a constant,

F' = dFldl and F is the polynomial on the left of (17.3) [so that F'(X) =(A — A2) ... (A -- Ad) if A 2 , . . . , Ad are f/re roofs o/ (17.3) distinct from A].

Proof. Write (17.1) as the system (17.4), (17.5) and make the change ofvariables £ = Yq, where Y = Y(G) is the constant matrix given in ExerciseIV 8.2 and having (A*-1,..., A, 1) as its first column. Then (17.4) becomes(17.16), where E — diag [7(1),..., J(g)] and 7(1) is the 1 x 1 matrix A.Since Y is a constant matrix, (17.19) implies that the /rth power of theabsolute values of the elements of Y~lG(t)Y are integrable over 0 t < oo.Hence, it follows from Theorem 11.1 that (17.16) has solutions y(t) suchthat if r) = (if1 , . . . , *7d), then ^(f) 5* 0 for large t and i^(f) = od^^OI)as / -»• oo for j = 2,. . . , d. Furthermore, by Exercise 11.2, any suchsolution satisfies

where g(t) is the element in the first row and first column of Y~lG(i) Y.In order to calculate #(/), note that since the first column of Y is

(A*1-1,..., A, 1), the element in the first row, first column of G(t) Y is—SA*-*/^/). All elements of G(t)Y not in the first row are 0. Hence,the upper left corner element of Y~lG(t) Y is — SAd~*/?d(0 times the corre-sponding element of y-1. This element of Y~l is the cofactor A of thecorresponding element of Y divided by det Y. If the distinct roots of(17.3) and their multiplicities are A, A(2),.. . , Afe) and 1, A(2),..., %),

respectively, then

see Exercise IV 8.2. The determinant which is the cofactor A has the sameform as det Y, except that A does not occur. It follows that A is the secondof the two products above. Hence

i.e., (17.21) holds. The relations f = Yy, f fc = ««*-*> and the fact that thefirst column of Y is (A*"1,..., A, 1) completes the proof of Theorem 17.5.

As an illustration of Theorem 17.5, consider the second order equation

foo(17.17) in which Re A * 0 and \q(t)\* dt < oo for some />, 1 ^ p ^ 2*

Then (17.17) has a pair of solutions satisfying

Exercise 17.4. Let <jr(f) be real-valued and continuous for t ^ 0,q(f) -»• 0 as / -* oo, and q(t) of bounded variation for t 0 [e.g., letq(t) be monotone or let q(t) have a continuous derivative such that

/•oo

|^'(OI * < °°]- Show that (a) u" + [1 + q(t)]u = 0 has solutions v(i}satisfying

and that (b) u" + [— I + q(t)]u = 0 has solutions u(f) satisfying

(c) State an analogous result for (17.17) where A ?* 0 and it is not assumedthat A or ^(f) are real-valued; cf. Exercises XI 8.4(b).

Exercise 17.5. In the differential equation

let /(/) be a continuously differentiate, complex-valued function for/ ^ 0 such that

(a) Show that if/'/I/I • |Re f^\ -^ 0 as f -* oo, then (17.23) has solutionssatisfying

(Z>) Show that if |Re/*(Ol1~*l/'(0//(OII'<* < oo for some p, 1

p^2, then (17.23) has solutions satisfying (17.25) and

(17.26) 40~/-*(Oexp±f/H(r)<fr as t-+co'.If / > 0, condition (17.24) is redundant.(c) Show that if /'(0//3/*(0 is of bounded variation, i.e.,

/ '(0//3/i(0-* 0 as f -»• oo, then (17.23) has a pair of solutions satisfying(17.25) and

For other results of this type, see § XI 9. For analogous results whenRe/* = 0, see Exercise XI 8.5.

Exercise 17.6. As a simple application of the last exercise, considerWeber's equation

where X is a constant, (a) By introducing the new independent variables = \t*, deduce from Theorem 17.4 that (17.28) has a pair of solutionsw0(0> «i(0 which do not vanish for large t and satisfy MO' «-^ — tu0, M/ =o(tu^ as /-»-QO. (b) Show that (17.28) has a pair of solutions w0»

wisatisfying u0 /~1~2V~'2/2, M! /^ /2* as / —»- oo. (c) Find asymptoticrelations for derivatives u' of solutions u of (17.28) by differentiating (17.28)and applying (/>). (See also Exercise XI 9.7.)

Notes

For references and other treatments of the topics in this chapter, see Cesari [2] andBellman [4].

SECTION 1. The main results, Theorems 1.1 and 1.2, are due to Wintner [3], [7], [8],who gave the existence assertions essentially in the form stated in Exercise 1.2. Linearcases, where F(t, £) = G(t)£ for a matrix G(t\ are much older; see Dunkel [1]. ForExercise 1.1, see Hale and Onuchic [1]; cf. § XII 9. For Exercise 1.4, see Wintner [21].

SECTION 2. Theorem 2.1 was formulated by Wazewski [5] and is a very useful toolin the study of differential equations. Special cases of this theorem and the argumentsin its proof had been used earlier; cf. Hartman and Wintner [1] or Nemytzkil and


Stepanov [1, p. 93]. For another type of topological argument, useful for similarpurposes, cf. Atkinson [2]. Exercise 2.1 is due to Plis [1J.

SECTION 3. The results of this section are due to Wazewski [5].SECTIONS 4-7. Lemmas 4.1 and 4.2 are related to results of Wazewski [6], Szmydt6wna

[1], Lojasiewicz [1], and Hartman and Wintner [17], [19]. The proofs in the text areadapted from those of Wazewski and his students just mentioned; for other proofs,see the papers of Hartman and Wintner. Lemma 4.3 and applications were given in thepapers of Hartman and Wintner. Conditions of the type (4.24) were introduced byHartman [5]. For Exercise 4.6, see Levinson [3] (for the part dealing with boundedness,see Cesari [1]); an analogous result (see Exercise 17.4) on a second order equation wasgiven by Wintner [10]. See Cesari [2, pp. 38-42], for related results and references.For results related to Exercises 4.8, 4.9 and applications, see Coffman [2].

SECTIONS 8-12. Results related to those occurring in § 8 for analytic systems are theoldest in this chapter and go back to Poincare and to Lyapunov [2]. For particularcases for linear differential equations, see PoincarS [4] and Perron [2]. Cotton [1] andthen Perron [9], [10], [12] systematically investigated nonanalytic, nonlinear cases,but under conditions heavier than those in the text. Their results depended on the methodof successive approximations. See also Bellman [1], who used fixed point theorems toobtain an analogue of Theorem 8.1, and the references above for §§ 4-7 to Wazewski,Hartman, and Wintner, etc. The relaxation of the condition "X')-*0 as / -»• oo" to(4.24) is due to Hartman [5] and to Hartman and Wintner [19]. A form of Theorem8.2 involving stronger hypothesis and weaker assertions was given by Petrovsky [1].The last two parts of Corollary 8.1 are proved in Coddington and Levinson [2, Chap. 13]by the method of successive approximations; cf. Exercise 8.1. For another applicationof a related method of successive approximations, see Lillo [1]. The comparativelysimple proofs in the text for Theorems 8.2,8.3 and Corollary 8.1 are new. Theorem 11.1is a slight improvement of a result of Lettenmeyer [2]. Theorem 11.2 is given byHartman and Wintner [19]. Results of the type in Exercise 11.1 go back to Bocher [2]and Dunkel [1]; cf. notes on §§ 13-16 below. Exercise 11.2 was first given by Hartman[5] for the case of a second order equation (see Theorem 17.S with d = 2) and generalizedto the situation in Exercise 11.2 by Hartman and Wintner [17].

SECTIONS 13-16. Results of the type in Theorem 13.1 were first given by Bocher [2]for a second order, linear equation. Using successive approximations similar to thoseof Exercise 8.1, Dunkel [1] generalized Bocher's result to arbitrary linear systems (13.1),where F(t, f ) = G(t)£, but his results are not as sharp as those given here. Theorems13.1, 13.2 and their corollaries in § 16 are due to Hartman and Wintner [19]. Theproofs in the text, which take full advantage of Wazewski's principle of § 2, depend inan essential way on the change of variables (14.1)-(14.2) similar to those introduced byHartman and Wintner [17] and simplified by Coffman [2]. See also Olech [1].

SECTION 17. When a = 0, Theorem 17.1 is due to Bocher [2] for d = 2 and, in aweakened form, it is contained in Dunkel's result [1 ] for arbitrary d. For a = 0, it isgiven by Faedo [1] and Ghizetti [2]. Theorem 17.2 and a less precise form of Theorem17.3 with a = 0, are also contained in Dunkel [1]; Faedo [1], [2]; and Ghizetti [1].Theorem 17.4 is a generalization of results of Poincare [4] and Perron [2] and is con-tained in Hartman and Wintner [17]. For Exercise 17.3, see Hartman [5]. Theorem17.5 for d *s 2 is due to Hartman [5]; the result formulated in the text is new. For ageneralization of the case d « 2, see Bellman [3]. Exercise!7.4 is due to Wintner [10),[13] and is contained in the more general result of Exercise 4.6; cf. also Exercise XI8.4(6). Results of the type in Exercise 17.5(a) go back to Wiman [1], [2]; for both parts(a) and (b)* see Hartman a,nd Wintner [17].

*For related results, see Coppel [SI ], Chapter IV.

Chapter XI

Linear Second Order Equations

1. Preliminaries

One of the most frequently occurring types of differential equations inmathematics and the physical sciences is the linear second order differentialequation of the form

or of the form

Unless otherwise specified, it is assumed that the functions /(f)> g(i), W0»and XO j4 0, q(t) in these equations are continuous (real- or complex-valued) functions on some /-interval /, which can be bounded or un-bounded. The reason for the assumption XO 0 WM soon become clear.

Of the two forms (1.1) and (1.2), the latter is the more general since(1.1) can be written as

if XO is defined as

for some a e/. As a partial converse, note that if XO *s continuouslydifferentiate then (1.2) can be written as

which is of the form (1.1).When the function p(t) is continuous but does not have a continuous

derivative, (1.2) cannot be written in the form (1.1). In this case, (1.2)is to be interpreted as the first order, linear system for the binary vector

322

Linear Second Order Equations 323

(1.5)

In other words, a solution u = u(i) of (1.2) is a continuously differentiablefunction such that p(t)u'(t) has a continuous derivative satisfying (1.2).When XO ?* °> ?(0> h(t) are continuous, the standard existence anduniqueness theorems for linear systems of §IV 1 are applicable to (1.5),hence (1.2). [We can also deal with more general (i.e., less smooth) typesof solutions if it is only assumed, e.g., that \lp(t), q(t), h(i) are locallyintegrable; cf. Exercise IV 1.2.]

The particular case of (1.2) where p(i) s 1 is

When />(/) 0 is real-valued, (1.2) can be reduced to this form by thechange of independent variables

for some a e J. The function s = s(t) has a derivative ds/dt = 1//KO 3^ 0and is therefore strictly monotone. Hence s = s(t) has an inverse functiont = t(s) defined on some 5-interval. In terms of the new independentvariable s, the equation (1.2) becomes

where / in p(t)q(t) and p(t)h(t) is replaced by the function f = t(s). Theequation (1.8) is of the type (1.6).

If g(t) has a continuous derivative, then (1.1) can be reduced to anequation of the form (1.6) also by a change of the dependent variableM —>• 2 defined by

for some a € J. In fact, substitution of (1.9) into (1.1) leads to the equation

which is of the type (1.6).In view of the preceding discussion, the second order equations to be

considered will generally be assumed to be of the form (1.2) or (1.6). Thefollowing exercises will often be mentioned.

Exercise 1.1. (a) The simplest equations of the type considered in thischapter are

where a 5* 0 is a constant. Verify that the general solution of theseequations is

respectively. (ft) Let a, b be constants. Show that w = e** is a solution of

if and only if A satisfies

Actually, the substitution u = ze~MIZ [cf. (1.9)] reduces (1.13) to

Hence by (a) the general solution of (1.13) is

according as (1.14) has a double root A = \b or distinct roots Al5 A2 =~-|6 ± (i*2 - a)H. When fl, ft are real and #>2 — a < 0, nonrealexponents in the last part of (1.15) can be avoided by writing

(1.16) u = «-w/2[c! cos .(a - \b*)*t + cz sin (a - #•)**].

(c) Let JM be a constant. Show that u — i* is a solution of

if and only if A satisfies

Thus if [i T£ I, the general solution of (1.17) is

If /* is real and /M > J, the nonreal exponents can be avoided by writing

Actually, the change of variables u = tl/*z and / = e* transforms (1.17)into


Thus by (a) the general solution of (1.17) is

according as fj. = J or p y& J.Exercise 1.2. Consider the differential equation

The change of variables


For a given constant p, consider the sequence of functions

defined by t*[qn(t) - l/4f2] = q^s) if r = e\ so that ^n(r) = rz[± +?«-x(log /)] or

l°gi t = log r, log, t = log (log,_i f), and the empty product is 1. Ifq(t) = qn(t), n > 0, in (1.23), then the change of variables (1.24) reduces(1.23) to the case where t, qn(t) are replaced by s, qn-i(s). In particular, ifju is real and q = qn(i), n 0, then real-valued solutions of (1.23) haveinfinitely many zeros for large t > 0 if and only if p > J.

2. Basic Facts

Before considering more complicated matters, it is well to point out theconsequences of Chapter IV (in particular, § IV 8) for the homogeneousand inhomogeneous equation

To this end, the scalar equations (2.1) or (2.2) can be written as the binaryvector equations


where x = (x1, a;2), y = (yl, y2) are the vectors x — (u,p(t)u), y = (w, p(t)w')and A(i) is the 2 x 2 matrix

Unless the contrary is stated, it is assumed that p(t) j& 0, q(t), h(t\ andother coefficient functions are continuous, complex-valued functions on a/-interval J (which may or may not be closed and/or bounded).

(i) If /0 e J and w0, «0' are arbitrary complex numbers, then the initialvalue problem (2.2) and

has a unique solution which exists on all of/; Lemma IV 1.1.(ii) In the particular case (2.1) of (2.2) and w0 = HO' = 0, the correspond-

ing unique solution is u(t) = 0. Hence, if u(t) ^ 0 is a solution of (2.1),then the zeros of u(t) cannot have a cluster point in J.

(iii) Superposition Principles. If u(t\ v(t) are solutions of (2.1) andclt c2 are constants, then ctu(t) + c2

y(0 is a solution of (2.1). If wQ(t) is asolution of (2.2), then wt(t) is also a solution of (2.2) if and only if u =wi(0 ~ wo(0 is a solution of (2.1).

(iv) If u(t\ v(t) are solutions of (2.1), then the corresponding vectorsolutions x = (u(f), p(i)u'(f)), (v(t), p(t)v'(t)) of (2.3) are linearly independent(at every value of /) if and only if w(f), v(t) are linearly independent in thesense that if clt cz are constants such that cvu(t) + czv(t) = 0, thenCl = c2 = 0; cf. §IV8(iii).

(v) If M(/), v(t) are solutions of (2.1), then there is a constant c, dependingon M(/) and v(t), such that their Wronskian W(t) = W(t; u, v) satisfies

This follows from Theorem IV 1.2 since a solution matrix for (2.3) is

detA-(0=XO^(0 and tr ,4(0 = 0; cf. §IV 8(iv). A simple directproof is contained in the following paragraph,

(vi) Lagrange Identity. Consider the pair of relations

where/ = /(f), g = g(0 are continuous functions on /. If the second ismultiplied by «, the first by v, and the results subtracted, it follows that

since [p(uv' — u'v)]' = u(pv')' — v(pu')'. The relation (2.9) is called theLagrange identity. Its integrated form

where [a, t] <= /, is called Green's formula.(vii) In particular, (v) shows that u(t) and v(t) are linearly independent

solutions of (2.1) if and only if c 0 in (2.7). In this case every solutionof (2.1) is a linear combination c^u(t) + c2y(0 of "(0> y(0 with constantcoefficients.

(viii) If />(/) = const, [e.g., p(f) s 1], the Wronskian of any pair ofsolutions w(f), v(i) of (2.1) is a constant.

(ix) According to the general theory of § IV 3, if one solution of u(t) & 0of (2.1) is known, the determination (at least, locally) of other solutionsv(t) of (2.1) are obtained by considering a certain scalar differential equa-tion of first order. If u(t) ^ 0 on a subinterval /' of /, the differentialequation in question is (2.7), where u is considered known and v unknown.If (2.7) is divided by «2(f), the equation becomes

and a quadrature gives

if a, tej'\ cf. § IV 8(iv). It is readily verified that if clt c are arbitraryconstants and a, t eJ', then (2.12) is a solution of (2.1) satisfying (2.7) onany interval J' where u(t) j& 0.

(x) Let u(t), v(t) be solutions of (2.1) satisfying (2.7) with c 0. For afixed s £ J, the solution of (2.1) satisfying the initial conditions u(s) = 0,p(s)u(s) = 1 is c~l[u(s)v(t) — u(t)v(s)]. Hence the solution of (2.2)satisfying w(f0) = w'(tQ) = 0 is

cf. § IV 8(v) (or, more simply, verify this directly). The general solution of(2.2) is obtained by adding a general solution cXO + c2v(t) of (2.1) to(2.13) to give

If the closed bounded interval [a, b] is contained in J, then the choice

reduces (2.14) to the particular solution

This can be written in the form

where

Remark. If /r(f) is (not necessarily continuous but) integrable over[a, b], then w(t) is a "solution" of (2.2) in the sense that M>(/) has a con-tinuous derivative w' such that p(f)w'(i) is absolutely continuous and(2.2) holds except on a /-set of measure 0.

Exercise 2.1 Verify that if a, ft, y, 8 are constants such that

then the particular solution (2.15) of (2.2) satisfies

An extremely simple but important case occurs if p = 1, q = 0 so that(2.1) becomes u = 0. Then «(/) = t — a and v(t) — b — t are thesolutions of (2.1) satisfying u(a) = 0, v(b) = 0, and (2.7) with c = a — b.Hence

(2.18)

is the solution of w" = h(t) satisfying w(a) = w(b) = 0.Exercise 2.2. Let [a, £] c /. Show that most general function (/(/, s)

defined for a < s, t b for which (2.16) is a solution of (2.2) for a <11 6for every continuous function /j(f) is given by


where A = (a,*), B — (bik) are constant matrices such that

and M! = u(t), «2 = v(t) are solutions of (2.1) satisfying (2.7) with c 0.In this case, (?(/, 5) is continuous for a s, t 6.

Exercise 2.3. Let a (and/or 6) be a possibly infinite end point of /which does not belong to J, so that />(f), q(t\ h(t) and u(t), v(t) need nothave limits as t -»> a + 0 (and/or t-+b — 0). Suppose, however, thatA, w, y have the property that the integrals in (2.15) are convergent (possibly,just conditionally). Then (2.15) is a solution of (2.2) on J. [This followsfrom the derivation of (2.15) or can be verified directly by substituting(2.15) into (2.2).]

(xi) Variation of Constants. In addition to (2.1), consider anotherequation

where pQ(i) ^ 0, q0(t) are also continuous in J. Correspondingly, (2.19) isequivalent to a first order system

where

0(0, f0(0 be linearly independent solutions of (2.19) such that

is a fundamental matrix for (2.20) with det Y(t) = 1; i.e.,

Hence

Consider the linear change of variables

for the system (2.3). The resulting differential equation for the vector y is

Let u0


cf. Theorem IV 2.1. A direct calculation using (2.5), (2.21), (2.22), and(2.23) shows that

In the particular case, p0(t) = /?(0, so that (2.19) reduces to

the matrix C(t) depends on uQ(i), vQ(t) but not on their derivatives. Here,(2.1) or equivalently (2.3) is reduced to the binary system

Exercise 2.4. In order to interpret the significance of y, i.e., of thecomponents y1, yz ofy in (2.28) for a corresponding solution «(/) of (2.1),write (2.1) as (pwj + q0w = h(t\ where w = «(/), h = [qjfy - q(t)]u(t).Then it is seen that the solution u(t) of (2.1) is of the form (2.14) if c — 1and w(f), v(t) are replaced by u0(t), v0(t). Using (2.24), where p =,/>„ andx is the binary vector (u(t),p(t)u'(tj), show that the coefficients of «0(f),v0(t) in this analogue of formula (2.14) are the component y1, yz of thecorresponding solution y(t) of (2.28).

(xii) If we know a particular solution «0(f) of (2.27) which does notvanish on J, then we can determine linearly independent solutions by aquadrature [cf. (ix)] and hence obtain the matrix in (2.28). Actually thisdesired result can be obtained much more directly. Let (2.27) have asolution w(t) T£ 0 on the interval /. Change the dependent variable fromM to z in (2.1), where

The differential equation satisfied by 2 is

If this is multiplied by w, it follows that

or, by (2.27),

i.e., (2.29) reduces (2.1) to (2.30) or (2.31). Instead of starting with adifferential equation (2.27) and a solution w(t), we can start with afunction H>(/) 7* 0 such that w(t) has a continuous derivative w'(t) andp(t)w'(t) has a continuous derivative, in which case qQ(i) is defined by (2.27),

sofjfo = — (pw'y/w. The substitution (2.29) will also be called a variationof constants.

(xiii) Liouville Substitution. As a particular case, consider (2.1) with/>(<) = 1,

Suppose that q(t) has a continuous second derivative, is real-valued, anddoes not vanish, say

is independent of t. Consider the variation of constants

Then (2.32) is reduced to (2.30), where/) s 1, i.e., to

A change of independent variables t-+s defined by


where

and the argument of q and its derivatives in (2.38) is t — t(s), the inverse ofthe function s = s(t) defined by (2.36) and a quadrature; cf. (1.7). Inthese formulae, a prime denotes differentiation with respect to f, so thatq' = dqldt.

The change of variables (2.34), (2.36) is the Liouville substitution. Thissubstitution, or repeated applications of it, often leads to a differentialequation of the type (2.37) in which /($) is "nearly" constant; cf. Exercise8.3. For a simple extreme case of this remark, see Exercise l.l(c).

(xiv) Riccati Equations. Paragraphs (xi), (xii), and (xiii) concern thetransformation of (2.1) into a different second order linear equation orinto a suitable binary, first order linear system. (Other such transfor-mations will be utilized later; cf. §§8-9.) Frequently, it is useful totransform (2.1) into a suitable nonlinear equation or system. In thisdirection, one of the most widely used devices is the following: Let

so that r' = (j>u')'/u — p-l(pu'/u)2. Thus, if (2.1) is divided by u, the resultcan be written as

This is called the Riccati equation of (2.1). (In general, a differentialequation of the form r' = a(t)rz + b(t)r + c(t), where the right side is aquadratic polynomial in r, is called a Riccati differential equation.)

It will be left to the reader to verify that if u(t) is a solution of (2.1)which does not vanish on a /-interval J' (c /), then (2.39) is a solution of(2.40) on J'; conversely if r — /•(/) is a solution of (2.40) on a /-interval J'(c /), then a quadrature of (2.39) gives

a nonvanishing solution of (2.1) on /'.Exercise 2.5. Verify that the substitution r = u'ju transforms

into the Riccati equation

(xv) Prufer Transformation. In the case of an equation (2.1) with real-valued coefficients, the following transformation of (2.1) is often useful(cf. §§ 3, 5): Let u = i/(/) 0 be a real-valued solution of (2.1) and let

Since u and u' cannot vanish simultaneously a suitable choice of <p atsome fixed point /„ e/and the last part of (2.42) determine a continuouslydifferentiable function <p(t). The relations (2.42) transform (2.1) into

The equation (2.43) involves only the one unknown function <p. If asolution 9? = 9?(/) of (2.43) is known, a corresponding solution of (2.44)is obtained by a quadrature.

An advantage of (2.43) over (2.40) is that any solution of (2.43) exists onthe whole interval / where p, q are continuous. This is clear from therelation between solutions of (2.1) and (2.43).

Exercise 2.6. Verify that if T(/) > 0 is continuous on J and is locally ofbounded variation (i.e., is of bounded variation on all closed, boundedsubintervals of J) and if u = u(t) & 0 is a real-valued solution of (2.1),then

and a choice of <p(t0) for some t0eJ determine continuous functionsp(t), gp(f) which are locally of bounded variation and

The relations (2.46), (2.47) are understood to mean that Riemann,Stieltjes integrals of both sides of these relations are equal. Conversely(continuous) solutions of (2.46)-(2.47) determine solutions of (2.1), via(2.45). Note that if q(t) > 0, p(t) > 0, and q(i)p(t) is locally of boundedvariation, then the choice T(?) = plA(t)qlA(t) > 0 gives q\r — rjp =p*lq* and reduces (2.45) and (2.46), (2.47) to

and

3. Theorems of Sturm

In this section, we will consider only differential equations of the type(2.1) having real-valued, continuous coefficient functions p(i) > 0, q(t)."Solution" will mean "real-valued, nontrivial (^ 0) solution." Theobject of interest will be the set of zeros of a solution u(t). For the study ofzeros of «(/), the Priifer transformation (2.42) is particularly useful sinceM(/0) = 0 if arid only if <p(t0) = 0 mod TT.

Lemma 3.1. Let u(t) &Qbea real-valued solution of(2.1) on tQ < t < t°,where p(t) > 0 and q(f) are real-valued and continuous. Let u(t) haveexactly n (^ 1) zeros t1<tz<-- <tn on t0 < t t°. Let y(t) be acontinuous function defined by (2.42) and 0 <p(t0) < TT. Then <p(t^ — k-nand <p(f) > k^ for tk < t t°for k = 1,. . . , n.

Proof. Note that at a /-value where u = 0, i.e., where 9? == 0 mod -IT,(2.43) implies that <p' = \{p > 0. Consequently <p(t) is increasing in theneighborhoods of points where <p(t) — JTT for some integer/ It followsthat if /0 a t° andj-rr < <p(a\ then y(t) > JTT for a < t t°; also ifjtr j> 9>(fl), then 9?(/) < JIT for f0 < t < a. This implies the assertion.

In the theorems of this section, two equations will be considered

where />,(0» ft(0 are real-valued continuous functions on an interval /,and

In this case, (3.12) is called a S/wr/w majorant for (3.10 on / and (3.1j) is aStorm minorant for (3.12). If, in addition,

or

holds at some point f of/, then (3.12) is called a -sfr/cf SVww majorant for(3.10 on/.

Theorem 3.1 (Sturm's First Comparison Theorem). Let the coefficientfunctions in (3.1,) be continuous on an intervalJ:t0 ^ t ^ t° and let (3.12)be a Sturm major ant for (3.10- Let u = w^/) Obea solution of (3.10 andlet w^O /iaw? exactly n (^ 1) zeros f = fx < / , < • • • < / „ on f0 < / ^ f°.Le/ u = uz(t) ^ 0 be a solution of (3.\2) satisfying

fl/ / = /0. (77re expression on the right [or left] of (3 A) at t = t0 is consideredto be + oo ifu2(t0) = 0 [or w^/,,) = 0]; w particular, (3.4) /ro/cfe at t — t0

f/wi('o) == 0-) ^e/* "2(0 Afl* fl/ / -s' n zeros ont0< t tn. FurthermoreM2(f) has at least n zeros ont0<t<tn if either the inequality in (3.4) holdsat t = tQ or (3.12) is a strict Sturm majorant for (3.10 on fo = t = tn.

Proof. In view of (3.4), it is possible to define a pair of continuousfunctions ^(r), 9>2(0 ont0<t^t° by

Then the analogue of (2.43) is

Since the continuous functions^/, <p,-) are smooth as functions of thevariable <pjt the solutions of (3.6) are uniquely determined by their initialconditions. It follows from (3.2) that/^/, (p) <fz(t, <p) for /„ t ^ f ° andall (p. Hence the last part of (3.5) and Corollary III 4.2 show that

In particular, (O — nir implies that nrr ^ 9>2(f „) andtne fifst Part oftne

theorem follows from Lemma 3.1.In order to prove the last part of the theorem, suppose first that the

sign of inequality holds in (3.4) at t = f0. Then <pv(t0) < <pz(t9). Let (Obe the solution of (3.62) satisfying the initial condition qpM(;0) = 9>i('o)> so

that 9>2o('o) < <pz(t0)- ^ince solutions of (3.6^) are uniquely determined byinitial conditions, <p2o(0 < ^zCO f°r 'o = f = *°- Thus the analogue of(3.7) gives <pv(t) ^ ^(O < <p2(f), and so <pz(tn) > nir. Hence w2(f) has nzeros on r0 < / < /„.

Consider the case that equality holds in (3.4) but either (3.3t) or (3.3^)holds at some point of [/0, *»]• Write (3.62) as

where

If the assertion is false, it follows from the case just considered that<Pi(0 = ^(0 f°r 'o ' = tn- Hence, ^(i) = <p2'(0 and so €(t) = 0 for/0 ^ / ^ /„. Since sin <p8(/) = 0 only at the zeros of u2(t), it follows that^(0 = ?i(0 f°r 'o ^ = *n and that (p^1 — f1) cos2 <p2 = 0. Hence,p%\t) — Pil(t) > 0 at some t implies cos <pa(f) = 0; i.e., «2' = 0. If (3.3^does not hold at any / on [f0, /„], it follows that (3.32) holds at some t andhence on some subinterval of [f0, tn]. But then «/ = 0 on this interval,thus (pzUz'y = 0 on this interval. But this contradicts q£t) ^ 0 on thisinterval. This completes the proof.

Corollary 3.1 (Sturm's Separation Theorem). Let (3.12) be a Sturmmajor ant for (3.1j) on an intervalJ and let u = uj(t) & 0 be a real-valuedsolution o/(3.1,). Let u^t) vanish at a pair of points t = tlt tz (>/0 ofJ.Then uz(t) has at least one zero on [flf /2]. In particular, ifpl s pz, q± = q2and Ui, uz are real-valued, linearly independent solutions of (3A^) = (3.12),then the zeros o/Ui separate and are separated by those ofuz.

Note that the last statement of this theorem is meaningful since thezeros of ult «2 do not have a cluster point on /; see § 2(ii). In addition,Mi(0» wa(0 cannot have a common zero t — tv\ otherwise, the uniqueness of

the solutions of (3.1!) implies that uv(t) = cuz(t) with c = Mi'COM'C'i)[so that M1(r), M2(0 are not linearly independent].

Exercise 3.1. (a) [Another proof for Sturm's separation theoremwhen/^(f) = p2(t) > 0, q2(t) ^ q^i).] Suppose that u^t] > 0 for ^ < t <t2 and that the assertion is false, say w2(f) > 0 for tt ^ / tz. Multiplying(S.lj) where u = MX by w2

and (3.12) where u = «2 by MJ, subtracting, andintegrating over [tlt t] gives

2t

where/; = Pi = p2; cf. the derivation of (2.9). This implies that (uju^' ^0; hence w^a > 0 for < f ^ tz (£) Reduce the case/^/) />2(f) to thecase p][t) = />2(0 by the device used below in the proof of Corollary 6.5.

Exercise 3.2. (a) In the differential equation

let q(t) be real-valued, continuous, and satisfy 0 < m q(t) ^ M. If« = w(^) ^ 0 is a solution with a pair of zeros / =tlt /2(> ^), thenTT\mA ^ /2 — /! TT/M1^. (6) Let ^(/) be continuous for / ^ 0 andq(t) -> 1 as r —*- oo. Show that if w = «(?) ^ 0 is a real-valued solution of(3.8), then the zeros of u(t) form a sequence (0 ) /x < /z < ... such that/„ — ?n_i -*• TT as « -*• oo. (c) Observe that real-valued solutions u(t) ^ 0of (1.17) have at most one zero for t > 0 if ju, J and have infinitelymany zeros for f > 0 if /u > J. In the latter case, the zeros cluster at/ = 0 and / = oo. (d) Consider the Bessel equation

where p is a real parameter. The variations of constants M = t^v trans-forms (3.9) into

Show that the zeros of a real-valued solution v(t) of (3.9) on t > 0 form asequence tl < /2 < ... such that tn — tn_^ -> TT as n -*• oo.

Theorem 3.2 (Sturm's Second Comparison Theorem). Assume theconditions of the first part of Theorem 3.1 and that w2(/) a/lyo has exactly nzeros on tQ < t f°. 77ie« (3.4) Ao/rfs at t — t° (where the expression onthe right [or left] o/(3.4) at t — t° is taken to be — oo if M2(f°) = 0 [oru^t0) = 0]). Furthermore the sign of inequality holds at t — t° in (3.4) ifthe conditions of the last part of Theorem 3.1 hold.

Proof. The proof of this assertion is essentially contained in the proofof Theorem 3.1 if it is noted that the assumption on the number of zeros of

w2(/) implies the last inequality in mr ^ q>i(t0) ^ 9>2(/°) < (« + !)«•• Also,the proof of Theorem 3.1 gives 9>i(f°) < q^O0) under the conditions of thelast part of the theorem.

4. Stunn-Liouville Boundary Value Problems

This topic is one of the most important in the theory of second orderlinear equations. Since a full discussion of it would be very lengthy andsince very complete treatments can be found in many books, only a fewhigh points will be discussed here.

In the equation

let />(0 > 0» ?(0 oe real-valued and continuous for a t b and A acomplex number. Let a, ft be given real numbers and consider the problemof finding, if possible, a nontrivial (^0) solution of (4.1 A) satisfying theboundary conditions

Exercise 4.1. Show that if A is not real, then (4.1 A) and (4.2) do nothave a nontrivial solution.

Exercise 4.2. Consider the following special cases of (4.1 A), (4.2):

Show that this has a solution only if A = (« + I)2 for n — 0, 1,. . . andthat the corresponding solution, up to a multiplicative constant, isu = sin (n + I)/.

It will be shown that the results of Exercise 4.2 for the special case (4.3)are typical for the general situation (4.1 A), (4.2).

Theorem 4.1. Let p(t) > 0, q(i) be real-valued and continuous fora ^ t b. Then there exists an unbounded sequence of real numbers A0 <A! < .. . such that (i) (4.1 A), (4.2) has a nontrivial (^0) solution if andonly zf A = An/or some n; (ii) if X = An and u = un(t) & 0 is a solution of(4.1An), (4.2), then un(t) is unique up to a multiplicative constant, and un(t)has exactly n zeros on a < / < b for n = 0, 1 , . . . ; (iii) // n m, then

(iv) //A is a complex number A 5* AB/or n — 0, 1 , . . . , then there exists acontinuous function G(t, s; A) = G(s, t; I) for a ^ s, t ^ b with theproperty that ifh(t) is any function integrable on a t ^ b, then

has a unique solution w =w(t) satisfying(4.2') w(a) cos a - p(d)w'(d) sin a = 0, w(b) cos p - p(V)w'(b) sin ft = 0

*m</ w(f) is given by

also G(t, s; X) is real-valued when X is real; (v) if X = Xn a«</ A(f) is afunction integrable on a t b, then (4.5AJ, (4.2') /KM a solution if andonly if

in this case, ifw(t) is a solution of(4.5Xn), (4.2'), then w(t) -f cwn(f) is alsoa solution and all solutions are of this form; (vi) if the functions un(t) arechosen real-valued (uniquely up to a factor ±1) so as to satisfy

then «„(/), «!(/),... form a complete orthonormal sequence for L?(a, b);i.e., ifh(t) e Lz(a, b), then h(t) has the Fourier series

and7

interpreted as in the Remark in § 2(x).Note the parallel of the assertions concerning the solvability of (4.5A),

(4.2') with the corresponding situation for linear algebraic equations(XI — L)w = h, where L is a d x d Hermitian symmetric matrix, / is theunit matrix and w, h vectors: (XI — L)u = 0 has a solution u j£ 0 if andonly if A is an eigenvalue A x , . . . , Ad of L\ A 1 } . . . , Xd are real; if A Xn,then (XI — L)w — h has a unique solution w for every h; finally, if X = An,then (XI — L)w = h has a solution w if and only if h is orthogonal (i.e.,u • h = 0) to all solutions u of (XI — L)u — 0.

Proof. This proof will only be sketched; details will be left to thereader.

On (i) and (U). In view of Exercise 4.1, it suffices to consider only realX. Let u(t, X) be the solution of (4.1 X) satisfying the initial condition

so u(t, X) satisfies the first of the two conditions (4.2). It is clear that

If h(t) is not continuous in (iv) or (v), then a solution of (4.5 ) is to be

(4.1 A), (4.2) has a solution (^ 0) if and only if u(t, A) satisfies the secondcondition in (4.2).

For fixed A, define a continuous function q>(t, A) of t on [a, b] by

Then ??(/, A) has a continuous derivative satisfying

cf. § 2(xv). If follows from Theorem V 2.1 that the solution (t, A)of (4.13) is a continuous function of (/, A) for a / ^ b, — <x> < A < oo.The proof of the Sturm Comparison Theorem 3.1 shows that <p(b, A) is anincreasing function of A. Without loss of generality, it can be supposedthat a satisfies 0 a < TT. Note that

In order to see this, introduce the new independent variable defined byds = dtlp(t) and s(d) — 0, so that (4.1 A) becomes

If M > 0 is any number, A > 0 can be chosen so large that p(t)[q(t) +A] M2 for a / ^ b. Sturm's Comparison Theorem 3.1 applied to

shows that if n is arbitrary and M is sufficiently large, then a nontrivialreal-valued solution of (4.15) has at least n zeros on the j-interval, 0

f6

s dt/p(t)', i.e., <p(&, A) « if A > 0 is sufficiently large by Lemma 3.1.Ja

It will be verified that

By Lemma 3.1, <p(b, A) ^ 0. Let -A > 0 be so large thatXOfa(0 + ^] -M2 < 0. The solution of

satisfying the analogue of (4.11), where a = 0 and/? = 1, is

The analogue of (p(t, A) is

(4.15) amd

For any fixed s > 0,

Phence y>(60, M) -> 0 as A/ -> oo, where 60 = I dtlp(t). By Sturm's Com-Ja

parison Theorem 3.1, <p(b, X) ^(60, M). This proves (4.16)The limit relations (4.14), (4.16) and the strict monotony of q>(b, X) as

a function of X show that there exist A0, A1 } . . . such that

where it is supposed that 0 < ft ^ ir. Furthermore (p(b, X) j£ ft mod TTunless X — An. This implies (i) and (ii).

On (HI). In order to verify (iii), multiply (4.Un) by um, (4.1ATO) by un,subtract and integrate over a t b; i.e., apply the Green identity(2.10) to/= -lnun(t),g = -lmum(t).

On(iv). See §2(x) and Exercise 2.1. Choose u ==«(/, A), and v(t) as asolution of (4.1 A) satisfying the second condition in (4.2).

On (v). Suppose first that (4.5An), (4.2') has a solution w = w(t).Apply the Green identity (2.10) in the case where q is replaced by q +^«>/= h, w = u, v = unt g = 0 in (2.8) in order to obtain (4.7).

Conversely, assume that (4.7) holds. Let u(i) = un(t) and let v(t) be asolution of (4.1AJ linearly independent of un(t), say p(f)[uv' — u'v] =c T£ 0. Then (2.15) is a solution of (4.5An). Furthermore w(t) satisfies the•first of the boundary conditions in (4.2') since u — un does; cf. Exercise2.1. On the other hand, (4.7) and (2.15) show that w(b) = w'(b) = 0.Hence w(t) is a solution (4.5A) satisfying the boundary conditions (4.2').

On (vi). Although the assertion (vi) is the main part of Theorem 4.1,it is a consequence of elementary theorems on completely continuous,self-adjoint operators on Hilbert space. For the sake of completeness, theproof of the necessary theorems will be sketched and (vi) will be deducedfrom them. A knowledge of Fourier series (involving, e.g., Bessel'sinequality, Parseval's relation, and the theorem of Fischer-Riesz) will beassumed. In order to minimize the required discussion of topics on Hilbertspace, some of the definitions or results, as stated, will involve redundanthypotheses.

Introduce the following notation and terminology:

where/, g e L*(a, b). Thus \(ftg)\ ^ \\f\\ - \\g\\ and ||/+ £|| II/H + |;|by Schwarz's inequality. A sequence of functions f^t^f^t),... in L\a, b)will be said to tend to/(r) in L?(a, b) if ||/n - /|| -»- 0 as n -* oo. They will

be said to tend to/(r) weakly in L?(a, b) if the sequence I/J, ||/2||,... isbounded and, for every <p(t) e L?(a, b), (/„, 9?) -> (/, <p) as n -» oo. (In thislast definition, the condition on H/J, ||/2||,... is redundant but this factwill not be needed below.) A subset H of L2(a, b) is called a linear manifoldif/, g e H implies that c-^f + c%g e H for all constants clt c2 and it is calledclosed if/n e H for « « 1,2,... ,/E L2(a, />) and ||/n -/|| -* 0 as n -» ooimply that/E //. A linear manifold H of L2(a, b) will be called weaklyclosed if/M e // for n = 1,2,.. . ,/E L2(a, £) and/n -*/weakly as « -* ooimply that feH. (The fact that the notions of "closed" and "weaklyclosed" are equivalent for linear manifolds will not be needed here.)

Lemma 4.1. Letfltfz,... be a sequence of elements ofL\a, b) satisfyingH/JI ^ 1. Then there exists an f(t) E L2(a, b) and a subsequence fn(l)(t),

/n(2)(0> • • • of the given sequence such that ||/|| ^ 1 andfn(i} ->/(0 weaklyas j-+ oo.

Proof. Without loss of generality, it can be supposed that [a, b] =[0, IT], Thus each/n(f) has a sine Fourier series

where, by Parsevai's relation, 2 |cBfc|2 = ||/J|2 1. It follows from

*Cantor's diagonal process (Theorem I 2.1) that there exists a sequence ofintegers 1 n(l) < n(2) < ... such that

(4.18) ck == lim cn(i}k exists as y-* oo for k = 1, 2

Note that

Hence £ |cfc|2 < 1 and so, by the theorem of Fischer-Riesz, there exists an

/(O E L\a, b) such that

It follpws from (4.18) that (/n(j), 9?) ->• (/, 9?) asy -»• oo holds if 9? = sin tofor k = 1,2, Hence it holds for any sine polynomial p(t) = al sin / +• • • 4- flm sin m/. For any 9>(f) E L2(0, «•), there exists a sine poly-nomial p(t) such that ||9> — p\\ is arbitrarily small and \(fn(j) — /, 9>)| =l(/«0) -/.T')! + K/«0> -//> - 9>)l, While |C/n(,-) -/./>- ?)l ^ ll/n(/, -

/||-||/> — 9?|| < 2 ||/> — 9?||. Hence the lemma follows.Lemma 4.2. Let G be a self-adjoint, linear operator defined on a weakly

closed linear manifold H of L2(a, b) satisfying (Gh, A) = 0 for all he H.Then Gh = Ofor allh<eH.

To say that G is a linear operator on H means that to every he H thereis associated a unique element w = Gh e H and that if H>, = Ghf forj — 1,2, then clwl + czw2 = (/(c^ + c^) for all complex constantsct, c2. The assumption that G is self-adjoint means that (Gh,f) = (h,Gf)for all/, h e f f .

Proof. If/, he H and c is a complex number, then 0 = (G(h + c/),£ + c/) = 2 Re c(G/z,/) since ((//,/) = (Gh, h) = 0. By the choices c = 1and c = /, it follows that (Gh,f) — 0. On choosing/= GTz, it is seen thatC?y* = 0.

Lemma 4.3. Let G be a completely continuous, self-adjoint linearoperator on a weakly closed linear manifold H ofL?(a, b) and let Gh 5* Qforsome he H. Then G has at least'one (real) eigenvalue p, j£ 0; i.e., thereexists a (real) number p ^ 0 and an h0 e H, h0 5* 0, such that Gh0 — fihQ.

A linear operator G on H is called completely continuous if hn, h e Hand hn-+h weakly as n -> oo imply that \\Ghn — Gh\\ -> 0 as n -> oo.

Proof. It follows from Lemma 4.1, the complete continuity of G, andthe fact that H is weakly closed that G is bounded, i.e., that there exists aconstant C such that \\Gh\\ < C for all h e H satisfying \\h\\ ^ 1.

By Schwarz's inequality, \(Gh, h)\ ^ \\Gh\\ • \\h\\ ^ Cif \\h\\ ^ 1. Hencesup (Gh, h) and inf (Gh, h) for all \\h\\ ^ 1 exist and are finite. SinceGh 5^ 0 for some he H, it follows from Lemma 4.2 that at least one ofthese two numbers is not zero. For the sake of definiteness, let [A =sup (Gh, h) T£ 0. The choice h = 0 shows that /* 0, hence fi > 0.

It will be shown that there exists an h0 e H such that (Gh0, /z0) = ^ andll^oll = 1- F°r there exist elements hlt hz,. .. in H such that \\hn\\ ^ 1 and(Ghn, hn) -+ [i as n -> oo. In view of Lemma 4.1, we can suppose thatthere exists an h0 e L?(a, b) such that hn -*• A0 weakly as n -*- oo and\\h0\\ ^ 1. Since ./f is weakly closed, h0 e H. The complete continuity ofG shows that \\Ghn - Gh0\\ -+ 0 as n -* oo. Also (G7i0, /i0) = (G/in, /in) +2 Re (G(h0 - hn), hn) + (G(hQ - hn), h0 — hn). From the boundednessof G and Schwarz's inequality, we conclude, by letting n ->• oo, that(Gh0, h0) = /*.

Note that p ^ 0 implies /r0 7^ 0. Also, since p > 0, it follows thatHA0|| = 1, otherwise (Gh, h) = /*/||/i0||

2 > H for /i = /j0/||A0|| and ||Aj| = 1.In order to verify that Gh0 = fth0, let h be any element of H satisfying

\\h\\ = 1 and (h0, h) = 0. Let he = (A0 + «A)/(1 + c2)1^ for a real e, so thatH/ya __ i xhen the function

of e has a maximum at e = 0 and hence Re (GhQ, h) = 0. Since h can bereplaced by ih, it follows that (Gh0, h) = 0 for all h e H satisfying (h0, h) =0. In particular, (Gh0, h) = 0 if h — Gh0 — ph0. This implies that

f = ||GA0||2 and hence \\Gh0 - /*/i0||

2 = ||GA0||2 - 2(Gh0, A0) + f = 0.

This proves Gh0 = ph0 and completes the proof of the lemma.Completion of Proof of (vi). A standard theorem on Fourier series

implies that (vi) is false if and only if there exist functions h(t) e L2(a, b),||h || 516 0, having zero Fourier coefficients (A, «B) = 0 for n — 0,1,. . . .Suppose, if possible, that (vi) is false and let //denote the set of all elementsh(t) e L\a, b) satisfying (h, un) = 0 for n = 0,1, Then H is aweakly closed, linear manifold in L2(a, b) and contains elements h 3* 0.

Choose a real number A 5^ An for n = 0,1, . . . . Then (4.6) defines alinear operator G, w = G/i, on L2(a, A). This operator is self-adjoint since

follows from the fact that G(t, s; A) is real-valued and G(t, s; X) = G(.y, f; A).Also G is completely continuous. In order to verify this, let hn-*h

weakly as n -»> oo and u>B = GAn, w = G/*, then

tends to 0 as n -> oo for every fixed /. Furthermore, by Schwarz'sinequality

if ||/in||2 C ||/i(5)||2 < C. Thus |kn - w||2 = J kn(0 - w(OI2 dt -> 0 as

n -*- oo by Lebesgue's theorem on dominated convergence. [Actually, byTheorem I 2.2, wn(t) -* >v(0 as « -*• oo uniformly for a t b since it iseasily seen that the sequence M^, n> 2 , . . . is uniformly bounded andequicontinuous.]

Finally, note that if h e //, then w = Gh is in //. In fact, (h, un) = 0implies that (w, wn) = 0 as can be seen by applying the Green identity(2.10) to u = un,f= — Anwn, v = w, g = — Aw + /». Thus the restrictionof G to the weakly closed linear manifold H gives a completely continuous,self-adjoint operator on H.

From (4.5A) and (4.6), it is seen that h & 0 implies that w = Gh & 0.Since // contains elements h 0, Lemma 4.3 is applicable. Let Gh0 =/xA0, where /i0 e #, ||/i0|| = 1, 76 0. Thus, if w0 = Gh& it follows from(4.5A), (4.6) that u = H>O(/) =£ 0 is a solution of (4.1 A — I//*) satisfying theboundary conditions (4.2). Hence, by part (i), there is a non-negativeinteger k such that A — \ln = Afc and vv0 = CM* for some constant c 7* 0.But this contradicts (H>O, «n) = 0 for « = 0, 1,. . . and proves the theorem.

Exercise 4.3. Let p0(t) > 0, r0(f) > 0 and q0(t) be real-valued con-tinuous functions on an open bounded interval a < t < b. Let A0 < Aj <. . . . Suppose that

has a (real) solution un(t) on a < / o(wo"n+i - "o'««+i)= 0 at a, &.(a) Show that if p^t) = l/r0(0w<>"2(0 > 0, r^f) = l//>0(/)«o2(0 > 0 and?i(0 = -*olpo(tW(t)> then yn(/) =/>0("o"«+i - Wo'"«+i) is a solution of

having at most n zeros on a < t < & and such that the analogue of(H ), t -> a and t -»• 6, holds. (&) Show that there existpositive continuous functions a0(/)>

ai(0> • • • > a*-i(0 on a < / < 6, suchthat w0(0, • • • , «fc-i(0 are solutions of the A:th order linear differentialequation

Exercise 4.4 (Continuation), (a) Let /?0, r0, ^0, AB, wre be as in Exercise4.3. Let a < / ! < • • • < tk+l < 6 and a x , . . . , afc+1 be arbitrary numbers.Then there exists a unique set of constants c0 , . . . , ck such that

Use induction on k (for all systems w0, MJ, ...) or use Exercise IV 8.3.[This result is, of course, applicable to (real-valued) un(t) in Theorem 4.1.If the functions p0, r0, q0 have derivatives of sufficiently high order, then theinterpolation property (4.22) can be generalized, as in Exercise IV 8.3(J).](b) Let a < tQ < • - • < tn < b. Then D(/0, . . . , / „ ) = det («,(/*)), wherey, k = 0 , . . . ,« , is different from 0. (c) Let c0,. .., cn be real numbersand Un(i) = c0M0(0 + • • • + cnun(t). Then Un(t) = 0 if Un(t) vanishes atn + 1 distinct points ofa<t<b, and if Un(t) ^ 0 vanishes at n distinctpoints, then it changes sign at each, (d) Every real-valued continuous

f*function v(t) orthogonal to «„, . . . , un on [a, b] (i.e., vu} dt = 0 forJ«

y = 0 , . . . ,«) changes sign at least n + 1 times, (e) For any choice ofconstants cm,..., cn, the function cOTww(f) + • • • + cnun(f) changes signat least m times and at most n times, where m n.

5. Number of Zeros

This section will be concerned with zeros of real-valued solutions of anequation of the form

Theorem 5.1. Let q(t) be real-valued and continuous for a t b.Let m(t) ^ 0 be a continuous function for a ^ t b and

If a real-valued solution u(t) ^ 0 of (5.1) has two zeros, then

where q+(t) = max (q(t), 0); in particular,

Exercise 5.1. Show that the inequality (5.3) is "sharp" in the sensethat (5.3) need not hold if ym is replaced by ym + e for e > 0.

Proof of Theorem 5.1. Assume that (5.1) has a solution (^ 0) with twozeros on [a, b]. Since q+(t) ^ q(t), the equation

is a Sturm majorant for (5.1) and hence has a solution u(f) ^ 0 with twozeros t = a, /? on [a, 6J; cf. Theorem 3.1. Since u" — —q+u, it followsthat

cf. Exercise 2.1, in particular (2.18). Suppose that a, (I are successivezeros of u and that u(i) > 0 for a < / < /?. Choose / = /0 so that w(^0) =max u(i) on (a, /?). The right side is increased if u(s) is replaced byU(IQ). Thus dividing by u(t0) > 0 gives

where t = tQ. Since /3 — t < (i — j for / ^ J and / — a s — a forJ ^ r ,

Finally, note that (t - a)(b - t)l(b -a)°£(t- <x)(0 - 0/05 - a) fora ^ a.^ t ^ ^ b; in fact, differentiation with respect to /3 and a showsthat (/ — a)08 — t)/(ft — a) increases with 0 if / ±2: a and decreases witha if / ^ /8. Hence (5.4) follows from the last display. The relation (5.3)is a consequence of (5.2) and (5.4). This proves Theorem 5.1.

Since (t — d)(b — t) < (b — a)2/4, the choice m(t) = 1 in Theorem 5.1gives the following:

Corollary 5.1 (Lyapunov). Let q(i) be real-valued and continuous ona / b. A necessary condition for (5.1) to have a solution u(t) =£ 0possessing two zeros is that

Exercise 5.2. Let q(i) ^ 0 be continuous on a t b and let (5.1)have a solution u(i) vanishing at / = a, b and u(t) > 0 in (a, b). (a) Use

f*(5.7) to show that q(t) dt > 2M/A, where M = max u(t) and A =rb Ja

u(t) dt. (b) Show that the factor 2 of M/A cannot be replaced by aJalarger constant.

Exercise 5.3. (a) Consider a differential equation u" + g(t)u' + /(/)« =0 with real-valued continuous coefficients on 0 / b having a solutionu(t) & 0 vanishing at t = 0,6. Show that

(b) In particular, if \g\ < M± and J/| < Afz, then 1 < MJ/2 + MJPfi.But this inequality can easily be improved by the use of Wirtinger's

[b rbinequality I u2 dt (b/ir)2 I w'2 dt (which can be proved by assuming

Jo Job = rr, expanding u into a Fourier sine series, and applying Parseval'srelation for u, u'). Show that 1 M^ir + MJPlir*. (c) The result ofpart (b) can further be improved to 1 2A/1*/7r2 + Af^/ir2. See Opial[3]. (d) An analogous result for a dth order equation, d 2, is as follows:Let the differential equation «<d) + / (O"**"" + ' ' ' + pd(t)u = 0 havecontinuous coefficients for 0 t b and a solution w(0 ^ 0 with ofzeros on [0, b]. Let |/>,.(OI M,. Then 1 < M£ + M^\2\ + • • • +M^b^Kd- 1)! + (MdM/</!)[(</- l)*-1/^]-

When ^(0 = q+(t) is a positive constant on [0, T], the number JV ofzeros of a solution (00) of (5.1) on (0, T] obviously satisfies

where the last inequality follows from Schwarz's inequality. It turns outthat a similar inequality holds for nonconstant, continuous q(t):

Corollary 5.2. Let q(t) be real-valued and continuous for 0 / 5: T.

Let u(t) & 0 be a solution of (5.1) and N the number of its zeros on 0 <t < T. Then

Proof. In order to prove this, let N 2 and let the N zeros of u on(0, X] be (0 <) rx < r2 < • • • < tN (^ T). By Corollary 5.1,

for k = 1, . . . , N — 1. Since the harmonic mean of N — 1 positivenumbers is majorized by their arithmetic mean,

Thus adding (5.10) for k = I,..., N — 1 gives

hence (5.9).Exercise 5.4. Show that N also satisfies

To this end, use (5.3) with m(t) = t — a in place of (5.7).Note that if q(t) is a positive constant, then

An analogous inequality holds under mild assumptions on nonconstant q:Theorem 5.2. Let q(t) > 0 be continuous and of bounded variation on

0 ^ t T. Let u(t) & 0 be a real-valued solution of (5.1) and N thenumber of its zeros on 0 < / ^ T. Then

Proof. In terms of u(i) define a continuous function <p(i) by

Then [cf. Exercise 2.6; in particular (2.49) where p(t) = 1]

By Lemma 3.1, N is the greatest integer not exceeding (p(T)J7r, so thatirN < 9>(r) <; TT(N + 1). This implies (5.12).

Exercise 5.5. (a) Let q(t) be continuous on 0 t T. Let u(t) ^ 0be a real-valued solution and N the number of its zeros on 0 < t T.Show that

(b) If, in addition, q(t) > 0 has a continuous second derivative, then

Corollary 5.3. Let q(i) > 0 be continuous and of bounded variation on[0, T] for every T > 0. Suppose also that

e.g., suppose that q(t) has a continuous derivative q'(t) satisfying

Let u(f) ^ 0 be a real-valued solution of (5.1) and N(T) the number of itszeros on 0 < t T. Then

This is clear from (5.13) and the formula (5.12) in Theorem 5.2. Itshould be mentioned that if, e.g., q is monotone and ^(0 -*• oo as / -> oo,then (5.14) imposes no restriction on the rapidity of growth of q(t) but is acondition on the regularity of growth. This can be seen from the fact thatthe integral

tends to a limit as T-* oo; thus, in general, q'lq^ is "small" for large /.The conditions of Corollary 5.3 for the validity of (5.15) can be lightened

somewhat, as is shown by the following exercises.Exercise 5.6. (a) Let q(i) > 0 be continuous for t ^ 0 and satisfy

Let u(i) & 0 be a solution of (5.1) and N(T) the number of its zeros on0 < / < T. Then (5.15) holds, (b) Necessary and sufficient for (5.16) is

/*00the following pair of conditions: q^ dt — oo and q(t + cq~lA(t))l

q(f) -> 1 as t -*- oo holds uniformly on every fixed bounded c-interval on— 00 < C < OO.

Exercise 5.7. Part (b) of the last exercise can be generalized as follows:Let q(t) > 0 be continuous for t 0. Let m(t) > 0 be continuous fort > 0 and satisfy [m(t)lm(s)]±l < C(tls)v for 0 < s < t < oo and somepair of non-negative constants C, y. Necessary and sufficient for

/»oois that m(q(tj) dt = oo and that q(t + c/m[q(t)])lq(t) -»> 1 as min [t,

t + clm(q(t))] -> oo holds uniformly on every bounded c-interval on— 00 < C < OO.

An estimate for Af of a type very different from those just given is thefollowing:

Theorem 5.3. Let p(i) > 0, q(t) be real-valued and continuous for0 t T. Let u(t\ y(0 be real-valued solutions of

satisfying

Let N be the number of zeros ofu(t) on 0 < t <; T. Then

Proof. Let a be an arbitrary real number. Consider the solutions

u*(i) = u(t) cos a + KO sm a> v*(t) — —"(0 sin a -f- v(t) cos aof (5.17). They satisfy

Choose a so that w*(0) = 0 and let N* be the number of zeros of u*(t) on0 < t < T.

Since (5.20) implies that «*, t?* are linearly independent, they have nocommon zeros. Hence it is possible to define a continuous function by

This function is continuously differentiable and, by (5.20),

Hence y(t) is increasing; also y(t) = 0 mod IT if and only if w*(0= 0-Thus N* is the greatest integer not exceeding <p(T)/7r and a quadrature of(5.22) gives

Sturm's separation theorem implies N* N N* + 1, thus (5.19)follows.

Exercise 5.8. LetXO> <7(0> M(0» K0» and be as in Theorem 5.3 and,in addition, let q(t) > 0. Show that

(If q > 0, the relations (5.19) and (5.23) are particular cases of "duality" inwhich (w, w', q, dt) are replaced by (pu', — u, \lq, q dt); cf. Lemma XIV 3.1.)

Exercise 5.9. (a) Let q(t~) be continuous for ? 0. Using (5.9) and(5.19), show that if all solutions of u" + q(t)u = 0 are bounded, then, forlarge /,

Replacing u, v in (5.19) by uje, ei>, show that if, in addition, a nontrivialsolution «(f) —»• 0 as / -> oo, then

(A) Let ^(/) > 0 for t > 0. Using (5.9) and (5.23), show that if the firstderivatives of all solutions of u" + q(t)u = 0 are bounded, then, for large /,

If, in addition, u'(t) -»• 0 as / -> oo for some solution u(t) ^ 0, then

(c) Generalize (0) [or (/>)] for the case when u + ^w = 0 is replaced by(pu'Y + qu = 0 and the assumption that solutions [or derivatives ofsolutions] are bounded is replaced by the assumption that all solutionssatisfy «(/) = O(l/O(r)) [or u'(t) — 0(1/(/))], where O(/) > 0 is con-tinuous.

6. Nonoscillatory Equations and Principal Solutions

A homogeneous, linear second order equation with real-valuedcoefficient functions defined on an interval J is said to be oscillatory on J

if one (and/or every) real-valued solution (?£ 0) has infinitely many zeroson J. Conversely, when every solution (^ 0) has at most a finite numberof zeros on J, it is said to be nonoscillatory on J. In the latter case, theequation is said to be disconjugate on J if every solution (^ 0) has at mostone zero on J. If t = co is a (possibly infinite) endpoint of J which doesnot belong to /, then the equation is said to be oscillatory at t = o> if one(and/or every) real-valued solution (& 0) has an infinite sequence of zerosclustering at t = to. Otherwise it is called nonoscillatory at t = CD.

Extensions of many of the results of this section to higher order equationsor more general systems will be indicated in §§ 10, 11 of the Appendix.

Theorem 6.1. Let />(/) > 0, q(t) be real-valued, continuous functions ona t-interval J. Then

is disconjugate on J if and only if, for every pair of distinct points tlt tzeJand arbitrary numbers wl5 w2, there exists a unique solution u — «*(/) of(6.1) satisfying

or, equivalently, if and only if every pair of linearly independent solutions"(0X0 of (6.1) satisfy

for distinct points tly tz e J.Proof. Let u(t), v(t) be a pair of linearly independent solutions of

(6.1). Then any solution «*(/) is of the form u* = cxi/(f) + czv(t). Thissolution satisfies (6.2) if and only if

These linear equations for cx, c2 have a solution for all ult «2 if and only if(6.3) holds. In addition, they have a solution for all ulf «2 if and only ifthe only solution of

is d = c2 = 0; i.e., if and only if the only solution u*(t) of (6.1) withtwo zeros / = tly tz is u*(t) = 0.

Corollary 6.1. Let p(t) > 0, q(t) be as in Theorem 6.1. IfJ is open or isclosed and bounded, then (6.1) is disconjugate on J if and only if (6.1) has asolution satisfying u(t) > 0 on J. If J is a half-closed interval or a closedhalf-line, then (6.1) is disconjugate on J if and only if there exists a solutionu(t) > 0 on the interior ofJ.

The example u" + u = 0 on /: 0 / < TT shows that, in the last partof the theorem, there need not exist a solution w(f) > 0 on /.

Exercise 6.1. Deduce Corollary 6.1 from Theorem 6.1 (another prooffollows from Exercise 6.6).

Exercise 6.2. Let p(t) > 0, q(t) ^ 0 be continuous on an interval/•«>

J: a t <. co (< ao) such that I dtfp(t) = oo, then (6.1) is disconjugate

on J if and only if it has a solution u(t) such that u(t) > 0, u'(f) ^ 0 fora < t < co.

A very useful criterion for (6.1) to be disconjugate is a "variationalprinciple" to be stated as the next theorem. A real-valued function rj(t)on the subinterval [a, b] of/ will be. said to be admissible of class A^a, b)[or A2(a, b)] if (i) r)(a) = r)(b) = 0, and (iix) ri(t) is absolutely continuousand its derivative »/(/) is of class L2 on a t b [or (ii^ »?(0 is contin-uously differentiate and XOV(0 is continuously differentiable ona£t£b]. Put

If rj is admissible A2(a, b), the first term can be integrated by parts and itis seen that

Theorem 6.2. Let p(t) > 0, q(t) be real-valued continuous functions on at-intervalJ. Then (6.1) is disconjugate on J if and only if, for every closedbounded subinterval a^t^bofj, the functional (6.4) is positive-definiteon At(a,b) [or A2(a, b)]; i.e., /(»?; a, b) 0 for r)EAj[a,b) [or yeA2(a, b)] and /(»?; a, b) = 0 if and only iff] = 0.

The "only if" half of the theorem is stronger for A^(a, b) and the "if"half is stronger for A2(a, b).

Proof ("Only if"). Suppose that (6.1) is disconjugate on a :Sj / b.Then, by Corollary 6.1, there is a solution u(t) > 0 on a t b. Ifrfc) E A^a, b), put £(/) = r)(t)lu(t). Then

An integration by parts [integrating u' and differentiating (pu')t?] showsthat the first term is

The integrated terms vanish since rj(a) = rj(b) — 0 imply that £(a) =mb) = 0. The last two formula lines and t,zu[(pu'Y + qu] — 0 give

It is clear that /(»?; a, b) ^ 0 and I(f]\ a, b) = 0 if and only if £(/) = 0.This proves the "only if" part of the theorem.

Proof ("If"). Suppose that I(rj; a, b) is positive definite on Az(a, b)for every [a, b] <=• J. Let rj(t) be a solution of (6.1) having two zerost = a, b E J. It will be shown that /?(/) = 0. In fact ??(/) 6 Az(a, b); thus(6.5) holds. Hence, /(??; a, b) = 0 because 77 is a solution of (6.1). Since(6.4) is positive definite on Az(a, b), it follows that rj(t) = 0. This impliesthat (6.1) is disconjugate on J and completes the proof of the theorem.

Exercise 6.3. Suppose that J is not a closed bounded interval. Showthat, in Theorem 6.2, (6.1) is disconjugate on J if /(»?; a, b) ^ 0 for all[a, b] cr J and all v\ e A2(a, b).

Exercise 6.4. Deduce Sturm's separation theorem (Corollary 3.1)from Theorem 6.2.

If P is a constant positive definite Hermitian matrix, then there existsa positive definite Hermitian matrix P-^ which is the "square root" of Pin the sense that P = Pf = Pf P^\ cf. Exercise XIV 1.2. An analogueof this algebraic fact will be obtained for the differential operator

Note that (6.5) can be written as

cf. (4.17). Also, (6.7) can be written as

In addition to the quadratic functional (6.4), consider the bilinear form

for r)lt r)z 6 A^(a, b). If ^x 6 2(a, b), an integration by parts shows that

If w(f) is a solution of (6.1) and u(t) > 0 on [a, b], it is readily verifiedthat, for ifc, >y2 e Az(a, b) and = yju, ^ = ?y/M,

or


Thus if the first order differential operator Lx is defined by

then it follows that

Consequently, if L [i.e., (6.4)] is positive definite on ^2(a> *)» so that there

exists a positive solution u(t) > 0 of (6.1) on [a, b], then formally

In'fact this relation is not only formally correct but is correct in thefollowing sense:

Corollary 6.2. Let p(t) > 0, q(t) be continuous on J and let (6.1) have asolution u(t) > 0 on J. Let A be defined by (6.8) and A* its formal adjoint

cf. § IV 8 (viii). Then

/or all continuously differentiable functions rj for which p(t)rf is absolutelycontinuous (i.e., for all qfor which L[r)] is usually defined).

This can be deduced from the identity (6.9) or, more easily, by astraightforward verification. See Appendix for generalizations of thisresult.

Theorem 5.3 and its proof have the following consequence.Theorem 6.3. Let p(t) > 0, q(t) be real-valued and continuous on an

intervalJ.Then(6.l)isnonoscillatory on J if and only if every pair of linearlyindependent solutions u(t\ v(t) of (6.1) satisfy

Furthermore, (6.1) is disconjugate on J if and only if

for every pair of real-valued solutions u(t), v(t) satisfying p(u'v — uv') —c ^ 0 and every interval [a, b] c /.

If J is a half-open interval, say J:a ^ t < <o (51 oo) and (6.1) is non-oscillatory at t = to, then (6.1) has real-valued solutions u(t) for which

/•to

<#//>w2 is convergent and solutions for which it is divergent. The

latter type of solution will be called a principal solution of (6.1) at / = co.Theorem 6.4. Let p(t) > 0, q(t) be real-valued and continuous on J:

a ^ t < <w (^ oo) and such that (6.1) is nonoscillatory at t = o>. 7%e/iMere exitf.? a real-valued solution u = w0(f) o/ (6.1) vf/jic/i w uniquelydetermined up to a constant factor by any one of the following conditions inwhich Mi(0 denotes an arbitrary real-valued solution linearly independentofu0(t): (i) M0, "i satisfy

(ii) M0, M! jflf/j/y

(iii) if TeJ exceeds the largest zero, if any, ofu0(t) and ifu^T) ^ 0, thenwi(0 ^fl>s on* °r no zero on T < t < o> according as

AoWs at t = T; in particular, (6.12j) holds for all t (6 J) near CD.It is understood that in (6.10) and (6.11) only /-values exceeding the

largest zeros, if any, of «0, wt are considered. A solution w0(f) satisfyingone (and/or) all of the conditions (i), (ii), (iii) will be called a principalsolution of (6.1) (at t = o>). A solution u(t) linearly independent of «0(/)will be termed a nonprincipal solution of (6.1) (at / = o>). In view of (6.10),(6.11), the terms "principal" and "nonprincipal" might well be replacedby "small" and "large." The expressions "small," "large" will not beused in this context because of the relative nature of these terms. Consider,e.g., the. equations u" — u = 0, u" = 0 and u" + w/4/2 = 0 for t 5: 1.Examples of principal and nonprincipal solutions at t — oo for the firstequation are u = e~* and u = e*; for the second, u — 1 and u = t; forthe third, u•= /* and w = r^log/ ; cf. Exercise 1.1. The proof of(ii) will lead to the following:

Corollary 6.3. Assume the conditions of Theorem 6,4. Let u = u(i) ^ 0be any real-valued solution of (6.1) and let t = T exceed its last zero. Then

is a nonprincipal solution of (6.1) on T ^ t < CD. If, in addition, u(t) is anonprincipal solution of (6.1), then

is a principal solution on T t < a>.

Proof of Theorem 6.4 and Corollary 6.3

On (/). Let u(t), v(t) be a pair of real-valued linearly independentsolutions of (6.1) such that

If T exceeds the largest zero, if any, of v(t), then (6.15) is equivalent to

for T t < co. Hence ufv is monotone on this /-range and so

exists if C = ± oo is allowed.It will be shown that u, v can be chosen so that C = 0 in (6.17). If this

is granted and if u(t) is called w0(/), then (i) holds. In fact, a solution «x(/)is linearly independent of u0(t) if and only if it is of the form u^t) =CoU0(t) + Cjv(t) and Ci^O; in which case, C — 0 implies that ut =[GI 4- o(l)]v(t); thus M0 = o(uj) as / -»• o>.

If C = ±00 in (6.17) and if M, y are interchanged, then (6.17) holdswith C = 0. If |C| < oo and if u(t) — Cv(t) is renamed «(f), then (6.15)still holds and (6.17) holds with C = 0. This proves (i).

On (it). Note that (6.16), (6.17) give

whether or not |C| = oo or |C| < oo. If u, v is a pair MO, ult so thatC = 0, then (6.11J holds. If u, v is a pair ult u0, so that C s= ± oo, then(6.110)holds.

On Corollary 6.3. Note that if u(t) is a solution of (6.1) and u(t) 0for T / < CD, then (6.13) defines a solution ut(t) linearly independent of

«(/) and that the same is true of (6.14) when the integral is convergent;see § 2 (ix). By (i), this implies Corollary 6.3.

On (HI). Since MO, «x can be replaced by —«0, — ult respectively, with-out affecting the zeros of u^ or the inequalities (6.12), it can be supposedthat

Multiplying (6.12) by u^T)uj(T) > 0 shows that the case (6.15), where(u, v) = (M!, MO) holds with c < 0 or c > 0 according as (6.120) or (6.12^holds. Hence Mi(0/Mo(0~* T00 as /-> co according as (6.120) or (6.12^holds. Since M1(7')/w0(r) > 0 and, by the Sturm separation theorem, ulhas at most one zero on T < f < co, the statement concerning the zerosof M! on T < f < co follows.

It remains to show that property (iii) is characteristic of a principalsolution; i.e., if M0(0 has the property (iii) for every solution u^t) linearlyindependent of u0(t), then i/0(r) is a principal solution. In particular (6.12j)holds for / (e/) near co. Consequently |M0(OI = const. |MI(?)| for t-*<o.This is a contradiction if «„(/) is not a principal solution and u^f) ischosen to be a principal solution.

Exercise 6.5. Assume (i) that the conditions of Theorem 6.4 hold;(ii) that (6.1) has a nonvanishing real-valued solution for (a ) T f < co;and (iii) that w0r(0 is the unique solution of (6.1) satisfying u^r(T) = 1,Mor(r) = 0, where T < r < co; cf. Theorem 6.1. (a) Show that t/0(r) =lim MO^/) exists as r -> co uniformly on compact intervals of / and is theprincipal solution of (6.1) at / = co satisfying u0(T) = 1. (b) Show that(a) is false if condition (ii) is relaxed to the condition that (6.1) isdisconjugate on T t < co.

Exercise 6.6. Let p(t) > 0, q(t) be real-valued and continuous func-tions such that (6.1) is disconjugate on a /-interval J having t = co (^ oo)as right endpoint. Let w0(f) be a principal solution of (6.1) at t = co.Then u0(t) 5^ 0 on the interior of/.

Sturm's comparison theorem implies that "q(t) ^ 0 on /" is sufficientfor (6.1) to be disconjugate on J. In this case, we can give some additionalinformation about a principal solution.

Corollary 6.4. Let p(t) > 0, q(f) ^ 0 be continuous on J : a t < co.Then (6.1) has a principal solution satisfying

and a nonprincipal solution wa(/) such that

Exercise 6.7. (a) In Corollary 6.4, the conditions (6.19) uniquelydetermine w0(/), up to a constant factor, if and only if

(b) Assume the first part of (6.21). Using Corollary 9.1, show that aprincipal solution in Corollary 6.4 satisfies MO(/) -*• 0 as / -> co if and only

if- J"rto(JW»M )#-«>.For generalizations, related results, and a different proof of Corollary

6.4, see XIV §§1,2.Proof. Assume first that/<0 = 1, so that (6.1) is of the form

where q 0. Hence the graph of a solution u = u(t) of (6.22) in the(/, w)-plane is concave upwards when u(t) > 0. Let u(t) be the solution of(6.22) determined by u(d) = 1, u'(a) = 1. Then u = u(t) has a graphwhich is concave upward for a rSs / < co. In particular, u(t) > w(<z) = 1,

/»o>w'(0 "'(«) = 1; so that u(t) ^ 1 + /• Thus dt/u*(t) is convergent,

and so u(t) is a nonprincipal solution of (6.22). By Corollary 6.3,

is a principal solution of (6.22). Differentiating this formula gives

Since u'(t) is nondecreasing,

This gives (6.19). The case/>(r) > 0 can be reduced to the case/>(0 — 1by the change of independent variables (1.7). This completes the proof.

Exercise 6.8. Give a proof of the part of Corollary 6.4 concerningM0(0 along the following lines: Let a < T < co and let UjJ(t) be the solutionof (6.1) satisfying uT(a) = 1, uT(T) = 0; cf. Theorem 6.1. Show thatu0(t) = lim uT(t) exists as T-+CO uniformly on compact intervals of[a, co), is a principal solution of (6.1) and satisfies (6.19); cf. Exercise 6.5.

Corollary 6.5. In the two differential equations

where j = 1,2, let pfc) > 0, qj(t) be real-valued and continuous on J : a <t < to; let (6.238) be a Sturm majorantfor (6.23J, i.e.,

let (6.232) be disconjugate [so that (6.23i) is also]. Let u2(t) & 0 be a real-valued solution of (6.232). Then (6.230 has principal and nonprincipalsolutions, w10(/) and un(t), which satisfy

for all t beyond the last zero, if any, ofu2(t).The rough content of this corollary is that the principal [nonprincipal]

solutions of (6.23j) are smaller [larger] than the principal [nonprincipal]solutions of (6.232). If pi = pz and w2, w10, «n are normalized by suitableconstant factors, (6.25) implies that w10 M2 =

un f°r t near °>-Exercise 6.9. In Corollary 6.5, the principal solutions w10 of (6.23j)

/*«!

satisfy u\Q(qz — q^dt < oo. In particular, if q < 0 in (6.1), then aJ p<o

principal solution MO of (6.1) satisfies j w02 \q\ ds < oo.

Proof.

Case 1 (/»! = /»2). Suppose that u2(t) > 0 for T < t < o>. Make thevariation of constants u «= M^ in (6.23!). Then (6.23X) is transformed[cf. (2.31) of §2(xii)]into

where qt — q% S 0 and

By Corollary 6.4, (6.26) has solutions z0(/), «i(0 satisfying z0 > 0, 20' 0,and zl > 0, 2/ > 0 for T f < o>. The desired solutions of (6.23^ are"10 = U&Q, MU = itfr.

Case 2 (pl & p2). The function r = p^u^u^ satisfies the Riccati equa-tion r' + r*lpz + ^2 = 0 belonging to (6.232); cf. § 2 (xiv). This equationcan be written as

where q0 = q2 + (\lpz — l//>iXftUi7"s)x ^ <Jz ^ ?i- But (6.28) is theRiccati equation belonging to

which is a Sturm majorant for (6.23!). In addition, (6.29) has the solution[cf.§2(xiv)J

satisfying

Thus application of the Case 1 to (6.23J), (6.29) gives the desired result.Exercise 6.10. In the differential equations

where j = 1,2, let fjt gf be continuous for 0 t < (a (<j oo); let 0 /i(0 /2(0 and gi(t) ^ £2(0 J let u^t) be a solution of (6.30/) satisfyingWl(0) = 1 and uj(t) > 0, ii/fr) ^ 0 for 0 t < co; cf. Corollary 6.4.Then (6.30^ has a solution t/2(f) satisfying «2(0) = 1, «2'(0 = 0 and

0 < M2(0 «i(0 for 0 / < co [in fact, satisfying w2(0) = 1 and0 < Wa/Wi 1, (M2/Wl)' 0 for 0 / < to].

The following is a "selection" or "continuity" theorem for principalsolutions:

Corollary 6.6. Let p^t), />2(/),..., /»«(/) fl/J^/ i(0, fc(0, • • • , ^ooCO continuous functions for a / < o> satisfying

a/i</

uniformly on every closed interval of a f < co. For I ^ y < oo, letUjo(t) be a principal solution of

satisfying (6.19) am/

Then there exists a sequence of positive integers j(l) <j(2) < • • • such that

«xww uniformly on every closed interval ofa^t<.co and is a solution of(6.33J satisfying (6.19) and (6.34).

Of course, a selection is unnecessary (i.e., j(ri) — n is permitted) if(6.33«>) has a unique solution satisfying (6.19) and (6.34); cf. Exercise 6.7.Note that ««, need not be a principal solution. E.g., let a = l,.cu =00,p; = t2/(l+tif), Pto = t2,qj = = 0, M,0= 1, but "coo -J/r.

Exercise 6.11. This corollary is false if the condition q^i) < 0 isreplaced by the assumption that (6.33,) is nonoscillatory a*nd (6.19) isdeleted from both assumption and assertion.

Proof. Let ufl(t) be the solution of (6.33,) determined by

Then (6.20) holds and wn(f) is a nonprincipal solution of (6.33,); cf. theproof of Corollary 6.4. Hence, by Corollary 6.3, the principal solutionw,o(0 of (6.33,)- satisfying (6.34) is given by

where

Differentiation of (6.37) gives

so that, if / = a,

Thus the sequence />,(fl)wj0(fl), y = 1, 2 , . . . , is bounded if

In order to verify ,(6.39), note that (6.36) and the assumption on (6.32)imply that un(t) -> Wooi(0 as j —>• oo uniformly on closed intervals ofa ^ t < co. Thus, by (6.38),

for any fixed T, a < T < co. This implies (6.39).Since the sequence of numbers ui0(a) = 1 and wj0(a) fory = 1, 2 , . . . ,

are bounded, there exist subsequences which have limits. If y(l) <y(2) < • • • are the indices of such a subsequence and

then the assumption on (6.32) implies (6.35) uniformly on every interval[a, T] c [a, w), where u^t) is the solution of (6.33«) satisfying M00(a) = 1,«a/(a) = w'ooo- The solution ««(*) clearly satisfies (6.19) and (6.34). Thisproves Corollary 6.6.

7. Nonoscillation Theorems

This section will be concerned with conditions, necessary and/orsufficient, for

to be nonoscillatory. In view of the Sturm comparison theorem, thesimplest (and one of the most important) sufficient conditions for (7.1)to be nonoscillatory [or oscillatory] is for (7.1) to possess a nonoscillatory[or oscillatory] Sturm majorant [minorant]. For example, if q(t) ^ 0 [sothat u" — 0 is a Sturm majorant for (7.1)], then (7.1) is nonoscillatory. Ifq(t) — jut~z, then (7.1) is nonoscillatory or oscillatory at / = oo accordingas fj, ^ I or ft > J; see Exercise 1.1 (c). This gives the following criteria:

Theorem 7.1. Let q(t) be real-valued and continuous for large t > 0.V

then (7.1) is nonoscillatory [or oscillatory] at t = oo.If, e.g., tzq(t) -*• \ as / -> oo, then Theorem 7.1 does not apply. In this

case, Exercise 1.2 shows that (7.2) can be replaced by

or

In fact, the sequence of functions in Exercise 1.2 gives a scale of tests for(7.1) to be nonoscillatory or oscillatory at / = oo.

The criterion given by Sturm's comparison theorem can be cast in thefollowing convenient form:

Theorem 7.2 Let q(t) be real-valued and continuous for J : a / <<° (= °°)' Then (7.1) is disconjugate on J if and only if there exists acontinuously differentiable function r(t)for a < t < o) such that

Exercise 7.1. Formulate analogues of Theorem 7.2 when / is open orJ is closed and bounded.

Remark. It is clear from § 1 that analogues of Theorem 7.2 remainvalid if (7.1) is replaced by an equation of the form (pu'Y + qu == 0or u" + gu' +fu ss 0 provided that (7.3) is replaced by the corresponding

Riccati differential inequality /•' + r*jp + q 0 or r' + r2 + gr + f^ 0,respectively.

Proof. First, if (7.1) is disconjugate on J, then (7.1) has a solutionu — w0(r) > 0 for a < t < co; see Corollary 6.1. In this case, r = w07«osatisfies the Riccati equation

for a < t < co. This proves the "only if" part of the theorem.If there exists a continuously differentiable function r(t) satisfying

(7.3), let q0(t) < 0 denote the left side of (7.3) for a < t < co, so thatr' + r2 + q - q0 = 0. Then

is a Sturm majorant for (7.1) on a < t < co and, by § 2 (xiv), possesses thert

positive solution u = exp r(s) ds, where a < c < co. This shows thatJc

(7.1) is disconjugate on a < / < co. In order to complete the proof, wemust show that if u^(t) ^ 0 is the solution of (7.1) satisfying u^a) = 0and Wi'(fl) = 1» tnen Mi(0 T* 0 for a < r < w. Suppose that this is notthe case, so that Ma(f0) = 0 for some t0, a < t0 < co. Since MJ changessign at / = /0 and solutions of (7.1) depend continuously on initialconditions, it follows that if c > 0 is sufficiently small, then the solutionof (7.1) satisfying u(a + e) = 0, u'(a + e) = 1 has a zero near /0. Thiscontradicts the fact that (7.1) is disconjugate on a < t < co and provesthe theorem.

Exercise 7.2. (a) Using the Remark following Theorem 7.2, show thatif, in the differential equations

where j =1,2, the coefficient functions are real-valued and continuous onJ:a < t < co (^ oo) such that

and if (7.52) has a solution w(f) satisfying u > 0, «' 2: 0 for a < r < co,then (7.5j) is disconjugate on /. [For an application in Exercise 7.9, notethat the conditions on (7.5^ hold if (7.5^ is disconjugate on /, /2(f) > 0

f. Exercise 6.2.] (b) Let f ( t ) be

continuous and g(f) continuously differentiable real-valued functions ona ^ t < b. Then

and

is disconjugate on [a, b] if there exists a real number c such that

for a < t b.Corollary 7.1. Let q(t) be real-valued and continuous on J:a ^ t < eo,

C a constant, and

If the differential equation

w disconjugate on J, then (7.1) & disconjugate on J.Exercise 7.3. Show that this corollary is false if the 4 in (7.8) is

replaced by a constant y < 4.Proof of Corollary 7.1. In the Riccati equation (7.4) belonging to (7.1),

introduce the new variable

so that p = r' + q, and (7.4) becomes

Since 2Qp ^ />8 + C2, a solution of

on some interval satisfies

The differential equation (7.11) can be written as

Finally, (7.13) is the Riccati equation for (7.8).Thus if (7.8) has a solution u(t) > 0 on /, then a = u'ju satisfies (7.13).

Hence p = \a satisfies (7.12) and r =* p + Q is a solution of the differentialinequality (7.3) on /. In virtue of Theorem 7.2, this proves the corollary.

Exercise 7.4. A counterpart of Corollary 7.1 can be stated as follows:Let q(i) be real-valued and continuous for 0 / ^ b. Let a be fixed,0 a < b. Suppose that

has the properties that Q(t) ^ 0 for a / ^ 6 and that if z(/) is a solutionof 2" + 62(0z = 0, z'(a) = 0, then z(t) has a zero ona<t^b. Then asolution w(0 of (7.1) satisfying «'(0) = 0 has a zero on 0 < t b.

One of the main results on equations (7.1) which are nonoscillatory at/ = oo will be based on the following lemma.

Lemma 7.1. Let q(t) be real-valued and continuous on 0 jSs t < oo withthe property that (7.1) is nonoscillatory at t = oo. Then a necessary andsufficient condition that

holds for one (and/or every) real-valued solution u(t) & 0 of (7.1) is that

(as a finite number).Remark. For the application of this lemma, it is important to note

that the proof will show that condition (7.15) can be relaxed to

In other words, when (7.1) is nonoscillatory at t = oo, then (7.16) implies(7.15); in fact, it implies the stronger relation

Exercise 7.5. Let q(t) be as in Lemma 7.1. Show that

holds for one (and/or every) real-valued relation u(t) 0 of (7.1) if andonly if

[Note that (7.19) holds if, e.g., q(t) -» 0 as t -» oo or \q(s)\v ds < oo forsome y ^ 1.]

Proof. Suppose first that (7.14) holds for a real-valued solutionu(t) & 0 of (7.1). Let / = a exceed the largest zero, if any, of u(t). Putr = u'lu for t a, so that r satisfies the Riccati equation (7.4). Aquadrature gives

s

for / ^ a. Then (7.14) implies that (7.20) can be written as

where C

Hence (7.21) implies (7.17) [by virtue of the inequality (a + /S)2 <2(a2 + £2) for real numbers a, 0]. Since Schwarz's inequality [cf. (7.22)]shows that (7.15) is a consequence of (7.17), it follows that (7.15) isnecessary for (7.14).

In order to prove the converse, assume (7.16), that u(t) ^ 0 is any real-valued solution of (7.1), and that u(t) > 0 for t a. Then (7.20) holdsfor r — VL\U and a quadrature of (7.20) gives

The assumption (7.16) implies that the right side is bounded from above.Suppose, if possible, that (7.14) does not hold, then the second term on theleft tends to oo as t ->• oo, thus

Schwarz's inequality implies

and, consequently,


as / -»> co. A quadrature gives

This contradiction shows that the hypotheses that (7.14) fails to hold isuntenable and proves the theorem.

Theorem 7.3. Let q(t) be real-valued and continuous for 0 < t < oo.A necessary condition for (7.1) to be nonoscillatory at t = oo is that either

or that (7.15) holds [and, in the latter case, (7.17) holds].It follows, e.g., that if q(t) ^ 0, then, in order for (7.1) to be non-

f°°oscillatory at t = oo, it is necessary that q(t) dt < oo. In fact, as is

seen from Exercise 7.8, it is necessary that tvq(t) dt < oo for everyy < l . J

Proof. Suppose that (7.1) is nonoscillatory at / = oo and that (7.23)fails to hold, so that (7.16) holds. The validity of (7.17) must be verified.But this is clear from the proof of Lemma 7.1 which shows that, on the onehand, (7.16) implies (7.14) for every real-valued solution u(t) ^ 0 of(7.1) and, on the other hand, that (7.14) for some solution assures (7.17).

Exercise 7.6. Let q(t) be as in Theorem 7.3 and, in addition, satisfy

or, more generally, (7.19). Then a necessary condition for (7.1) to benonoscillatory at t — oo is that either

or that

(possibly conditionally).Exercise 7.7. (a) Give examples to show that (7.15) in Theorem 7.3 is

compatible with each of the possibilities

(b) Show that if, in Theorem 7.3, q(t) is half-bounded or, more generally,if there exists an e > 0 such that

then a necessary condition that (7.1) be nonoscillatory is that either(7.15) or (7.25) hold. See Hartman [10].

Changes of variables in (7.1) followed by applications of Theorem 7.3(and its consequences) give new necessary conditions for (7.1) to be non-oscillatory. This is illustrated by the following exercise.

Exercise 7.8. (a) Introduce the new independent and dependentvariables s = tv, y > 0 and z — t(v~l)lzu, and state necessary conditionsfor the resulting equation and/or (7.1) to be nonoscillatory at t = oo.(b) In particular, show that if </(/) 0 and (7.1) is nonoscillatory at

/* 00

/ = oo, then tl~vq(t)dt < oo for all y > 0.

The next result gives a conclusion very different from (7.17) in Theorem7.3 in the case (7.15).

Theorem 7.4. Let q(t) be as in Theorem 7.3 such that (7.1) is non-oscillatory at t = oo and (7.23) does not hold [so that (7.15) does]. Then

where

In applications, interesting cases of this theorem occur if (7.26) holds,so that

It is readily verified from q(t) = fift*, t > 1, that the "4" in (7.28) cannotbe replaced by a larger constant. It is rather curious that the proof ofCorollary 7.1 and Theorem 7.4 depend on the inequality 2Qp ^ pz + Qz.In the proof of Corollary 7.1, this inequality is used to deduce (7.12) from(7.11); in the proof of Theorem 7.4, it is used to deduce

from (7.10).Proof. Let u = «(/) 0 be a real-valued solution of (7.1) and suppose

T is so large that u(t} ^ 0 for t T. Since it is assumed that (7.15) holds,the relation (7.14) holds. Thus if r = u'ju, a quadrature of the corre-sponding Riccati equation gives (7.21) as in the proof of Lemma 7.1.Rewrite (7.21) as r(t] = p(t) + Q(t), where

Since p = —r2 = — (p + 02, the equation (7.10) holds. This gives(7.31). In particular, if Q(t) ^ 0, then

Note that if Q < 0, then (7.33) holds since p' 0, p 0. Hence(7.33) holds for / ^ T. Since the result to be proved is trivial if q(t) = 0for large t, it can be supposed that this is not the case. Hence r £ 0 forlarge t and so, p(f) > 0. Consequently, (7.33) gives

Suppose, if possible, that (7.28) fails to hold, then (7.32), (7.34) showthat

holds for r — u'ju, where u(t) ^ 0 is an arbitrary real-valued solution of(7.1). It will be shown that this leads to a contradiction. To this end, notethat

Thus Schwarz's inequality gives

Consequently there exist constants c0, c such that \u(t)\ ^ c0 exp c(log t)1'*for large /. It follows that

for all real-valued solutions (^0) of (7.1). This contradicts the existence(Theorem 6.4) of nonprincipal solutions and completes the proof.

Exercise 7.9. In the differential equations

where j =1,2, let q(i) be real-valued and continuous for large / and suchthat }converges (possibly conditionally) and (7.352) is nonoscillatory at t = oo.Show that (7.350 is nonoscillatory at t = oo if | (^(Ol ^ 02(0> or if crc2 are constants, 0 <c2< 2, c*^ + (l-c2to2^ maxfO.c2^ + (l-Cjfoj)and c1Q1^c2Q2

8. Asymptotic Integrations. Elliptic Cases

In the next two sections, we will consider the problem of the asymptoticintegration of equations

where q(t) is continuous for large t. Except for the last part of this section,the main interest will center around the situations where the coefficientq(t) is nearly a constant or (8.1) can be reduced to this case. The last partof this section (see Exercises 8.6, 8.8) deals with bounds for \u'\ whenq(t) is bounded from above.

When q(t) is a constant, say A, and A is real and positive, then the solu-tions are, roughly speaking, of the same order of magnitude. On theother hand, if X is not real and positive, then essentially there is one smallsolution, as t -> oo, and the other solutions are large. These facts indicatethat different techniques will be needed when q(t) is nearly a constant A,and A is or is not real and positive. In this section, the first case will beconsidered.

Theorem 8.1. In the differential equations (8.1) and

let q(t), q0(t) be continuous, complex-valued functions for 0 rSs t < oosatisfying

for every solution w(t) of (8.2). Let u0(t), v0(t) be linearly independentsolutions o/(8.2). Then to every solution u(t) o/(8.1), there corresponds atleast one pair of constants a, (3 such that

as t —*• oo; conversely, to every pair of constants a, /?, there exists at leastone solution u(t) o/(8.1) satisfying (8.4).

Note that for a given u(t), (8.4) might hold for more than one pair ofconstants (a, 0). This is true, e.g., if y0(f) = o(u0(tj) as t -»• oo.

An interesting aspect of Theorem 8.1 is the fact that the main condition(8.3) does not involve the derivatives w'(t) of solutions w(t) of (8.2). Thisadvantage is lost if (8.1) or (8.2) is replaced by a more complicated equationas in the Exercise 8.4 below.

Proof. It can be supposed that det Y(t) = 1, where

Write (8.1) as a first order system x' — A(t)x for the binary vectorsx =s (u, w'); cf. (2.5). Then the variations of constants x = Y(t)y reducethe system x' = A(t)x, say to y' = C(t)y, in (2.28); cf. §2(xi). Thus

Theorem 8.1 follows from the linear case of Theorem X 1.2, cf. ExerciseX'1.4.

Corollary 8.1. Let q(t) be a continuous complex-valued function on0 5: t < oo satisfying

Then if a, ft are constants, there exists one and only one solution u(t) of(8.1) satisfying the asymptotic relations

The relations (8.6) can also be written as u = d cos [r + y + 0(1)],«' as — 6 sin [r + y + 0(1)] as f -*• oo for some constants y and b.

Exercise 8.1. Show that if a, /? are constants, there exists a uniquesolution v(t) of the Bessel equation t*v" + tv' + (t2 - n*)v = 0 for / >0such that u(t) — tl/iv(t) satisfies (8.6) as t -* oo.

Exercise 8.2. Show that the conclusion of Corollary 8.1 is correct if(8.5) is relaxed to the following conditions in which/(/) = 1 — q(t): theintegrals

exist as (possibly conditional) improper Riemann integrals I = lim

) and |&(/)/(/)| * < oo for A: = 0, 1, 2.

Exercise 8.3. (a) Let </(0 be a positive function on 0 <^ f < oo possess-ing a continuous second derivative and such that

Then the assertion of Corollary 8.1 remains valid if (8.6) is replaced by

(o) Show that (8.7) in (a) holds if 0 =* a > —\, q(t) ^ const. > 0 for0 t < oo, and/(f) = q*(t) ^ 0 has a continuous second derivative such

/•oothat |/"(OI < °°- [In fact> f°r tne valid.ity of the conclusion of (a), it

ss a

as T

can be merely supposed that/(r) has a continuous first derivative which is

/•ooof bounded variation on 0 t < oo, i.e., \df'(t)\ < oo; e.g:,/'(/) is

monotone and bounded. This last refinement follows from the first partby approximating q(t) by suitable smooth functions.]

Exercise 8.4. (a) In the differential equations

let p;(t) j* 0, qf(t), rJ(t) be continuous complex-valued functions for0 t < oo such that

hold for all solutions u — w(t) of (8.80). Let u0(t), v0(t) be linearly in-dependent solutions of (8.80). Then to every solution u(t) of (S.S , therecorresponds at least one pair of constants a, ft such that (8.4) holds;conversely, if a, {$ are constants, then there is at least one solution u(t) of(8.8t) satisfying (8.4). (b) In the differential equation (8.1), let q(t) ^ 0 bea continuous, complex-valued function for 0 t < oo such that q(t) is

of bounded variation over 0 t < oo ji.e., I |dy|<oo); c0 =

lim^(f), / -> oo, is a positive constant; and the solutions u(t) of (8.1) are/•oo

bounded (e.g., if q(t) is real-valued or, more generally, I |Im (f)| dt < oo,

then solutions u(t) and their derivatives u'(t) are bounded). Let a, ft beconstants. Then (8.1) has a unique solution «(/) satisfying, as t -+ oo,

where q** (t) is any fixed continuous determination of the square root of q.Exercise 8.5. Let/(f) be a nonvanishing (possibly complex-valued)

function for t 0 having a continuous derivative satisfying

Suppose further that

as t -> oo. Then the differential equation

has a pair of solutions satisfying, as t -*> oo,

Here all powers of/(r) that occur can be assumed to be integral (positiveor negative) powers of a fixed continuous fourth root/!/i(/) off(t). Condi-tion (8.11) is trivially satisfied if f(i) is real-valued and positive and0 < y 2 < 1.

The object of the next exercise is to obtain bounds for derivatives ofsolutions of (8.1) or, more generally, the inhomogeneous equation

Exercise 8.6. Let q(t\ /(/) be continuous real-valued functions on0 < t :£ f0. Let the positive constants e, I/O > 1, C be such that

and

and 0 a < A I0. [The inequality (8.15) holds, e.g., is q(t) < Czfdfor 0 r < f0.] Let M = «(r) be a real-valued solution of (8.13). Considerthe case

or the case

(a) Show that, in either case,

(b) Show that (8.16) holds if \u(U)\ is replaced by 2e~* \ \u\ dr . (c) Part(a) implies that if 0/C / t0 - 0/C, then

where C0(c) = max (C/(l - 6), C + l/«). (</) Put

Show that there exists a nondecreasing function J£(A) for 0 < A <min (0,1 - 0)/C such that

if £ - a = A and 0 a < b /„.The results of the last exercise can be extended to an equation of the

form

see Exercise 8.8. In fact, the results for (8.20) can be derived from thoseon (8.13) by the use of the lemma given in the next exercise which hasnothing to do with differential equations.

Exercise 8.7. Let h(t) ^ 0 be of bounded variation and g(t) continuouson an interval a / b. Then

where the integrals are Riemann-Stieltjes integrals and var h denotes thetotal variation of h(t) on a t < b.

Exercise 8.8. Let />(/), q(t),f(t) be continuous real-valued functions on0 < / < t0 and u(t) a real-valued solution of (8.20). Let 1/0 > 1, C bepositive constants such that (8.15) holds. Consider the two cases inExercise 8.6, with (8.14) replaced by

(a) Then parts (a), (b) of Exercise 8.6 hold if \u'(T)\ in (8.16) is replaced by\u'(T)\IE. (b) Part (c) of Exercise 8.6 holds if |w'(OI in (8.17) is replaced by\u(t)\/E, where

(c) Part (d) of Exercise 8.6 holds if A is restricted to 0 < A <min (0,1 - 0)/C£2, where £ is defined in (8.23).

9. Asymptotic Integrations. Nonelliptic Cases

Asymptotic integrations of u + q(t)u — 0, where q(t) is "nearly" a real,but not positive constant, can be based on Chapter X as were the resultsof the last section. Instead a different technique will be used in this section;this technique takes greater advantage of the special structure of thesecond order equation

This equation is equivalent to a binary system of the form

in which the diagonal elements vanish. [This system cannot be reducedto an equation of the form (9.1) unless either fi(t) or y(t) does not vanish.]The main results on (9.1) will be based on lemmas dealing with (9.2).

A system of the form (9.2) on 0 t < co (< oo) will be called of type Zat t — co if , . .. ..

for every solution (v(t), «(/)), and z(eo) 0 for some solution. It is easyto see that (9.2) is of type Z if and only if there exist linearly independentsolutions (i>XO» 2AO)> j' — 0> 1> such that lim z0(/) = 0 and Urn z^t) — 1.

Lemma 9.1. Let 0(/), y(t) be continuous complex-valued functions for0 t < a) (^ oo). Suppose that

or, more generally, that

(possibly conditionally) and that

T/i£/i (9.2) is of type Z.Unless ]8(0 == 0, the condition (9.32) implies (9.30. If the order of

integration is reversed, it is seen that (9.32) is equivalent to

This shows that (9.3) implies (9.4). Lemma 9.1 has a partial converse.

Lemma 9.2. If f$(i) and y(t), where 0 5j t < o> (r^oo), are continuousreal-valued functions which do not change sign (i.e., ft ^ 0 or ft 0 andy^Qpry^Q)and if (9.2) is of type Z, then (9.31)-(9.32) /roW.

Exercise 9.1. Generalize Lemma 9.1 to the case where (9.2) is replacedby a {/-dimensional system of the form vj> = y^,t)vi+1, where j = 1,..., dand v*+l = u1.

Proof of Lemma 9.1. Two quadratures of (9.2) give

On interchanging the order of integration, the last formula becomes

If t T, then T r ^ f and the definition of T in (9.4^ imply that

Consequently

where

By Gronwall's inequality (Theorem III 1.1),

for T t < <y. Hence (9.42) implies that z(/) is bounded. The relations(9.7) and (9.42) then show that z(a)) = lim z(t) as t -»• ey exists.

The limit 2(o>) is obtained by writing / = at in (9.7). In order to showthat z(o>) ^ 0 for some solution of (9.2), choose the initial conditionsCl = V(T) = 0 and c2 = z(T) = 1 in (9.5), (9.6). Thus C = 1 in (9.9)and (9.10) and so (9.7), (9.8), and (9.10) give

Since the right side tends to 0 as T-+w, it follows that if T is sufficientlynear to o>, then z(w) 76 0. This proves Lemma 9.1.


Proof of Lemma 9.2. Let (y(/), z(f)) be a solution of (9.2) such thatz(co) = 1. It can also be supposed that v(T) = 0 for some T. Otherwiseit is possible to add to (v(t\ z(t)) a suitable multiple of a solution (ve(t),z0(/)) 0 for which z(o>) = 0. In fact v0(t) = 0 cannot hold, for then(9.2) shows that z0(0 s z0(co) = 0.

Thus cl = 0 in (9.6) and z(o>) = 1 shows that (9.32) holds (since ft, y donot change sign). If /5(/) ^ 0 for t near o>, (9.3X) follows. If, however,/3(f) = 0 and (9.2) is of type Z, then (9.3t) holds when y does not changesigns. This completes the proof.

Let (y(f), z(0), ("i(0> *i(0) be solutions of (9.2). Then

is a constant. This follows from Theorem IV 1.2 (or can easily be verifiedby differentiation). Ifz^t) 7* 0 and (9.11) is multiplied by y(0/Zi2(0> it isseen from (9.2) that (z/z^' = Coy/Zj2, and hence there is a constant ct suchthat

if zl ^ 0 on the interval [T, t]. Similarly, if vl j* 0 for [T, t] then(vjvj' = -c0/3/yi

2 and

Conversely, if zl ^ 0 [or yx ^ 0] in the ^-interval [T, t], then (9.12) [or(9.13)] and (9.11) define a solution (v(t), z(t)) of (9.2).

Exercise 9.2. Suppose that (9.2) is of type Z and that (y^f), z^f)) isa solution of (9.2) satisfying z^co) = 1. (a) Show that

and that (9.2) has a solution (v0(t\ z0(0) in which

for t near to. (6) If (v(t), z(0) is any solution of (9.2), then

If, in addition, (9.3) holds, then (9.14) satisfies


c = lim v0(t) exists as t -> to, and

Also, if y(f) is real-valued and does not change signs, then (9.15) can beimproved to

Lemma 9.3. Let /8(f), y(t) be as in Lemma 9.1. In addition, suppose that0(0 0 and that

Then (9.2) /KM a pair of solutions (v,(0> ZXO) for j = 0, 1, satisfying, ast-*a>,

*«/ '

This has a partial converse.Lemma 9.4. Lef /S(f), y(0 continuous real-valued functions such that

0(/) 0 satisfies (9.18) a/irf y(/) does not change signs. Let (9.2) have asolution satisfying either (9.190) or (9.190- 7%«* (9.3) holds [so that (9.2)has solutions satisfying (9.190) and (9.190].

Exercise 9.3. Prove Lemma 9.4.Proof of Lemma 9.3. By Lemma 9.1, (9.2) has a solution Mt), zx(0)

such that 21(o>) = 1. Thus the first part of (9.19!) follows from the firstequation in (9.2). Note that

tends to const, as / — v c o by (9.18). Consequently, the integral cl =r«>I 0(5) dsfvfts) is absolutely convergent (for T near eo). It follows from

JT(9.13) with the choice CQ = 1, that (9.2) has a solution (v, z) = (y0»2o)satisfying (9.11) with c0 = 1 and

Then i'0/~ 1 follows from the first part of (9.190 and (9.20). Letting

(v, z, c0) — (v0, z0, 1) in (9.11) and solving for 20 gives the last part of(9.190). This completes the proof.

Theorem 9.1. Let p(t) be a positive and q0(t) a real-valued continuousfunction for 0 fs / < o> such that

is nonoscillatory at t = co and let x0(t), xv(t) be principal, nonprincipalsolutions of (9.21); cf. § 6. Suppose that q(t) is a continuous complex-valuedfunction satisfying

or, more generally,

Then (9.1) has a pair of solutions u0(t), Mx(r) satisfying, as t -> o>,

forj = 0, 1.Exercise 9.4. Verify that if ^(/) is real-valued, ^(/) — 0(0 does not

change signs, and (9.1) has a solution w,(/) satisfying (9.24)-(9.25) foreither y = 0 ory = 1, then (9.22) holds.

Condition (9.22) in Theorem 9.1 should be compared with (8.3) inTheorem 8.1. The analogue of (8.3) is the stronger condition

Remark. It will be clear from the proof of Theorem 9.1 that if q0(t)is complex-valued but has a pair of solutions asymptotically proportionalto real-valued positive functions «„(/), x^t) satisfying (9.22) [or (9.23)] and

dsfpxj2 < oo, dslpx<f = oo, Xt^xQi ds/pxQz, then Theorem 9.1

remains valid.Exercise 9.5. Let p(t) 9* 0, ^(/), ^o(0 oe continuous complex-valued

functions for 0 r < w (^ oo) such that (9.21) has a solution x±(t) whichdoes not vanish for large t and satisfies

s s s

exists,

where

since

exists

and

Then (9.1) has a pair of nontrivial solutions uQ(t), u±(t) such that

Proof of Theorem 9.1. The variations of constants u = xQ(t)v reduces

for/near co; cf. (2.31). Write this as a system (9.2), where

It will be verified that Lemma 9.3 is applicable. Note that condition(9.18) holds since #„(/) is a principal solution of (9.21); Theorem 6.4. Anonprincipal solution x-^t) of (9.21) is given by

and any other nonprincipal solution is a constant times [1 + 0(l)]#i(0 ast-><o; Corollary 6.3. The.condition (9.4) is equivalent to (9.23).

Thus Lemma 9.3 is applicable. Let (y0, z0), (t^, zx) be the correspondingsolutions of (9.2) and w0 = XQVO, MX = x0vl the corresponding solutions of(9.1). Then the first part of (9.19y) fory = 0, 1 gives (9.24) for/ = 0, 1.Note that u = x0v implies thatpu'/u = px0'/x0 + pv'jv, so that, by (9.27),

pu'ju = px0'lx0 + 2/ar02i?. Sincez0/v0 — oil J /5(j)dsI,thecasey = 0of

(9.25) follows. Also, zjxfa = [1 + <?(l)]/a;02 f P(s) ds = [1 + ^l)]/*^

and, from (9.28), pxl'/x1= pxQ'lx0 •}- 1/XgX^ Consequently, the casej =. 1 of (9.25) holds. This proves the theorem.

Corollary 9.1. In the equation

let q(t) be a continuous complex-valued function for large t satisfying

v s


or, more generally,

Then (9.29) has a pair of solutions «0(f), «i(0 satisfying, as t -* oo,

Conversely, ifq(t) is real-valued and does not change signs and if (9.29) hasa solution satisfying (9.32) or (9.33), then (9.30) holds.

The first part of the corollary follows from Theorem 9.1, where (9.29)and x" = 0 are identified with (9.1) and (9.21), respectively. The latter hasthe solutions z0(f) = 1, xv(t) — t. [Under the condition (9.30), theexistence of «„, u± is also contained in Theorem X 17.1.] The last part ofthe corollary follows from Lemma 9.4 or Exercise 9.4.

Corollary 9.2. In the equation

let A > 0 and q(i) be a complex-valued continuous function for large tsatisfying

or, more generally,

Then (9.34) has solutions uQ(t), u^t) satisfying

Conversely, ifq(t) is real-valued and does not change signs and if (9.34) hasa solution u0(t) or u^t) satisfying the corresponding conditions in (9.37),then (9.35) holds.

The first part follows from Theorem 9.1 if (9.34) and x" — fix = 0 areidentified with (9.1) and (9.21), respectively. The latter has solutions^oCO = *~A'» x\(f) = *At- [Under condition (9.35), the existence of w0, u^ isalso implied by Theorem X 17.2.]

Exercise 9.6. Let q(t) > 0 be a positive function on 0 / < oopossessing a continuous second derivative and satisfying

Then u" — q(t)u = 0 has a pair of solutions satisfying

as / —> oo. (Compare this with Exercise X 17.5.)Exercise 9.7. Find asymptotic formulae for the principal and non-

principal solutions of Weber's equation

(where X is a real number) by first eliminating the middle term using theanalogue of substitution (1.9) and then applying Exercise 9.6 to theresulting equation; cf. Exercise X 17.6.

Corollary 9.3. In equation (9.29), let q(t) be a continuous complex-valued function for large t such that Q(t) in (9.31) satisfies

Then a sufficient condition for (9.29) to have solutions «0(f), u^t) satisfying

as t -> oo, «• .f/rar

r/r/j condition is also necessary ifq(t) is real-valued.Proof. It is easily verified that

is a solutipn of

One of the conditions on Q implies that lim x9(i) exists as t —*• oo and isnot 0. Correspondingly, the solution of (9.42) given by

is asymptotically proportional to /, as t -*• oo.


Thus (9.42) has solutions asymptotically proportional to the (positive)functions 1, /. Hence if (9.29) and (9.42) are identified with (9.L), (9.21),respectively, and if (9.40) holds, the Remark following Theorem 9.1 showsthat the conclusions of that theorem are valid. Consequently (9.29) hassolutions MO, M! satisfying MO ~ XQ, «! ~ x± as t -*• oo. The analogues of(9.25) are

Since Q(t) -> 0 as t -*• oo, it is clear that certain constant multiples ofM0, M! satisfy (9.38), (9.39). The last part of the theorem follows from thefact that q + Qz q when q is real-valued; cf. Exercise 9.4.

By the use of a simple change of variables, a theorem about (9.29) for"small" q(t) can be transcribed into a theorem about (9.34) for "small"q(t), and conversely:

Lemma 9.5. Let q(t) be a continuous complex-valued function for large t.Then the change of variables, where A > 0,


while the change of variables


Exercise 9.8. Verify this lemma.Exercise 9.9. (a) Let A > 0 and q(t) be a continuous complex-valued

function for large / such that

Then u" — [A2 + ^(f)]" = 0 has a pair of solutions satisfying, as t -*• oo,

(b) Let q(t) be a continuous complex-valued function for 0 such thatf*oo

tzv~l \q(t)\v dt < oo for some p on the range 1 p ^ 2. Then w" —

q(t)u — 0 has a solution satisfying

and a solution satisfying

APPENDIX: DISCONJUGATE SYSTEMS

10. Disconjugate Systems

This appendix deals with systems of equations of the form

or, more generally, systems of the form

Here x, y are ^/-dimensional vectors; A(t), B(t), C(t),P(t\ Q(t), R(t) ared x d matrices (with real- or complex-valued entries) continuous on a/-interval J. The object is to obtain generalizations of some of the resultsof § 6. The difficulty arises from the fact that the theorems of Sturm in§ 3 do not have complete analogues.

In dealing with (10.1), it will usually be assumed that

If the vector y is defined by

then (10. i) is of the form (10.2), where

so that


The motivations for the assumptions (10.3), (10.4) are the following:Condition (10.4) makes the system (10.1) or, equivalently, (10J2) non-singular in the sense that the usual existence theorems for initial valueproblems are applicable. Condition (10.3) makes (10.1) formally "self-adjoint" in the sense that if — L[x] denotes the vector on the left of (10.1),whether or not L[x] = 0, then we have Green's formula

for all suitable vector functions x(t), z(t\In particular, if x(t), x0(t) are solutions of (10.1), so that L[x] = 0,

L[x0] = 0, then

When this constant is 0, the solutions x, x0 will be called conjugate solutionsof (10.1).

A system (10.1) is called disconjugate on J if every solution x(t) ^ 0vanishes at most once on /. Correspondingly, a system of the form (10.2)will be termed disconjugate on J if, for every solution (x(t), y(f)) ^ 0, thevector x(t) vanishes at most once on /.

Instead of (10.2), it will be more convenient to deal with the matrixequations

where £/, V are d x d matrices. Note that U(t), V(i) is a solution of(10.9) if and only if x(t) = C/(f)c, y(t) = V(t)c is a solution of (10.2) forevery constant vector c. Thus all solutions of (10.2) are determined if weknow two solutions (U(t\ V(t)\ (U0(t), K0(0) of (10.9) such that

is a Id x Id matrix. In fact, Y(t) is a fundamental matrix for the system(10.2)

(i) If £7(0, V(t) is a solution of (10.9) and if (10.8) holds, then V(t)is determined by U(t); in fact,

(ii) If (10.7) holds and (£/(/), K(/)), (C/0(f), K0(0) are solutions of (10.9),then

where AQ is a constant matrix. This is readily verified by differentiating


(10.12). If KQ = 0, the solutions (U, V\ (U0, F0) of (10.9) will be calledconjugate solutions.

(iii) In particular, if U0 = U, F0 = V, then U*V - V*U is a constant.When this constant matrix is 0, then

In this case, the solution (U, V} of (10.9) is called self-conjugate.(iv) Let (£7(0, F(0) be a solution of (10.9) such that

on some /-interval and let

Consider the "variation of constants"

where (UQ, F0) is a solution of (10.9) satisfying (10.12). Then

or, by (10.15),

Since U0' = AU0 + BV0 and U' = AU + ^F, it follows from (10.16),(10.17) that

Thus, if J(/) is the fundamental solution of the homogeneous part of thisequation, satisfying T(s) = I,

then solutions Z are of the form

where ATX is a constant matrix; see Corollary IV 2.1. Correspondingly,by (10.16),

Conversely, it is readily verified that if (10.7}-(10.8) and (10.14) hold, then(10.19) and (10.11) determine a solution of (10.9) satisfying (10.12).

(v) If, in this discussion, (U, V) is a self-conjugate solution of (10.2),so that K = 0 in (10.15), then T(t) = / and (10.19) reduces to

The corresponding solution (t/0(0» Po(0) °f 00-2) is self-conjugate if andonly if KSK0 = #0*^; i.e., (K^K^* = #i**o-

Let (t/0(0, ^o(O) be self-conjugate, det UQ(t) * 0, and det K^ ^ 0.Interchanging (U, K), (£/„, K0) in (10.12) changes the sign of KQ. Thusthe argument leading to (10.20) gives

since t/0(s) = (/(j)^.Theorem 10.1 Let A(i), £(/), C(f) be continuous on J. Then (10.2) is

disconjugate on J if and only if, for every pair of points t = tlt tz E J andarbitrary vectors xlt x2, the equation (10.2) has a (unique) solution (x(t),y(t)) satisfying xfa) = xlt x(tz) = xz.

The proof will be omitted as it is similar to that of Theorem 6.1. Itdoes not depend on the structure of (10.2) and applies equally well to anysystem x' = A(i)x + B(t)y, y' = C(t)x + D(t)y, where A, B, C, D arecontinuous.

In order to avoid an interruption of the proofs to follow, we now provea lemma (which has nothing to do with differential equations).

Lemma 10.1 (F. Riesz). Let AltAz,... be Hermitian matrices satisfying

Then A = lim An exists as n -> oo.If A, B are two Hermitian matrices, the inequality A > B [or A 1> B]

means that A — B is positive definite [or non-negative definite]. To saythat "A = lim An exists" means that "Atf, Azr),... is convergent forevery fixed vector 77."

Proof. Note that if A 2: 0 and £, rj are two arbitrary vectors, then thegeneralized Schwarz inequality

holds. In order to see this, let e be real and £c = £ 4- e(At- • rj)v], so that

Since the right side is a quadratic polynomial in e which is non-negative forall real <•, its discriminant is non-negative. This fact gives the desiredinequality.

Let Amn — An — Am where n > m, so that Amn ^ 0. Hence thegeneralized Schwarz inequality implies that

Since 0 Amn ^ I, it follows that the norm of Amn is at most 1, andso (A*mnr} - Amn^ ^ \\ri\\* and

The sequence of numbers Anrj • r), for n = 1, 2, ... is nondecreasing andbounded, hence convergent, so that AHTJ — Amrj is small for large m and n.Therefore, lim Anr) exists, as was to be proved.

Theorem 10.2. Let A(t), B(t) = B*(t), C(t) = C*(t) be continuous on Jand let B(t) be positive definite, (i) IfJ is a half-open bounded interval or aclosed half-line, then (10.2) is disconjugate on J if and only if there exists aself-conjugate solution (U(t), V(t)) o/(10.9) such that det V(t) ^ 0 on theinterior ofJ. (ii) IfJ is a closed bounded interval or an open interval, then(10.2) is disconjugate on J if and only //(10.9) has a self-conjugate solution(U(t), F(0) satisfying det U(t) ^OonJ.

Proof of (i). For the sake of definiteness, let J = [a, at), co < oo. LetY(t) in (10.10) be the fundamental matrix for (10.2) such that Y(t) is theidentity matrix at / = a. In particular, U(a) — I, V(a) = 0, and U0(a) — 0,yo(a) = /; thus both (U(t), V(t}), (U0(t), F0(/)) are self-conjugate solutionsof (10.9), since (10.13) and the analogue for U0, F0 hold at t = a. Thegeneral solution of (10.2) vanishing at / = a is given by x = U0(t)c,y = y0(t)c for an arbitrary vector c. If (10.2) is disconjugate on /, thendet t/0('o) 5* 0 for a < /„ < at. Otherwise there is a c0 5* 0 such thatx = U0(t)c0 vanishes at / = a and at t = /0. But then x(t) = 0. Hencey = y0(t)c0 = 0 since det 5 5* 0.

Conversely, let (U(t), V(t)) be a self-conjugate solution of (10.9) suchthat det U(t) ¥> 0 on a < / < co. Define a solution U0(t), VQ(t) of (10.9)by (10.20), where a < s < CD, K^ = 0, K0 = /. Thus U0(s) = 0. SinceB, hence U~*BU*-\ is positive definite, it follows that det U0(t) ^ 0 fors <t <co. Clearly, if (a</), y(/)) 5* 0 is a solution of (10.2) such that xvanishes at / = s, then it is necessarily of the form (x, y) as (U0(t)c,F0(/)c) and x does not vanish for s < t < co.

It remains to show that if (x(t), y(t)) j£ 0 is a solution of (10.2) suchthat x(a) = 0, then x(t) j* 0 for a < t < co. To this end, it suffices toshow that if a < s < co, then there exists a self-conjugate solution(U0(t), y0(tj) of (10.9) such that t/0(a) = 0 and det t/0(0 9* 0 for a < t < s.Put

so that, by the analogue of (10.20), Ult Kx = B~\U^ - AUJ is a self-conjugate solution of (10.9). Since B is positive definite, the factor{. ..} of £/(f) in the last formula is negative definite for a < t 5. Hencedet t/j(0 ^ 0 for a < t < 5. By the analogue of (10.21),

Consequently,

Since the first factor is negative definite for a < / ^ s, the second factor(which is the inverse of the first) is negative definite. Hence

is positive definite for a < t < s, increases with decreasing / (in the sensethat S(r) - S(t) > 0 for a < r < t 5), and S(t) ^ 7. It follows fromLemmalO.l that

For fl f ^ 5, put

which can also be written in the form

Hence, by (10.20), i/0, V0 = B~\UQ — AU0) is a self-conjugate solutionof (10.9). It is clear that U0(a) = 0 and det t/0(f) ^ 0 for a < t <i j. Thiscompletes the proof for the case of [a, w).

Proof of (ii). If J is a closed bounded interval [a, b], we can extend thedefinitions of the continuous A(t), B(t), C(t) to an interval J' = [a — d,b •+ d] => /, so that B = B*, C = C*, and B(t) is positive definite. It isclear that if d > 0 is sufficiently small, then (10.2) is disconjugate on[a — d, b + 6] if and only if it is disconjugate on /. Since [a, b] [a — d, b + d), the case / = [a, b] of the theorem follows from the case[a, o>) just treated.

Consider J open. The arguments used in the case [a, co) show that if(10.9) has a self-conjugate solution (U0(t), VQ(tJ) with det C/0 0 on J,then (10.2) is disconjugate on /. The converse is implied by Exercise 10.6.


Consider the functional

where the matrices P, Q, R are determined by (10.6). A vector function77(7) on a subinterval [a, b] of / will be said to be admissible of classAâ, b) [or A2(a, b)] if (i) ij(a) = rj(b) ~ 0, and (iij) r/(t) is absolutelycontinuous and its derivative rf(i) is of class Lz on a / b [or (iij) j?(f )is continuously differentiable and Prf + Ry is continuously differentiableona t b]. If rj(t) e A2(a, b), then an integration by parts [integratingr)' and differentiating P(t)rf + R(t)rj] gives

If — L[x] denotes the vector on the right of (10.1), whether or not L[x] = 0,then

Theorem 10.3. Let A(t), B(t) - £*(/), C(t) = C*(/) be continuous on J,and B(t) positive definite. Then (10.2) is disconjugate on J if and only if forevery closed bounded interval [a, b] in J, the functional I(rj',a, b) is positivedefinite on Aâ, b) [or A2(a, b)]; i.e., I(rj; a, b) Qfor every v\ e Aâ, b)[or r) e Az(a, b)] and /(»?; a, b) = 0 if and only ifrj = 0.

Proof ("Only i/")- The proof of "only if" is similar to that of the"only if" portion of Theorem 6.2. Suppose that (10.2) is disconjugate on/. Then, if [a, b] ^ J, there is a self-conjugate solution (U(t), F(/)) of(10.9) such that det U(t) ^ 0 on [a, b].

For a given »?(0 e Ai(a> *)» define £(/) by rj(t) — U(i)t,(t). Using theanalogue V = PU' + RUof (10.5) and the analogue V = R*U' - QUof(10.1), it is seen that the integrand of (10.22) is

Since PU? • U'£ = U? • PU't, and R*U? - U£ = • /?C/^ the integrandcan be written as

In addition, the second term is V*U? • t, = U*Y? ' t = *T • Ut, since(I/, K) is self-conjugate and satisfies (10.13). Hence the integrand in(10.22) is PU? • U? + (n' U£)', and so

and £(a) = £(£) = 0. Since B(t) is positive definite, P = B~l is alsopositive definite. Hence Ity, a, b) 0 and /(?/; a, b) = 0 if and only ifC e O .

Proof ("//")• This proof is identical to the corresponding part of theproof of Theorem 6.2 and will be omitted.

Exercise 10.1 (Jacobi). Let /t3/i(a, b) denote the set of vector-valuedfunctions on [a, b] such that (i) r)(a) = r)(b) = 0; (ii) rj(t) is continuouson a < f ^ 6; and (iii) the interval [a, b] has a decomposition (dependingon 17) a = /0 < / ! < • • • < rm = ft, such that Y\ is continuously differenti-able and Pr\ + /ty is continuously differentiable on each interval[tf, tj+i\ for j = 0 , . . . , m - 1. Thus Aâ, b] => A*A[a, b] => A2[a, b].Let 4(0, 5(0, C(t) be as in Theorem 10.3 and suppose that J is not aclosed bounded interval. Show that (10.2) is disconjugate on J if and onlyif I(ri\ a, b) 0 for all r? e A3/i(a, b) and all [a, b] <= /.

Exercise 10.2. Let P,(0» 0X0 ê continuous d x d matrices for/ = 1 , 2 on J:a<t^b such that (i) P2 = A* and £2 = Q2* areHermitian; (ii) 0 < P2 < Re P^ and Re gt ^ Q2 if the inequality P2 > 0means that Pz is positive definite and, e.g., Pt ^ Re Pl means that7*2(0*?' n = ê IîCO7?' n] f°r a^ constant vectors r\; finally (iii) let(/V)' + Q& = 0 be disconjugate on J. Then (P ')' + d* = 0 isdisconjugate on /.

If f(r), ??(f) are of class L2 on [a, b], introduce the "scalar product"

Thus, in this notation, (10.24) can be written as /(??; a, b) = (L[r)], rj) forrj e A2(a, b). Correspondingly (10.25) and rj = Ut, imply that

where

and where PH(f) is the unique positive definite Hermitian matrix suchthat(pK(f))2 = P(0, so thatP(f) is continuous on/; cf. Exercise XIV 1.2.The bilinear relation corresponding to (10.26) is

Formally, then, L = L^*L. Actually, this is correct in the followingsense: (10.27) can be written as

for (U~1)' = —U^U'U'1 as can be seen by differentiating the relationU~1U = I. The formal adjoint of Lr is therefore

cf. IV§8(viii); i.e.,

Corollary 10.1. Let A, B, C be as in Theorem 10.3. Let (10.9) have aself-conjugate solution (U(t), V(t)) such that det U(t) y^ 0 on J. ThenL = LI "A; i.e.,

is given by

for every continuously differentiate Y}(t) such that Prf + Rrj is continuouslydifferentiable on J.

This can be deduced from (10.28) or, more easily, by straightforwarddifferentiation (using the relations F = PU' + RU, V = R*U' - QU\

Theorem 10.4 Let A(t), B(t) = B*(t), C(t) = C*(t) be continuous onJ:a ^ t < (o (^ oo) and let B(t) be positive definite. Let (10.2) be dis-conjugate on J and (U(t), V(t)) a solution o/(10.9) such that det U(t) ^ 0on s / < cofor some s e [a, cu), and let T(t) be defined by (10.18). Then

is nonsingular for s < t < a) and

where M depends on s and the matrix function U(t).If, in Theorem 10.4, M = 0, the solution (£/(*), K(/)) of (10.9) will be

called a principal solution of (10.9). It will turn out that a principalsolution is necessarily self-conjugate. In this case T~\r) — I, and M = 0can be expressed as

in the sense that

uniformly for all unit vectors c. [Compare (10.36), where B — P~l, andthe definition (6.11) for principal solution in § 6.]

Theorem 10.5. Let A, B, C and J be as in Theorem 10.4. (i) Then (10.9)possesses a principal solution (t/0(0, F0(0). 00 Another solution (U(t),K(0) is a principal solution if and only if(U(t\ K(0) =* (U&)Klt K0(0*i),where K^ is a constant nonsingular matrix, (iii) Let (t/(0» K(0) be asolution o/(10.9). Then the constant matrix K0 in (10.12) is nonsingular ifand only J/det U(t) ^ 0/or t near co and

in which case M in (10.35) is nonsingular.The proofs of Theorems 10.4 and 10.5 will be given together.Proof of Theorem 10.5 (i). This proof is essentially contained in the

proof of Theorem 10.2(i). Since (10.2) is disconjugate on 7, there exists aself-conjugate solution (U^(t\ Vv(i)) of (10.9) such that det U^t) 5* 0 fora < t < co. Let a < a < co and put

Then Uz, Vz = B~l(U2' - AUJ is a self-conjugate solution of (10.9),det C/2(r) 0 for a t < co, and

by (10.21). Thus

As in the proof of Theorem 10.2(i), it follows that

if 5(0 is denned by

The limit Af2 is nonsingular since M2 > 5(0 and 5(0 is positive definitefor a < t < co.

In terms of C/2(0. define

or, equivalently,

therefore, (70, V0 = B~l(U0' — AU0) is a self-conjugate solution of (10.9)and det U0(t] 5* 0 for a ^ t < a>. Thus, by (10.21),

Consequently,

and hence

so that

Thus (£/0(0, ^o(O) is a principal solution of (10.9); cf. (10.35).Proof of Theorem 10.4. Let ({/(/), V(t)) be a solution of (10.9) such that

det U(t) T* 0 for s < t < a>. Thus the matrix Ss(t) in (10.34) is definedfor / > s.

It will first be shown that det Ss(t) ^ 0 for s < t < aj. Put

and F°= ^-^C/0' - /4£/°). This defines a solution (£7°, F°) of (10.9);cf. (10.19). Suppose that det Ss(t0) = 0 for some /0 > s, then there is aconstant vector c0 5^ 0 such that #(0 = U°(t)c0 vanishes at t = s, t0.Since (10.2) is disconjugate on J, it follows that £/°(/)c0 = 0. Hence5*s(/)c0 = 0, and so Ss'(t)c0 = 0. Since St' = T^U^BU*'1 is nonsingular,this is a contradiction and shows that det 5S(0 5^ 0 for 5 < / < to.

It will next be shown that the limit (10.35) exists. Let (UQ(t), F0(f)) bethe principal solution of (10.9) just constructed, so that U0(t) is given by(10.42) in terms of a self-conjugate solution (Uz(t), F2(0) with det Uz(t) ^ 0for a -< t < a>. Let a 5 < r < w and consider the function

It is clear that (U0r, F0r), where F0r = B~\U^ - AUQr) is a solution of(10.9), for C/0r(/) can be written as

cf. (10.19). It follows that if A, is the constant matrix

then, by the analogue of (10.19), where U0(s) = U(s)Kl,

Since U0r(t) = 0 when / = r, it is seen that, in the last line, {. . .} = 0when t — r. Hence

It is clear from (10.42) and (10.44) that U9r(t) -* £/0(0 as r -» <o; henceKr -*• AT0 as r -*• oo, where

Thus

This proves Theorem 10.4.Proof of Theorem 10.5 (ii). Let (£/(/), V(t)) be a principal solution of

(10.9). Let 5 be such that det U(t) ^ 0 for s < t < co and that the limitM in (10.35) is 0. In view of (10.48), M = 0 holds if and only if K0 « 0in (10.47). Since (U0, K0) is self-conjugate, it follows that F0 = U^~1VQ*U^Hence K0 = 0 gives

Let ^ = UQ\S)U(S). Then ((7(0^^(0) satisfies the initial conditions

at t — s. Hence (10.49) holds for all t. This proves (ii).Proof of Theorem 10.5 (iii). Let (£/(/), ^(0) be a solution of (10.9) and

let K0 be given by (10.47). Then

Since (10.36) holds with U replaced by £/0, it is clear from (10.36) that ifK0 is nonsingular, then U(t) is nonsingular for t near o>. In this case,

and (10.37) follows from (10.36), where U is replaced by C/0.Conversely, if, U(t) is nonsingular for rnear to and (10.37) holds, then

the last formula line shows that K0 is nonsingular. Thus M is nonsingularby virtue of (10.48). This completes the proof of Theorem 10.5.

Exercise 10.3. State an analogue of Corollary 6.3.Exercise 10.4 (Analogue o/(6.1l!) in Theorem 6.4). Let the conditions

of Theorem 10.5 hold. Let (U(t), V(t)) be a self-conjugate solution of(10.9) such that det U(t) ^ 0 for t near co and the limit M in (10.35) isnonsingular. [Note that T(t) = I.] Let c 0 be a constant vector and

x(t)=U(t)c a solution of (10.1). Then [*"[/> (*XO ' aft)]"1 dt < oo.

Exercise 10.5 (Analogue of Exercise 6.5). Let the conditions of Theorem10.5 hold and let a < T < co. Let (C/oXO, ôr(0) be the solution of(10.9) satisfying U0r(T) = I, U0r(r) = 0; cf. Theorem 10.1. Thenlim (U0r(t), V0r(t)) = (U0(t), y0(t)) exists as r-+.co and is a principalsolution.

Exercise 10.6. Let 7 be an open interval; A(t), B(t) — B*(t), C(t) =C*(/) continuous on /; and B(t) positive definite. Let (10.2) be dis-conjugate on J. Let co (^ oo) be the right endpoint of J and [a, co) c: /.Let (t/0(0, ô(O) be a principal solution of (10.9) on [a, co). Thendet U0(t) 5* 0 on J.

Exercise 10.7 (Analogue of (Hi) in Theorem 6.4). Let A, B, C, and / beas in Theorem 10.4. Let (t/0(0» ô(O) be a principal solution of (10.9) andlet det t/0(0 5* 0 for a f < co. Let (£/(/), F(/)) be a self-conjugatesolution of (10.9) satisfying U (0)9*0. Let A = V0(a)UQl(d) - V(a)U~l(a\so that A = A*. Then A < 0 (i.e., Ax • x ^ 0 for all vectors x) if andonly if det U(t) ^ 0 for / ^ a.

11. Generalizations

The methods of the last section are applicable to more general situationswhich will be indicated here. The material of the last section can beconsidered from the following point of view: First, what are conditions(necessary and/or sufficient) for the functional l(r\\ a, b) in (10.22) to bepositive definite for all [a, b] <= J on certain classes of functions A^(a, b)or A2(a, b)l Second, if this is the case, what are some consequences forthe solutions of the corresponding Euler-Lagrange equations (10.1) ortheir Hamiltonian form (10.2)?

In this section, a similar problem is considered, but the assumption thatPis positive definite is relaxed and the classes Aâ, b), A2(a, b) are replacedby more restricted classes of function y(t). In particular, it will be requiredthat the competing functions rj(t) satisfy certain side conditions, namely,certain linear differential equations (as in Bolza's problem in the calculusof variations).

Let P(t) = P*(t), Q(t) = Q*(t)> &(*) be continuous d x d matrices on/ and consider the functional

)

V

In addition, consider a set of e first order linear differential equations

where A/(f), N(t)are e x d, e < d, matrices of complex-valued continuousfunctions on /. It will be assumed that the (d + e) x (d -f e) Hermitianmatrix

In particular, det P0(f) 5^ 0 and the rank of M(t) is e.For the variational problem (11.1) subject to the side conditions (11.2),

the Euler-Lagrange equations are

where z is an e-dimensional vector. [The derivation of (11.4) and signifi-cance of z need not concern us here.]

The matrix inverse to (11.3) is of the form

Eisad x d,G ane x e, and Fane X d matrix. Introduce the variables

Then equations (11.5), (11.7) can be written as PQ(x', z) = (y — Rx, —Nx)or (x, z) = P0~% - Rx, -Nx); i.e., x' = Ey - (ER + F*N)x andz = Fy -(FR + GN)x. Hence (11.4), (11.5) become

where

In particular, (11.8) implies (11.5).The assumptions on P, Q, R will be P = P*, Q = Q*,

Correspondingly, B = B*, C = C*, and, by (11.3) and (11.6),

5(0 is non-negative definite and of rank d — e (for BPr\ • Pq = Prj • 77 ifMr] = 0).

There is a complication for the system (11.8) which did not arise in thelast section. If (*(/), y(t)) is a solution of (11.8), it does not follow thaty(t) is determined by x(t). This difficulty will be avoided by an extraassumption:(11.12) (11.8) is identically normal,

where (11.12) means that if (x(t),y(t)) is a solution of (11.8) such thatx(t) = 0 on some subinterval of/, then x(t) = 0, y(t) = 0 on /.

Notions of "disconjugate," "conjugate solutions of (10.9)," "self-conjugate solutions of (10.9)" can be defined as before.

Exercise 11.1. Verify the validity of analogues of Theorem 10.1 andTheorem 10.2. (As in the proof of Theorem 10.2, postpone one half of theproof of the statement concerning open /.)

Let the classes A^a, b) [or Az(a, b)] of vector functions rj(t) be definedas before with the additional condition: (iii) r)(t) satisfies the differentialequations (11.2) except for a /-set of measure 0.

Exercise 11.2. Verify the validity of the analogue of Theorem 10.3.For the proof, note that if (z(t), y(t)) is a solution of (11.8), then v\ — x(t)is a solution of (11.2). Hence if (£/(/), V(t)) is a solution of (10.9), thenMU' + NU = 0. Thus if rj(t) is a solution of (11.2) and r) = U£, thenM(Ut,') = 0, so that PUt,' • Ut,' = 0 only if r\ = 0.

Exercise 11.3. If P(t) is of rank d — e [so that P(t) is non-negativedefinite], let P^(t) denote the unique, non-negative Hermitian squareroot of/»(/); thus />'••*(/) is continuous; cf. Exercise XIV 1.2. If P(t) is ofrank > d — e, let £"0(0 be the orthogonal projection of the vector spaceonto the null manifold of M(t) and let P*(t) = (E9(t)P(t)E0(ty*.Actually, P(t) can be replaced by E0(t)P(t)E0(t) in (11.1HH-12), sinceE0rj = rj if Mr) = 0. Using Pl^(t), state and prove the analogue ofCorollary 10.1. [An additional condition for the validity of (10.33) willbe that rj(t) satisfy (11.2).]

Exercise 11.4. Verify the validity of the analogues of Theorem 10.4and 10.5 with an analogous definition of principal solution.

As an example and application, consider a formally self-adjointdifferential equation of order 2d for a scalar function u,

where

where p0(t),... ,/>2<j(0 are real-valued functions on an interval /, i =(-1)*,

and />2fc(0» Pwc-i(t) nave ^ continuous derivatives. For a given functionu(t) of class C2d on /, define a vector y = (yl,..., y*) by

for A: = ! , . . . , < / , where /J^+i = 0. The operator L{u} = 0 is termedformally self-adjoint because of the validity of the Green relation

where z = (z1 , . . . , zd) belongs to v as y belongs to u. In particular ifM, M0 are solutions of (11.14), then

where yQ belongs to MO. When this constant is 0, the solutions M, w0 will becalled conjugate. If u = «0 and w, M are conjugate solutions, then w iscalled a self-conjugate solution, [When p± = /;3 = • • • = /?2d_1 = 0, sothat (11.14) is a real equation, then all real-valued solutions are self-conjugate.]

Consider the functional

A formal integration by parts (ignoring the integrated terms which vanishif M = u' = • • • = u(d~u = 0 at t = a, b) gives

Then /(»?; 0, 6) = /{M; a, 6} is of the type (11.1) for 77 = (u, u',..., «<*-»)and the conditions (11.2) become

Here

/> = diag [0,..., 0, (- \Ypu], Q = diag [p0, -plt...,(-1)"-1/^-!!]

/{ = / diag [-/>!,/>„ . . . , (-l)d/>2d_i];

also M, N are the (d — 1) X d matrices

Correspondingly, A is the matrix with ones on the superdiagonal, diag-onal elements (0,. . . , 0, —ip2d-i/P&i)> and other elements zero; B =diag [0,. . ., 0, (— \)dlpza\', and C is the Hermitian matrix with superdiag-onal elements (ip^ —ip3,. . ., (— \Y~lipzd-i)> diagonal elements (p0, —p2,. . .,(-iy-2p2,i-4,(-l)d~lp2d-2 + \p2d-i\z/Pzd\ and the elements not onthe main diagonal and superdiagonal are zero.

When (x(t), y(t)) is a solution of (11.8), then x(t) = (u, u', . . ., u(d-1}),where u is a solution of (11.13) and the components of y are given by(11.16). Conversely, if M is a solution of (11.13), this choice of x, y gives asolution of (11.8). The condition (11.15) assures (11.3), (11.10), and(11.12).

Note that if (U(t), V(t)} is a solution of (10.9) and the kih column ofU(t) is (uk, uk',..., w].''"1'), where u = uk is a solution of (11.13), thendet U(t) is the Wronskian of the solutions ult. . ., ud. The solution(V(t), V(tJ) is self-conjugate, if and only if «,, uk are conjugate solutionsof (11.13) for j, k = ! , . . . ,< / .

Exercise 11.5. State the analogue of Theorem 10.3, specifying theclasses Av(a, b), A2(a, b) in terms of scalar functions u.

Consider finally the analogue of Corollary 10.1; cf. Exercise 11.3.Since P(t) = diag [0, .. ., 0, (-1)%*], the matrix PlA(t) is P'A =diag (0,. . ., 0, \pzd\'A). The vector rj(t) satisfies the analogue of (11.2) ifand only if q(t) is of the form •/? = (v(t), v'(t),..., v(d~l)(t)) for a scalarfunction v(t). In this case, (10.29) is a vector of the form (0, . . . , 0, L^v}),where L^v} = <x.Q(t)v{d} + • • • + ad(/)y is a differential operator of orderd and a0(f),.. ., ad(/) are continuous complex-valued functions. In fact,it is clear from (10.29) that <x0(f) = |/?2d(0|^ > 0. It is also clear thatLi{v} = 0 if v = uk and (uk, uk,.. ., u(

kd~l}) is the £th column of U(t).

Since the Wronskian of wls .. ., ud is det U(t) ^ 0, it follows that v =«!, . . ., ud are d linearly independent solutions of L-^v} — 0.

Consequently, if W(w^ . .. , w,) denotes the Wronskian of the jfunctions wlf..., wp then

In order to see this, note that the expression on the right is a differentialoperator of order d with leading coefficient |/»8d(0|1/* an^ solutions u =«!,..., ud. Thus if either side of (11.19) is written as a linear homogeneoussystem rf = Q(0»? for r] — (v, v', .. ., vd~l) in the obvious way, then thesystem has a fundamental solution U(t) and so Q = U'U'1. This provesthe identity (11.20).

Exercise 11.6. If (11.13) has solutions w^r) , . . . , ujt) on /, which arepairwise conjugate (and self-conjugate) and which have a nonvanishingWronskianW(u^..., ud)^ 0 on /, then{u]==LfiLM}fofunctions u of class Cu on J. Here, if L^u] = S at.k(t)u(k},thenL/M = 2 (-l)fc{afc(0«}(/c), where the sum 2 is over 0 < k ^ rf.

Notes

SECTIONS 1 AND 2. See notes on relevant portions of Chapter IV. For the substitution(2.34) in §2 (xiii), see Liouville [1, II, pp. 22-23]. The substitution r = u'ju, whichtransforms (2.32) into a Riccati equation, was used in special cases by Euler (circa 1765)and Liouville (1841) for dth order linear differential equations. For the transformation(2.42) in § 2(xv), see Prufer [1, p. 503].

SECTION 3. The results of this section are due to Sturm [1]; see Bocher [1], [3]. Theproofs in the text are suggested by Prufer's work [1 ] and are given in detail by Kamke[3]. The proof for Sturm's separation theorem (Corollary 3.1) given in Exercise 3.1(a)for the case^ = pt goes back to arguments of Sturm; a similar proof for the generalcase is due to Picone [1].

SECTION 4. Theorem 4.1 goes back to the work of Sturm [1] and Liouville [1]. Theproof of (i)-(iii) in the text follows that of Prufer [1]. The proof of (vi) is based onresults of Hilbert and E. Schmidt on integral equations with "Hilbert-Schmidt" kernels;cf. Riesz and Sz.-Nagy [1, pp. 239-242]. For useful and interesting results on the asymp-totic behavior of the eigenvalues An, see Borg [1 ] and references there to Weyl. For acomplete characterization of spectra of singular boundary value problem in terms ofzeros of solutions, see Hartman [6] and Wolfson [1]. For Exercises 4.3 and 4.4, seePrufer [1].

SECTION 5. Theorem 5.1 is due to Hartman and Wintner [9] and generalizes Corollary5.1, which is an interpretation of a result of Lyapunov [1]. The proof of Theorem 5.1in the text is that of Nehari [1]* That the factor 4 in (5.7) cannot be increased wasfirst proved by van Kampen and Wintner [1]. The proof in Hints for Exercise 5.1 isgiven in Hartman and Wintner [9] and is adapted from Borg [2]. For Exercise 5.2see Hartman and Wintner [10]. For Exercise 5.3(a), see Hartman and Wintner [21];for part (c), see Opial [3]; part (d) is a slight improvement of a result of de la ValleePoussin [1] (cf. Sansone [1,1, p. 183]; see also Nehari [2]). Corollary 5.2 and Exercise5.4 are due to Hartman and Wintner [5]; for generalizations, see Hartman [7]. Theorem5.2 is a result of Hartman and Wintner; see Hartman [7, p. 642]. Corollary 5.3is a result of Wiman [1]; for the generalization in Exercise 5.6(o), see Hartman andWintner [2]. For Exercises 5.6(6) and 5.7, see Hartman [18]. For Theorem 5.3, seeMilne [1]; cf. Hartman and Wintner [5]. Generalizations of Exercise 5.9(a)*and (6)to binary systems of first order were obtained by Petty [1] by different methods.

SECTION 6. The use of the term "disconjugate" here is suggested by Wintner [20].Theorem 6.2 is a classical result in the calculus of variations (Jacobi, Weierstrass,*See footnotes, page 403.


Erdmann); cf. Bolza [1, chap. 2 and 3] or Morse [2, chap. 1]. The proof in the text isbased on Clebsch's [1] transformation of the second variation; cf. Bolza [1, p. 632].Corollary 6.2 is a particular case of a result of Heinz [2]; cf. Exercise 11.6. The notionof a "principal solution" for a disconjugate equation was introduced by Leighton andMorse [1]; cf. Leighton [1] for the use of the term "principal." The proof in the textof Theorem 6.4 is adapted from Hartman and Wintner [18, Appendix]. Corollary 6.4is a particular case of a theorem of A. Kneser [2] on second order (not necessarilylinear equations); cf. Chapter XII, Part I in this book. The proof in the text is that ofHartman [3]; Kneser's proof is similar to the one suggested in Exercise 6.8. ForCorollary 6.5, see Hartman and Wintner [6, p. 635]. For Corollary 6.6, see Hartmanand Wintner [3]. (For an application of Theorem 6.4 and Corollary 6.5 to differentialgeometry, see E. Hopf [1].)

SECTION 7. Theorem 7.1 is due to A. Kneser [1]. The remark following Theorem 7.1on the use of the functions in Exercises 1.2 is due to Hille [1] and to Hartman [4].Theorem 7.2*was given by Wintner [20]. For Corollary 7.1 and Exercise 7.4, seeHartman [9] and [25], respectively. Exercise 7.2(6) is a result of Hartman and Wintner[20] and generalizes a result of Picard [4, p. 8]. Hartman [10] contains Lemma 7.1,Theorems 7.3-7.4, and Exercises 7.5-7.8. Theorem 7.4 is a generalization of a result ofWintner [20], Exercise 7.8 is related to results of Wintner [9], [15], [20]; Hille [1]; andLeighton [2]. Exercise 7.9 may be new (it was first given by Hille [1] under the additionalassumption that q^t) Si 0, qz(t) j£ 0 and then by Wintner [24] under the conditions0 ^ Qi(t) 5i (?»(0; the proof suggested in the Hints is much simpler than the proofsof these authors). For some results related to this section, see Wintner [14], Zlamal [1],Olech, Opial and Wazewski [1], and Opial [4]. For a study of zeros of solutions ofcertain fourth order equations, see Leighton and Nehari [1].

SECTION 8. Theorem 8.1 is a variant of a result of Wintner [10]. Corollary 8.1 is aresult of Bocher [2]. For Exercise 8.2, see Prodi [1 ]. For Exercise 8.3, see Wintner [12].Exercise 8A(b) is due to Wintner [10] and sharpens a result of Cesari [1]. For Exercise8.5, see Hartman and Wintner [17]; for related results, see Atkinson [1] and referencesthere. For Exercises 8.6 and 8.8, see Hartman [25]. For Exercise 8.7f see Ganelius[1]; this result was first used in connection with linear, second order differentialequations by Brinck [1].

SECTION 9. The general procedure in this section is suggested by unpublished notesof Hartman and Wintner. Lemma 9.1 is related to results of Wintner [13], [17] on asecond order equation. Analogues of Lemmas 9.1 and 9.2 for (9.1) when/?(f) > 0 andq(t) ^ 0 go back to Weyl [1]. Theorem 9.1 is an unpublished result of Hartman andWintner. The first part of Corollary 9.1 is a result of Bocher [2] under the condition(9.30) and of Wintner [17] under condition (9.31). Similarly, the first part of Corollary9.20 is due to Bocher [2] under condition (9.35) and to Hartman and Wintner [12] undercondition (9.36). For Exercise 9.6, see Wintner [12]. For Corollary 9.3 and Exercises9.7 and 9.8, see Hartman and Wintner [12], where analogues and generalizations aregiven. For results related to this section, see Opial [2], [5], [6]; Rab [1]; Zlamal [1].

SECTION 10. The use of the term "conjugate solutions" is the same as that suggestedby von Escherich [1] for the case of real systems (10.1); the analogous Lagrangerelation [see displays following (10.8)] on which this definition is based is due toClebsch [1]. In relation to (10.18H10.19), see Kaufman and R. L. Sternberg [1],Barrett [1], and Reid [3]. The remarks above concerning Theorem 6.2 are applicableto Theorem 10.3 in the case of real matrices P, Q, R. For the complex case, see Reid[2], [3], [5], [6]. The proof in the text is based on Clebsch's transformation of thesecond variation. Exercise 10.1 is a special case of Jacobi's classical theorem on*See footnotes, page 403.


conjugate points. Exercise 10.2 when Pt = /V and Qx = d* is the simplest case of aresult of Morse [1]; see Hartman and Wintner [22], where Pif Q, are assumed real.Corollary 10.1 is suggested by Heinz [2]; cf. Exerdse 11.6. The proof in the text ismuch simpler than that of Heinz.

The concept of a "principal solution" for systems (10.1) with /?(/) & 0, was intro-duced by Hartman [12] who proved an analogue of Theorem 10.5 dealing however,with only self-conjugate solutions (U(t), V(t)). The definition of principal solution inthe text, and Theorems 10.4, 10.5 are due to Reid [3]. Although a principal solution inReid's sense turns out to be self-conjugate and hence identical with a principal solutionin Hartman's sense, the handling of nonprincipal solutions is more convenient byReid's definition. The proof in the text for the existence of principal solutions [Theorem10.5(i)] follows Hartman [12]. Reid's existence proof is outlined in Exercise 10.4.The proofs of Theorem 10.4 and the other parts of Theorem 10.5 are based on Reid[3]. Lemma 10.1 and its proof are due to F. Riesz; cf. Riesz and Sz.-Nagy [1]. ForExercise 10.3, see Hartman [12].

SECTION 11. The results of this section given in Exercises 11.1-11.5 are due to Reid[3]; see also Sandor [1] and Reid [6] for related results and generalizations. ForExercise 11.6, see Heinz [2] (and, for d = 1, cf. Brinck [1]).

For a correct proof and generalization of Nehari s result [2] on n-th orderequations; see Hartman [S2] ; for related results, see references there toA.Yu. Levin and Hukuhara. For further discussion and bibliography, seeCoppel [S2].

The first part of Exercise 5.9(a) is due to Wintner [S I ] .

Theorem 7.2 goes back to Bocher [3] and de la Vallee Poussin [1].

The case ck = 1 of Exercise 7.9 was given by Taam [SI].

For Exercise 8.7, see also \Viist [Si].

For an extension of some of the results of Sections 4-7 (e.g., Sturm com-parison theorems, principal solutions, disconjugacy criteria, etc.)to n-th order equations, see Hartman [SI] and Levin [SI]. Coppel s book[S2] also deals with these subjects among others.

Chapter XII

Use of Implicit Function and

Fixed Point Theorems

Many different problems in the theory of differential equations are solvedby the use of implicit function theory—either of the classical type or of amore general type involving fixed point theorems and/or functionalanalysis. This will be illustrated in this chapter. Part I deals with theexistence of periodic solutions of linear and nonlinear differential equations.Part II deals with solutions of certain second order boundary value prob-lems. In Part III, a general abstract theory is formulated. Use of thisgeneral theory is illustrated by an application to a problem of asymptoticintegration.

Although Parts I and II are applications of the general theory of Part III,there are several reasons for giving them separate treatments. The firstreason is the importance and comparative simplicity of the situationsinvolved. The second reason is that Parts I and II serve as motivation forthe somewhat abstract theory of Part III. The third and most importantreason is the fact that, as usual, a general theory in the theory of differentialequations only provides a guide for the procedure to be followed. Its usein a particular situation generally involves important problems of ob-taining appropriate estimates in order to establish the applicability of thegeneral theory.

Two general theorems will be used. The first is a very simple fact:Theorem 0.1. Let 35 be a Banach space of elements x,y,... with norms

\x\, \y\,.... Let TQ be a map of the ball \x\ ^ p in D into 3) satisfying\T9[*] ~ TM £0\x- y\for some 0, 0< 0 < 1. Let m = \TJfS\\ andm ^ p(l — 6). Then there exists a unique fixed point x0 of TQ, i.e., aunique point XQsatisfyingT0[x0] = XQ. In fact, x0 can be obtained as thelimit of successive approximations x± = T0[Q],xz = TQ[Xj],x9= T0[xz],....

Remark. If T0 maps the ball \x\ (! — 6) can be omitted.

Exercise 0.1. Verify this theorem and the Remark.

404

Use of Implicit Function and Fixed Point Theorems 405

A much more sophisticated fixed point theorem is the following:Theorem 0.2 (Tychonov). Let £> be a linear, locally convex, topological

space. Let S be a compact, convex subset 0/3) and T0 a continuous map ofS into itself. Then T0 has a fixed point x0 e S, i.e., T0[x0] — x0.

The following corollary of this will be used subsequently.Corollary 0.1. Let I) be a linear, locally convex, topological, complete

Hausdorff space (e.g., let D be a Banach or a Frechet space). Let S be aclosed, convex subset of T) and TQ a continuous map of S into itself suchthat the image T0S of S has a compact closure. Then T0 has a fixed pointx0eS.

Theorem 0.2 was first proved by Schauder under the assumption thatD is a Banach space and this case of the theorem is usually called "Schauder'sfixed point theorem." For a proof of Theorem 0.2, see Tychonov [1 ].

Parts I and II will use the cases of Corollary 0.1 when D is the Banachspace C°, C1. Part III will use the case when £> is a simple Frechet space,namely, the space of continuous functions on J : 0 t < co (^ oo) withthe topology of uniform convergence on closed intervals in J.

Corollary 0.1 is obtained from Theorem 0.2 in the following way: LetD, S, T0 be as in Corollary 0.1. Let S: be the closure of T0S, so that iscompact. Also St <= S since S is closed. Under the assumptions on D,the convex closure of Si (i.e., the smallest closed convex set containingSj) is compact since Si is. (This is an immediate consequence of Arzela'stheorem in the applications below; cf., e.g., the Remark following theproof of Theorem 2.2.) Let 5° denote this convex closure of Sv Since Sis convex 5° <= S. Thus TQ is a continuous map of the convex compact5° into itself (in fact, T0S° c T0S <= Sl c 5°) and the corollary followsfrom Theorem 0.2.

Part HI will depend on the "open mapping theorem" in functionalanalysis. This theorem will be used in the following form:

Theorem 0.3 (Open Mapping Theorem). Let XltXbe Banach spacesand T0 a linear operator from X± onto X2 with a domain 9>(T.^, which isnecessarily a linear manifoldinXlt and range @(T0) = X2. Let T0 be aclosed operator, i.e., let the graph of T0, &(T0) = {(xlt 7^) : x^ E0(T0)} bea closed set in the Banach space Xl x X2 = {(xlt x^) : xt e Xlt xz e X2}with norm $xlf x^\ = max ((zj, \xz$. Then there exists a constant Kwith the property that, for every xz e Xz, there is at least one xl e @(T0)such that T<fCi = xz and \X}\ <j K \xz\. [Inparticular, when T0 is one-to-one,so that xt is unique, then \xt\ <i K IT^jJ holds for all xt e&(T0).]

For a proof of the open mapping theorem in the form that "if P is acontinuous, linear map from a Banach space X to another Banach spaceX2 with domain ^(P) = X and range (%(P) = X2, then P maps opensets into open sets," see Banach [1, pp. 38-40]. Theorem 0.3 results by

applying this to the projection map P : &(T0) —*• X2, where P(x^ T^) —T^i and noting that a sphere about the origin in ^(r0) has a P-imagewhich contains a sphere about the origin in X2.

As a motivation for the procedures to be followed consider the problemof finding a solution of the differential equation

in a certain set S of functions y(t). Write this differential equation as

for some choice of A(t). Suppose that for every x(t) E 5, the equation

has a solution y(t) e S. Define an operator TQ : S -> 5" by putting y(t) =TolXOL where y(t) e 5 is a suitably selected solution of (0.2). It is clearthat a fixed point y0(t) of T0 [i.e., T0\y0(t)] = y0(t)] is a solution of (0.1)in S.

For the applicability of the theorems just stated, it will J>e assumed thatS is a subset of a suitable topological vector space D. It will generally beconvenient to introduce another space % and two operators L and 7\.The operator L is the linear differential operator L\y] — y' — A(t)y, sothat£(/) = L[y(0]if

It will also be assumed that if x(t) e S, then g(t) — f(t, x(t)) is in % andTj : S-»-93 is defined by g(t) — T^t)]. Investigations of 7*0 are thenreduced to examinations of the linear differential operator L and of thenonlinear operator 7^.

The applicability of Theorem 0.1 can arise in the following type ofsituation: Suppose that $3, X), are Banach spaces and that \g\%, \y\^denote the norms of elements g G 93, y e D, respectively. Assume that forevery g(r) e S, the equation (0.3) (i.e., L(y] = g) has a unique solutiony(t) e 5" <= t), that y(t) depends linearly on g(t), and that there exists aconstant K such that \y\^ ^ K \g\%. Suppose that, for the map 7\ : S -»• 93there is a constant 6 such that l^fo] — rjajgJta ^ 6Ja^ — a?2|D forxlt x2eS. Then T0 satisfies \T0[xi(t)]-T9[xJtt)]\v<sOK\xl--xt\j>.According to Theorem 0.1, the sequence of successive approximations

will converge to a fixed point of T0 (under suitable conditions on 5, xltand OK).

In some situations, the equation L\y] = g may have solutions y satisfy-ing \y\t = Klgl* although y is not unique; cf., e.g., Theorem 0.3. Inthis case, y need not depend linearly on g but it might be possible to formconvergent successive approximations in the following way: For a given#!, let x2 = y be a solution of L\y\ — T^x^t)]. If xlt x2,. .., xn^ havebeen defined for n > 2, determine an xn from the equation L[xn — xn^] =Tfcn-i} ~ 7\K-2] and the inequality \xn - xn_^ £ KlT^x^] -ITifrn-zllsB- This situation will not arise below.

When the inequality |7\fo(0] - 7\[*2(0]|<B ^ 0 \xi - *il» is n°tavailable, Theorem 0.2 may still be applicable to assure the existence ofa fixed point of T0.

PART I. PERIODIC SOLUTIONS

1. Linear Equations

In this section, unless otherwise specified, the components of the d-dimensional vectors y, z are real- or complex-valued. Let/? > 0 be fixed.Consider an inhomogeneous system of linear equations

and the corresponding homogeneous system

where A(t) is a continuous d x d matrix and g(t) a continuous vector-valued function for Q t ^p. In addition, consider a set of boundaryconditions

where M, N are constant d x d matrices. For example, if M — N = Iand A(t), g(t) are periodic of period p, then a solution y(t) of (1.1) or (1.2)satisfying (1.3) is of period p.

Lemma 1.1. Let A(t) be continuous for 0 t p and Af, # constantd x d matrices. Let Y(t) be a fundamental matrix for (1.2). Then anecessary and sufficient condition for (1.2) to have a nontrivial (^ 0) solutionsatisfying (1.3) is that MY(G) — NY(p) be singular. In fact, the number k,0 ^ k ^ d, of linearly independent solutions of (1.2), (1.3) is the number oflinearly independent vectors c satisfying

i.e., d-k = rank [MY(Q) - NY(p)\.This is clear since the general solution of (1.2) is y = Y(i)c.


Exercise 1.1. Let A(t) be periodic of period p and

and R is a constant matrix; cf. the Floquet theory in § IV 6. Then (1.2)has a nontrivial (& 0) solution of period p if and only if A = 1 is acharacteristic root of (1.2); i.e., eRp — /is singular. In fact, the numberof linearly independent solutions of period p is the number of linearlyindependent solutions c of

For algebraic linear equations, the inhomogeneous system Cy = g has asolution y for every g if and only if the only solution of Cy = 0 is y = 0.The analogous situation is valid here.

Theorem 1.1. Let A(t) be continuous for 0 t ^p', M, N constantd x d matrices such that the d X Id matrix (M, N) is of rank d. Then (1.1)has a solution y(t) satisfying (1.3) for every continuous g(t) if and only if(1.2), (1.3) has no nontrivial ( 0) solution; in which case y(t) is unique andthere exists a constant K, independent ofg(i), such that

Proof. The general solution of (1.1) is given by

Corollary IV 2.1. This solution satisfies (1.3) if and only if

Assume that (1.2), (1.3) has no nontrivial solution. Then, by Lemma1.1, the matrix V= MY(Q) — NY(p) is nonsingular, thus (1.9) has aunique solution. Substituting this value of c in (1.8) gives the uniquesolution of (I.I), (1.3):

It is clear that there exists a constant K satisfying (1.7) for 0 t ^p.This proves one-half of Theorem 1.1 (and this part did not use the

assumption that rank (M, N) = d). The converse follows from Theorem1.2.


Exercise 1.2. What is the Green's function G(f, s) in the last part ofTheorem 1.1, i.e., what is the function G(t,s), 0 s, t ^p, such that

is the unique solution (1.10) of (1.1), (1.3)?Consider the equations adjoint to (1.1), (1.2)

where A* is the complex conjugate transpose of A; cf. § IV 7. Consideralso a set of boundary conditions

where P, Q are constant d x d matrices. If y(t) is a solution of (1.1) andz(/) a solution (1.11), the Green formula (IV 7.3) is

When do the boundary conditions (1.3) and (1.13) imply that

(1.15) 3^)-*(/>)-y(0)-z(0) = 0,

i.e., that the right side of (1.14) is 0? Note that if M, Q are nonsingular,then this is the case if and only if 0 = y(p) - Q~lPz(Q) — M~lNy(p) - z(0) =(P*Q*-i _ M~lN)y(p) • z(0) = [M~\MP* - NQ*)Q*-i]y(p) • z(0). Inthis case, necessary and sufficient for (1.3), (1.13) to imply (1.15) is that

Lemma 1.2. LetM, Nbeconstantd x d matrices such that rank(M, N) =d. Then there exist d x d matrices P, Q satisfying rank (P, Q) = d,(1.16), and having the property that the relations (1.3), (1.13) imply(1.15). The pairs of vectors z(0), z(p) satisfying (1.13) are independent ofthe choice ofP, Q.

Proof. Since rank (Af, N) = d, there exist d X d matrices Mlt NI suchthat the 2d x 2d matrix

is nonsingular. Write the inverse of W as

so that (1.16) holds and rank (jP, Q) = d.

Let ylt 1/2, «i, zz be ^-dimensional vectors and r\ — (ylt yz), £ = (zlt zz)

be corresponding 2</-dimensional vectors. Then

thus

The choices y^ — y(Q), yz = y(p), z1 = z(0), z2 = —*(p) show that (1.3),(1.13) imply (1.16). This completes the existence proof.

The formulation (1.19) of the implication (1.3), (1.13) => (1.16) makesthe last part of the lemma clear. For if rj — (ylt yz) ^ 0 satisfies My—Ny* = 0, then A/i fA^ 5* 0. In fact, since rank (P, Q) = d, the set ofvectors £ = (z(0), -z(p)) satisfying Fz(G) — Qz(p) = 0 is the set ofvectors satisfying r\ • £ = 0 for all r\ = (ylt yz) such that Myr — Nyz = 0.Since this set of vectors £ = (z(0), — z(p)) is determined by M, N, theproof of the theorem is complete.

Boundary conditions (1.13) satisfying the conditions of Lemma 1.2 willbe called the adjoint boundary conditions of (1.3). Correspondingly, theproblems (1.2)-(1.3) and (1.12)-(1.13) will be called "adjointproblems"(Note that the adjoint of the "periodic boundary conditions" y(p) = y(0),i.e., M = N = /, are equivalent to the "periodic conditions" z(p) = z(0),i.e., P = Q = /.)

There is an analogue of the algebraic fact that if C is a d x d matrix,then the number of linearly independent solutions of Cy = 0 and of the"adjoint" equation C*z = 0 is the same:

Lemma 1.3. Let A(t) be continuous for 0 < t ^p', M, N constantd X d matrices such that rank (M, N) = d; and (1.13) boundary conditionsadjoint to (1.3). Then (1.2)~0-3) am/(1.12)-(1.13) have the same number oflinearly independent solutions.

Proof. Since the relationship between (1.2)-(1.3) and (1.12)-(1.13) issymmetric, it suffices to show that if (1.12)-(1.13) has k linearly independentsolutions, where 0 k ^ d, then (1.2)-(l-3) has at least k linearlyindependent solutions.

Let Y(i) be a fundamental matrix of (1.2), then Y*~l(t) is a fundamentalsolution of (1.12) by Lemma IV 7.1. In terms of (1.17), define a constant2d x Id matrix

so that U is nonsingular and

Thus, if c0 is a constant ^/-dimensional vector such that z(r) = Y*~l(t)c0is a solution of (1.12)-(1.13), then U*~l(c0, -c0) = (6,0). Here b is arf-dimensional vector, and if c0 varies over a set of k linearly independentvectors, then b varies over a set of k linearly independent vectors, sinceU*~l is nonsingular. From (1.20), it is easy to see that the equation(c0, -<:„) -17*0, 0) gives

so that

Hence the matrix Y*(G)M * — Y*(p)N* annihilates k linearly independentvectors b\ therefore, the same is true of its complex conjugate transposeMY(Q) — NY(p). In view of Lemma 1.1, this proves Lemma 1.3.

Remark. For the purpose of the next proof, note that the lemma justproved implies that (1.22) holds if and only if the vector c0 in (1.21) issuch that the solution z = Y*~l(t)c0 of (1.12) satisfies (1.13).

Another algebraic fact is that if C is a singular matrix, then Cy = g hasa solution y if and only if g is orthogonal (i.e., g • z — 0) to all solutions zof the homogeneous "adjoint" system C*z = 0. Again an analogoussituation is valid here:

Theorem 1.2. Let A(i) be continuous for 0 t ^ p, M and N constantd x d matrices such that rank (M, N) =• d, and let (1.2)-(l-3) and (1.12)-(1.13) be adjoint problems. Suppose that (1.2)-(1.3) has exactly k linearlyindependent solutions Vi(t),..., y k(t) and letz t(/),. . . , zk(t) be linearlyindependent solutions of (1.\2)-(l.13). Let g(t) be continuous for 0 t ^p.Then (1.1) has a solution y0(t) satisfying (1.3) if and only if

In this case, the solutions of (1.1), (1.3) are given by y0(f) + a^^/) + • • • +a*2/fcO)> where al5 . . . , afc are arbitrary constants.

Proof. Note that, by the proof of Theorem 1.1, the problem (1.1), (1.3)has a solution if and only if (1.9) has a solution c. This is the case if andonly if

for all solutions b of (1.22). In view of (1.21), this is equivalent to thecondition that


for all solutions z = Y*~l(s)c0 of (1.12)-(1.13), i.e., that (1.23) holds.This proves the theorem.

The next theorem is a rather particular result for the case that A(t), g(t)are of period p.

Theorem 1.3. Lei A(t) be continuous and of period p. Then, for a fixedcontinuous g(t) of period p, (1.1) has a solution ofperiodp if and only if (I.I)has at least one bounded solution for t 0.

Proof. The necessity of the existence of a bounded solution is clear.In order to prove the converse, assume that (1.1) has a solution y(t)bounded for t 0. Let Y(t) be the fundamental matrix of (1.2) satisfying7(0) = /. Then (1.1) has a solution of period/? if and only if the equationc = Y(p)c + b, where

has a solution c; cf. (1.9) in the proof of Theorem 1.1.If c = y(G) in (1.8), then y(p) = Y(p)y(Q) + b holds for every solution

y(t) of (1.1). Since y(t + p) is also a solution, y(2p) = Y(p)y(p) + b =Yz(p)y(Q) + Y(p)b + b, or more generally,

Suppose, if possible, that [/ — Y(p)]c = b has no solution. Then [Y(p) —/]* is singular and there exists a vector c0 such that [Y(p) — I]*c0 = 0and b • CQ * 0. Thus c0 = Y*(p)c0 and c0 = ( Yk(p))*c0 for k = 0, 1,Multiply the equation in the last formula line scalarly by c0 to obtain

since Yk(p)y(G) • c0 = y(G) • (Yk(pj)*c0. As b • CQ 0 and the sequencey(p)t y(2p),... is bounded, a contradiction results. This proves thetheorem.

2. Nonlinear Problems

This section deals with the existence of periodic solutions for non-linear systems. With very minor changes, the methods and results areapplicable to the situation when the requirement of "periodicity" isreplaced by boundary conditions of the type (1.3). The results depend onthose of the last section for linear equations and, in particular, on the "apriori bound" for certain solutions of (1.1) given by (1.7). The first twotheorems concern a nonlinear system of the form

in which y is a vector with real- or complex-valued components.

Theorem 2.1. Let A(t) be continuous and periodic of period p and suchthat (1.2) has no nontrivial solution of period p. Let K be as in (1.7) inTheorem 1.1, where M = N = L Let f(t, y) be continuous for all (f, y),of period p in t for fixed y, and satisfy a Lipschitz condition of the form

for all t, yl5 y2 with a Lipschitz constant 6 so small that Kdp < 1. Then (2.1)has a unique solution of period p.

Actually, it is not necessary that f(t, y) be defined for all y. If m =max ||/(f, 0)||, it is sufficient to require that f(t,y) be defined for||y|| ^ r, where

Proof. Introduce the Banach space D of continuous periodic functionsg(f) of period p with the norm |g| = max ||g(OII- Thus convergence of£i(0> £a(0>... in 3) is equivalent to the usual uniform convergence over0 ^ t < p.

Let g(t) be a continuous function of period p satisfying ||g(OII = r.Thus by Theorem 1.1 the equation

has a unique solution y(t) of period/?. Define an operator T0 on the set ofall such g(t) by putting y(t) = T9[g\. Note that (1.7), (2.4) and (2.2) showthat if z(f) = T0[h], then

where \y\ = max ||y(OII for 0 / S /?. In addition, if m = max ||/(r, 0)||,then|r0[0]| ^ /:/?m.

Thus Theorem 2.1 follows from Theorem 0.1, for y0(t) is a fixed point ofT0, T0[yQ] = 2/0, if and only if y0(t) is a solution of (2.1) of period p; cf.(2.4) where y = TQ[g].

In Theorem 2.1, we can omit assumption (2.2) when ||/(r, y)\\ is "small,"at the cost of losing "uniqueness."

Theorem 2.2. Let A(t), K be as in Theorem 2.1. Letf(t, y) be continuousfor all t and \\y\\ 5: r, of period p in t for fixed y, and satisfy

Then (2.1) has at least one periodic solution of period p.Proof. As in the last proof, define y(t) = TQ[g] as the unique solution of

(2.4) of period p, where g(t) is of period p and |g| 51 r. In order to provethe theorem, it suffices to show that T0 has a fixed point y0, T9\y0] = y0.This will be proved by an appeal to Corollary 0.1 of Tychonov's theorem.

It follows from (1.7) and (2.6) that y = T0[g] satisfies \y\ r. In otherwords, if D is the same Banach space as in the last proof, then T0 mapsthe sphere \g\ <; r of D into itself. Also, (1.7) gives

Since/is continuous, it is clear that if \g — h\ — max \\g(t) — h(t)\\ -> 0,then T0[g] —. T0[h] -> 0. Thus T0 is a continuous map.

If y = T0[g], then ||y(/)|| < r and (2.4) show that there is~a constant C,independent of g, such that ||y'(OII ^ C. This implies that the set offunctions y(t) = T0[g] in the range of T0 is bounded and equicontinuous.Hence, by Arzela's theorem, it has a compact closure in X> (i.e., anysequence ylt yz> • • • has a uniformly convergent subsequence). Conse-quently, Corollary 0.1 implies that T0 has a fixed point y0. Clearlyy = y0(t) is a periodic solution of period p. This proves the theorem.

Remark. In the deduction of Corollary 0.1 from the TychonovTheorem 0.2, it is necessary to know that the convex closure of therange {%(T0) of T0 is compact. This is clear in the proof just completed,for y(t) in the range of Tsatisfies the conditions: (i) y(t) is continuous ofperiod/?; (ii) \\y(t}\\ ^ r; and (iii) \\y(t) - y(s)\\ ^C\t- s\. The convexhull of ^(r0)[i.e., the smallest convex set containing ^(Tfunctions y(t) representable in the form ^y^t) + * • • + ^n(t), wheren = 1, 2, . . . ; A,. 0 and + • • • + Aw = 1. It is clear that functionsin this set satisfy (i)-(iii). The'closure of this set of functions under thenorm of D (i.e., under uniform convergence over 0 / ^ p) gives a setpf functions satisfying (i)-(iii). Thus the compactness of this set in £> isclear from Arzela's theorem. (A remark similar to this can be made for theother applications of Corollary 0.1 in this chapter; see Theorem 4.2 andTheorem 8.2.)

Consider now a system of nonlinear differential equations depending ona parameter /*,

where F is continuous, of period p in t for fixed (x, //), and x, F are real^/-dimensional vectors. Suppose that for p = 0, (2.7) has a periodicsolution x = g0(t). Write y = x — gQ(t); then (2.7) becomes

If F has continuous partial derivatives with respect to x and A(t) =dxF(t, g0(t), 0), where dxF is the Jacobian matrix of F with respect to x,then the last equation is of the form (2.1), where

and \\f(t,y)\\l\\y\\ -*0 as (y,//)->0 uniformly in t for 0 ^ t <p. Inparticular, when \u\ is small, (2.6) holds for small r > 0; in fact, (2.2)holds for small UyJ, \\yt\\ with arbitrarily small 0 and /(/, 0) = 0. Itfollows from Theorem 2.1 that if (1.2) has no nontrivial periodic solutionof period p, then (2.7) has a unique solution x(t) = x(t, //) of period pfor each small \u\. The proof of Theorem 2.1 can also be used tb showthat if F depends smoothly on //, then x(t, ju) depends smoothly on (i.All of these assertions can, however, be proved more directly by the use ofthe classical implicit function theorem.

Theorem 2.3. Let x, F be real vectors. Let F(t, x,- u) be continuous forall t, small \u\t andx on some d-dimensional domain. Let Fbe of period p int for fixed (x, p) and have continuous partial derivatives with respect to thecomponents ofx. Let (2.7), where p = 0, have a solution x s= g0(r) of periodp with the property that ifA(t) — dxF(t,gQ(t), 0), then (1.2) has no nontrivialsolution of period p. Then, for each small |/*|, (2.7) has a unique solutionx = x(t, u) of period p with initial point x(Q, u) near g0(Q); x('> i") & a

continuous function of (t, u), and x(t, 0) = g0(t). # addition, F has acontinuous partial derivative with respect to /*, then x(t, u) is of class C1.

It will be clear from the proof that if more smoothness is assumed forF (e.g., F e C* or F analytic), then x(t, p) is correspondingly smoother (e.g.,x(t, fji) e Ck or x(t, pi) analytic).

Proof. Let x = |(r, x0, u) be the unique solution of (2.7) satisfying theinitial condition x(0) = x0. Then £(t, a;0, u) is continuous and has con-tinuous partial derivatives with respect tb t and the components of xn;see Corollary V 3.3. Also, if z0 is near to g0(0)> then g(t, x0, u) exists on theinterval 0 / ^ /?; see Theorem V 2.1. The solution x = |(r, a?0, ^) isperiodic of period p if and only if

Since £(/, £o(0), 0) - £o(0, the equation (2.8) is satisfied if (x0, /i) = (^0(0), 0).Hence it can be solved for a?0 = x0(u) if the Jacobian matrix of the leftside, BXo£(p, x0, u) — I, is nonsingular at (z0, u) = (g0(0)» 0). The partialderivatives of !(/, ^o» A*) w'th respect to a component of x0, when (a:0, u) =(^0(0), 0), is a solution of the equations of variation (1.2); see TheoremV 3.1. In fact, Y(t) = dx^(t,g0(G), 0) is a fundamental matrix for (1.2)satisfying 7(0) = 7. Hence the assumption that (1.2) has no periodicsolution is equivalent to the assumption that Y(p) — / is nonsingular;cf. Lemma 1.1, where M •==• N = 7. Thus the implicit function theorem isapplicable to (2.8) and gives a continuous function x0 = x0(^). Corre-spondingly, x = £(/, ZO(AO> u) is a periodic solution of (2.7) of period pand the only such solution with initial point XQ near £0(0)- The otherassertions of Theorem 2.3 also follow from the implicit function theorem.


The question of the existence of periodic solutions when

has a vast literature and will not be pursued here.Note that if Fin (2.7) does not depend on t and g0(f) const., then the

conditions of Theorem 2.3 cannot be satisfied since x = g0'(t) is a non-trivial periodic solution of the equations of variation (1.2). Here, however,we have the following analogue.

Theorem 2.4. Let x, F be real vectors. Let F(x, fj) be continuous forsmall \/LI\ and for x on some d-dimensional domain and have continuouspartial derivatives with respect to the components of x. When fi = 0, let

have a solution x = g0(t) ^ const, of period />0 > 0 such that if A(t) =dxF(g0(t), 0), then exactly one of the characteristic roots of(\.2) is 1 [i.e.,eRp° has A = 1 as a simple eigenvalue; cf. (1.5) where p — p0]. Then, forsmall \p\, (2.9) has a unique periodic solution x = x(t, u) with a period p(u),depending on p, such that x(t, //) is nearg0(t) and the period p(u) is near p0;furthermore x(t, u), p(u) are continuous, x(t, 0) = g0(t), and />(0) = /?„.

Remarks similar to those for Theorem 2.3 concerning the smoothness ofF and corresponding smoothness of x(t, u), p(u) hold.

The geometrical considerations in the proof to follow are clarified byreference to Lemma IX 10.1, which shows that we obtain all solutions of(2.9) near x = g0(t) by considering solutions with initial points x(0) — x0near to £0(0) and x0 restricted to be on the hyperplane TT normal to^Ofo(0)» 0) and passing through g0(G).

Proof. Let x = j-(t, x0, u) be the unique solution of (2.9) satisfyingz(0) = x0. This solution is of period p if and only if (2.8) holds. Theequation (2.8) is satisfied when (p, x0, u) = (p0, g0(G), 0).

Since solutions of (2.9) are uniquely determined by initial conditions and£0(0 const., it follows that F(g0(t), 0) 5* 0 for all t. Suppose that thecoordinates in the #-space are chosen so that g0(G) = 0 and F(Q, 0) =(0,.. . , 0, a), a ^ 0, and let IT denote the hyperplane x* — 0 through thepoint g0(0) ss 0 normal to F(Q, 0). Consider ar0 on this hyperplane,x0 = (xQ

l,..., rrjjj"1, 0). Then for small \u\, the equation (2.8) has aunique solution for/?, x0, in terms of p if the Jacobian matrix of !(/, xo> A*)— #0 with respect to a:,,1,.. ., a^""1 and t is nonsingular at (t, x0, /*) =(Po, 0, 0).

The matrix Y(t), in which the columns are the vectors dSldxJ,...,df/foo"1 and £' at (xo> A*) = (0» 0), is a fundamental matrix for (1.2) andits last column is F(g0(t), 0). At / == 0,

Use of Implicit Function and Fixed Point Theorems417

by (1.5). Since (1.2) has, up to constant factors, only the last columngo'(0 of 7(0 at (x0, /j) = (0, 0) as a periodic solution of period p0, thematrix Y(p0) — 7(0) annihilates vectors c of the form c — (0,..., 0, cd)and no others.

The Jacobian matrix J of £(t, XQ, /*) — x0 with respect to x0l, ..., x$~l,

and t at (t, x, p) = (p0,0, 0) is

and the last column of 7(/>0) is F(£0(0), 0) = (0,. . . , 0, a), so that J =[7(/>0) — 7(0)] + diag [0,.. . , 0, a]. If J is singular, then there exists avector c — (c1,..., cd) 9* 0 such that Jc = 0; i.e.,

In view of (2.10) and Z(0) = Z(p0), this is the same as

ZCO){(«*"— 7)c + (0,. . ., 0, cd)} = 0 or(e«»o_/)c + (0, . . . ,0,cd) = 0.

If cd = 0, then c = 0 for eRp* — I only annihilates vectors of the form(0,. . . , 0, cd). If cd 7* 0, then (eRp* - I)*c = 0. But this implies thatA = 1 is at least a double eigenvalue of eRp*. This contradiction shows that/ is nonsingular.

Hence the implicit function theorem is applicable to (2.8) and gives thedesired functions x0

l(ji),..., X^~I(JLI), and p(j*). Correspondingly, ifXM = (*oV)» • • • . aÔ), 0), then x(t, /^) = |(r, a;0(//), //) is a periodicsolution of (2.9) and is the only periodic solution having an initial pointx0, with xQ

d = 0, near to #0(0) and a period near to /v This provesTheorem 2.4.

Exercise 2.1. Let dim # = 2; F(f, x) continuous for all t and x,periodic of period p in t for fixed x. Let the solution a; = x(t, t0, x0) of

satisfying x(t0) = x0 be unique for all /0» ô an<i exist f°r * = fo- Finally,for some (f0, »0), let a:(/, /0, a:0) be bounded for t j£ /0. Then (2.11) has atleaTst one periodic solution of period/?. See Massera [1].

Exercise 2.2. Let a(0 = (aÔ, • • • , «d(0), <3(0 = W). • • • . W)) bepiecewise continuously differentiable for 0 t ^p; aj(r) 0*(f) fory = 1,..., rf; and o(0) = aO»), /3(0) = /8(p). Let

be continuous on an open set containing U° = {(t, y): a.\i) yi ^ /3'(0 for0 f ^ p} and let/(f, y) be uniformly Lipschitz continuous with respectto y. Suppose finally that the functions u'(t, y) = <x''(r) —fj(t,y1,...,y'-Sa'XO,^1,...,/) and v\t,y) = p\t)-f(t,y\ ... ,y*-\ ?(t\

yj+1,..., y1) do not change signs (e.g., uj ^ 0 or uj ^ 0) and that MV < 0for all (t, y) E &°. Then y' =f(t, y) has at least one solution y = y(r),0 t p, such that (t, y(i)) e Q° and y(0) = y(p). See Knobloch [1].

PART H. SECOND ORDER BOUNDARY VALUE PROBLEMS

3. Linear Problems

This part of the chapter concerns boundary value problems involvinga system of second order equations. Consider first a linear inhomogeneoussystem of the form


for a ^/-dimensional vector x (with real- or complex-valued components).The problem involves solutions satisfying boundary conditions

when p > 0, XQ, xv are given. For the inhomogeneous equation (3.1), theconditions (3.3) are not more general than

for if a;— [(xv — x^tjp + x0] is introduced as a new dependent variable,the equation (3.1) goes over into another equation of the same form withh(t) replaced by h(t) + B(i)(xv - x0)tfp + B(i)xQ + F(t)(xv - *„)//>.

Actually, the theory of the boundary value problem (3.1), (3.4) iscontained in § 1. In order to see this, write (3.1) as a first order system

where y = (x, x') is a 2rf-dimensional vector, g(t) = (0, h(tj), and A(f) is a2d x 2d matrix

The boundary conditions (3.4) can be written as

where M, N are the constant 2d x 2d matrices


Note that

Instead of restricting M, N to be of the type (3.8), it is possible to choosemore general matrices; in this case, (3.4) is replaced by conditions of theform

where Mik, Njk are constant d x d matrices such that

is of rank 2d. For the sake of simplicity, only the choice (3.8), i.e., onlythe boundary conditions (3,4), will be considered.

Lemma 1.1 implies the following:Lemma 3.1. Let B(t), F(t) be continuous d X d matrices for 0 t /»;

U(t) the d X d matrix Solution of

Then (3.2) has a nontrivial solution (^ 0) solution satisfying (3.4) if and onlyifU(pfy is singular. In fact, the number k, 0 k ^ £/, of linearly independentsolutions of (3.2), (3.4) w //ze number of linearly independent vectors csatisfying U(p)c = 0.

The corresponding corollary of Theorem 1.1 isTheorem 3.1. Let B(t\ F(i) be continuous for 0 t ^ p. Then (3.1) has

a solution x(t) satisfying (3,4) for every h(t) continuous on [0,^?] if and onlyif (3.2), (3.4) has no nontrivial (*£ 0) solution. In this case, x(t) is unique andthere exists a constant K such that

Exercise 3.1. Verify Theorem 3.1.The homogeneous adjoint system for (3.5) is y' — —A*(t)y which is not

equivalent to a second order system without additional assumptions on Bor F. The simplest assumption of this type is that F(t) is continuouslydifferentiable. In this case, the homogeneous adjoint system y' = —A*(i)yis equivalent to

and the corresponding inhomogeneous system is

[Actually, the differentiability condition can be avoided by writing theterms involving F* as (F*z)', and interpreting (3.11), (3.12) as first ordersystems for the 2</-dimensional vector (—z' — F*z, z).]

In order to obtain the corresponding Green's relation, multiply (3.1)scalarly by z, (3.12) by x, subtract and integrate over [Q,p] to obtain

Thus, if x satisfies (3.4) and z satisfies

then

so that (3.4) and (3.14) are adjoint boundary conditions.Exercise 3.2. Verify that (3.2), (3.4) and (3.11), (3.14) are adjoint

boundary problems in the sense of § 1.Lemma 3.2. Let B(i) be continuous and F(t) continuously differentiable

for 0 t 5j p. Then (3.2), (3.4) have the same number of linearly inde-pendent solutions as the adjoint problem (3.11), (3.14).

Finally, a corollary of Theorem 1.2 isTheorem 3.2. Let B(t) be continuous and F(t) continuously differentiable

on [O,/?] and such that (3.2), (3.4) has k, 1 k ^ d, linearly independentsolutions. Let %(/)> • • • > zfc(0 be k linearly independent solutions o/(3.11),(3.14). Let h(t) be continuous on [Q,p]. Then (3.1), (3.4) has a solution ifand only if

The next uniqueness theorem has no analogue in § 1.Theorem 3.3. Let B(t), F(i) be continuous d x d matrices on 0 if; / p

such that

for all vectors x (i.e., let the Hermitian part of the matrix B — %FF* be non-negative definite). Let g(t) be continuous for 0 / ^ p. Then

has at most one solution satisfying given boundary conditions x(G) = XQ,x(f) = x»-

Remark 1. Actually, Theorem 3.3 remains valid if (3.17) is relaxed to

(3.19) 2 Re ((B(t) - iF(r)F*(0)* • x] > -(trip? ||*||2

for all vectors x -^ 0; cf. Exercise 3.3.Proof. Since the difference of two solutions of the given boundary value

problem is a solution of

it suffices to show that the only solution of (3.20) is x = 0.Let x(t) be a solution of (3.20). Put r(?) = ||a{r)||2. Then r' — 2 Re x • x

and r" = 2 Re (x • x" + ||*;||2), so that r" = 2 Re [(B(t)x + F(t)x') • x +||z'||2]. It is readily verified that

Re (B(t)x + F(t)x') - x + \\x'\\* = ||a;' + $F*x\\* + Re (Bx - ±FF*x) • x.Thus

Hence (3.17) implies that r" > 0. Since the last part of (3.20) means thatr(0) = r(p) = 0, it follows that r(t) = 0 for 0 t p. This provesTheorem 3.3.

Exercise 3.3. (a) Show that if there exists a continuous real-valuedfunction q(t), 0 t p, such that the equation

has no solution r(t) & 0 with two zeros on 0 £1 / ^ p [e.g., q(t) < (rr/p)2]and (3.17) is relaxed to

for all vectors x, then the conclusion of Theorem 3.3 remains valid, (b)Let there exist a continuously differentiable d X d matrix K(t) on [0, p]such that

for all vectors x and 0 ^ / < p, where KH = K^ + ^*)- Then theconclusion of Theorem 3.3 is valid. [Note that (3.23) reduces to (3.17) ifK(t) = 0, so that (b) generalizes" Theorem 3.3, but not part (d) of thisexercise.] The 2 in (3.22), hence in (3.19), is not needed if F = 0.

Remark 2. If F(f) has a continuous derivative, then (3.20) implies thatx = 0 if and only if z = 0 is the only solution of

cf. Lemma 3.2. Hence, the conclusion of Theorem 3.3 is valid if B, F inthe criteria (3. \ 7), (3.22), (3.23) are replaced by B* — F*', -F*, respectively.

4. Nonlinear Problems

Let x and / denote vectors with real-valued components. This sectiondeals with second order equations of the form

and the question of the existence of solutions satisfying the boundaryconditions

or, for given x0 and xv,

The equation (4.1) will be viewed as an "inhomogeneous form" of

The problem (4.2), (4.4) has no nontrivial solution. Thus, by Theorem3.1, an equation

has a unique solution satisfying (4.2). In fact, this solution is given by

This can be verified by differentiating (4.6) twice; cf. (XI2.18). Therelation (4.6) can be abbreviated to

where

according &sQ^s^t:£porQ^t<s^p. Thus

where Gt = dG/dt. Thus (4.6) or (4.7) and its differentiated form imply

where the max refers to 0 s p.

Theorem 4.1. Let f(t, x, x') be continuous for 0 / ^ p and all (x, x')and satisfy a Lipschitz condition with respect to x, x' of the form

with Lipschitz constants 00, 0X so small that

Then (4.1) has a unique solution satisfying (4.2).Remark 1. Instead of requiring/to be defined for 0 / ^ p and all

(x, x'), it is sufficient to have/defined for 0 t <p, \\x\\ R, \\x'\\ ^ 4Rjp,where R satisfies either

if m = max \\f(t, 0, 0)|| for 0 / ^ /?, or merely

if Af = max ||/(r, x, x'}\\ for ||x|| ^ /?, ||a:'|| ^ 4%.Proof. Let 3) be the Banach space of functions h(t), 0 t ^ p, having

continuous first derivatives and the norm

Consider an h(t) in the sphere \h\ R of 35. Let x(t) be the uniquesolution of

satisfying z(0) = x(p) = 0. Define an operator T0 on the sphere \h\ ^ rof D by putting ro[/i(/)] = a:(r).

If *0 = r0[0] and ||/(f, 0,0)|| ^ w, then

by the case h =/(/, 0,0) of (4.10). Thus the norm z0(f) = r0[0] eDsatisfies

Also, i fa r j = T^hi], xz = T0[hz], then, by (4.10) and (4.11),

If the last inequality is multiplied by p/4 and 0i(/>2/8) max ||V — A2'll iswritten as (0i/>/2)[(>/4) max ||V - /r,'|| j, it follows that

Thus the inequalities (4.12), (4.13) and (4.18) show that Theorem 0.1 isapplicable and give Theorem 4.1.

Similarly, if ||/(/, x, x')\\ < M for \\x\\ R, \\x'\\ ^ 4Rfp, then thederivation of (4.17) shows that if \h\ ^ R, then x = T0[h] satisfies \x\ <Mpzj$. Thus if (4.14) holds, T0 maps the sphere \h\ < R into itself and theRemark following Theorem 0.1 is applicable in view of (4.12). Hence theproof of Theorem 4.1 and Remark 1 following it is complete.

Corollary 4.1. Let f(t, x, x) be continuous for 0 / , ||z|| ^ R0,\\x'\\ < /?! and satisfy (4.11), (4.12) and \\f(t, x, x')\\ < M. Let

7%e« (4.1) Aa5 a unique solution satisfying

Exercise 4.1. (a) Prove Corollary 4.1. (A) In Corollary 4.1, let!!/(', *, *')H ^ M be relaxed to (/(/, tx0fp, x0jp)\\ ^ m for 0 ^ / < /> and/? be defined by replacing "^" by "=" in (4.13). Show that the con-clusion of Corollary 4.1 remains valid if R + \\XQ\\ < R0,4R/p + ||#oll//> =/?! replaces (4.20).

Theorem 4.2. Let f(t, x, x') be continuous and bounded, say,

for Q t ^p and all (x, x'). Then (4.1) has at least one solution x(t)satisfying ar(0) = x(p) = 0 and

It is sufficient to require that/(f, x, x') be defined only for ||*|| /w/>2/8,\\x\\ < mpl2.

Proof. Let D be the Banach space of continuously differentiablefunctions h(t), 0 / < />, with norm \h\ defined (4.15). Consider h(t) in

the sphere \h\ w/>2/8 of X). For such an h, put x = TQ[h], where x(i) isthe unique solution of (4.16) satisfying a;(0) = x(p) = 0. Then \\x(t)\\ <m/?2/8 and ||a;'(OII ^ w/>/2, so that T0 maps the sphere \h\ ^ /w/?2/8 intoitself.

If |Ail, I/»2I ^ "y>2/8 and a;! = TM, xz = r0[/»2], then (4.7) and (4.9)imply that

Since / is a continuous function, it follows that if \hi — hz\ -> 0, thenki — xz\ -*- 0- Thus T0 is continuous.

For any x(t) in the range of 7"0, i.e., x = 70[/i] for some h, (4.16) impliesIk'TOII = m- It follows that the set of functions x(t) in the range of2T0[/z], |/i| ^ w/?2/8, are such that a;(f), a?'(0 are bounded and equicontinuoussince

Hence Arzela's theorem implies that the range of T0[h] has a compactclosure. Consequently, Tychonov's theorem is applicable and givesTheorem 4.2.

Corollary 4.2. Let f(t, x, x') be continuous and satisfy ||/|| M forO^t^T, \\x\\ R0, \\x'\\ ^ R!. Let p and x0 satisfy 0 0 SMC/I //wf //|x0|| (5, //ie/i (4.1) Aaj a solution satisfying (4.21)/or p

Exercise 4.2. Prove Corollary 4.2.Exercise 4.3. Let /(/, #, a?') be continuous for 0 f 5! /?, ||a:|| ^ /?0,

and arbitrary a;'. Let there exist positive constants a, b such that||/(/, x, x')\\ ^ a \\x'\\z + b for 0 f ^ />, ||x|| ^ /?0. Assume thata, b, \\x0\\ are such that a(bpz + 2 ||a?0||) < 1 and r* = (a/?)-1!1 — tt —a(bp2 + 2 ||a?0||)]^} satisfies r*p + 3 ||*0|| ^ 4/?0. Then the boundaryvalue problem (4.1),(4.21)has a solution.

Note that Corollaries 4.1 and 4.2 are similar except that in Corollary4.1 there is the extra assumption that (4.11) and (4.12) hold; corre-spondingly, there is the extra assertion that the solution of (4.1), (4.21) isunique. We can prove another type of uniqueness theorem.

Theorem 4.3. Letf(t, x, x') be continuous for 0 / 5: p and for (x, x')on some Id-dimensional convex set. Letf(t, x, x') have continuous partialderivatives with respect to the components of x and x'. Let the Jacobianmatrices off with respect to x, x'

satisfy

for all (constant) vectors z ^ 0. Then (4.1) has at most one solutionsatisfying given boundary conditions x(G) = x0, x(p) = xp.

By the use of Exercise 3.3(a), condition (4.24) can be relaxed to

where q(t) satisfies the conditions of Exercise 3.3(a). Here and in(4.24), "2" is not needed if / is independent of *'.

Proof. Suppose that there exist two solutions x^t), x2(t). Put x(t) =xz(i) — x^t), so that


where

and the argument of B, F in (4.25) is

This is a consequence of Lemma V 3.1.For any constant vector z, an application of Schwarz's inequality to

the formula in (4.25) for each component of Fy*(t)z gives

where the argument of F* is (4.26). Hence,

Thus by (4.24)

for all vectors z 0. Consequently, Theorem 3.3 and Remark 1 followingit imply that x(t) = 0. This proves the theorem.

Exercise 4.4. Let/(f, x, x') be continuous for 0 <j f ^ p and (#, x') onsome 2</-dimensional domain and satisfy a Lipschitz condition of the form(4.11), where

Then (4.1) has at most one solution satisfying given boundary conditions*(0) = x0, X(P) = X

P-Exercise 4.5. Let/(f, x, x') be continuous for 0 / ^ p and (x, x') on

a 2rf-dimensional domain. Let Ax = xz — x^ A#' = xz' — x^', A/ =f(t,xz,xz')—f(t,xl,xl'), where xlt x2, x/, xz' are independent variand assume that

Then the boundary value problem x" — f(t, x, x), x(0) = x0, x(p) = xphas at most one solution.

Exercise 4.6. (a) Let a; be a real variable. Let/0, *, *') be continuousand strictly increasing in x for fixed (f, x'). Then (4.1) can have at mostone solution satisfying given boundary conditions z(0) = x0, x(p) = xv.(b) Show that (a) is false if "strictly increasing" is replaced by "non-decreasing." (c) Show that if, in part (a), "strictly increasing" is replacedby "nondecreasing" and, in addition, / satisfies a uniform Lipschitzcondition with respect to x\ then the conclusion in (a) is valid. [For anexistence theorem under the conditions of part (c), see Exercise 5.4.]

Exercise 4.7 (Continuity Method). Let a; be a real variable. Let <x(/,»'),/5(r, x') be real-valued, continuous functions for — oo < t, x' < oo withthe properties that (i) a, /? are periodic of period p > 0 in / for fixed *';(ii) a > 0; (iii) |/J(f, x')\ -> oo and |a(f, *')/#), *')! -^ 0 as |a?'| -* oouniformly in /. (a) Show that

has at most one solution of period p,

(6) Show that if C = max |$>, 0)|/a(f, 0) and is so large that Ca(f, a;') <^ |,8(f, aj')| and |0(f, 0)| |)9(r, x')l/4 when |x'| ^ /iT, then any periodicsolution x(t) of (4.28) satisfies \x(t)\ ^ C, \x'(t)\ ^ /:. (c) Assume thata, /3 are of class C1. By showing that the set of A-values on 0 A < 1 forwhlch

has a periodic solution is open and closed on 0 A 1, prove that(4.28) has a unique periodic solution, (d) Show that the assumption in(c) that a, /3 are of class C1 can be omitted.

Exercise 4.8 (Continuation). Let a(f, xy x'), fi(t, x, x') be continuousfor — oo < t, x,x'<cc with the properties that (i) a, ft are periodic ofperiod p > 0 in f for fixed (x, x'); (ii) a > 0; (iii) there is a constant Csuch that |/8(r, x, 0)| < C<x.(t, x, 0) for - oo < t, x < oo; (iv) |/J(r, ar, z')| -»oo and |a(r, x, x')fp(tt x, x')\ ->• 0 as |a:'| -*• oo uniformly on bounded

(/, z)-sets. Show that

has at least one periodic solution.

5. A Priori Bounds

The proofs for the existence theorems for solutions of boundary valueproblems in the last section depended on finding bounds for the solutionand its derivative. This section deals with more a priori bounds and theirapplications. The main problem to be considered is of the following type:Given a ^/-dimensional vector function x(t) of class C2 on some interval0 < / /?, a bound for ||z(/)||, and some majorants for ||ar"||, find a boundfor ||«'||. The following result holds for the case when x is a real-valuedfunction:

Lemma 5.1. Let <p(s), where 0 s < oo, be a positive continuous func-tion satisfying,-

Let R 0 and r > 0. Then there exists a number M [depending only on<p(s), R, T] with the following property: Ifx(f) is a real-valued function ofclass C2/or 0 S t p, where p 5: r, satisfying

then \x'\ Mfor Q<t<p.Proof. In view of (5.1), there exists a number M such that

It will be shown that M has the desired property. [Instead of assumption(5.1), it would be sufficient to assume the existence of an M satisfying (5.3).]

Let \x'(t)\ assume its maximum value at a point t = a,Q ^ a ^p.We can suppose that x'(a) > 0, otherwise x is replaced by —a;. If x'(a) >2R]r, then there exists a point / on 0 t £ p where x'(t) ^ 2Rjp ^ 2/?/r.Otherwise x(p) — a;(0) > 2R which contradicts |a;| R. Assume x'(a) >2Rjr and let / = b be a point nearest t = a where x'(t) = 2Rfr. For sakeof definiteness, let b > a. Thus 0 2Rjr = x'(b) < x'(t) < x'(a) fora^t<b.

If the second inequality in (5.2) is multiplied by x'(t) > 0, a quadratureover a t b gives

Even though it is not assumed that x" *£ 0, the formal change of variabless = x'(t) is permitted on the left and gives

cf. Lemma 14.1. From (5.3), it is seen that x\d) ^ M. Thus it followsthat either x'(d) ^ 2R/r or x'(d) ^ M. In either case x'(a) <± M. Sincex'(a) = max \x'(t)\ for 0 t /?, the lemma follows.

Lemma 5.1 is false if a; is a (/-dimensional vector, d 2, and absolutevalues are replaced by norms in (5.2). In order to see this, note that(p(s) = ysz + C> 0, where y and C are constants, satisfies the conditionof Lemma 5.1. Let x(t) denote the binary vector x(t) = (cos nt, sin nt).Thus \\x\\ == 1, ||x'(OII = M> II*"(OI1 = «2 = M2. Thus the inequalitiesanalogous to (5.2),

hold for R = 1, 93(5) = s2 4- 1. But there does not exist a number M suchthat ||*'(OII = M for all choices of n. The main result for vector-valuedfunctions will be the next lemma.

Lemma 5.2. Let 99(5), where 0 s < oo, 6e a positive continuousfunction satisfying (5.1). Lef a, J£, /?, T be non-negative constants. Thenthere exists a constant M [depending only on y(s), a, R, r, K] with the

following property: If x(t) is a vector-valued function of class Cz on0 / ^ />, where p^r, satisfying (5.4) and

Proof. The first step of the proof is to show that (5.5) alone implies theexistence of a bound for ||x'(OII on anv interval LM,/> — u], 0 — /", then

This inequality and the analogue of (5.6) in which * is replaced by r give

hence

Similarly, for ju t p, the relation

implies that

Let

The choice p. = %p in (5.7) and (5.8) gives

Adding (5.10), (5.11) for / = />/2 shows that

The assumption (5.4) and (5.10)-(5.11) imply that

where ± is required according as t |/> or / ^ J/?. Let (s) be defined by

Then, by Lemma 14.1,

where the integral is taken over the r-interval with endpoints t and p/2.In view of (5.13), the integral is majorized by

Hence

In view of (5.12) and the fact that $ is an increasing function, ||a?'(/)|| <M(p), where

and O"1 is the function inverse to $. If/? >: T, then / e [0, p] is containedin an interval of length r in [O,/?]. Thus the considerations just completedshow that/? can be replaced by r, and the lemma is proved with M(r) asan admissible choice of M.

Exercise 5.1. Show that an analogue of Lemma 5.2 remains valid if(5.5) is replaced by

where p(t) is real-valued function of class C2 on 0 t p such thatlp(OI = ^i- 1°- this case, M depends only on g)(s), a, R, r, and Kr

The choice 99(5) = ys2 + C in Lemma 5.1 gives the following:Corollary 5.1. Let y, C, a, K, R, r be non-negative constants. Then

there exists a constant M [depending only on y, C, a, R, T, K] such that ifx(t) is of class C2 on 0 t /?, where p ^ T, satisfying (5.5) am/

Remark 1. If y in (5.16) satisfies yR < 1, then (5.5) holds with

Thus assumption (5.5) is redundant in Corollary 5.1 when yR < 1 (butthe example preceding Lemma 5.2 shows that (5.5) cannot be omitted ifyR = 1). Also if a in (5.5) satisfies 2oc# < 1, then (5.16) holds with

so that (5.16) is redundant in this case. Even if d = 1 (so that x(t) isreal-valued), condition (5.16) cannot be omitted if 2a.R > 1).

In order to verify the first part of Remark 1, note that

Hence (5.16) shows that r" 2[(1 - yR) \\x'\\z - CR]. Another application of (5.16) gives yr" ^ 2[(1 - y/?)(||*1 - C) - CRy] = 2[(1 yR) \\x"\\ - C]. This is the same as (5.5) with the choices (5.17). Theproof of the remark concerning (5.18) is similar.

Exercise 5.2. Show that if 2<x.R > 1, then assumption (5.5) carinot bedropped in Corollary 5.1.

The following simple fact will be needed subsequently.Lemma 5.3. Letf(t, x, x') be a continuous function on a set

and let f have one or more of the following properties:

Let M > 0. Then there exists a continuous bounded function g(t, x, x')defined for 0 5s / 5j p and arbitrary (x, x') satisfying

and having the corresponding set of properties among the following:

Proof. We can obtain such a function g as follows: Let d(s), where0 s < oo, be a real-valued continuous function satisfying 6=1,0<6<l,d = Q according as d < M, M < s 2M, s > 2M. Put

On E(p, R), the identity

makes it clear that g has the desired properties on E(p, R). Furthermorethe validity of any of the relations (5.21')-(5.24') for ||#|| = R implies itsvalidity for ||z|| > R. This proves the lemma.

Note that inequalities of the type (5.23), (5.24) imply that solutions of

satisfy (5.4), (5.5), respectively; cf. (5.19).

Theorem 5.1. Letf(t, x, x') be a continuous function on the set E(p, R)in (5.20) satisfying

(5.24) and (5.23), where (p(s), 0 5j s < oo, is a positive continuous functionsatisfying (5.1). Let \\XQ\\, \\x9\\ ^ R. Then (5.26) has at least one solutionsatisfying z(0) = x0, x(p) = xp.

It will be clear from the proof that assumption (5.23) can be omittedif 2ouR < 1. Furthermore, if/satisfies

where y, C are non-negative constants and yR < 1, then both assumptions(5.23) and (5.24) can be omitted.

If the vector x is 1-dimensional, Lemma 5.1 can be used in the proofinstead of Lemma 5.2. This gives the following:

Corollary 5.2. Let x be a real variable andf(t, x, x') be a real-valuedfunction in Theorem 5.1. Then the conclusion of Theorem 5.1 remainsvalid if condition (5.24) is omitted.

Note that, in this case, condition (5.27) becomes simply f(t, +/?, 0) 5: 0and/(f, -R, 0) 0 for 0 t p.

Proof of Theorem 5.1. The proof will be given first for the case that/satisfies (5.22) instead of (5.27). Let M > 0 be a constant (with p = T)supplied by Lemma 5.2. Let g(t, x, x') be a continuous bounded functionfor 0 t p and arbitrary (x, x') satisfying (5.25), (5.22'), (5.23'), and(5.24'). By Theorem 4.2, the boundary value problem

has a solution x(t). Condition (5.22') means that r = ||tf(OH2 satisfiesr" > 0 if r' - 0 and r > Rz; cf. (5.19). Hence r(t) does not have amaximum at any point t, 0 < t < p, where r(t) ^ R2. Since r(0) =llaj2, r(p) = ||*J2 satisfy r(0), r(p) ^ R\ it follows that r(t) ^ R* (i.e.,||a<r)|| < R) for 0 t ^p. By virtue of x" = g and (5.23'), (5.24'),Lemma 5.2 is applicable to x(t) and implies that ||a;'(OII = M for 0 / 5j p.

Consequently, (5.25) shows that x(t) is a solution of (5.26). This provesTheorem 5.1 provided that (5.27) is strengthened to (5.22). In order toremove this proviso, note that if e > 0, the function /(/, x, x) + exsatisfies the conditions of Theorem 5.1 as well as (5.22) if <p, K in (5.23),(5.24) are replaced by <p + eR, K + e/?, respectively. Hence

has a solution x = xf(t) satisfying the boundary conditions. It is clear that||aje(OII ^ R and that there exists a constant M (independent of e, 0 < e 1) such that \\x€'(t)\\ ^ M. Consequently, if N = max \\f(t, x, x')\\ + 1

for 0 t ^p, \\x\\ R, \\x'\\ ^ M, then ||*/(OH ^ N. Thus the familyof functions x€(t), xe'(t) for 0 t p are uniformly bounded and equi-continuous. By Arzela's theorem, there is a sequence 1 > el > e2 > • • •such that en -> 0 as n -> oo, and #(f) = lim a;e(r) exists as e = ew -»• 0 andis a solution of (5.26) satisfying x(G) = XQ, x(p) = xv. This completes theproof of Theorem 5.1.

Exercise 5.3. Show that if (5.27) in Theorem 5.1 is strengthened to

then (5.26) has a solution x(t) satisfying x(G) — x0, x(p) — 0, and

Exercise 5.4. Let M be a real variable. Let h(t, u, u') be real-valuedand continuous for 0 / ^ p and all (u, u'), and satisfy the followingconditions: (i) h is a nondecreasing function of M for fixed (t, u'); (ii) \h\ (p(\u'\) where <p(s) is a positive, continuous, nondecreasing function fors^.0 satisfying (5.1); (iii) u" =• h (t, u, u') has at least one solutionu0(t) which exists on 0 < t < p [e.g., (ii) and (iii) hold if \h\ ^ a \u'\ + Kfor constants a, K ]. Let w0, uv be arbitrary numbers. Then u" = h (t, u, u1)has at least one solution u(t) satisfying w(0) = «0, u(p) = uv. [For a relateduniqueness assertion, see Exercise 4.6(c).]

Theorem 5.2. Letf(t, x, x) be continuous in

For every p > 0, let f satisfy the conditions of Theorem 5.1 on E(p, R) in(5.20), where <p(s) and the constants a, K in (5.23), (5.24) can depend on p.Let ||£0|| < R. Then (5.26) has a solution x(t) which satisfies x(0) = x0and exists for t 0.

Exercise 5.5. (a) Prove Theorem 5.2. (b) Show that if, in addition,(5.27) is strengthened to (5.29) in Theorem 5.2, then the solution x(t) canbe chosen so that (5.30) holds, (c) Furthermore, if (5.29) is strengthened to* '/+ Ik'll2 0, then r > 0, r' ^ 0, r" 0 for t^ 0. (d) If x is 1-dimensional, show that condition (5.24) can be omitted from Theorem 5.2and parts (b) and (c) of this exercise.

Exercise 5.6. Let f(t, x, x') be continuous on the set E(R) in (5.31).For every m, 0 < m < /?, let there exist a continuous function h(t) =

h(t, m) for large t such that th(f) dt = oo and x •/(/, x, x') ^ h(f) l> 0

for large /, 0 < m £ ||z|| /?, a;' arbitrary. Let x(f) be a solution of(5.26) for large t. Then x(t) -+ 0 as t -> oo.

Exercise 5.7. Let/(f, x, a:') be continuous on £"(/?) in (5.31) and havecontinuous partial derivatives with respect to the components of x, x';let the Jacobian matrices (4.23) satisfy \(B + B*) - IFF* ^0; cf. (3.17).

s

Let H£O|| ^ R- Then (5.26) has at most one solution satisfying x(Q) = XQand IKOH ^ R for t ;> 0.

Remark 2. The main role of the assumptions involving (5.23) and/or(5.24) in Theorems 5.1, 5.2 is to assure that the following holds:

Assumption (Ap). There exists a constant M = M(p) with the propertythat if x(t) is a solution of x" = f(t,x, x ) for Q<t^p satisfyingIKOII ^ R, then ||*'(OII ^ M for 0 t p.

Exercise 5.%. Let f(t, x, x') be continuous on E(R) in (5.31) andsatisfy assumption (Av) for all p ^ p0 > 0. Suppose that, for each x0 in||x0|| ^ /?, (5.26) has exactly one solution x(t) = x(t, z0) satisfying z(0) =x0 and existing for / ^ 0 (cf., e.g., Theorem 5.2 and Exercise 5.7.) (a) Showthat x(t, a;0) is a continuous function of (t, x0) for t 0, ||a;0|| ^ R.(b) Suppose, in addition, that /(f, x, x') is periodic of period p0 in t forfixed (x, x'). Then (5.26) has at least one solution x(t) of period p0.

PART III. GENERAL THEORY

6. Basic Facts

The main objects of study in this part of the chapter will be a linearinhomogeneous system of differential equations

the corresponding homogeneous system

and a related nonlinear system

Let J denote a fixed /-interval J:0 r < w(< oo). The symbolsx, y,f,g,... denote elements of a ^-dimensional Banach space Y overthe real or complex number field with norms ||x||, ||y||, ||/||, ||g||,....(Here ||a;|| is not necessarily the Euclidean norm.) In (6.1), g = g(t) is alocally integrable function on J (i.e., integrable on every closed, boundedsubinterval of J). A (t) is an endomorphism of Y for (almost all) fixed tand is locally integrable on /. Thus if a fixed coordinate system is chosenon Y, A (t) is a locally integrable d x d matrix function on J.

When y(t) is a solution of (6.1) on the interval [0, a] <= /, the fundamen-tal inequality

follows from Lemma IV 4.1. If this relation is integrated with respect to/.' over [0, a], we obtain

Let L = Lj denote the space of real-valued functions <p(t) on J withthe topology of convergence in the mean Ll on compact intervals of J.Thus L is a Frechet (= complete, linear metric) space. For example, thefollowing metric, which will not be used below, can be introduced onL: let 0 = /„ < tl < tz < . . . , / „ -*• (o as n-+ oo, and let the distancebetween 9?, y> e L be

Correspondingly, let C = Cj denote the space of continuous, real-valued functions <p(t) on J with the topology of uniform convergence oncompact interval of J. Thus C is also a Frechet space. A metric on C,e.g., is

The symbols Lv = Ljp, 1 p ^ c», denote the usual Banach spaces ofreal-valued functions y(t) on /: 0 t < to (^ oo) with the norm

JL0°° is the subspace of L°° consisting of functions y(t) satisfying y(t) -> 0as t -»• co. For other Banach spaces 5 of real-valued, measurable functionsg?(/) in,/, the notation \q>\B will be used for the norm of q>(t) in B.

Remark. Strictly speaking, the spaces L, L°°, L0°°,... are not spacesof "real-valued functions" but rather spaces of "equivalence classes ofreal-valued functions," where two functions are in the same equivalenceclass if they are equal except on a set of Lebesgue measure zero. Since noconfusion will arise, however, over this "abuse of language," theabbreviated terminology will be used. In this terminology, the meaningof a "continuous function in L" or the "intersection L C\ C" is clear.

L(Y), LV(Y\ B(Y),... will represent the space of measurable vector-valued functions y(t) on J: 0 ^ / < CD (£j oo) with values in Y such that(p(t) = \\y(t)\\ is in L, Lf, B, With L? or B, the norm |<p|p or \<p\B

will be abbreviated to \y\v or |y|B.A Banach space D will be said to be stronger than L(Y) when (i) t> is

contained in L( 7) algebraically and (ii) for every a, 0 < a < a>, there is anumber a = aE(a) such that y(t) £ X> implies

[It is easily seen from the Open Mapping Theorem 0.3 that condition (ii) isequivalent to: "convergence in t> implies convergence in L(7)."]

If I) is a Banach space stronger than L(Y), a ^-solution y(t) of (6.1) or(6.2) means a solution y(t) e i. Let Y% denote the set of initial pointsy(0) e r of D-solutions y(t) of (6.2). Then Y^ is a subspace of Y. Let Ft

be a subspace of Y complementary to 7$; i.e., Yl is a subspace of Y suchthat 7 = KD © Fj is the direct sum of 7^ and Ylt so that every elementy E 7 has a unique representation y = y0 + yx with y0 6 Fj,, yl 6 7t

(e.g., if K is a Euclidean space, Yl can be, but need not be, the subspaceof Y orthogonal to Y%). Let />0 be the projection of Y onto Y$annihilating 7^ thus ify = y0 + yl with y0 6 1 , yi e Kj, then Poy = y0.

Lemma 6.1. Let A(i) be locally integrable on J and let D be a Banachspace stronger than L (7). Then there exist constants C0, Q such that ify(t) is a ^-solution of (6.2), then

Proof. Yf, is a subspace of the finite dimensional space Y. In addition,there is a one-to-one, linear correspondence between solutions y(t) of(6.2) and their initial points y(0). Thus the set of 3)-solutions of (6.2) is afinite dimensional subspace of X) which is in one-to-one, linear corre-spondence with Tjy It is a well known and easily verified fact that if twofinite-dimensional, normed linear spaces can be put into one-to-onecorrespondence, then the norm of an element of one space is majorized bya* constant times the norm of the corresponding element of the otherspace. [For example, an admissible choice of Q is a

for any a, 0 < a < w. This follows from (6.6) and the choices t = 0,g(s) = 0 in (6.5).]

Let 93, D be Banach spaces stronger than L(X). Define an operatorT = TW from D to 23 as follows: The domain ,0(7) c D of Fis the setof functions y(t\ teJ, which are absolutely continuous (on compact

subintervals of J), y(t) e £>, and y'(i) — A(t) y(t) e 93. For such a functiony(t), Ty is defined to be y'(t) — A(t) y(t). In other words, Ty — g, whereg(t) e 93 is given by (6.1).

Lemma 6.2. Let A(f) be locally integrable on J and let 93, D be Banachspaces stronger than L( Y). Then T = T^ is a closed operator; that is thegraph of T, &(T) = {(y(t),g(t)):y(t) e ®(T),g = Ty}, is a closed set ofthe Banach space D X $5.

Proof. In order to prove this, it must be shown that if yi(f)» ya(0» • • •are elements of ®(T),gn = Tyn, y(f) = limyn(0 exists in t> and g(t) lim gn(t) exists in 93, then y(t) e @(T) and g(t) = Ty.

The basic inequality (6.5) combined with (6.6) and the analogue of(6.6) for the space % give

y(t) is the uniform limit of y^f), y*(f),... on any interval [0, a] c J.The differential equation (6.1) is equivalent to the integral equation

Since the convergence ofglt gz,... in 93 implies its convergence in L(Y), itfollows that (6.1) holds where y = \irnyn(t) in £>, g = lim£n(/) in 93.Finally, y e I), g e 58 show that y e (7). This proves Lemma 6.2.

The pair of Banach spaces (93, D) is said to be admissible for (6.1) or forA(t) if each is stronger than L( Y), and, for every g(t) e 93, the differentialequation (6.1) has a D-solution. In other words, the map T == T^^:2(T)^-% is onto, i.e., the range of T is 93. (For example, if /:0 < / < oo,-4(0 is continuous of period p, and 33 = X) is the Banach space of con-tinuous functions y(i) of period p with norm \y\x> — sup ||y(OII» then(93, 3)) is admissible for (6.1) if and only if (6.2) has no nontrivial solutionof period p; see Theorem 1.1.)

Lemma 6.3. Let A(t) be locally integrable on J, let (93, £>) be admissiblefor (6.1), and let y0 6 Y$. Then, ifg(t)e"$>, (6.1) has a unique ^-solutiony(t) such that Pyy(Q) = ya. Furthermore, there exist positive constantsC0 and K, independent ofg(i), satisfying

Proof. Consider first the case that y0 = 0, so that we seek t)-solutionsy(t) with y(0) 6 yt. For any g 6 93, (6.1) has a solution y(/)et>, byassumption. Let y(0) = y0 + ylt where y0 = P0y(Q) 6 Y$, y± e IV Lety0(t) be the solution of the homogeneous equation (6.2) such that y0(0) =yQ, so that y0(t) e D. Then yt(r) = y(0 — y0(t) e D is a solution of (6.1)and 2^(0) = yt e 7^

Hence y(t) is tghe uniform

It is clear that y^(t) is a unique 3)-solution of (6.2) with initial pointin IV Thus there is a one-to-one linear correspondence between g e 93and t>-solutions ^(r) of (6.2) with ^(0) e IV The proof of Lemma 6.2shows that if 7\ is the restriction of T = J^ with domain consisting ofelements y(t) e ®(T) satisfying y(Q) e Ylt then Tl is closed. Thus 7\ isa closed, linear, one-to-one operator which maps its domain in D onto %$.By the Open Mapping Theorem 0.3, there is a constant K such that ifT-gj = g, then \y\$ ^ K \g\^. This proves the theorem for y0 = 0.

If y0 5* 0, let «/i(0 be the unique D-solution of (6.2) satisfying ^(0) e IVLet y0(f) be the unique D-solution of the homogeneous equation (6.2)satisfying y0(0) = y0. Then y(t) = y0(t) + y^f) is a D-solution of (6.1),^<>3/(0) = 2/o, and lyfo <j |y0(0lj> + WOls- By the Part of the lemmaalready proved, MOIs ^|^|» and, by Lemma 6.1, |y0(0lx ^ C9 \\y0\\.This completes the proof of Lemma 6.3.

7. Green's Functions

Let /Zoo(0 be the characteristic function of the interval 0 t 2s a, sothat A0a(0 = 1 or 0 according as 0 ? a does or does not hold.Similarly, let ha(i) be the characteristic function of the half-line t a, sothat ha(t) = 1 or 0 according as t 5: a or / < a.

A Banach space © of functions on/: 0 ^ f < c o ( ^ s o o ) will be calledlean at t = eo if y<0 6 SB and 0 < a < co imply that /^(OvCO. Aa(0v(0 e23; |Ao.vl»» !AaVl» ivl»; and |/2avl»->0 as fl->ta. Since ha(t)y(t} =vKO ~ ^oo(0v(0 on J> tne property "lean at t = o>" implies that theset of functions /z0a(0v(0 of 93 vanishing outside of compact intervals[0, a] c J is dense in 93.

Let £> be a Banach space stronger than L( 7). As above, let yx = y1X) bea subspace of Y complementary to Y$. Let P0 = P9% be the projectionof Y onto Xj, annihilating Y^ and Pl = I — P0 the projection of K ontoF! annihilating 7$. In terms of a fixed basis on 7, P0 and Pj arerepresentable as matrices.

Let U(t) be the fundamental matrix for (6.1) on 0 t < <o satisfyingt/(0) =s /. For 0 <£ 5, f < w, define a (matrix) function <7(f, j) by

For a fixed /, G(t, s) is continuous on 0 5 < co, except at J = t, whereit has left and right limits, U(t)P0U~l(t) and -U^P^-^t).

Theorem 7.1. Let A(t) be locally integrable on J. Suppose that 93, Dare Banach spaces stronger than L(Y); that 93 is lean at to; and that

3) has the property that ify(t), y t(t) are continuous functions from J to Yand y(t) — yx(t) = 0 near t = CD (i.e., yl — yz = 0 except on an interval[0, a] cr /), then y(t) e 3) implies that y^t) e D. Then (93, 3>) is admissible

for (6.1) if and only if, for every g(t) e 93,

exists in 35. /« this case, the limit is uniform on compact intervals of J andis the unique ^-solution of (6.1) with y(0) e Y^

Proof. "Only if". Let g(t) e 93, ga(t) = h^(t)g(t). Then (7.2) becomes

where the integral exists as a Lebesgue integral for every fixed f, sinceG(t, s) is bounded for 0 s a and a(j) is integrable over /. In view ofthe first part of (7.1), the contribution of 0 < s t to (7.3) is

Hence, by the second part of (7.1), (7.3) is

where

It follows from (7.4) and Corollary IV 2.1 that ya(t) is a solution of (6.1)when g(t) is replaced by ga(t).

An analogue of the derivation of (7.4) gives

Hence

Thus for a t < co, ya(t) is identical with the solution U(t) U~l(a)ya(a)of the homogeneous equation (6.2). Since the initial point of the lattersolution is in Y%, the property assumed for D implies that ya(t) e 3X

Since ya(0) e Yl by (7.5), it follows that ya(t) is the unique solution of(6.1), where g — gj[t)t satisfying ya(Q) e Yv Hence, by Lemma 6.3,|yJ»^*l*J«.

Let 0 < a . Then, since 3$ is lean at / = o>,

Thus y = lim ya(t) exists in £> as a -*• a>. Also £ = lim ga(t) in 23. SinceT = !r81) in Lemma 6.2 is closed, y(f) is a D-solution of (6.1). The proofof Lemma 6.3 shows that y = lim ya(t) uniformly on compact intervalsof J. Hence, y(G) = lim ya(G) e Yv This proves "only if" in Theorem7.1. The "if" part is easy.

Corollary 7.1. Let co = oo; B and D be Banach spaces of class &~#;B' be the space associate to B; cf. § XIII 9. For the admissibility of(B(Y),D(K)),(i) it is necessary that \\G(t, Oil e B' for fixed t—thus the integralsin (7.2) are Lebesgue integrals; (ii) when B is lean at oo, it is necessary andsufficient that (7.2) define a bounded operator g -^y from B(Y) to D(Y);(iii) it is sufficient that r(t) ED where r(t) = \ \\G(t, -)|| |a«J Ov) wnen

D — L°°, it is necessary and sufficient that r(t) eL00.Exercise 7.1. Verify this corollary.

8. Nonlinear Equations

Lemmas 6.1-6.3 will be used to study the nonlinear equation

Let 58, X) be Banach spaces stronger than L( Y) and Sp the closed ball

Theorem 8.1. Le/ /; 0 ^ / < w (^oo); ,4(0 a locally, integrabled x d matrix function on J, and (95, 35) admissible for (6.1). Let f(t, y(/))6e an element of$>for every y(t) 6 2P flni/ satisfy

/or a// yi(t),yz(t)el,pand some constant 0; r = |/(r, 0)1^; y0 e 1^.Suppose that if C0, /^ are f/ie constants in Lemma 6.3, f/ie/i 0, r, ||ycll ^^so small that

Then (8.1) /K« a unique solution y(t) E 2P satisfying

It will be clear from the proof that the first part of (8.3) can be replacedby the assumption


In fact, the role of the assumption in (8.3) is to assure (8.5). In (8.4), P0is the projection of Y onto Y% annihilating a fixed subspace Y^ whereY- Y9@Ylf

Proof. Theorem 8.1 is an immediate consequence of Theorem 0.1 andLemma 6.3. Since /(/, x(t)) e % for any x(t) e 2P, Lemma 6.3 and theassumption that (33,X>) is admissible imply that

has a unique D-solutiony(r)satisfying (8.4) and (6.8), whereg(/) = /(/, #(?)).Define the operator T0 from Sp into £> by y(t) = T0[x(t)]. In particular,i fw = irjOJIs, then

If ^(0,^(0 eSp and Vl = TJxJ, yz = T9[xt], it follows that ^(0 -yz(t) is the unique ID-solution of

satisfying P0y(Q) = 0. Hence, by Lemma 6.3 and by (8.2),

Consequently, Theorem 0.1 is applicable, and so T0 has a unique fixedpoint y(t) E 2P. This proves Theorem 8.1.

The statement of the next theorem involves the space C( Y) of continuousfunctions y(t) from / to Y with the topology of uniform convergence oncompact intervals in /. The theorem will also involve an assumptionconcerning the continuity of the map Ti(y(/)] =f(t, y(t)) from the closureof the subset Sp n C( Y) of C( F) into %. This condition is rather naturalin dealing with Banach spaces 93, D of continuous functions on / withnorms which imply uniform convergence on /. This is the case in PartsI and II, where J is replaced by a closed bounded interval 0 / ^ p.This continuity condition will also be satisfied under different circumstancesin Corollary 8.1.

Theorem 8.2. Let A(i) be locally integrable on /; 33, D Banach spacesstronger that L( Y); Sp the closed ball of radius p in 3); and S the closureo/2p n C(7) in C(Y). Let A(t)andf(t, y) satisfy (i) (33, 2>) is admissiblfor (6.1); (ii) y(t)-+f(t, y(t)) is a continuous map of the subset S of thespace C( F) into S; (iii) there exists an r > 0 such that

and (iv) there exists a function A(/) e L such that

Use of Implicit Function and Fixed Point Theorems443

Let C0, K be the constants of Lemma 6.3 and let y0 e Y^. Let r, ||yj be sosmall that

Then (8.1) has at least one solution y(t) e Sp satisfying /*0y(0) = VQ-Proof. As in the last proof, define an operator r0 of S into D by putting

y = T0[x], where z(0 6 5 and y(t) is the unique D-solution of (8.6)satisfying (8.4). Thus, by Lemma 6.3,

Hence assumption (8.11) implies that T0 maps S into itself, in fact, into£ pnC(7)cS.

Note that the basic inequality (6.5) implies that

for 0 <; t < a if g(t) =/(/, a(0). Since D is stronger than L(Y\ (6.6)holds. Also there is a similar inequality for elements g E SB with a suitableconstant oc^ (a). Hence, for 0 / < a,

It will first be verified that T0: S —> S is continuous where S is consideredto be a subset of C(7). Let xjf) e 5, ^(0 =/(r, x^O), Vy(0 = ^o[*X')lfor y = 1, 2, then y^t) — yz(t) is the unique D-solution of (6.1), whereg = gl — g2, satisfying P0\yj(Q) — y2(°)] = °- Hence Lemma 6.3 impliesthat

Also, (8.12) holds if y = yl — y2 and g = gl — gz. Thus, for 0 r a

Since, by assumption (ii), (O-^^CO 'n ^(^) implies £i-*£2 »n ®» rtfollows that y^t] -> y2(0 uniformly on intervals [0, a] of /; i.e., yx(r) -*y2(r) in C(y). This proves the continuity of T0: S -+ S.

It will now be shown that the image T0S of 5 has a compact closure inC(Y). It follows from (8.12), where g(t) =/(/, x(tj) and y(t} = T0[x(t)]that, for 0 < / ^ a,

Thus the set of functions y(t) 6 T0S are uniformly bounded on everyinterval [0, a] of/. If c(d) is the number on the right of the last inequality,

443

then (8.6) and (8.10) show that

Therefore, the functions y(t) in the image T0S of 5 are equicontinuous onevery interval [0, a] <^ J. Consequently, Arzela's theorem shows thatT0S has a compact closure in C(Y). Since S is convex and closed inC(Y), it follows from Corollary 0.1 that T0 has a fixed point y(t)eS.Thus Theorem 8.2 is a consequence of the fact that y(i) = T0\y(t)] espnc(r).

It is convenient to have conditions on 23, D, f(t, y), A(0 which imply(ii), (iii), (iv) in Theorem 8.2.

Assumption (Ho) on 23 = B(X): Let 23 = B(X) (cf. § 6), where X is asubspace of Y and B is a Banach space of real-valued functions on J suchthat (i) B is stronger than L; (ii) B is lean at t = co (cf. § 7); (///) B containsthe characteristic function h^(t) of the intervals [0, a] c /; and (iv) ifg>t(t) e B and <p2(t) is a measurable function on J such that |<p2(OI ^ l99i(OI>then cpz(t) e B and \<p2\9 ^ M®.

It is important to have 23 = B(X) rather than S = B( Y) for applicationsto higher order equations. If such equations are written as systems ofdifferential equations of the first order, the "inhomogeneous term/(f, y)"will generally belong to a subspace X of Y; e.g.,/(/, y) might be of theform (h, 0, . . ., 0).

Examples of spaces B satisfying the conditions in (H0) are B — Lv,1 ^ p < oo, and B — L0°° (but not B = L00). Other such spaces B canbe obtained as follows: Let y(/) > 0 be a measurable function such thaty(f) and l/y>(f) are bounded on every interval 0 t a (<o>). Denoteby B = L*0 the space of functions (p(t) on / such that 9?(OMO e 0°°with the norm \(p\<g = l^/yU. The space B = L^0 satisfies conditions(i)-(iv). For this space, A(0 e B holds if

Assumption (H,) on/f/, y): Letf(t, y) be continuous on the product setofJ and the ball \\y-\\ ^ p in Y, let f have values in X, and let there exist afunction Mf) e L such that

Corollary 8.1. Let A(f) be locally integrable on J, (S, X)) admissible for(6.1), 23 satisfies (H0), 2) = L°°(r) (or D = L0™(Y)], f(t, y) satisfies (HOand Ji(t) e B with r = |A|B. Let y0 e 7 . Then, if (8.11) AoWy, (8.1) has

at least one solution y(t) on 0 / < co satisfying P^/(G) = yQ, \\y(t)\\ ^ p[and y(t) -> 0 as t -*• to}.

Exercise 8.1. Verify Corollary 8.1.Exercise 8.2. Let Y be expressed as a direct sum Y$® Y±\ let /*0

be the projection of Y onto Y$ annihilating Ylt and P1 = I — P0 theprojection of 7 onto Yl annihilating 7$. Let /*(f) be locally integrableon /: 0 t < oo. Define G(t, s) by (7.1) and suppose that there existconstants N, v > 0 such that \\G(t, j)|| ^ Afe-'"-'1 for j, f ^ 0. Let/(r, y) be continuous for 0 f < oo, ||y|| ^ p, and let \\f(t, y)\\ ^ r. Let3/0 e 7$. Show that if ||yj and r > 0 are sufficiently small, then (8.1) hasa solution y(t) for 0 / < oo satisfying ||y(/)|| ^ /> and P<&(0) = y0.(For necessary and sufficient conditions assuring these assumptions on G,see Theorems XIII 2.1.and XIII 6.4.)

9. Asymptotic Integration

In this section, let /be the half-line J: 0 t < oo (so that to = oo). Asa corollary of Theorem 8.2, we have:

Theorem 9.1. Let A(i) be continuous onJ: 0 / < oo. Let f(t, y) becontinuous for t 0, ||y|| ^ p, satisfy

arm/ /wye yfl/wes in a subspace X of Y. Assume either (i) that X(t) e L1 a/jrf//ia/ (LMXXD), w/iere D = L00^) [orD = Lo*(K)], w admissible for

or (ii) /Aa/ there exists a measurable function y(t) > 0 on J such thattf}(t) and l/v(0 ore locally bounded, that

anrf that for every g(t) e L(X),for which

(9.2) Afl^ a ^-solution. Then ///„ w sufficiently large, the system

has a solution for t ^ /„ SMC/I rAa/ ||y(/)|| ^ p [flnrf y(r) -»• 0 as t -* oo].Remark 1. Assumption (ii) merely means that (L^(X), !D) is admis-

sible for (9.2). Actually, assumption (i) is a special case of (ii) but is isolatedfor convenience. For a discussion of conditions necessary and sufficient

for (Ll(X), L°°(Xy) or (L\X), L^(X)) to be admissible for (9.2), whereX = Y, see Theorem XIII 6.3.

Remark 2. Let U(t) be the fundamental solution for

satisfying t/(0) = /. Let y0 e Y^. Then if ||y0j| is sufficiently small andt0 ^ 0 is sufficiently large, the solution y(t) in Theorem 9.1 can be chosenso as to satisfy

Let C0, K be the constants of Lemma 6.3 associated with the admissibilityof the appropriate pairs of spaces (Ll(X), D) or (L™0(X), 35). Accordingas (i) or (ii) is assumed, the conditions of smallness on ||y0|| and largenessof /0 are

Proof. Let S = Ll(X) or 93 = L™0(X) according as (i) or (ii) isassumed. Then Theorem 9.1 is a consequence of Corollary 8.1 obtainedby replacing /(/, y), A(/) by the functions ha(t)f(t, y), ha(t)X(t), wherea = /„ and ha(t) is 1 or 0 according as / ^ a or t < a.

Exercise 9.1. The following type of question often arises: Let y^t) bea solution of the homogeneous linear system (9.6). When does (9.5) havea solution y(t) for large t such that y — yt —>• 0 as f —»• oo? Deducesufficient conditions from Theorem 9.1.

As an application of Theorem 9.1, consider a second order equation

for a real-valued function u. Assume that h(t, u, u'} is continuous fort ^ 0 and arbitrary (M, u). Let a, ft be constants and consider the questionwhether (9.7) has a solution for large / satisfying

Introduce the change of variables u -> v, where

then (9.7) becomes

and (9.8) is v, v' -> 0 as / -*• oo. Theorem 9.1 implies the following:Corollary 9.1 Let h(t, u, M') be continuous for t 0 and arbitrary (u, u)

such that

where A(f) is a function satisfying

Then (9.7) has a solution u(t)for large t satisfying (9.8).Exercise 9.2. (a) Verify Corollary 9.1. (b) Apply it to the case that

h =f(t)g(u), where a ^ 0 or a = 0. (c) Generalize it by replacing (9.7by ««*> = h(t, u, « ' , . . . , M*"-1').

Actually Corollary 9.1 is a special case of Theorem X 13.1, but TheoremX 13.1 can itself be deduced from Theorem 9.1; cf. Exercise 9.3 below

Many problems involving asymptotic integrations can be solved by theuse of Theorem 9.1. Often these problems, can be put into the followingform: Let Q(t) be a continuously differentiable matrix for / 0. Doesthe nonlinear system (9.5) have a solution y(t) such that if

then c = lim x(t) exists as r -*• <x> ? The differential equation for x(t) is

The change of variables(9.13) z = x - ctransforms (9.12) into

where

The problem is thus reduced to the question: Does (9.14) have a solutionz(t)for large / such that z(f)->0 as t—»• oo? Clearly, Theorem 9.1 isadapted to answer such questions.

We should point out that if the answer is affirmative, then (9.11) andthe conclusion x(t) — c -»• 0 as f -»• oo need not be very informative unlessestimates for ||a;(/) — c|| are obtained [e.g., if Q(t) is the 2 x 2 matrix2(0 = (?*(<)), where qkl = (- l)*«r<, qu . e* for k = 1, 2 and c = (1,then we can only deduce y(t) = o(e'), but not an asymptotic formula ofthe type y(t) = (-1 + 0(1), 1 + 0(1))*-' as / -». oo.]

Exercise 9.3. Follow the procedure just mentioned and deduceTheorem X 13.1 by using Theorem 9.1 (instead of Lemma X 4.3).

Notes

INTRODUCTION. The use of fixed point theorems in function spaces was initiated byBirkhoff and Kellogg [I]. For Theorem 0.2, see Tychonov [1]. For Schauder's fixed


point theorem, see Schauder [1]. For the remark at the end of the Introduction, seeGraves [1]. As mentioned in the text, Theorem 0.3 is a result of Banach [1).

SECTION 1. Results analogous to those of this section but dealing with one equationof the second order, e.g., go back to Sturm. Boundary value problems for systems ofsecond order equations were considered by Mason [1J. The results of this section(except for Theorem 1.3) are due to Bounitzky [1]; the treatment in the text followsBliss [1]. These results are merely the introduction to the subject which is usuallyconcerned with eigenfunction expansions; see Bliss [1] for older references to Hilde-brandt, Birkhoff, Langer, and others. For an excellent recent treatment for the singular,self-adjoint problem; see Brauer [2]. Theorem 1.3 is given by Massera [1], who at-tributes the proof in the text to Bohnenblust.

SECTION 2. Theorems 2.1 and 2.2 are similar to Theorems 4.1 and 4.2, respectively.Exercise 2.1 is a result of Massera [1] and generalizes a theorem of Levinson [2]; itsproof depends on a (2-dimensional) fixed point theorem of Brouwer. Exercise 2.2 is aresult of Knobloch [1], who uses a variant of Brouwer's fixed point theorem due toMiranda [1]; cf. Conti and Sansone [1, pp. 438-444].

Theorems 2.3 and 2.4 are due to Poincare [5,1, chap. 3 and 4]; see Picard [2, III,chap. 8]. Problems concerning "degenerate" cases of Theorems 2.3 and 2.4 when theJacobians in the proofs vanish were also treated by Poincare and since then by manyothers, including Lyapunov. For some more recent work and older references, seeE. Holder [1], Friedrichs [1], and J. Hale [1]; for the problem in a very general setting,see D. C. Lewis [4].

SECTION 3. The scalar case of Theorem 3.3 is a result of Picard [4]; the extension tosystems is in Hartman and Wintner [22]. In the scalar case, (3.17) can be relaxed to thecondition Re B(t)x • x > 0, Rosenblatt [2]; see also Exercise 4.5(c). The uniquenesscriterion in Exercise 3.3(6), among others, is given by Hartman and Wintner [22].Sturm types of comparison theorems for self-adjoint systems have been given by Morse[11-

SECTION 4. Theorem 4.1 and its proof are due to Picard [4, pp. 2-7]. For relatedresults in the scalar case, see Nagumo [2], [4], references in Hartman and Wintner [8]and Lees [1 ] to Rosenblatt, Cinquini, Zwirner, and others. Theorem 4.2 is a result ofScroza-Dragoni [1]. The uniqueness Theorem 4.3 is due to Hartman [19]. For Exercise4.6(6), see Hartman and Wintner [8]; for part (c), with the additional condition that/ has a continuous partial derivative df/dx 5: 0, see Rosenblatt [2]. For Exercises 4.7and 4.8, see Nirenberg [1].

SECTION 5. Lemma 5.1 and Corollary 5.2 are results of Nagumo [2]. The examplefollowing Lemma 5.1 is due to Heinz [1]. The other theorems of this section arecontained in Hartman [19]. Exercise 5.4 is a generalization of a result of Lees [1] whogives a very different proof from that in the Hints. For the scalar case in Exercise5.5(d), see Hartman and Wintner [8]; this result was first proved by A. Kneser [2] (seeMambriani [1]) for the case when /does not depend on x'. For related results, seeExercises XIV 2.8 and 2.9. A generalization of Exercise 5.9 involving almost periodicfunctions is given in Hartman [19] and is based on a paper of Amerio [1].

SECTION 6. Part III is an outgrowth of a paper of Perron [12], whose results werecarried farther by Persidskii [1], Malkin [1], Krein [1], Bellman [2], Kucer [1], andMaizel' [1]. Except for Kucer, these authors deal, for the most part, with the case93 = jL°°(y), X> = L°°(y). (For a statement concerning the results of these earlierpapers, see Massera and Schaffer [1,1].) The results of this section are due to Masseraand Schaffer [1] who deal with the more general situation when the space 7 need not befinite-dimensional.


SECTION 7. For the notion of "lean at eo," see Schaffer [2, VI]. The Green's functionsG of this section occur in Massera and Schaffer [1, I and IV]. Theorem 7.1 andCorollary 7.1 may be new.

SECTION 8. Theorem 8.1 is a result of Corduneanu [1]. Theorem 8.2 is a correctedversion of a similar result of Corduneanufl] (see Hartman and Onuchic [1]); alsoMassera [8]. For Corollary 8.1, see Hartman and Onuchic [9]. For Exercise 8.2, seeMassera and Schaffer [1,1 or IV].

SECTION 9. This application of the results of § 8 is given by Hartman and Onuchic[1]. For Corollary 9.1, see Hale and Onuchic [1].

Chapter XIII

Dichotomies for Solutions of Linear Equations

For / 2£ 0, consider an inhomogeneous linear system of differentialequations


or, more generally, an inhomogeneous, linear system of equations of(m + l)st order


Suppose that SB, D are Banach spaces of vector-valued functions anthat (0.1) [or (0.3)] has a solution y(t) e $ [or u(t) e 3>] for every g(t) e 93[or/(0 6 IB]; i.e., that (93, t>) is admissible in the sense of § XII6. Then,under suitable conditions on the coefficients and the spaces 93 and £>,this implies a [an exponential] dichotomy for the solutions of the homo-geneous equations, roughly in the sense that some of the solutions aresmall [or exponentially small] and that others are large [or exponentiallylarge] as / -»• oo. This type of assertion and its converse will be the subjectof this chapter.

In particular, for (0.2), we shall obtain conditions necessary and/orsufficient in order that there exist Green's matrices G(t> s) defined as in§ XII 7, satisfying

for s, t 0, where K, v > 0 are constants. The main results for (0.2) aregiven in § 6; corresponding results for (0.4) are given in § 7. The main

450

Dichotomies for Solutions of Linear Equations 451

tools will be the Open Mapping Theorem XII 0.3 (in fact, analogues ofthe lemmas of § XII 6) and the following basic inequality for solutions of(0.1): Let y be a point of the (real or complex) vector space Y with norm||y|| and let \\A(t)\\ = sup \\A(t)y\\ for ||y|| = 1, then solutions of (0.1)satisfy

for arbitrary s, t. When Y is the Euclidean space, this can be strengthenedto

where //(/) = sup |Re A(t)y • y\ for \\y\\ = 1; see § IV 4.It will be convenient to write (0.1), (0.2) as equations Ty = g, Ty — 0,

where T is the operator Ty = y' — A(t)y. In order to avoid a specialtreatment of (0.3), this equation will be written as (0.1), where y = (M, M',. . . , M(m))- It will be advantageous, however, to introduce a "projection"operator P, which in the general case of (0.1) is the identity Py = y butin the case (0.3) of (0.1) is Py = M.

This chapter will be divided into two parts: Part I deals with analoguesof (0.1), (0.3) and Part II with the analogues of the adjoint equations.

PART I. GENERAL THEORY

1. Notations and Definitions

(i) Below y,z,... [or M, v,...] are elements of a finite dimensional realor complex Banach space Y [or U] with given norms |y||, ||z||,... [orIIw| | i IMI, • • • ]• It will not be assumed that these spaces are Euclidean.For example, in dealing with a product space X x Y, it is often moreconvenient to use the norm ||(x, r/)|| = max(||a;||, \\y\\). It is also moreconvenient to work with the angular distance between two nonzero ele-ments y, z E Y defined by

than to assume that Y is Euclidean and to deal, e.g., with the Euclideanangle between y, z or with |sin (y, z)|. (If Y is Euclidean, then y in (1.1) is2|sinHy,z)|.)

Note that ||y|| • ||z|| y[y, z] is the norm of y \\z\\ — \\y\\ z and hence is\\(y - z} M - (112/11 - W)z|l ^ 2 ||z|| • lit/ - z\. Interchanging y and zshows that

If A' is a linear manifold in Y and y e F, let d(X, y) = dist (X, y) —inf ||a; — y\\ for x e X. In particular,

The condition

will frequently occur, where is a subspace (i.e., a closed linear manifold).Note that the inequality (1.4) can hold for y E X only if y = 0. If y ?& 0and X is an admissible number in (1.4), then I/A can be interpreted as a"rough measure of the angle between y and the linear manifold Ar." Thisis clear if (1.4) is written as I/A d(X, y \\y\\~1) = inf ||* - y \\y\\-l\\ forxeX.

Y* denotes the space dual to Y and (y, y*) is the corresponding pairing(i.e., "scalar product") of y E 7, y* e Y*.

(ii) It will be supposed that a coordinate system in Y [or U] is fixed.Thus an element yeY can be represented as y = (y\ ..., y*), whered = dim Y, and a linear operator from Y to Y is a d x </ matrix ^4 withthe norm ||y4|| = sup \\Ay\\ for ||y|| — 1. (This is only for the purpose ofmaking the theorems of Chapter IV, as stated, available here.)

(iii) Let J denote the closed half-line Q < t <. ao and J' a boundedsubinterval of /. The characteristic function of J' will be denoted byhj,(t\ so that hj.(t) = 0 or 1 according as t$J' or t e/'. Correspond-ingly, ha(t) is the characteristic function of the half-line s / < oo andha€(t) the characteristic function of y' = [s, s + e].

<pg€(t) will always denote a non-negative, integrable function on / withsupport on [s, s + e] [i.e., vanishing for / < s or t > s + e, so thatMO = ywoyuoi.

(iv) Let «^" denote the set of normed spaces O whose elements are(equivalence classes modulo null sets of) real-valued measurable functions(f(t) on / satisfying the following conditions: (d) <E> -f0}; (b) the elements<p(t) of O are locally integrable and for every bounded /' there exists anumber a = a(J', $) such that

the least number a satisfying this relation will be denoted by \hj,\9,, sothat

cf. § 9. (c) if 9? e O and y is a real-valued measurable function on / suchthat |y>(0| < |y(OI» then v e O and IvU |^U; (^) if <f e O, j > 0, andy>(/) = 0 or y(t) — <f(t — s) according as 0^t<s or t^s, then

y> e O and (yl^ = l^f^; (e) the characteristic functions hj.(t) of boundedintervals /' are elements of O.

Unless the contrary is implied, B and D below denote Banach spacesin 3T. It is clear that all of the spaces Lv on /, 1 p S oo are in $~. Also,the subspace LQ™ of L°°, consisting of functions <p(t) E L™ satisfying<p(t) -*• 0 as t -> oo, is in y. x

When 0 = 1*, 1 < /» ^ oo, the norm \<p\LV will be abbreviated to

M,-(v) A/ will denote the Banach space of (equivalence classes modulo null

sets of) locally integrable functions ?>(/) on J with the norm

Clearly , M e .7".(vi) If 4>e^", Ooo denotes the linear manifold of functions 9?(*)eO

with compact support, i.e., functions <p(t) e <£ vanishing for large t. If,in addition, O is a Banach space (i.e., complete), then O^ is the completion(closure) of O^ in O.

(vii) If <D e 3~ and Y is a finite-dimensional Banach space (over thereal or complex numbers), O(7) will denote the normed vector space of(equivalence classes modulo null sets of) measurable functions y(t) fromJ to Y [i.e., functions y(t) with components which are measurable functions]such that <p(r) = \\y(t)\\ is in with the norm |y(/)|<r>(y) defined to bel^la,. For brevity, the norm of y(f)eO(y) will be denoted by jyl^.It is easy to see that if O is Banach space, then so is O(IO.

(viii) Let L denote the space of (equivalence classes modulo null sets of)real-valued measurable functions <p(t) on / with the topology of con-vergence in the mean L1 on bounded intervals. Correspondingly, L(Y) isthe space of locally integrable functions y(t) from /to Y with the topologyof convergence in the mean L1 on bounded intervals.

Condition (b) in (iv), cf. (1.5), on spaces O 6 y means that O is strongerthan L, so that convergence in O implies convergence in L; see § XII 6.

(ix) A space 0 e «^~ is called quasi-full if it has the property that 97(0 e L,<p(t) $ $ implies that either /^('MO £ $ for some A > 0 or thatl^oA^U -»- co as A r-> (30. Clearly, the spaces O = Lv for 1 p ^ oo arequasi-full.

(x) Dichotomies. Let K, M^be Banach spaces and^f a linear manifoldof functions y = y(i) from J to 7. With each y(i) e ^T, let there beassociated a non-negative function py(t) on J and an element y[Q] of W.We shall assume that the map Q from^T to W given by Qy(t) = y[0] islinear and one-to-one; y[Q] will be called the "initial value" of y(t). LetWQ be a linear manifold in the range of 0.

M

When WQ is a subspace (i.e., closed linear manifold), it is said to inducea partial dichotomy for (Jf, pv, y[Q]) if there exist a positive constant MQand a non-negative number 6° such that

(a) if y(t) e JT with y[0] e JK0, then

(*) if 2(0 e with ||z[0]|| ^ A </(W0, z[0]) and A > 1, then

A subspace W^ is said to induce a total dichotomy for (^T, y[0]) if itinduces a partial dichotomy for(v/f, p,,, y[OJ), where pv(t) — Hy(0i, and inaddition the following holds:

(c) there exists a constant y0 > 0 such that if y(t), z(t) are as in (a) and(6), respectively, then

and y(0 5* 0, z(t) * 0.A subspace W9 is said to induce an exponential dichotomy for (^T,

P*> [0]) if tner^ exist a non-negative number 0°, positive numbers Mltv, V and, for every A > 1, a positive number Afx' == Afx'(A) such that

(a) if y(/) e rf with y{0] e FT0, then

A subspace W% is said to induce a /o/a/ exponential dichotomy for(i/f, y[0]) if it induces an exponential dichotomy for ( ", /»„, y[0]), wherePtXO = lly(OII> afld condition (c) of a total dichotomy holds (i.e., W0induces a total dichotomy for {yf\ y[0]) and an exponential dichotomy for(^./•wyloDwithftM-lrtOI).

A manifold If0 (not necessarily closed) is said to induce an individualpartial [or exponential] dichotomy for (./f, py, y[0]) if

(a) for every y(t) e ,/f" with y[0] e >f0, there exist constants 6° S£ 0,Af0 > 0 [or Aflf v > 0] depending on y(0 such that (1.7) [or (1.10)]holds;

(b) for every z(/) e */T with z(0) ^ W* there exist contants 0° 0,A^o' > 0 [or M I , v' > 0] depending on z(t) such that

[or (1.11)] holds.If no confusion results (JV, pyt y[0]) will be shortened to (Jf, />„); also,

if />„(/) = |j?/(/)||,^f" will be written in place of (JT, pv).


2. Preliminary Lemmas

The notation Y, */T, W, W0, Q, py(t) is the same in this section as inparagraph (x) of the last section. It will sometimes be assumed thatpv(t), for fixed /, is a "semi-norm,"

c an arbitrary constant; and/or that there exist 01 > 0 and K0 > 0 suchthat

whenever

respectively.Note that if WQ is a subspace, a sufficient condition for W0 to induce a

partial dichotomy for (c/f, pv(0) is that there exist 0° 0, Af0 > 0 suchthat (2.4), (2.5) imply that

In fact, conditions (a), (b) follow from the cases z s 0, arbitrary A > 1,and the case y = 0 of (2.6).

The first lemma will be useful and will illustrate the meaning of "totaldichotomy."

Lemma 2.1. Let W0 be a subspace in the range of Q, Let y(t), z(t)denote arbitrary elements ofN satisfying (2.4), (2.5), respectively. If thereexist 6° 0, M0 > 0 such that

ands ^ 0°, then W0 induces a total dichotomy for ^V (with the corresponding0°, M0, y0 sas 1/A/0). Conversely, if WQ induces a total dichotomy for *AT,then

and s 0°.Remark 1. The factor I/A of ||z(/)|| on the left of (2.8) can be removed

under some additional conditions: If WQ induces a total dichotomy for^T, and there exist 01 0, K0 > 0 such that pv(t) = ||y(OII satisfies (2.2),

(2.3) whenever (2.4), (2.5) hold, then there exists a constant M0' > 0 suchthat

Proof of Lemma 2.1. Assume (2.7). The cases z = 0, A > 1 arbitrary,and y s 0 give conditions (a), (b) of a partial dichotomy. Replacingy(t),z(t)e^ by the elements y(t)l\\y(s)\\, z(t)l\\z(s)\\ E .^ for a fixed5^0° for which y(s) ^ 0, z(s) &. 0 gives (1.9) when r = s or r = j withXo = l/^o-

Conversely, assume that WQ induces a total dichotomy for «tf. Then(1.9) and (1.2) show that

if s 0° and z(.s) ^ 0, yfa) 5"* 0. Conditions (a), (b) of a total dichotomygive (2.8) if s j> 6° and 2(5) 7* 0, y(j) =* 0.

The proof of the last part of this lemma can obviously be modified togive

Corollary 2.1. Let W0 be a subspace of Q inducing a total exponentialdichotomy for Jf. Let y(t\ z(i) be as in Lemma 2.1. Then, for 0 r s< lands'^ 0°,

where v, v', A// = Mj'(A) > 0, Aflf a/w/ y0 occur m the definition of totalexponential dichotomy.

Instead of proving Remark 1 following Lemma 2.1, the following moregeneral assertion will be proved.

Lemma 2.2. Let W0 be a subspace of W with the property that there exist6° 0, Af0 > 0, A0 > 1 such that (2.4), (2.5) with X = A0 imply (2.6) withX = A0. Assume that pv(t) satisfies (2.1) and that there exist 01 0,K0 > 0 j«c^r to (2.4), (2.5) wy»(y (2.2), (2.3). Then there exists an Af0' > 0such that (2.4), (2.5) imply

w^re 00 = max (0°, 01).Proof. The definition of d(W0,z[Q\) shows that there exist elements

y° 6 W* such that ||y° - z[0]|| is arbitrarily near to d(W0,2[()]), Since^o > 1» y° can be chosen so that z° — y° — z[Q] satisfies ||2°|| A0d(W0,z[Q]) = A0f/(H^0, 2°). As FFo is in the range of Q, there is a3/°(0 E T such that /[O] = y° and «°(0 = y°(0 -• «(f) with 2°[0] =2°. Since ||2[0]|| A d(W0, z[Q]) = A d(W^ 2°) A ||2°[0]||, it follows that

From pz(r) ^ />zo(r) + pvo(r) and the case z = 0 and A = A0 of (2.6),py0(r) <: A0M0/v(0o). The inequality (2.2) implies that py0(60) ^#o lly°[0]!l, so that Pz(r) ^ Pzo(r) + M0K010 ||</°[0]||. By (2.3) applied to2 = z°(0 with A = A0,

Hence the last two displays and A > 1 give

(2.12) Pz(r) < A(l + 2M0/i:02A0>X'>).

Thus (2.10) with A/0' = (1 + 2A/o/i:02A0

2)A0M0 follows from (2.6), where(A, 2, y) are replaced by (A,,, z°, y° + y).

The following lemma, which is of interest in itself, will be used severaltimes. (It is false if the assumption that ff is finite dimensional is omitted.)

Lemma 2.3. Let W be finite dimensional, WQ a subspace of W in therange of Q with the property that there exist 0° 0, M0 > 0 such that(2.4), (2.5) imply (2.6). Assume also that py(t) is a continuous function oftfor each y(t) e Ji', that (2.1) holds and that there exist 01 > 0, KQ > Qsuchthat (2.4), (2.5) imply (2.2), (2.3). Let X be the set of initial values y[0] ofelements y(t) e Jf satisfying py(i) -> 0 as t -*• oo. Then X «= W0 is asubspace of W and there exists a constant M0' > 0 such that the conditions

imply that

w/iere 00 = max (0°, 01).This clearly has the following corollary:Corollary 2.2. Let W be finite dimensional and WQ a subspace of W

which induces a total dichotomy for ^V. Assume that \\y(t)\\ is a continuousfunction of t for y(t) e JV* and that (2.4), (2.5) imply (2.2), (2.3) for pv(t) =\\y(t)\\ and 6l = 0 (e.g., ify[Q] = y(0)). Let X be the set of initial valuesy[Q] of elements y(t) E Jf satisfying \\y(t}\\ -> 0 as t -> oo. Then X <^ W0induces a total dichotomy for '.

Proof of Lemma 2.3. If the norm || w\\ in W is replaced by an equivalentnorm ||w||0 (i.e., if Cj ||w|| ^ (|w||0 c2 \\w\\ for constants cl9 cz > 0), thenassumption and assertion of this lemma remain unchanged. Thus,without loss of generality, we can suppose that W is a Euclidean space.


Let W-L be the subspace of W orthogonal to W0. Then, if y(t) E V withy[0] E WQ and z(t) e jV with z[0] E Wlt (2.6) implies that

since (2.6) holds for all A > 1.It is clear from (2.6) that X c ^0. Let JIT1 be the subspace of WQ

orthogonal to X. First we will show that there exists a constant a > 0such that if x\t) e .A^ with xl[Q] X1, then

To this end, it is sufficient to show the existence of a constant ax > 0satisfying

For then (2.16) follows from (2.2) and (2.17) with a = ^KQ. In order toverify the existence of an al5 note that, by (2.2), y[0] = 0 gives />„(/) = 0for / 00. In particular pxi(t) is uniquely determined by [0]. Letp(xl[Q]) = inf px,||(/)|| for / > 00. It is clear that /^[O]) > 0 unless^[0] = 0 (otherwise pxi(t)-+Q, t -»> oo, but 0 j± ^[0] E X1). Since (2.2)shows that convergence of ^[O] in W implies the convergence of px\(t) inthe norm "sup pxi(t) for / ^ 00," it follows that ^(^[O]) is a continuousfunction of [0]. Thus if X1 ^ {0}, ^(^[O]) has a positive minimum 1/otjon the sphere \\xl[Q]\\ = 1. This gives (2.17) if X1 * {0} while (2.17) istrivial if X1 = {0}.

It will next be verified that if x(t) E jV with a;[0] e X, x\t) E Jf with^[O] E X\ then

Suppose that (2.18) is false. Then there exist x(t), xl(t) as specified and ans 00 such that

Since y = xl(t) — x(t) E with y[0] 6 WQ, it follows that M0pxi,x(s) ^pxi-x(t) for t s 60. By (2.1), (2.16), and the fact that px(t) -»> 0 as t -* ooit follows that, for large t,

where the last inequality is a consequence of (2.16). Since the last twoformula lines hold, max (pxi(s), px(s)) = px(s), and so px(s) ^ 3pxi(s)-Thus by (2.1) i. ) 2px(s)/3. Hence (2.19) implies Px(s) > 2v.M0px(s\which is impossible since the constants a, MQ in (2.15), (2.16) must satisfya ^ 1, M0 ^ 1. Hence (2.18) holds. An immediate consequence of (2.18)

is that

The subspace X° of W orthogonal to X is the direct sum of X1 and W^.It will now be shown that if x(t) e Jf with a;{0] e X, x°(t) 6 JV withx°[0] e X°, then

if Ml = 6aW03. To this end, let z°[0] = zJO] + ^[0] be the decomposi-

tion of 30[0] into orthogonal components zJO] £ Wlt xl[Q] e X1. Since^[O] € X1 ^ WQ is in the range of 0, there is an x\i) e ./T with Qx\i) =ar^O]. Let x^t) = x°(r) - ^(0 e JT> so that C^(0 « *i[0]. By (2.15)applied to z = — xlt y — x\t) — x(t),

Hence <2.20) gives

and (2.21) follows with MI = 6a2A/03 since px* < /v + pXl by (2.1).

Lemma 2.3 now follows from Lemma 2.2 (or its proof), where A0 = 1 ispermitted here since orthogonal decomposition can be used in theEuclidean space W,

The proof of the existence of exponential dichotomies below willgenerally depend on proving first the existence of a dichotomy and thenthe applicability of the following:

Lemma 2.4. Let o(i) be non-negative for a / < oo with the propertiesthat there exist positive constants 6 < 1, M0, 6 such that a(t) ^ MQa(s)fora s t <: s + 6 and a(t + <5) < 0<r(f) for t a. Then a(t) <;M$-le-r(t-'}a(s)for a < s < t < oo, where v = — 8~l log 6 > 0.

If, in this lemma, the assumption a(t + S) < 6a(t) holds only fort = b (= fl)» then the main inequality in this assertion is valid forb j t < oo. It can, however, be replaced by a(0 Jt' e""""*' )valid for a < s r < oo if K' = Mo"1 '"-'11 and m = 1 + b - a.

Proof of Lemma 2.4. Clearly, crfa + nS) ^ 0no-(^) for 5 a andn = 0,1,.... Hence,

Since r^"-" > e-"<n+1)') = fl«+», the assertion follows.Applications of Lemma 2.4, in proving the existence of exponential

dichotomies, generally lead to an exponent v' in (1.11) which depends onA > L In order to get a v independent of A, the following will be used.It is derived by the arguments used in the proof of Lemma 2.2.

a


Lemma 2.5. Let WQ be a subspace of W in the range of Q. Letthere exist 6° £ 0, M0' > 0, v' > 0 such that if z(t) e jV with ||z[0]|| ^^o d(WQ, z[Q])for a fixed A0 > 1, then

Assume that py(t) satisfies (2.1) and that there exist 01 > 0, K0 > 0, A > 0such that (2.4), (2.5) imply (2.2), (2.3), and

Finally, suppose that condition (b)for a partial dichotomy holds; cf. (1.8).77ze« condition (b) of an exponential dichotomy holds with the given v' (forain> 1); cf. (1.11).

Proof. Let (2.5) hold and let z(t) = y\t) — z°(0 be the decompositionused in the proof of Lemma 2.2, so that (2.11) holds. Then (2.2) and (2.3)with 2 = 2°, A = A0 give

Since z(t) = y°(t) - z°(t), (2.1) implies that

Applying (2.22) for z = z°, with s and t interchanged, gives

so that if j is replaced by j — A,

By (2.23), with 2, f, y, A replaced by 2°, $ — A, y°, A0,

Thus (2.24) and the last two inequalities give

if K' = (1 + 2VWo*o*'''AW > °- Finally, (1.8) in condition (b)for a partial dichotomy shows that

and s 0° + A. Another use of condition (&) of a partial dichotomyallows the removal of the restriction t 01 by a suitable alteration of thefactor XK'Mtf?'^. This proves Lemma 2.5.

The preceding lemmas and their proofs can be used to obtain a charac-terization of "total dichotomy" or "total exponential dichotomy" for thelinear manifold jV of solutions of the homogeneous equation (0.2), whereY is a finite dimensional (Banach) space. Let U(t) be the fundamental

matrix of (0.2) satisfying l/(0) = /. Let W0, W^ be complementarysubspaces of Y (i.e., Y = W0 @ W^, and let P0 [or Pj] be the projectionof Y onto W0 [or W^\ annihilating Wv [or W0]. Define a Green's matrixG(t, s) by

according a s 0 ^ . s < ^ < o o o r O f i f < s < o o ; cf. (XII 7.1).Theorem 2.1 Lef A(t) be a matrix of locally integrate (real- or

complex-valued) functions ont^Q,jV the set of solutions of (0.2), W — Y,and y[Q] = y(G) £ Y = W. A subspace WQ of Y induces a total dichotomy[or total exponential dichotomy} for (s¥°, \\y(t)\\, y(ty) if and only if, forone ahdjor every subspace Wôf Y complementary to W0, the norm of theGreen's matrix (2.25) satisfies

for some constant K = K(W^ [or

for some constants K — K(W^ v = v(W^ > 0].Proof ("Oflfyi/")- Let induceatotaldichotomy for(^, \\y(t)\\,y(Q))

and W-L be a subspace of Y complementary to WQ. There exists a A0 > 1such that if z(0) e Wlt then ||z(0)|| J.0d(W0,z(Q)). Hence Lemma 2.implies that if y(t\ z(i) are solutions of (0.2) and y(G) e W0, z(0) E W^ then(2.7) holds with A = A0.

Let c e Y be arbitrary. Then y(i) = U(t}PQU~l(s)c, for a fixed s, is asolution of (0.2) and y(0) = ./W'OX e ^0, y(5) = t/^Po^-^^c. Alsoz(0 = —U^PiU'^c, for a fixed 5, is a solution of (0.2) and z(0) =-PÛ-\s)c € Wlt z(s) = - U(s)PlU-\s)c. Thus (2.7) and P0-\-P1 = Igive

for Q < r <s t if K= A0M0. In view of (2.25), this is equivalent to(2.26).

Similarly, if W0 induces a total exponential dichotomy for^, then a useof (2.9) instead of (2.7) leads to (2.27) with K, v in (2.27) given, e.g., b2t.0yol max (Afl5 l/Mj'ô)), min (v, v'(A0)), respectively, in terms of theconstants in (2.9).

Exercise 2.1. Prove the "if" portion of Theorem 2.1.

3. The Operator T

The general theory will be presented in a somewhat abstract form whichcan then be applied to (0.1) or (0.3) or other situations. In what follows,


B and D are Banach spaces in 3~. The results of this section are analoguesof the lemmas of § XII 6.

Let Y, F be finite dimensional Banach spaces. The objects of studywill be a linear operator T from L( Y) to /.(/"),

and the elements y — y(t) of its null space JV(T)

The domain of definition of Twill be denoted by £>(T).Let U be a finite dimensional Banach space and W another Banach

space. (It will not be assumed that W is finite dimensional although in theapplications below this will be the case. In applications, e.g., to difference-differential equations, this need not be the case.) P will denote an operatorfrom L(Y) to L(U) and Q an operator from L(Y) to W with the samedomain of definition as T, &(P) = 2(Q) = 2>(T\ The element w =Qy(t) 6 W will be called the "initial value" of y(t) and denoted by y[0].There will be no confusion even though || . . . || denotes norm in eithery, F, U, or W.

Remark. It will be convenient to illustrate various statements in thegeneral theory from time to time by references to (0.1). In such references,it is always assumed that A(t\ g(t) are defined on J and are integrable.over all finite intervals J' c j and that y(t) is an absolutely continuoussolution (0.1). In this case, the spaces Y, F, U, Ware taken to be identical;Ty(t) is defined by Ty = y'(t) — A(t)y(t) and 3>(T) is the set of functionsy(t) from J to Y which are absolutely continuous (on every J'); Py(t) =y(t) is the identity operator; and Qy(t) = y(0). In applying the generaltheory to (0.3), where u can be a vector, it is supposed that (0.3) is writtenas a system (0.1) for y = (u, u, . . ., u(m)), but Py(t) = u(t), Qy(t) = y(0).

Definition. PD-Solutions and WD. Let De^~ be a Banach space.y(t) is called a PZ)-solution of (3.1) for a given/(/) e L(F) if (3.1) holdsand Py(t) 6 D(U). WD = WD(P) denotes the linear manifold in Wconsisting of initial values w = y[0] of P-D-solutions of (3.2).

Definition. P-Admissibility. The pair (B, D) of Banach spaces in yis called P-admissible for (3.1) if, for every /(/) e B(F), (3.1) has at leastone PZ>-solution y(t).

Various assumptions (A0), (A^,... or (Bj), (B2),. . . concerning T willbe made from time to time. These and some of their consequences willnow be discussed.

(A0) If y(t) e 9(T\ then u(i) = Py(t) is (essentially) bounded onevery bounded interval J' of/.

This assumption implies that the answer to the question whether or noty(t) is a PD-solution of (3.1) depends only on the behavior of u(t) = Py(t)for large f, cf. conditions (c), (e) for 4> = D e in § 1 (iv).

(A!> Uniqueness for Q. (a) If (3.2) holds and y[0] = 0, then y(t) = 0.(b) There exist positive constants a, Cx such that (3.1) implies that

This assumption is, of course, suggested by the inequality (0.5).(A/) Same as (Ax) except that (3.3) is replaced by

(A2) Normality for P. If (3.1) holds, then y(t) is uniquely determinedby Py(t) and/(0; furthermore, the linear map from L(U) x L(F) toL( 7) defined by (Py(t\ Ty(t)) -+ y(t) is continuous in the following sense:if yn(t) E @(T) and the two limits u(t) = limPyn(0 in L(U),f(t) = limryn(r)in L(F) exist, then y(t) = iimyn(t) in L(K) exists and y(t)E@(T),Py(t) = U ( t ) , T y ( t ) = f ( t ) .

The main role of (A2) is the following (cf. Lemma XII 6.2):Lemma 3.1. Assume (Ag). The operator T0from D(U) to B(F) defined

by T0[Py(t)] = Ty(t), with a domain 2(T^ consisting of those elementsu - Py(t) in the range of P for which u = Py(t) e D(U)and Ty(t) E B(F),is closed. Also, for every t > 0, there exists a constant C2 = C2(0 suchthat

Proof. In order to prove that T0 is closed, let (Wi,/i), Ov/g), • - • be aconvergent sequence of elements in the graph of T0, where un = Pyn(t),/„= Tyn(f) and yn(f)e9(T). Thus u = lim «„(/) in D(U) and / =lim/n(/) in B(F) exist. Since convergence in B or D implies convergencein L [cf. condition (1.5) on O = B, De^}, (Aa) implies that y(t) =lim yn(t) exists in L( 7) and y(r) e J0(r), Py = M, Ty = f. Hence w e ^(70)and r0w =/; thus 7"0 is closed.

Let 7\ be the map from D(U)x B(F) to the space L\0it](Y) of F-valued functions which are integrable over the interval [0, t], where thedomain @(T^ is the graph of T0, and T^ is defined by 7\(Py, 7y) = y.Thus 7\ is defined on a subspace of D(f/) x B(F). (A,j) implies thatTj is continuous, hence bounded. The inequality (3.4) is equivalent to theboundedness of 7\.

Although the trivial space B = {0} is not in <^~, this choice of B ispermitted in Lemma 3.1. The fact that J0 is closed gives

Corollary 3.1. Assume (Aa). The set of u = Py(t), where y(t) variesover the PD-solutions of (3.2), is a subspace (i.e., closed linear manifold) ofD(U).

(A3) (B, D) is ^-admissible.Lemma 3.2. (Aj) [or (A/)], (A2), and(A3) imply that there exist constants

C3, C30 such that iff(t) E B(F), then (3.1) has a PD-solution y(t) satisfying

Proof. Let T0 be the operator from D(U) to B(F) occurring in Lemma3.1. Thus T0 is closed by Lemma 3.1 and onto by (A3). Hence theexistence of C3 follows from the Open Mapping Theorem XII 0.3.

If (Aj) is assumed, then (3.3), (3.4) with t = a, and (3.5^ give

Thus, by (1.5), (3.52) holds with C30 » C^C^CC, + 1) + |AJ*.}.If (A/) holds, it is similarly seen that (1.5), (3.3'), and (3.50 imply (3.5a)with C30 = C,{a~l lAJs-Q + \hJB.}.

(A4) WD is closed.This assumption is of course trivial if W, hence WD, is finite dimensional.Lemma 3.3. Assume (At) [or (A/)] and (A2). Let Wm be a subspace of

W contained in WD [e.g., //(A4) holds, let Wm — WD]. Then there existsa constant C4 = C^Wj^ such that if y(t) E^(T) and y[Q] e Wm, then

Proof. Let Q0 be the operator from D(U) to Wm defined byQo[Py(t)] = y[0], where ^((?0) is the set of u = Py(t) such that y(t) is aP/)-solution of (3.2) with y[Q] E Wm. Q0 is closed by (3.3) [or (3.3')] and(A2), one-to-one by (Axfl), and onto since Wm <=• WD. Thus the openmapping theorem is applicable to Q0 and implies Lemma 3.3.

Lemma 3.4. Assume (Ax) [or (A/)], (A2), (A3), and (A4). Then thereexists a constant C5 with the property that if A 1, Ty =f, Py e D(U),fE B(F), and \\y[Q]\\ -solution of (3.1) supplied by Lemma 3.3,so that 1/V.fo ^ C3 |/|B, ||y0[0]|| ^ C30 |/|B. Since t/(r) - y0(/) is aPD-solution of (3.2), (3.6) gives

As w = j/[0] — y0[Q] e WD, we have

Thus the second inequality in (3.7) holds. In addition,

Thus A 1 shows that the first inequality in (3.7) holds with C5 =^3 i ^v^Cgo.

4. Slices of \\Py(t)\\

Recall that <pse(t) always denotes a non-negative, integrable function onJ with support on [s, s + e] and that | . . . |t refers to the L1 norm on J.

(Bxe) Let 00 > 0, e > 0 and K > 0 be fixed. For every pair ofsolutions y(t\ z(t) of (3.2), with y(i) — z(t) ^ 0, and for any given function9Pse(f) as in § l(iii), with s 00, let there exist a function y^t) e 2(T) withthe properties that (i) yx[0] = (const.) z[0]; (ii) \\Pz(t)\\ < K \\Pyl(t)\\l\<pS€\lfor 0 t 5; (hi) ||Py(OII ^ H/Vi(OII/l^li for t*Z s + c; (iv) thereexists a constant #' (depending on y, zt <pse) such that ||/tyi(f)|| ^ A^'UPKOIIfor t^ s + ; finally, (v) 11 (011 ,(0 \\Py(t) - Pz(t)\\ for / 0.Also, if y(r) 0 is a solution of (3.2), then l/\\Py(t)\\ is integrable overany closed interval /' of /.

Remark 1. This assumption will be used only in the particular case

For (4.1), condition (v) becomes

and, by Holder's inequality,

Remark 2. Note that assumption (Bje) holds for all 00 > 0, e > 0 withK = 1 if T is the operator associated with (0.1) as in the Remark in § 3;thus Ty(t) = y'(t) - A(i)y(t\ Py(t) = y(t) and y[0] = y(0). In fact, let

Then Tfc = Vi - A (t)yi = <pl€(tMV - z(01 and (/) = |yse(0li y (0 fot^ s + , yi(r) = I^U 2(r) for 0 r ^ 5.

Theorem 4.1. Assume (A0), (AJ [or (A/)], (A2), (A3), (AJ, ow/B^c)/or a/x*/ . L*r y(/)e^(r), y[0] e , am/ z(/)e^T(r), ||z[0]|| ^A </(»!>, |2[0]ll)/^ « A > 1, «K/j £ 00. 7%«i

where K and C5 are the constants in (Bje) and Lemma 3.4, respectively.In particular, //A t is fixed and either

Men JFp induces a partial dichotomy for (^(T), />„(/)) w/fA 0° = 00 fl71^

In the applications of Theorem 4.1, it will be important that A/0 dependson €, but not on A. The inequalities (4.10), (4.12) in the following proofwill be used in the proofs of Theorems 4.2 and 4.3.

Proof. Apply (Bje) with the choice (4.1) of 9?, (0> so tnat (v) implies(4.2). Assumption (A0) and (iv) in (B1c) show that Pyx £/>((/). SinceHyJOlH < hd(WD, ll^fO]!) by (i) and the condition on z[0], Lemma 3.4and (4.2) give \Py-,\D ^ A/STC5 \h0e\B. It follows from (ii) and (iii) in (B^)that

Hence (4.4) is a consequence of (4.3).In order to prove the assertions concerning (4.5), note that \ht£jfy\D ^

\hs+fPy\D for / ^ * + e, A > 0, and \hrAPz\D < \h0gPz\D if r + A < s.Thus, for any A > 0 and s 00, the inequality (4.4) implies that, forr + A^ j , s + e ^ r ,

The relations (1.5) and A e show that

Thus, for A e, s 00, r 0, r + A s, s + e < t,

where F(e) is defined in (4.6).The case z = 0 and A = 1 of (4.10) combined with

gives condition (a) [i.e., (1.7)] for a partial dichotomy for py(t) = \ht£fy\D

with 6° = 00 and M0 given by (4.6), even with the factor 2 omitted fromthe second term. Similarly, the choice y = 0 in (4.10) combined with ananalogue of (4.11) gives condition (b) [i.e., (1.8)] with 0° = 00 and thesame M0.

In order to deal with the second function in (4.5), apply an analogue of(4.9) to the left side of (4.8) with A = e to obtain

if r + c s, t s + , s 00. For A e, let k ^ 1 be an integer suchthat ke < A < (k + l)e. Then it follows, by replacing t by t + je and s bys + jc fory = 0 , . . . , k — 1 and adding, that

In addition,

If the upper limits of integration on the right of the last two inequalitiesare replaced by s + A, we obtain the second of the inequalities containedin

for r + A s, t s + e, s 00. The other inequality, involving thefirst integral, is obtained similarly. Combining (4.13) with inequalities ofthe type

leads to a partial dichotomy for the second function in (4.5) with 0° = 00

and A/o given by (4.6). This proves Theorem 4.1.Corollary 4.1. In addition to the assumptions of Theorem 4.1, assume

that D is quasi-full (cf. (ix) in § 1) and that z(t) is a non-PD-solution o/(3.2)then

This follows from (4.4) with y(i) = 0 and the definition of a quasi-fullspace.

Corollary 4.2. In addition to the assumptions of Theorem 4.1, assumethat W is finite dimensional, that pv(t) is defined in (4.5) and, in the case ofthe first choice, that pv(t) is a continuous function oft. Let WQ be the set of

initial values y[0] of y(t) e^V(T) satisfying />„(*) -»• 0 as t -> oo. ThenWQ c: WD avid W^ induces a partial dichotomy for (^(T), py).

Exercise 4.1. Verify this corollary which follows from Lemma 2.3combined with Theorem 4.1 and its proof.

Theorem 4.2. Let the assumptions of Theorem 4.1 hold with B = Ll andD = L00 or D = L0°°; m addition, let (Bl ) fto/rf/or a// e, 0 < e 0, wi/A00, independent ofe. Let PJi~ be the set of functions Py(t) with y(t) &Jfand let the "initial value" y[0] be assigned to Py(t). Assume that \\Py(t)\\ is acontinuous function of t for y(t) e^V. Then WD induces a total dichotomyforPjV.

Proof. By the proof of Theorem 4.1, (4.12) holds for r + e s,t ^ s + €, s £ 00. Since B = L1 and D = L°° (or L0°°), r(e) in (4.6) ic - 2 - e - e = 1; cf. (1.5). Hence

Letting e -»• 0 gives

for 0 r * /, 5 2£ 0°. In view of Lemma 2.1, this proves Theorem 4.2.Theorem 4.3. Assume (A0), (A/), (A2), (A3), (A4), andE^for all

e 0 > 0 fl/w/ w/VA 00, J1^ independent of e. Assume also that either

Then WD induces an exponential dichotomy for the functions (4.5) forevery fixed A 0, >v/f/r 0° = 00 and constant MV'(X, A) depending on A,few/ A/!, v, v' > 0 independent of A.

Note that if (5, D) = (L*, La), where 1 <p,q ^ oo, then (4.14) holdswhen(/?,0) 5^ (1, oo).

Proof. Theorem 4.3 will be deduced from Lemmas 2.4, 2.5, andTheorem 4.1. In the proof consider only the first function py(t) == \ht^Py\D.in (4.5); the proof for the other function is similar. The condition (4.14)is equivalent to

In order to see this, note that if k ^ 1 is an integer such that 0 < kt ^f] < (k 4- IX then |Aoi?lz>' = (^ + 0 l^o lz>'- This is a consequence of thefact that \hj'\jy is the "best" constant in (1.5). Hence r)~l \h^\D> (A: + I)*-1*-1 \h0f\D, ^ 2c-J \h0e\D> for all r] e and so the secondfunction of A in (4.14) is bounded for e0. Also lim inf e~l \h0f\D> = 0as f -*• oo implies that e"1 (AoJ/y -> 0 as e -^ oo. Similar remarks applyto the first function in (4.14). This makes it clear that (4.14) and (4.15)are equivalent.

Let e (^ „ ) be fixed so large that

Let y(/)e./T(r), y[0] e WD, so that (1.7) in condition (a) of a partialdichotomy is applicable for A e0. By the case z = 0, A = 1 of (4.10),it follows that

and A e. Thus, for fixed A *± e, condition (a) [cf. (1.10)] for an exponen-tial dichotomy with 0° = 00, Mr = M06~l, and r = — e"1 log 0 followsfrom Lemma 2.4 applied to <r(r) = \ht£fy\D. (Hence Ml and v areindependent of A e.) If 0 ^ A < e, let A: 1 be an integer such that(k — 1)A < e < A: A; thus/: < 1 + e/e0. Then, by what has been proved,

for 00 j ?. By (1.7) in a partial dichotomy,

These two inequalities give condition (a) of an exponential dichotomywith 0° = 00, M! = kM0B-\ and v = -«-* log 0.

In order to obtain condition (6) [cf. (1.11)], consider first a fixed A0 > 1.Let e = (A 0) be so large that 0 = A0#

2C6r(e) < 1. Let z(t)e^(T),N0]ll = ô d(WDi z[0]). Then (1.8) in condition (6) of a partial dichotomyis applicable for A . The case y = 0, A = AQ, e = (A,, ) of (4.10) gives

An application of Lemma 2.4 to <r(f) = l/|/»tA.Pz|£) gives condition (b) foran exponential dichotomy for A = A0 with 0° = 00, Af/ = AoA/,,0-1, andv = -A log 0.

When A e, the corresponding condition (b) for all A > 1 with thesame v' will be deduced from Lemma 2.5. In fact, condition (2.22) hasjust been verified. (2.1) is clear and (2.2) follows from (3.6) with KQ = C4,6l'= 0. Condition (2.3) follows from (3.3') applied to y = z,/= 0 which,together with (1.5), gives

In fact, since (1.8) implies that IV/zl/j A£M0 \ht±Pz\D for t max (a, 00)if a < fcA, (2.3) follows with K6 = Qflr1 \h^\jy kM0, 01 - max (a, 00).Finally, (2.23) follows from (4.10) with K0 = A:2C6r(e). Consequently,condition (b) of an exponential dichotomy holds for A e with Af/ —MÂXr^vÂ).

As in case (a), it can be shown that (b) holds for eQ A < « with asuitable Af/(A, A) and v' = v'(c) independent of A. Finally, an analogousargument shows that it is possible to choose v' = v'(e) for all A > e0.This proves Theorem 4.3.

Theorem 4.4. Let the conditions of Theorem 4.3 hold; in addition, letthere exist a subspace WQ of W which induces a partial dichotomy forC/F, \\Py(t)\\). Then WQ = WD and WD induces an exponential dichotomyfor(^,\\Py(tW.

Proof. In view of Theorem 4.3, WD induces an exponential dichotomy/*<+A

for C/f, ft//)), where py(t) = \\Py(r)\\ dr for a suitable A > 0. Thismakes it clear that if Tz(i) = 0 and z[0] £ WD, then pz(i) is not bounded ast -> oo, and so z[0] £ J^. Thus W0 c WD. Also, if z[0] £ WQ, then"/>„(/) -> 0 as t -> oo" does not hold, and so z[0] £ W^,. Thus ffp c fp0

and, consequently, fF0 = WD.Using the fact that WD = WQ induces a partial dichotomy for (^T,

||Py(/)||) and an exponential dichotomy for(./f, />„(/))> it is easy to see thatit induces an exponential dichotomy for C/T, ||Py(/)||). Details will beleft to the reader.

Theorem 4.1 and 4.4 are immediately applicable to the operator Tassociated with (0.1); the results will be given in § 6. These theorems arenot applicable to operators associated with (0.3) without some boundednessconditions on the coefficients Pk(t). The difficulty arises from the fact that,in general, condition (l^e) does not hold. The next section leads totheorems applicable to (0.3) as well as (0.1).

5. Estimates for \\y(t)\\In this section, the role of (I^e) will be played by the following

condition:(B2c) Let e > 0, 00 > 0, Kz > 0 be fixed. For every s 00 and every

pair y(0,2(0 e./f (7), where y[Q] e WD, there exists a <pse(f) as in § l(iii) andayi(/) e 9(T) such that (i) yi[0] = (const.) z[0]; (ii) ||Pz(/)|| ^ Kz \\Pyi(t)\\for 0 t^ s; (iii) \\Py(t)\\ ^ K2 \\PVl(t)\\ for s + <• ^ t < oo; (iv)Pyv(t) e D(U); (v) Tyjf) satisfies

(vi) <f>Sf(t) e B and there exists a number b(f) satisfying

Note that (5.1) implies that Ty^t) = 0 unless s < / ^ s + e.(B38) Let 5 0 be fixed. There exists a constant K3(d) such that

(i) if y(t)^V(T) and y[Q] e WD, then y(t)&D(Y) and \h^y\D <K*(d) \hfy\D for 5>0; (ii) if y(t)eJT(T), then h9s(t)Py(t) E D (U),h,s(t)y(t) eD(Y) and \h0s^y\D <, K,(d) \h0aPy\D for s > 6.

It is clear that if (B3«5) holds for d = <50, then it holds for all d > 60.For convenient reference, the following variant of (B3<5), which will beneeded in Part II of this chapter, is stated here.

(8308) Let d 0 be fixed. There exists a constant KM(6) such that if/(/) e BJF) [cf. (vi) in § !],/(/) = 0 on an interval [s - d, s + A + d]for some s 6, and y(t) is a PZ>-solution of Ty = /, then y(t) e D( Y) and\ha±y\D£K30(d)\Py\D.

(B4A) Let A > 0 be fixed. The solutions y(t) of Ty = 0 are continuousfunctions of / (from J to Y) and there exists a constant ^(A) such that if*t) e^(T), then ||y(OII ^ A^(A) || )l! if k - /I ^ A.

It is clear that if (B4A) holds for some A > 0, then it holds for all A > 0.Remark. If A (?) satisfies

or if Y is Euclidean and

then the operator T belonging to (0.1), as described in the Remark of § 3,satisfies (B4A). This is clear from (0.5) or (0.6).

Theorem 5.1. Let e, d > 0 be fixed. Assume (Aj^AJ; (B2e) with b(c)independent ofy, z; (B3<5); anrf(B4A). Then WD induces a total dichotomyfor ^V as V(T) with 0° = 0. If, in addition, D is quasi-full and z(t) is anon-PD-solution ofTz = 0, then \\z(t)\\ -> oo as t -* oo.

If the assumption that b(e) can be taken independent of t/, z is omitted,then we obtain an individual partial dichotomy instead of a totaldichotomy.

Proof. In view of (5.1) and (B4e),

and *^00. Hence \Tyi\B < A^(e) |Vte|fl ||y('i) - KOII for J /! 5+ e. If ||z[0]|| ^ A^(W0,z[0]), then, since Pyx e £)(£/), Lemma 3.4gives

thus (ii) and (iii) in (B2e) imply that

By (B3<5),

maxdêb, \hs+(+6y\D} ^ *K3(d)C&K4(e) \<pS(\B ||y(^) - z(tl)\\.The inequality in (B4e) shows that, for 0 :fs T < /0 < r -f «,

and that a similar inequality holds for z. Consequently,

where 0 r < j - e - 6, t j + + d, s^B0, and K(f, 6) =KMCtKê). Finally, applications of (B4e + <5) and (B400) give

for 0 r ^ 5 < / if M0 = A#( , W)^( e -f <5)#4(00)/l/»o lz>- HenceTheorem 5.1 follows from Lemma 2.1.

Theorem 5.2. Assume (AjHA^; (B2e)/or a// « j> e0 > 0 w///z 00, AT,independent ofe, \<pse\B independent ofy, z, and

uniformly for large s; (B35); anrf (B4A). Then WD induces a totalexponential dichotomy for ^V =^(T) with 6° = 0.

Proof. Let z = 0 in (Bje). Then (5.1), t 5, Theorem 5.1 implyII7X011 ^ ô^seCO lly(j)l|. Arguing as in the last proof, it follows that\hs+€+dy\D ^ C5K3(d)Mo \<Ps£\B \\V(s)l Also, by condition (1.7) of a totadichotomy, M0 \hT€y\D ^ \\y(r + c)|| • \h0f\D. Hence if / s + 2« + «5,

In view of (5.6), e and s0 can be chosen so large that

so that |J2/(/)|| < 0 Hy(5)|| for s > s0, t s + 2c + (5. An application ofLemma 2.4 and the remark following it to a(t) = \\y(t)\\ give condition (a)for an exponential dichotomy with 01 = 0; cf. (1.10).

Let A0 > 1 and z(t) e^(T), ||2[0]|| A0 </(fFZ), 2[0]). Choosing y = 0in (B2«) and arguing as before shows that if e = e(A0), s = j(A0) are solarge that

then ||z(r)|| ^ 0 ||z(s)|| if r < s - « - d, s s9. Applying Lemma 2.4to a(t) = l/l|z(OII giyes condition (b) of an exponential dichotomy forA = A0with0° = 0; cf. (1.11).

Condition (b) for all A > 1, with v independent of A, can now bededuced from Lemma 2.5, where pv(/) = \\y(t)\\. In fact, (2.22) has justbeen verified and (2.1) is clear. In order to deduce (2.2), note that (3.6)and (B36) imply that

for any / ^ 6, A > 0. By condition (a) of a partial dichotomy,MO 1/r^lD llriOH I/»OA!Z>. This gives (2.2) with K0 = A:,(<5)C4M0/IVJ1>and 01 SB <5. To obtain (2.3), begin with (3.3) applied to y = z, / = 0.Then, by condition (b) of a partial dichotomy,

which is (2.3) with K0 - QM0 and 01 = a. Finally, (2.23) is implied by(5.5). Hence Lemma 2.5 implies condition (1.11) of an exponentialdichotomy for^T(r).

We can obtain results analogous to Theorems 5.1-5.2 under a conditionsomewhat weaker than (B4A):

(B5A) Let A > 0 be fixed. The solutions y(t) of Ty — 0 are continuousfunctions of t (from / to 7) and there exists a number #5(A) such that ify(Oe^(D,then

Condition (B5A) is useful for applications to second order equationsand is suggested by Exercises XI 8.6 and 8.8. For applications, seeExercises 7.1, 13.1, and 13.2, below.

Choosing s = / — $6 in (5.7) and integrating with respect to d over aninterval [0, d] gives

Introduce the Banach space K(2) = 7 x 7 with the norm of rj =(y,, y2) e F<2> defined by Nl = max (UyJ, ||ytH).

Theorem 5.3. Let e, A > 0 be fixed. Assume (AiXAJ; (B2e) with£(e) independent ofy, z; (B3j) ; and (B6A) w/rA A 2e. For (5 > 0, /e/^ftf be the manifold of functions r\ = rf(t) = (y(t), y(t -(- o))from J to F<2),wAere y(t) €JV(T), and let rf[Q\ — y[Q]. If € d A, r/ie/i W^ inducesa partial dichotomy for (jVd, \\rf(i)\\, rjd[Q]) with 0° = max (00, «) andM0 — M0(f, S). If3f^d^ A, then WD induces a total dichotomy for*V6 with 0° = max%(00 — f, 0) and y0 = y0(«, 6). If, in addition, D is

quasi-full, € d A, and y(t) is a non-PD-solution of Ty = 0, thenIl»rt0ll -^ <x> crs f -» oo.

Exercise 5.1. Prove Theorem 5.3.Theorem 5.4. £e/ e0 > 0, A0 ^ 2c0 and 0 6 A0. Assume (Aj)-

(A4); (B2)/o r e e0 H' ^o» #2 independent of e, |9>Je|B independent of

y, z, and (5.6) Ao/c/s uniformly for large s; (B3Jt0); am/ (B5A0). 7Vie/i WDinduces a total exponential dichotomy for the manifold ^s, defined inTheorem 5.3, with 0° = max (00, 0).

Exercise 5.2. Prove Theorem 5.4.

6. Applications to First Order Systems

As was pointed out in §§ 3-4, the assumptions (A0)-(A4) and (B^) aresatisfied by the operator T associated with (0.1) in the Remark of §3,with Py = y, Qy = y(0). Hence Theorems 4.1-4.3 imply

Theorem 6.1. Let A(t) be a matrix of locally integrable (real- or complex-valued) functions for t 0. Let B, D be Banach spaces in O and let (B, D)be admissible for (0.1) in the sense that for every g(t) e B(Y\ (0.1) has asolution y(t) e D (Y). Let JV denote the set of solutions y(t) of (0.2) and YDbe the set of initial conditions y(0) belonging to solutions y(t) &JV C\ D(Y).(i) Then YD induces a partial dichotomy for (^V, pv(t), y(Q)), where

and A > 0 fixed arbitrarily, (ii) If, in addition, D is quasi-full and z(t) eyT, but z(t) $D(Y), then,'for every A > 0,

(iii) If (4.14) holds, then YD induces an exponential dichotomy for (A^, pv(t),y(G)), where the exponents v, v can be chosen independent of A > 0.(iv) IfB = Ll and D = L°° or D — L0°°, then YD induces a total dichotomyfor./r.

Exercise 6.1. State the consequences of Corollary 4.2 for parts (i)and (iv) of Theorem 6.1.

The next theorem, except for condition (c) of a total dichotomy [cf.(1.9)], is easily deduced from Theorem 6.1 by virtue of the Remarkconcerning (5.3), (5.4). The entire theorem, however, will be deducedfrom Theorems 5.1 and 5.2.

Theorem 6.2. Let A(t) be a matrix of locally integrable, real- or complex-valued functions satisfying (5.3) or (5.4). Let B, D be Banach spaces in Oand (B, D) admissible for (0.1), and^V, YD as in Theorem 6.1. Then YD

induces a total dichotomy jor Jf. If, in addition, D is quasi-full andz(t) e^,but z(t)$D(Y), then

Finally, if

then YD induces a total exponential dichotomy for N'.Proof. It suffices to verify the conditions of Theorem 5.1 and/or 5.2

for the operator T, with Py = y and y[Q] = y(0), associated with (0.1).Conditions (A^-fAJ have already been verified. As pointed out before,

(B^) holds for all 00 > 0, e > 0 with K - 1 and arbitrary <j>S€(t). Choosing(jpse(t) = e~lhtf(t) in Remark 2 preceding Theorem 4.1 shows that (B2e)holds for all 00 > 0, e > 0 with K2 = 1 (since K = 1 and jyjj = 1) and

independent of s and y(t), z(t). Condition (B3<5) is trivial since y = Py.The Remark preceding Theorem 5.1 shows that (5.3) or (5.4) impliescondition (B4A). Thus Theorem 5.1 is applicable and gives the statementsconcerning a total dichotomy and (6.2).

In order to apply Theorem 5.2, the condition (5.6), which is

must be verified. But is readily seen that this is equivalent to (6.3); cf.,e.g., the beginning of the proof of Theorem 4.2. Hence the proof ofTheorem 6.2 is complete.

The next theorem is the main result on total dichotomies for solutionsof (0.2) and admissibility for (0.1). For the sake of brevity, let Yp andYXO denote the subspace YD of Y, when D — Lp and D = L0°°, respec-tively; i.e., Yv and y^o denote the set of initial points y(G) e Y ofsolutions y(t) of (0.2) of class Lv, 1 p oo, and L0™, respectively.

Theorem 6.3. Let A(t) be a matrix of locally integrable (real- or complex-valued) functions oj' t ^ 0 andi/V the set of solutions o/(0.2). Then thereexists a subspace WQ of Y = W which induces a total dichotomy forC^> IIKOII, 2/(0)) if and only if (L\ L°°) and/or (L\ L0°°) is admissible for(0.1). In this case, Y^o c: WQandboth Y and Y^^ induce total dichotomiesfor(JV,\\y(t)\\,ym.

Exercise 6.2. The proof of Theorem 6.3 will depend on Lemma 2.3.Without using this lemma, prove the parts of Theorem 6.3 which do notinvolve L0°° (an analogous result is applicable even if Y = W is notfinite dimensional and Lemma 2.3 is not applicable).


Proof. If (L1, £,0°°), hence (L1, Z,00), is admissible for (0.1), then Yx andYMQ induce total dichotomies for^F by (iv) in Theorem 6.1.

Conversely, if there exists a subspace W^ of Y which induces a totaldichotomy for./f", then Yô c ô a°d ^0 induces a total dichotomy for*V by Corollary 2.2. Thus, in order to complete the proof, it is sufficientto show that if yô induces a total dichotomy for jV, then (L1, L°°) or(L1, L0°°) is admissible for (0.1).

The proof of this part will use the easier ("only if") portion of Theorem2.1. Let Z be a subspace of Y complementary to Y^^, so that Y is thedirect sum YMO © Z (e.g., if Y is Euclidean, let Z be the orthogonalcomplement of Y^ in 7). Let P0 be the projection of Yon Y^ annihilat-ing Z, and Pj = 1 — P0 the projection of Y on Z annihilating Y&Q. LetU(i) be the fundamental matrix of (0.2) satisfying C/(0) = /, and G(t, s)the matrix function

Then, by Theorem 2.1, there exists a constant K such that

Let g(/) be an arbitrary element of L\Y). In order to show that (0.1)has an L0°°-solution, put

The integral is absolutely convergent [in view of (6.5) and g(t) e Ll( Y)]and defines a solution of (0.1). Also ||y(OII ~ K\g\\\ in particular,rtOeL-cr).

It will be verified that y(t) in (6.6) is in VCF). By (6.4) and (6.5),

The last integral tends to 0 as t -> oo. For fixed s, the solutionU^PoU-^g^s) of (0.2) tends to 0 as t -> oo, since its initial valueP0U-\s)g(s) E Y^. Furthermore,

Thus Lebesgue's term-by-term integration theorem (with majorizedconvergence) shows that the first integral tends to 0 as / -> oo.Consequently, y(t) e Z,0°°( Y). This proves the theorem.

The main result on total exponential dichotomies is the followinganalogue of Theorem 6.3.

Theorem 6.4. Let A(t) be a matrix of locally integrable (real- or complex-valued) functions oft>Q and^f the set of solutions o/(0.2). Then thereexists a subspace W0 of Y which induces a total exponential dichotomy for(-f, llyOII, y(0)) // and only if (L*, L") for (p, q) » (1, oo) and for some(p,q) j± (1, oo), 1 p < q co, are admissible for (0.1). In this case,W0 = Yv = Y^, and (L\ L0°°) and (Lp, I") are admissible for all (p,q),where 1 p q oo.

Proof. If (I1, L*) and (Lv, LQ) for some (p, q) (1, oo) and 1 p,q ^ oo are admissible for (0.1), then Y^ = YQ = YXO induces a totalexponential dichotomy for^T by (iv) in Theorem 6.1 and by Theorem 4.4.

Conversely, let a subspace W0 of Y induce a total exponential dichotomyfor-xT. Let W^ be a subspace of Y complementary to W9 and let G(t, s)be the Green's matrix (6.4) defined in terms of projections P0, Pl of Yonto WQ, Wv annihilating W^ W0, respectively. Then, by Theorem 2.1,

and some constants AT, v > 0. Let #(f) e Lp( Y), where 1 /? oo. Thenthe integral in (6.6) is absolutely convergent and defines a solution of (0.1).Thus it remains to show that y(t) e L"( Y) for p ^ q oo.

Consider first the case that p = 1. As in the last proof, it is easy to seethat y(f) E L0°°( Y). Also y(t) e Ll( F) for

Hence y(t) e L"(Y) for 1 < ^ oo. Also, if p = oo, then y(t) e Lco(7)for

Consider 1 0, a + 0 = 1, and <p(t) = ||g(0||.Then repeated applications of Holder's inequality show that ||y(/)||« is atmost

Hence (J||y(0ir dt)llq is finite and is at most

cf. (6.8). Consequently, y(t) e L"( Y) for p < q < oo. Since the majorantremains finite as q^-p or ^->oo, it follows that y(t)ELQ(Y) forp q oo. This proves the theorem.

Since (6.7) is a necessary condition for W0 to induce a total exponentialdichotomy, it is possible to conclude in this case that many pairs (B, D)are admissible. Such a result is given by the following:

Exercise 6.3. Let A (/) be a matrix of locally integrable functions fort |£ 0 and jV° the set of solutions of (0.2). Let B, D be Banach spaces iny such that

Let there exist a subspace WQ of F which induces a total exponentialdichotomy for^T. Then (5, D) is admissible for (0.1),

7. Applications to Higher Order Systems

In dealing with (0.3), it will be assumed that U is a finite dimensionalvector space with elements u and norm ||«]|. 0 is the correspondingBanach space of linear operators P0 of U into itself with the norm \\P0§ =sup \\P0u\\ for ||w|| = 1. P0 can, of course, be considered to be a matrix if acoordinate system is fixed on U.

A function «(/) from a /-interval to (/will be called an (m + l)st integral,m*z 1, if u(t) has continuous derivatives, u = M(O), u'(i),..., «<w)(f) suchthat M<m)(/) is absolutely continuous [with a derivative M<m+1)(/) almosteverywhere].

In this section, let Y = U(m+1) = U x • • • x U and if y = («(0),. ..,w<m)) 6 r, put

If/(O e L(£7) and /^(f) 6 L (0) for fc = 0,... , m, then u(t) is a solutionof (0.3) if it is an (m + l)st integral and y(i) = (w(0)(/),..., w(m)(0)satisfies the first order system (0.1) corresponding to g(t) — (0,..., 0,/(/)),and the m + 1 equations: wo)/ = wo+1) fory = 0,. . . , m — 1 and (0.3).It is clear that in this identification of (0.3) with (0.1), (7.1) implies thatthe norm of A (t) e Y satisfies

Correspondingly, the inequality (0.5) is applicable.

In dealing with (0.3), 7* is an operator from L( Y) to L(U). The domainD(r) of T is the set of y = («(/),«'(/),..., u(m}(tj), where u(t) is an(m 4- l)st integral for / 0, and u,..., t/(TO) are its derivatives. Also,f=Tyis defined by (0.3), Py(0 = u(t\ and y[0] = y(0); thus 7 = JP,F = U; cf. § 3. Finally, jV = ^T(r) is the set of y(t) where «(/) = Py(t)is a solution of the homogeneous equation (0.4); i.e., JV* is the set ofsolutions y(t) of (0.2).

The following preliminary lemma has nothing to do with differentialequations.

Lemma 7.1. Let m be a positive integer. Then there exists a constantcm > 0 with the property that if u(t) e U is an (m + l)st integral on aninterval r < f ! | T + A, A>0 , then its derivatives satisfy

for k = 0, 1, . . . , m.Proof. It is sufficient to verify the corresponding inequality

when (jp(t) is a real-valued function on [T, T + A] which is an (m + l)stintegral. If T /! T + A, let M* e U* be chosen so that ||u*|| = 1 anthe "scalar" product <Mw(fj), M*> = HM^I)!!- If <p(0 = Re (w(0, «*>,then (7.4) at / = ^ implies (7.3) at / = /x.

In order to prove (7.4), let <p(t) = /?(0 + v(0> where /)(/) is the poly-nomial of degree m satisfying p(t) = <p(t) for / = T -f yA/m,y = 0 , . . . , m;i.e.,

It is clear that there is a constant cm > 0 such that

and A: = 0 , . . . , m. Since y(0 — 9>(0 — XO vanishes at the m + 1points / = T +y"A/m,y = 0 , . . . , m, it follows that there is a /-valuer' = /»' such that v<;c)(/') = 0. Hence, for T < t < T + A,

Integrating over [T, T + A] gives

so that, by induction, (7.6) implies that

Since />(f) is a polynomial of degree m, y(m+1)(t) — <p(m+l}(t). Hencey =p + y> and (7.5) and (7.7) give (7.4).

Lemma 7.2. Let Pk(t) eZ,(#), k = 0,..., m, and let a = a(?0) be apositive integer satisfying

Lef /(O G L(U) and u = w(r) a solution of (0.3). T/ien, for 0 < j < /0,5 f ^ s + 1, #«</ A: = 0 , . . . , m,

wA^re C1 = 2eacmaw a«rf C2 = 3e".Remark. Note that /„ > 0 can be chosen arbitrarily and a (hence

C1, C2) independent of t0 ifPk(t) E Af (#), k = 0, . . . , m and

cf. (1.6) and (7.8).Proof. Let j 0. Then (7.8) implies that there exists an integer

i = i(s), 0 / < a, such that

By (0.3), ||«<TO+1)(f)|| ^ S ||Pfc(OII • ||«W(OII + II/WII- If the relations (7.3)are inserted into this inequality, an integration of the resulting inequalityover [T, T -f A] gives, by (7.11),

Thus, if y = (M, w(1),..., M<W)), (7.3) implies that

Therefore, (7.9) follows from (7.2), (7.8), and the corresponding inequality(0.5) with s replaced by T.

Corollary 7.1. Under the conditions of Lemma 7.2,

for 1 / < /0 ~ 1 <""* A: = 0,.. ., /«.This follows by integrating (7.9) with respect to s for / — 1 5j s /.

Corollary 7.2. Assume the conditions of Lemma 7.2. Letf^fyf^t),...be functions in L(U) and u = un(i) a solution 0/(0.3) whenf=/„. Supposethat f= lim/n, u = lim u exist in L(U). Then the function u = u(t)(up to an equivalence modulo a null set) is a solution of (0.3) and w|f\t) -*•u(k)(t), n ->• oo, uniformly on bounded subsets of J for k = 0, . . . , m.

This follows by replacing M,/ in (7.13) by uv — uq,fp — fq for p, q =1,2, . . . .

Theorem 7.1. Let Pk(t) e M(0)for k = 0, . . . , m and

Suppose that (B, D) is admissible for (0.3); i.e., that for every f ( t ) e £(£/),(0.3) /KW a solution u(t) e D(U). Let YD be the set of initial conditionsy(Q) = (w(0), w'(O),. . . , M

(m>(0)) o/ 50/u/ioiu u(t) e /)(t/) o/ (0.4). ThenYD induces a total dichotomy for JV (with 6° = 0). If, in addition, D is

quasi-full and u(t) $ D(U) is a solution o/(0.4), then \\y(t)\\ -*• oo as t -»• oo.The conditions (7.14) on Pk(t) are satisfied if, e.g., Pk(t) 6 L°°(f7).

Note that (7.14) is not required for k = 0. The condition that (£, D) isadmissible for (0.3) is the condition that (B, D) be P-admissible for theoperator T described at the beginning of this section. It does not corre-spond to the admissibility of (B, D) for the corresponding linear system(0.1) for we do not consider arbitrary g 6 B(Y) but only g of the form£-<0,... fO,/),/e *(£/)•

Proof. This theorem will be proved by verifying the applicability ofTheorem 5.1. (Aj) follows from (0.5) in view of the identification of (0.1)and (0.3); (A2) follows from Corollary 7.2; (A3) is an explicit assumptionof the theorem; (A4) is trivial since W = Y is finite dimensional.

In order to verify (B2) , let e 1 and u(t), v(t) solutions of (0.4). Lety>(t) be a non-negative function on — oo < t < oo of class Cm whichvanishes except on [0,1 ] and satisfies

Put y,e(0 = - 1v(«-1(? — J)) and

Then for k = 1,.. . , m + 1,

where CM is the binomial coefficient £! \j\ (k —j)\ It follows that u = u^is a solution of (0.3), where/is

and Pm+i(0 = / is the identity operator.Since yS€(t) = 0 for / < s or / j + e, it follows that yx(/) = z(t) for

0 <; f < s and y^f) = y(t) for / j> j + e if y = (u,..., M(m))> « =(t;,..., »<«>), yi = (ii l f . . . , WI<

TO>). Thus, if 5 > 0, yi(0) - z(0) and Pfc =«! ED(U). There is a constant c > 0 such that \^(t)\ ^ chs€(t)/e forA: = 0,. . . , m and s, t 0 and c 1. Thus if Ty^ =/1?

Thus (B2e) with e 1 holds for arbitrary 00 > 0 and Kz = 1 when

By virtue of (7.14), <psf(i) e B and

Lemma 7.2 with/= 0, and the Remark following it show that (B3<5)holds for d 1 if K3(S) = Cl(\ + /wa). Finally, (B4A) is a consequenceof (0.5) and (7.2), since P0(f), • . . , Pm(t) e M(#). Thus Theorem 7.1follows from Theorem 5.1.

Theorems 6.3 and 7.1 have a curious consequence. Let (0.2) be thefirst order system obtained in the standard way [as before (7.2)] from(0.4). Then we can consider (0.1) without the restriction that g is of theform (0,..., O,/) and we obtain from Theorem 6.3:

Corollary 7.3. Let the conditions of Theorem 7.1 hold. Then (L\Lm)and/or (L1, L0°°) is admissible for (0.3) // and only if it is admissible for(0.1).

For example, if Pk(t) e L°°(£7), k = 0,1, . . . ,m, and if some pair(B, D) is admissible for (0.3), then (L1, L0°°) is admissible for (0.1) [andfor (0.3)].

Theorem 7.2. Let the assumptions of Theorem 7.1 hold. In addition,assume either the condition

or the condition

uniformly for large s

as A -*• oo. Then YD induces a total exponential dichotomy for Jf (with6° = 0).

When |AOAIWA -* 0 as A -* oo and Pk(t) e L°°(£/) for k = 1,..., m,then (7.23) holds, since \hOA\B ^ (1 + A) \hol\B.

Proof. This is a consequence of Theorem 5.2. It suffices to verify theanalogue of condition (5.6). By (7.21), (5.6) follows either from (7.14),(7.22) or from (7.23).

Exercise 7.1. This exercise concerns real second order equations

for t 0. Let U be the (Banach) space of real numbers; W the productspace U(z} as U x U with norm ||H>1| = max (|w*|, iw*|) if w — (w1, w2);and Tthe product space £/(4) with norm ||y|| = max (jj^j, |y*|, jy8!, jy*|) ify = (yl> y2. y3* y4)- For 8 > 0, let JTt denote the linear manifold offunctions y\f) - (u(t)t u'(i), u(t + <5), «'(' + <$)) e y for r 0 wherew(0 is a solution of (7.25). Let the "initial value" of y*(f) be y*{0] = («(0),w'(0)) 6 IT. Assume that p(t) is a real-valued function in /: t 0 whichsatisfies

Let (0 be a real-valued element of L for which there exists a constant Csatisfying the one-sided inequality

Finally, suppose that (5, D) is admissible for (7.24); i.e., if/(f) 6 5, then(7.24) has a solution «(/) 6 Z>. Let be the 0-, 1-, or 2-dimensionalmanifold of initial conditions (u(0), «'(0)) of D- solutions of (7.25). (a) Then,for sufficiently small 6 > 0, WD induces a total dichotomy for./fV (b) If,in addition, D is a. quasi-full and u(t) $ D is a solution of (7.25), thenHsAOII -*• oo as 7 -> oo. (c) If, in addition to the conditions of (a), either(7.22) or

uniformly for large s

as A -»• oo, then WD induces a total exponential dichotomy for «/f"d ifd > 0 is sufficiently small.

8. P(B, Z))-Manifolds

The main role of assumptions (A3), (A4) is in the proof of Lemma 3.2.An analogous lemma follows from a similar pair of assumptions (A5), (A7).The notation in this section is the same as in §§ 3-4.

(A^ If Ty(i) = /(/) and/(f) — 0 for large t, then there exists a uniquesolution yao(r) of Ty — 0 such that VooO) = y(0 for large t.

Definition. P(B, D)-Manifold. Assume (A5). Let B, D be Banachspaces in &~ and B^ be defined as in (vi) of § 1. A submanifold X of WDis called a P(B, D)-manifold if there exists a constant C3 with the propertythat, for/(0 6 BJJF), there exists a PD-solution y(t) of Ty =/such that\Py\o ^ C3\f\B, and yj/) [defined by (A5)] satisfies ^[0] e X. [When(A0) is assumed, then even without the assumption y „,[()] E X, it followsthatyJO]60^]

Exercise 8.1. Assume (A5). Show that there exists a P(B, Z))-manifoldif and only if (B^, D) is P-admissible.

(Ag) X is a P(5, ZO-manifold.Lemma 8.1. Assume (Ax) [or (A/)], (A2), (A5), and (A6). Le/ /, y, y^

be as in the definition of a P(B, D)-manifold. Then there exists a constantC» such that Mm £C»\f\B-

This is a consequence of (3.3) and (3.4) [or (3.3')], (3.5^ and (1.5).(A7) X is a P(B, Z>)-subspace, i.e., a closed P(B, J9)-manifold.Lemma 8.2. Assume (Ax) [or (A/)], (A2), (A5) and (A7). Let A > 1,

/e500(F), y(t) any PD-solution of Ty=f with ym[0]eX, ||y[0]|| <hd(X, y[0]). Then there exist constants C5, C^ such that (3.7) holds (withA = 1 permitted ify[0] = 0).

The proof is identical to that of Lemma 3.4. If, in addition, to (Aj) [or(Ai')]-(A4), (A0) and (A5) are assumed, then Lemma 3.4 is contained inLemma 8.2, for X = WD is a P(B, Z))-subspace in this case.

Remark. In view of the uses of Lemmas 3.3 and 3.4, it follows that(A3), (A4) can be replaced by assumptions (A5), (A6), (A7) in the theoremsof §§ 4-5 (and their applications in §§ 6-7) if conclusions of the type " WDinduces a ... dichotomy" are replaced by "X induces a ... dichotomy."

When (A4) holds (e.g., if W is finite dimensional), then (A5) and (A«)imply, by Exercise 8.1, that (A4) holds with B replaced by I?*,. The pointin the Remark above, however, is that a subspace X [which can be smallerthan WD] can replace WD in the conclusions of the theorems of §§ 4-5.

PART II. ADJOINT EQUATIONS

9. Associate Spaces

In this part of the chapter, the concept of an associate space will beneeded. Let «^~# denote the set of normed spaces O e^" satisfying theadditional condition: (d#) if<p(t) e , s 0, andy(t) = <p(t + s)for t 0,then y(t) e O.

Hypothesis. In the remainder of this chapter, it is assumed that B, Dare Banach spaces in «^#.

Let O be a Banach space in «^"#. Let O' be the set of (equivalence classesmodulo null sets of) real-valued measurable functions y>(f), t 0, such

that for all <p(/) G O, <p(0y(0 e L1. It is easy to see that there exists aconstant a = a(y) satisfying

Otherwise, there are elements <pn e O such that |9>J,j> ^ 2~n, 9?n(0v(0 = 0,r°°

and <pn(f)v(0<# ^ 1- Tnen 9>(0 = S <pn(0 E O, but, by Lebesgue'sJo

theorem on monotone convergence,

Let the least constant a satisfying (9.0) be denoted by \y>\Q>, so that

This is clearly equivalent to

Lemma 9.1. Let O 6«^"# ^e a Banach space. Then O', wrt/i the normIvU' /or V G ^'» ^ a Banach space in 3~ (in fact, in &~#} and is quasi-full.

Exercise 9.1. Prove this lemma. The assumption 0 6 «^"#, rather than e .7", assures that 4>' 6 ".

It is clear that $' is isomorphic and isometric to a subspace of the dualspace O* of O.

Exercise 9.2. Let 1 p ^ ao,\fp+ \fq = 1. Show that the associatespace of Lp is La.

Lemma 9.2. Lef O e«^"# />e a Banach space and Y a (finite-dimensional)Banach space with Y* its dual space. Let y*(t)eL(Y*). Then y*(t)eO'(y*) if and only if there exists a constant a = a(y*) such that for everyy(t) e 000( y), (y(t), y*(/)> JL1, fl«^

/« /Aw caje, ?fo 7ea5? constant a satisfying (9.2) is |y*U<(r«) = |y*|®'.Recall that (y, y*} denotes the pairing ("scalar product") of elements

y e 7, y* 6 Y*. In the case under consideration (when Y is finite dimen-sional), the nontrivial part of this lemma can be reduced to the 1-dimen-sional case by the introduction of suitable bases in 7, Y* and examiningthe components of y*(t).

Exercise 93. Prove Lemma 9.2.

10. The Operator 7"

This part of the chapter concerns an operator T as in § 3, an associateoperator

and the corresponding null space jV(T').

In dealing with the pair 7* and 7", the following will be assumed:(C0) T is a linear operator from L(Y) to L(U\ i.e., F = U; the ele

ments y(t) e ^(T) are continuous (so as to avoid ambiguities on null sets);and y[0] = y(0), so W = Y. Correspondingly, T' is a linear operatorfrom L(Y*)to L(U*)\ the elements y*(t)e@(T') are continuous withy*[0] = y*(0); and u*(t) = P'y*(t) is a linear operator from L(Y*) toL(U*) with ^(P') « .0(7").

(C^ For each / ^ 0, there exists a bounded bilinear form Vt(y, y*) ony x K* such that for y(t) e (r), y*(r) e 0(7"), w(0 = /WO. u*(t) =

P'y*(t), and for 0 a < T < oo, the following "Green's relation" holds:

For t 0, ft(t) denotes a number satisfying(10.4) | Vt(y, y*)| < /?(/) |y| • |y*|| for all y e 7, y* 6 7*.

Note that (10.3) implies that Vt(y(i), y*(t)) is constant on any intervalwhere Ty = 0, T'y* = 0. In particular, Vt(y(t), y*(f)) is constant on/ for y(t) e (F), y*(0 6 ^(T).

Definition. If A' is a manifold in Y, let Xv be the subspace of Y*defined by Xv — {y*: V^y, y*) = 0 for all y e X}. Correspondingly, ifX* is a manifold in Y*, let X*v denote the subspace of Y defined byX*v = {y: y0(y, y*) = 0 for all y* E X*}.

Assumption (AJ* or (Bn«)* will mean the analogue of (An) or (Bne)with T, Y, F = U, P, B, D, WD = YD, constants Cf,... replaced byT', Y*, U*, P', D', B', Y&, constants C,*,..., respectively, whereB', D' are the associate spaces of B, D. Note the replacement of theordered pair (B, D) by (D', B').

11. Individual Dichotomies

The next theorem involves the following notion: A set S of real-valuedfunctions <p(t) e L will be called small at oo if every <p(t) e 2 is small atoo in the sense that

Theorem 11.1. Assume (Aj) [or (A{)]-(A4); (B30<5); (B3<5)*, (B4A)*;and (C0), (Cx) with a /3(/) e M m (10.4). Suppose also that f(t) or B' or Dis small at oo. Then ¥%., induces an individual partial dichotomy forc/nn ny*«u).

It will be clear from the proof that 00, A/0 in (1.7) of condition (a) ofan individual partial dichotomy can be chosen independent of y*(t).

The assumption that /?(/) or B' or D is small at oo will be used only inthe derivation of the condition (a) of an individual partial dichotomyinvolving y*(t) e ^V(T') with P'y* e B'. At the cost of allowing M0 todepend on y* in condition (a) and of making the additional hypothesis(A6) of § 8, the condition of "smallness at oo" will be eliminated in Theorem11.3.

In view of Theorem 12.3, it follows that the main point of the theoremsof this section is that no assumption of the type (B2e)* occurs. Theimportance of this can be seen by noting that assumption (7.14) occurs inTheorem 7.1 only to insure (B2c) for the operator T there.

Proof of Theorem 11.1.

Condition (a). The condition @(t) e M implies that, for 6 > 0,

The condition that p(t) or B' or D is small at oo will be used as follows:if three functions £(/), \\y(t)l ||y*(f)ll are in M[e.g., $(t)e M, y(t)e D(Y),y*(t) e B'(Y*); cf. (9.1)] and at least one is small at oo, then

In order to see this, note that if <p(t) ^ 0 is integrable on an intervalrt+i

[t, t + 1 ], the measure of the set of 5-values where q>(s) > 3 <p(r) dr isJt

less than 1/3. If this remark is applied to <p = ft, ||y||, \\y*\\, it follows thatfor any / ^ 0, there is a common t-value T e [t, t + 1] satisfying

By the assumption oo as n -»• oo. Hence (11.2) follows.

Let y*(t) E JV(T'} with P'y* E B'(U*). Let s > 0 and/(f) any elementof B(U) with compact support on t 5 + 3d. By Lemma 3.2, Ty —fhas a PD-solution y(t) satisfying (3.5), hence \Py\D < C3\f\B. By

(B30<5) and (B32<5)*, it is seen that y(t)eD(Y), y*(t) e D'(Y*). SinceT'y* = 0 and Ty = /, the Green relation (10.3) gives

where u*(t) = P'y*(i). The left side is independent of a s + 3<5 andof large r [since f(t) = 0 for t s + 3d and large t]. Thus (10.4) and(11.2) give for Q <> a < s + 3d,

The argument leading to (11.2) shows that for a suitable choice of a,s<>a£s + 3d,

The first factor on the right is at most (3S)~lft0(3S) by (11.1); the secondfact is at most (36)~l \h9iM\jy \hs3dy\D by (1.5), hence, at most

by (B30<5). Since \Py\D ^ C3 |/|B, the right side of (11.4) can be replacedby K' \f\B ||y*(<r)||, where K' — K'(S) is a constant and a is some pointof s a s + 3d.

Since f(t) is an arbitrary element of B(U) with compact support on/ ^ s -f 3d, Lemma 9.2 implies that

or by (B3<5)«,

Arguments involving (B4A)* similar to those used in the proof of Theoren5.1 give condition (a) of a partial dichotomy for(^T(r'), |jy*(r)ID-

Condition (b). Let z*(/) 6 rf(T\ P'z* = v*, and y* 5'( r*). Let/(/)be any element of£(U) with support on [0, s] and y(f) a solution ofTy=fsupplied by Lemma 3.2. Then T tZ s, Green's relation, and (10.4) give

The use of (3.5) and the arguments in the derivation of (11.5) show that

for s + d T s + 3d,where K' = AT'(^) is the same as in (11.5).

Since v* $ B'(Y*\ it follows from the fact that B' is quasi-full (Lemma9.1) that \h0iv*\B- ->• oo as s -*• oo. Thus there exists an s0 depending onthe solution z*(t) such that \H0sv*\B. 2C^(G) ||z*(0)|| for s s0. Thenby (11.7)

Using (B3<5)*,

The proof can be completed by the arguments in the proof of Theorem 5.1.Theorem 11.2. Let the conditions of Theorem 6.1 hold. In addition,

assume either

Then ¥%> induces an individual exponential dichotomy for (^V(T'\lly*(OII)> where 6°, Mlt v do not depend on y*(t) in condition (a).

Exercise 11.1. Prove this theorem.The elimination of the "smallness at oo" condition will depend on the

following lemma, which involves assumption (A5) of § 8. In this lemma,the notation of (A6) is used; i.e., if Ty =/and Ty^ =/i, where/,/! vanishfor large t, then y^, yloo are the corresponding solutions suppplied by (A5).

Lemma 11.1. Assume (A^-^Aj) and (CoMQ). Let y*(t) be a P'B'-solution of T'y* = 0. Then there exist constants C3 and Cso, depending ony*(0), such that for any f(t) 6 5w(t/), Ty =/ has a PD-solution y(t)satisfying (3.5) and V0(y„,($), y*(0)) = 0; i.e., if{y*(G)} is the l-dimensionalmanifold in Y* spanned by y*(0), then YD n {y*(0)}F is a PD-manifoldfor (3.1).

Proof. We can suppose that there is a .PD-solution y0(t) of Ty — 0satisfying a = K0(y0(0), y*(0)) 0. For if not, Lemma 11.1 follows fromLemma 3.2. Let y(t) be the solution of Ty =/ supplied by Lemma 3.2and put

Then yx(0 is a PD-solution of Tyl =/satisfying F0(y100(0), y*(0)) = 0.By Green's relation and T'y* — 0,

if M* = P'y*. Since/vanishes for large T, (A6) implies that yt(/) = y100(0for large ^, so that Vt(yv(t\ y*(t)) = Ft(ylao(0, y*(0) for large r. But the

latter expression does not depend on t (since 7V100 = 0, T'y* = 0) andis therefore VQ(y^(Q\ y*(0)) = 0. Thus (11.11) shows that

Thus, by (11.10),

The right side is at most \f\B(\P'y*\B- + /S(0)C30 ||y*(0)||), by (10.4)andLemma 3.2. Hence (11.10) shows that

\fyi\D ^ \py\D + \f\s(\P'y*\ir + NP)C» \\y*(m*-1 \Py0\D-By (3.5^ in Lemma 3.2, \Py\D < C3 \f\B. Thus the analogue of (3.5j)holds with y replaced by y^ and C3 replaced by the constant C3 +(\P'y*\ir + flOJCw lly*^)*-1 \PyQ\D. The analogue of (3.5,) follows asin Lemma 8.1.

Theorem 11.3. In Theorems 11.1 or 11.2, replace the assumption that"/3(f) or 5' or Z> w small at oo" iry Me assumption (A5). TTzen r/re conclusionsof these theorems remain valid.

Exercise 11.2. Prove this theorem. In view of the remarks followingTheorem 11.1, only condition (a) need be considered. The proof ofcondition (a) here is similar to (but simpler than) the proof of the corre-sponding condition in Theorem 11.1 or 11.2. The solution of Ty = fsupplied by Lemma 11.1 is used in place of that given by Lemma 3.2.

12. P'-Admissible Spaces for T

The object of this section is to show that, under suitable conditions, if(B, D) is -admissible for T, then (/V, B') or (£>', B') is P'-admissible forr.

Lemma 12.1 Assume (AÂg), (A5)-(A6); (A5)*; and (CoHQ).[In particular, X is the P(B, D)-manifold in (Ae).] Iff*(t) e DÛ*) andy*(t) is a solution of T'y* =/* with yÔ) e Xv, then y*(t) is a P'B'-solution and

where C3, C30 are the constants in the definition of a P(B, D)-manifold Xand Lemma 8.1. In particular Xv c y*,.

Proof. Let/ e BÛ) and ?/(/) the solution of 7y =/supplied by (A6),so that ya)(0) 6 A- and (3.5) holds. For large t, Vt(y(t\ y*(t)) = ^(yj/),y«,*(0) = ô(^(0), ôo*(0)) = 0- Thus Green's formula gives

where the right side is at most \Py\D \f*\D. + 0(0) |jy(0)|| • ||y*(0)|| ^l/la(Q I/*!D- + 0(0)C30 lly*(0)||) by (3.5). Hence the assertion followsfrom Lemma 9.2.

Let y* e Y*. Then (Q) implies the existence of a unique x* e Y* suchthat the linear functional K0(y, y*) on y is representable in the usualpairing of y, 7* as K0(y, y*) = (y, **> for all y e Y and ||z*|| ^ 0(0) ||y*||;i.e., there is a unique linear map x* = Sy* of 7* into itself satisfying

In this section, the following will be assumed:(Ca) The (unique) bounded linear map S: Y* -> Y* defined by (12.2) is

onto (and hence has a unique inverse S~* defined on all of Y*).It is clear that for a manifold X <= y, we have JTK = S'-1*'0, where A'0

is the usual annihilator of X\ i.e., A'0 = {y* e K*:(y,y*> = 0 for allyeJT}.

Lemma 12.2. Assume (AÂg); (A5)*; am/(CoMCa). Letf*(i)eDÛ*) and y*(/) a jo/wf/o* o/ Ty* =/* wiYA yw*(0)e YD

V. Thend(YD

v,y*(fy) ^ C4 US-1!! ' \f*\jy, where C4 is the constant in (3.6) ofLemma 3.3 a/w/ US"1!! is the norm of the operator S~lfrom Y* to Y*,

Proof. Let y(r) be a P/)-solution of Ty = 0. Then Vr(y(r), y*(r)) =K-(y(T)» y«*(T)) f°r ârge T- By Green's relation, the last expression is theconstant K0(y(0), yw*(0)) = 0. By Green's formula,

From the inequality l-Pyl^ C4 ||y(0)|| in Lemma 3.3 and from (12.2), itfollows that |<y(0), Sy*(0)>| ^ C4 ||y(0)|| • \f*\D, for all y(0) 6 F,,. ThusSy*(0) considered as a bounded linear functional on YD (i.e., as an elementof the dual space YD*) has a norm not exceeding C4 \f*\D-. Since y#*is the quotient space Y*{ YD° in which the norm of an "element" y* isd(YD°ty*) it follows that d(YD°, Sy*(0)) C4 \f *\D.. Hence F^ =S~1YD° implies that d(y/, y*(ty) ^ \\S~l\\ d(YD°, Sy*(0)), and soLemma 12.2 follows.

Theorem 12.1. Assume (\J-(Aj; (C0HC2); (A6)*; and that T'y = f*has a solution y*(t) for every /*(/) e D^Û*). Then (D^,Br) is P-admissible for T'; in fact, YD

V is a P'(D', B^-subspace for T (withpermissible constants analogous to C3, C30 being C3* = C3 •+ AC4C30 ||S~l\\and C3*0 = AC4 HS^H for any fixed A > 1).

Proof. Let /*(r) e />«/(£/*). Then Tz* =/* has a solution z*(t)which vanishes for large t. [For if z*(f) is any solution of T'z* =/*,then, since z^t) exists by (A5)*, the desired solution is z*(t) — z^t).]

Hence *<»*(*) = 0- In particular, 0 = Zoo*(0) e YDV, so that, by Lemma

12.2, d(YDv, 2*(0)) C4 US-1!! • \f*\D>.

Let X > 1 be fixed and decompose the element z*(0) e F* into «*(0) =*i* + y<>*, where a;,* 6 F/ and \\y0*\\ ^ A (F/, z*(0)). Consequently

Since a?!* e Fj/ c y*, by Lemma 12.1, Tx^*(t) = 0 has a solutionXi*(t) beginning at a^* for t = 0. Let y*(0 = «*(0 — #i*(0» so thatTy* =/* and y^t) = -^^(O- Hence yoo*(0) = -^* E YD

V. Inaddition y*(0) = z*(0) - x^* = y0*. By Lemma 12.1,

Consequently, \P'y*\B' = Q*!/*|JD'» where C3* is the specified constant.This proves the theorem.

Theorem 12.2. Assume (A^-CAg); (CoMQ); and that for everyy0*eY* and f*eD'(U*), T'y* =/* has a solution y*(t) satisfyingy*(Q) = y0*. Then(D',B')isP'-admissibleforT. (Furthermore.permissibleconstants analogous to C3, C30 are C3* = C3 + /3(0)C30C4 || 5^*11 an</C3*0 = C4 H5-1.)

Proof. Let /* 6 !>'(£/*). It must be shown that Ty* = /* has a P'B'-solution y*(t). Let y(/) erf(T) with y(0) G 7 ,. Put

Then ^(OÎÎPyl^l/^^C.IyCO)!!-!/*!^ by Lemma 3.3. Inother words (p(y) is a bounded linear functional on YD with norm ^Ct\f*\D- which, therefore, has an extension to Y with the same norm.Consequently, there is an element x0* e Y* such that \\x0*\\ ^ C4 \f*\yand (p(y) = (y, x0*) for all y e F. By (Cjj), y0* = ^~ô* exists, so that

for all y e F and

By assumption, Ty* —f* has a solution y*(f) satisfying y*(0) == y0*.Let/eB^C/), y(t) a PD-solution of Ty =/supplied by Lemma 3.2.

By Green's formula applied to ^^(0 and y*(f)»


for large t. Since y(t) '= yn(t) for large t, it follows from (12.3), (12.4) that,for large /,

Thus Green's formula applied to y(i) and y*(r) gives

Consequently, (3.5) and (12.5) imply that

By Lemma 9.2, P'y* e B' and |P'y*U ^ (C3 + m^C* \\S^\\) \ f * \ f f .In view of (12.5), this proves the theorem.

The usefulness of theorems like Theorems 12.1, 12.2 will be illustratedby an application of Theorem 12.1 and of Theorem 11.3 with T, T'interchanged. Note that D' = (Ax>)'» so that the second associate spaceD" = (/>')' of D is the same as (Z>7)'.

Theorem 12.3. Assume the conditions of'Theorem 12.1; (B^d}*; (B3$)for a fixed 6 > 0 with D replaced by D"; (B4A); |8(r) e M in (10.4).Then YD" induces an individual partial dichotomy for (T), and \\y(t)\\ -> ooas t -> oo ify(f) is a non-PD"-solution of Ty = 0. If, in addition,

then YD' induces an individual exponential dichotomy for ^V(T).If the condition "0(0 or B" or D' is small at oo" is assumed, then the

constants in condition (a) of the dichotomies do not depend on the solutiony(i) involved.

Exercise 12.1. Verify Theorem 12.3.

13. Applications to Differential Equations

The systems formally adjoint to (0.1), (0.2) are

where A*(i) is the complex conjugate transpose of A(t); cf. § IV 7. LetT be the operator associated with (0.1) with Py(t) = y(t) and y[0] = y(0)and T' is the negative of the corresponding operator associated with (13.1)

[i.e., T'y* = —(y*' + A*(t)y*) = g*], then T, T are associate operatorsin the sense of § 10. The corresponding Green identity is

as can be seen by differentiating with respect to r. Thus Vt(y, y*) —(y, y*>. Clearly, (C0)-(C2) hold [with £(/) = 1 and Sy* = y*] andTheorem 12.2 implies

Theorem 13.1 Let A(t) be a matrix of locally integrable functions fort ^ 0. Suppose that (B, D) is admissible for (0.1). Then (D\ B') is admis-sible for (13.1).

Thus the theorems of § 6 become applicable to (13.1).In order to consider the systems adjoint to (0.3) and (0.4), suppose that

Pt(t) is a Arth integral (i.e., has k — 1 absolutely continuous derivatives).The equations formally adjoint to (0.3), (0.4) are

where

an asterisk on a matrix denotes complex conjugate transposition, andindices in parentheses denote differentiation; cf. § IV 8(viii). For y =(M(O), M(I), . . . , M(TO)) e Y = U(m+» and y* = ( w * < ° > , . . . , M*(m>) e Y*, put

where Pm+i(t) = /is the identity operator. This is a bilinear form in y, y*.The following Green's formula is readily verified

if w(0, u*(t) are solutions of (0.3), (13.4), respectively.A rearrangement of the sums in (13.7) shows that

Thus, if y* = (v* (0),..., y*<m>) e 7* is arbitrary, the m + 1 equations

can be solved recursively for k = m, m — 1 , . . . , 0 since Pm+l = /.For t = 0, this implies the analogue of condition (C2) in § 12.

Let T', P' be the operators associated with (13.4) in the same way thatT, P are associated with (0.3) in § 7. The analogue of (0.5) shows that

(B4A)* holds for T' if Qk* e M(U*) [i.e., QteM(0)]; also, Lemma7.2 shows that the same condition on Qk implies (B3<5)*. Thus Theorem11.3 gives

Theorem 13.2. Let Pk(t)EL(U), k = 0,. . ., m be a kth integral;Qk(t) e M(V)for k = 0, . . ., m; and let there exist a p(t) E M such that(13.7) satisfies (10.4). Let (B, D) be admissible for (0.3). Then Y*K inducesan individual partial dichotomy for ^V(T'); and ifu*(t) is a non-B'-solutiono/(13.5), then ||y *(/)!! -»• oo as t -* oo. If, in addition, (11.9) holds, thenYU> induces an individual exponential dichotomy for ~W(T').

An immediate corollary of Theorem 12.2 is the following:Theorem 13.3. Let Pk(t) e L(U), k = 0,.. ., m, be a kth integral and

let (B, D) be admissible for (0.3). Then (D', B') is admissible for (13.4).Thus, under the appropriate conditions on the coefficients Qk*(t) of

(13.4), the theorems of § 7 become applicable. An application of Theorem12.3 gives individual dichotomies for (T) without a condition of thetype (7.14), but with a condition on Qk*(t).

Theorem 13.4. Let the conditions of Theorem 13.3 hold. Let Pk(t) e M(U),Qk e M(U)for k = 0, 1, . . . , m and let there exist a 0(0 e M such that(13.7) satisfies (10.4). Then YD~ induces an individual partial dichotomyfor ^(T); and if u(t) is a non-D"-solution o/(0.4), then \\y(t)\\ -> oo ast -*• oo. If, in addition, (12.6) holds, then YD~ induces an individual ex-ponential dichotomy for ^{T).

It is clear from (13.7) that a function satisfying (10.4) is

for a suitable constant cm depending only on m. Thus if

holds, then 0(0 6 M, also Qk e M(U) for k = 1 , . . . , m. Thus (13.10)and the condition P0, @0 6 M(0) imply that the conditions of the secondsentence of Theorem 13.4 hold.

Note that if P0 e M(0) and (13.10) hold with j = k also permitted,then £0 6 M(U). But in this case, Pk, Qk e L°°(C7) for k = 1, . . . , m,and Theorem 13.4 is contained in the theorems of §7. (This statementconcerning L°° follows from the fact that if cp(t) is absolutely continuousfor t ^ 0 and <p, <p' e M, then q> e L°°.)

Exercise 13.1. If p(t) is absolutely continuous, the equations formallyadjoint to the real equations (7.24), (7.25) are

The corresponding Green's relation is

where w(0 = (M, M'), w*(t) = (M*, w*')> and

Using the notation of Exercise 7.1, let ^rd* denote the linear set offunctions

where w*(r) is a solution of (13.12). Let the "initial value" of y*\i) bey*s[Q] = (w*(0), w*'(0)) 6 IV*. Let/<0 be a real-valued, absolutely con-tinuous function and q(t) E L such that

and, for some constant C',

Suppose, finally, that (B, D) is admissible for (7.24). (a) Then, for suf-ficiently small 6 > 0, W%- induces a total dichotomy for^,,* with 0° =0. (b) If, in addition, either \hOA\B, -+ oo or ^~l(\hOA\D- + \ha£kp\jy)-+Qas A -*• oo uniformly for large s, then W\. induces an exponential dichot-omy for ./-fa* for sufficiently small <5 > 0.

Exercise 13.2. Let/?(0e^ be a real-valued, absolutely continuousfunction on / and q(t) e L such that there exist constants C, C' satisfying(7.27) and (13.16). Let (B, D) be admissible for (7.24). Then, for small6 > 0, (a*)W% induces an individual partial dichotomy for^*; andif i/*(0 B' is a solution of (13.12), then \\y*\t)\\ -> oo as t -* oo; (a) Jf^induces an individual partial dichotomy for ^V6\ and if «(f) £ /)* is asolution of (7.25), then ||yd(OII -* co as t -»• oo; (b*) if (11.9) holds, thenW^ induces an individual exponential dichotomy for *Wt*\ (b) if(6.3) holds, then WD. induces an individual exponential dichotomy for*.

14. Existence of /V>-Solutions

Lemma 12.1 has the following consequence:Theorem 14.1. Let A(t)eL(0) and let (B, D) be admissible for (0.1).

If(Q.2) has no solution y(t) & 0 in D(Y\ then every solution y*(t) o/(13.2)isinB'(Y*).

For X = YD is {0}, hence YDV = 7* <= r*,; i.e., 7* = 7*-.

In some situations,'it is easy to deduce for (0.2) the existence of solutionsy(t) £ D( 7). Suppose that there is a dichotomy [or exponential dichotomy]for the solutions (e.g., let the theorems of § 6 be applicable). If all solutionsy(t) of (0.2) are in D(Y), then all are bounded [or exponentially small] ast —> oo. The same is then true of det £/(/) if U(t) is a fundamental matrixof (0.2). Since

the integral must be bounded from above [or bounded from above by arnegative constant times t]. Thus if tr A(s) ds does not satisfy this

condition, then not all solutions y(t) of (0.2) are in D( 7).Analogues of Theorem 14.1 and the remarks following it hold if (0.1)

is replaced by (0.3). This will be illustrated for scalar second orderequations, first in the formally self-adjoint form:

Let C denote the Banach space of complex numbers.Theorem 14.2. Let p(t), 1//>(0, ?(0 be locally integrable complex-valued

functions on t ^ 0. Suppose that (B, D) is admissible for (14.1). Theneither (14.2) has a solution u(t) ^ 0 in D(C) or every solution u(t) o/(14.2)is in B'(G).

This is a consequence of Lemma 12.1 for if u(t) is a solution of (14.1)and v(t) a solution of (14.1) with/(/) replaced by g(t), then the correspond-ing Green's relation is

cf. the proof of Theorem 14.3.A variation on the proof of Lemma 12.1 gives a result on non-self-adjoint

equations


Theorem 14.3. Let /?(/), q(t) be locally integrable complex-valuedfunctions on t ^ 0. Suppose that (B, D) is admissible for (14.3). Theneither (14.4) has a solution 0 & M(/) e £(£) or every soiution u(t) o/(14.4)

satisfies u(t) exp p(r) dr e B'(G).JoProof. Suppose no solution u(t) & 0 of (14.4) is in D(G). Let/(/) e

£«,(£). Then, by assumption, (14.3) has a solution u = u(0 6 D(C), whichis necessarily unique and vanishes for large t. Let (14.3) for u = v bewritten as

If u(t) is any solution of (14.4), a corresponding relation holds with£(0/(0 replaced by 0. Thus Green's relation gives

The analogue of inequalities (3.52) for y(t) = (v(t), v'(t)) gives an inequalityof the type

where C depends only on C30, £(0), («(0), w'(0)) and choice of norm onC x C. In view of Lemma 9.2, the assertion follows.

Notes

PART I. The idea that "admissibility" of some pair (B, D) leads to some sort of a"dichotomy" for solutions of the homogeneous equation occurs in a paper of Wintner[18] on a self-adjoint equation of the second order with B = D = L2 (see also Putnam[1] and Hartman [8]) and in a paper by Maizel' [1] dealing with first order systems andB — D — L°°. The main results (§6) on the system (0.2) are due to Massera andSchaffer [1, particularly, IV] and Schaffer [2, VI]. For the first order differentialoperator Ty = y' — A(t)y, these authors have written a series of papers treatingsystematically many of the questions considered in this chapter. An attempt to obtaina unified treatment for (0.1), (0.3), and for other problems (such as those involvingdifference-differential equations) led to the introduction of the more general operatorsT of § 3 in Hartman [25]. The results (§ 7) on solutions of the higher order system(0.3) are due to Hartman [25]. The procedures and arguments of Part I are based onthose of Massera and Schaffer. (The papers of Massera, Schaffer, and Hartman justmentioned deal with the case when dim Y oo.) The classes ", "* of linear spaces of§§1,9 are discussed by Schaffer [1]. The definitions of dichotomies in § 1 vary somewhatfrom those of Massera and Schaffer. The results of § 2 are adapted from discussionsin Massera and Schaffer [1, IV] and Schaffer [2, VI]. The notion of a (B, Z))-manifoldin connection with linear systems (0.1) (with P = I) is used in SchSffer [2, VI].


PART II. Most of this part of the chapter is an adaptation of results and methods ofSchafFer [2, VI] dealing with first order systems (on arbitrary paired Banach spacesY, Y'). The treatment follows that of Hartman [25]. The idea of obtaining an individualdichotomy for the "adjoint" equation as in Theorem 11.1 without using "test" functions[say as supplied by assumptions of the type (B,e)* or (B2«)*] is due to Hartman. (Itshould be mentioned that "associate spaces" have been discussed by Luxemburg andZaanen; see Schaffer [1] for references and pertinent results.)

Chapter XIV

Miscellany on Monotony

This chapter contains miscellaneous results related only by the fact thatone of the main features of either the assumptions, conclusions, or proofsdepends on the notion of "monotony."

Part I deals principally with linear systems of differential equations.Most of the conclusions of the theorems are to the effect that some functionsof particular solutions are monotone. Some of these results, in conjunctionwith the theorem of Hausdorff-Bernstein, imply that certain solutions canbe represented as Laplace-Stieltjes transforms of monotone functions.

Part II deals with a very special problem. It is concerned with a singular,boundary value problem related to a particular third order, nonlineardifferential equation. This problem had its origins in boundary layertheory in fluid mechanics.

Part III is a discussion of the stability in the large for a trivial or periodicsolution of a nonlinear autonomous system. An interesting feature of theproof of Theorem 14.2 is that it essentially reduces a {/-dimensionalproblem to 2-dimensional considerations by dealing only with 1-parameterfamilies of solutions at any one time.

PART I. MONOTONE SOLUTIONS

1. Small and Large Solutions

Consider a system of linear differential equations

for a vector y = (yl,..., yd) with real- or complex-valued components on0 ^ t < a> (^ oo). Let \\y\\ denote the Euclidean norm

This section concerns systems (1.1) with the property that, for everysolution y(t), either

500

Miscellany on Monotony 501

or

(For example, a sufficient condition for (1.30) or (1,3W), respectively, is thatthe Hermitian part AH(i) = \[A(i) + A*(t)] ofA(t) be nonpositive definiteor nonnegative definite forO ^ t < <w; so that \\y(t)\\ is nonincreasing ornondecreasing.)

When (1.30) [or (1.3W)] holds for all solutions, it is natural to ask whetherthere is a solution y0(t) satisfying

The next theorem gives an answer to this question.Theorem 1.10[00]. Let A(t) be a d x d matrix with (complex-valued)

continuous entries for 0 ^ / < to (^oo) such that (1.30) [°r (1-3^)] holdsfor every solution of (I.I), Then a necessary and sufficient [or sufficient]condition for (1.1) to have a solution y0(t) ^ 0 satisfying (1.40) [or (1.4^)]is that

Although (1.50) is necessary and sufficient for the existence of a solutiony0(t) satisfying (1.40), (1.5^) is not necessary for the existence of a solutionsatisfying (1.4^). The second order equation u" + 3«/16f2 = 0 for t > 0has the linearly independent solutions u = tA/* and u — tl/t; cf. ExerciseXI l.l(c). Thus if this equation is written as a system (1.1) for the binaryvector y = (u, u'), then every solution satisfies \\y(t)\\ —*• oo as f -*• oo. Butfor this system tr A(t) = 0, so that (1.5^) does not hold.

Proof of Theorem 1.10. In this proof, "limit" means "finite limit."Since it is assumed that (1.30) holds for every solution, it follows that ify^i), y2(0 is any pair of solutions of (1.1), then the limit of the scalarproduct 3^(0 • yz(t) exists as t -*• CD. This follows from the relations

where i2 = —I.Let 7(0 be a fundamental matrix of (1.1). Since the elements of the

matrix product Y*(t) Y(t) are complex conjugates of the scalar products ofpairs of solutions of (1.1), it follows that C = lim Y*(i)Y(t) exists ast -+ a>. In particular, det \Y*Y\ = |det Y\2 tends to a limit as t -+ a>.

If c is an arbitrary constant vector, the general solution of (1.1) isy — Y(t)c and

Hence

Since C is Hermitian and non-negative definite, Cc0 • c0 = 0 can holdfor c0 5^ 0 if and only if Cc0 = 0. (Note that for a Hermitian matrix C,the minimum of Cc • c, when ||c|| = 1, is the least eigenvalue of C.)The equation Cc0 = 0 has a solution c0 7* 0 if and only if det C = 0. Thus(1.1) has a solution y0(t) & 0 satisfying (1.40) if and only if

In view of Theorem IV 1.2,

Thus Theorem 1.10 follows.Exercise 1.1. Prove Theorem 1.1^.The next theorem concerns a linear system of second order equations

for a vector y — (y\ . .,, yd).Theorem 1.20[00]. Let A(t) be a d x d matrix of continuous, complex-

valued functions for Q t < CD (^ oo) with the properties that A(t) isHermitian, positive definite, and monotone (i.e.,

in the sense that A(t) — A(s) is non-positive [or non-negative] definite).Then, ify(t) is a solution of11.7),

and

If, in addition,

then (1.1) possesses a pair of(^O) solutions y0(t), y^(t) satisfying

When (1.7) represents Euler-Lagrange equations in mechanics, thenthe expression {...} in^(l.lO) is essentially "energy." The first part of thetheorem implies that if A(t) is positive definite and monotone, then the"energy" is monotone along every solution. It will be undecided if

y0(0, y\(t) are linearly independent solutions. For the 1-dimensional caseof these theorems, see § 3. The proofs will depend on some facts aboutHermitian, positive definite matrices given by the following exercise.

Exercise 1.2. Let A be an Hermitian positive definite matrix, (a)Verify that A has an Hermitian, positive definite square root A1^, i.e.,(Al*f - A. (b) Show that A* is unique, (c) Show that if A = A(t) isa continuously differentiate function of /, then A^(t) is continuouslydifferentiate.

Proof of Theorem 1.20[00]. Suppose first that 4(0 is continuouslydifferentiable. Write (1.7) as a system of first order differential equationsfor a 2</-dimensional vector (y, z), where

and A~l/* = (A1**)'1 = (A'1)1*. The resulting system is

It is clear that (1.7) and (1.15) are equivalent by virtue of (1.14).A 2df-dimensional vector solution (y(i), z(t)) of (1.15) has the Euclidean

squared length

Differentiation of F with respect to t gives F' = y' • y + y • y' + z' • « +z-z'\ or, by (1.15),

since 4^ and its derivative are Hermitian. Differentiation of {A^f = Ashows that A*(A*)' + (A*)'(A*) = A', or

Hence

Since 4' ^ 0 or A' ^ 0 according as A is nonincreasing or nondecreasing,(1.9OM) follows from (1.16) and (1.80[oo]).

In order to verify (1.100[oo]), write (1.7) as a system of first order for(x, y), where

This system is

The squared (Euclidean) length of (a:, y') is

Along a solution (x(t), y'(i}) of (1.19), the derivative of E is

so that, by (1.17),

Thus (1.100[00]) follows from (1.20) and (1.80[00]). This proves the firstpart of the theorem.

It follows that the squared Euclidean lengths \\(y, z)||, \\(x, y')\\ of solu-tions of (1.15), (1.19) tend to limits (^ oo) as t—> oo. The existence ofthe appropriate solutions yQ(t), y^(t} of (1.7) will be obtained by applyingTheorem l.lot«>] to the systems (1.15), (1.19).

Let 7X0 be the trace of the matrix of coefficients in (1.15). Then T(t)is the trace of -A-^A*)'. Hence


for 0 t < o).In order to prove (1.22), it is sufficient to consider /-values near a fixed

; = r0. If A(t) is multiplied by a positive constant, neither side of (1.22)is affected. Hence it can be supposed that there are constants e, 6 suchthat 0 < c < 0 < 1 and « ||y||2 ^ A(t)y -y^O \\y\\z for all vectors y andt near tQ. In particular, ||/ — 4(0 II ^ 1 — e < 1. Define a matrix, calledlog A(t), by the convergent series

[in analogy with log (1 — r) = — Sr"/w]; cf. §IV 6. This series can bedifferentiated term-by-term. Since [(/ - A(ty)n]' = -[A'(I — A^ +(/ — A)A'(\ — A)n~z + • • •] and tr CD — tr DC for any pair of matrices,


since

Hence, by (1.21),

It is readily verified from (1.23) that if, for a fixed /, A = A(0 is an eigen-value of A(t) and if y is a corresponding eigenvector of A(t\ then

Thus log A is an eigenvalue of log A and y is a corresponding eigenvector.Hence if Alt..., Ad are the eigenvalues of the (Hermitian) matrix -4(0.then log A!, . . . , log Ad are those of the (Hermitian) matrix log A(t).Hence tr log A(t) = log Aj + • • • + log Xd — log (A^ ... Ad) = log detA(t) and (1.22) follows from (1.24).

Consequently,

Thus the existence of y0(t) in Theorem l.20[ao] follows from Theorem

1-loofO]-If S(t) is the trace of the matrix of coefficients in (1.19), then S(t) is

lr(A*)'(A-lA). Thus Re 5(0 = -Re T(t) and the existence of the solutionyr(t) in Theorem 1.20[oo] follows from Theorem l-l^oo]-

This proves Theorem 1.20(00] under the extra assumption that A(t) has acontinuous derivative. If this is not the case, A(t) can be suitably approxi-mated by a sequence of smooth matrix functions A^t), A2(t\ . . . each ofwhich satisfies the assumptions of Theorem 1.20[00]. The approximationscan be made so that ^4(0 — An(t) are so "small" that the solutions of (1.7)and x" + Anx = 0 are "close"; cf. § X 1. Theorem 1.20[00] then followsfrom a limit process.

Exercise 1.3. Let A(t) satisfy the assumptions of Theorem 1.20[00].Let B(i) be a continuous matrix on 0 :$= t < co (^ oo) and consider

in place of (1.7). (a) The assertion (1.90[00]) remains valid if, in addition to(1.80[00]), it is assumed that B(t)A(t) + A(t)B*(t) < 0 [or 0]. (WhenA(t} is continuously differentiate, (1.80[oo]) and this condition on BA +AB* can be replaced by the single condition that A' + BA + AB* 0[or 0].) Also, the assertion concerning the existence of y0(t) is valid if(1.110[00]) is replaced by

(b) The assertion (1.100[oo]) remains valid if, in addition to (1.80[00]), it is

assumed that B + B* ^ 0 [or 0]. Also, the assertion concerning theexistence of y^t) is valid if (l.H0[oo]) is replaced by

2. Monotone Solutions

In contrast to the last section, the notation A ^ 0 or A > 0 for anarbitrary (not necessarily Hermitian) matrix A = (ajk) will mean thatajk ^ 0 or aik > 0 hold for j, k = ! , . . . ,< / . Similarly, y 0 or y > 0for a vector y == (y1,..., y*) will mean that yi 2i 0 or yj> 0 fory = 1,. . . , d.

Theorem 2.1. Lef A(f) be continuous for 0 t < co (^oo) o«^ satisfyA(t) ^ 0. 77ze« f/ze system

has at least one solution y(t) ^ 0 satisfying

Remark. If the interval 0 ^ / < c o ( ^ o o ) i s replaced by 0 < t < co(^ oo) in both assumption and assertion, this theorem (and its corollaries)remain valid. For, if 0 < a < co, then Theorem 2.1 implies the existenceof a solution satisfying 0 0 y(t) ^ 0, y'(t) ^ 0 for a / < co. But thenthese inequalities also hold for 0 < / < a; cf. the proof of the theorem.

Proof. Since A(t) ^ 0, it follows from (2.1) that a solution of (2.1)satisfies y'(t) ^ 0 on any interval on which y(t) > 0. In particular, if0 < a < co and y(d) > 0, then y(t) > y(d) > 0 on 0 < t < a.

Let y0 > 0 be fixed; e.g., y0 = (1, . . . , 1). Let ya0(t) be the solution of(2;1) satisfying the initial condition ya0(a) = y0, where 0 < a < co. Thusy«o(0 2/o > 0 on 0 / a. Let c(d) = ||ya0(0)||, so that c(a) ^ ||y0|| > 0.

Let ya(t) = yaQ(t)lc(d). Hence ya(t) is a solution of (2.1), ya(t) ^ y0/c(d) > 0 for 0 t a, and ||y0(0)|| = 1. Choose 0 < al < az < ...satisfying an -> co as n —>> oo and

Then ||y°|| = 1. In addition,

exists uniformly on closed intervals of [0, co) and is the solution of (2.1)satisfying y°(0) = y°; see Corollary IV 4.1. In view of ya(t) ^ 0 on0 / ^ 0, (2.3) implies that y°(t) ^ 0 on 0 <j t < co. Also, y°(0) =y° ^ 0. This proves the theorem.

Exercise 2.1. (a) Let y(t) be a solution supplied by Theorem 2.1.rio

Then y(eo) = lim y(t) as t -> eo exists and ||y'(OII * < °°- (*) !*>"*(«>) >

0 for some m,l^m^d, and y4(0 = (aik(t)) wherey, A; = ! , . . . ,< / , then

(c) Show that if (2.4) holds for some fixed m, it does not follow thatym(aj) > 0. (d) The condition (2.4) for m = 1 , . . . ,d is necessary andsufficient that y(/) in Theorem 2.1 can be chosen so that y(co) > 0 (i.e.,ym((o) > 0 for m = 1, . . ., d).

Exercise 2.2. The following is a theorem of Perron-Frobenius: LetR be a constant d x d matrix satisfying R 0. Then R has at least onereal, non-negative eigenvalue A 0 and a corresponding eigenvectory 0, y ?* 0. Furthermore, if R > 0, then A > 0 and y > 0. Deducethis from Theorem 2.1.

Corollary 2.1. Let A(t) be completely monotone on 0 t < oo [i.e.,/e/ ,4(0 eC^fort^Oand (- l)M(n)(0 0 /or n = 0, 1, . . . ; in otherwords, A(t) > 0, A'(t) ^ 0, A\t) ^ 0,...]. Then (2.1) /KW a solutiony(t) ^ 0 which is completely monotone [i.e., (— l)Vn)(0 0/or n = 0, 1,.. .] on 0 < f < oo.

It follows, therefore, from the theorem of Hausdorff-Bernstein thatthere exist monotone nondecreasing functions o'(t) ont ^ Ofory = 1 , . . . ,d, such that the components y\t) of y(t) have representations of the form

j— 1, . . ., d. (For the theorem of Hausdorff-Bernstein, see, e.g., Widder[1].)

Exercise 2.3. (d) Prove Corollary 2.1. (b) If p = (pl, . . . >pd) is avector use the notation py =• (plyl,.. . ,pdyd). Show that the conclusionof Corollary 2.1 is true if (2.1) is replaced by

where ,4(0 satisfies the conditions of Corollary 2.1 and /j(OeC°° for0 / < oo satisfies

[i.e., p(t) > 0 and p(t) has a completely monotone derivative p'(t) forr^O] .

Corollary 2.2. In the linear differential equation,

let the coefficients p0(t),... ,pd(t) be continuous (real-valued) functions on0 :Ss f < co (^ oo) satisfying

(while Pi(t) is arbitrary). Then (2.7) has at least one solution u(i) satisfying

for n = 0,. .. ,d — 1. If, in addition, p^t) ^ 0, then (2.9) holds also forn = d.

Exercise 2.4. Deduce Corollary 2.2 from Theorem 2.1.For a different proof in the case d = 2, see Corollary XI 6.4.Corollary 2.3. In (2.7), let d > 2 #«</ /ef //re coefficient functions be of

class C00 /or / > 0, />0(f) > 0 and -p9'(t), -pM, p2(t) pd(t) com-pletely monotone for t 0 [so that

for k = 2,...,d and n = 0, 1 , . . . , 0 t < oo]. T/ie» (2.7) fou asolution u(i) satisfying (2.9) for n — 1, 2 , . . . on 0 / < oo.

There is no condition on /?„' or /?t. In view of the theorem of Hausdorff-Bernstein, Corollary 2.3 implies that if p0 > 0 and —p0"t —pi, p2,..., pdhave representations of the form

where a(s) is nondecreasing for s 0, then (2.7) has a solution M(J) > 0representable in this form.

Exercise 2.5. Prove Corollary 2.3.Exercise 2.6. (a) The differential equation for the associated Legendre

functions

is transformed into the differential equation for the toroidal functions

by the substitution v =* n — \, p = m, x = cosh f. If n2 J, show thatthis last equation has a solution «(/) > 0 completely monotone for t > 0

(and that this solution is unique up to constant factors if and only ifn2 > i)- (b) In the hypergeometric equation

make the change of independent variables 2x — I = cosh /, where 1 <x < oo, or 2x — 1 = —cosh /, where — oo < x < 0, so that 0 < t < oo.The resulting equation is of the form

Show that if ab 0 and a + b max (2c — 1, 0), then there exists acompletely monotone solution u > 0 on t > 0 (and that this solution isunique up to constant factors if and only if either ab < 0 or a — b = 0).(c) Kummer's form of the confluent hypergeometric equation

has a completely monotone solution u(t) > Oforf > Oifa ^ 0, c arbitrary;this solution is unique up to constant factors. Also, if t is replaced by— t, the new equation has a completely monotone solution for t > 0 ifa < 0 and c 0 (and this solution is unique up to constant factors if andonly if a < 0). (d) Whittaker's normal form of the confluent hypergeo-metric equation

has a completely monotone solution u — Wkm(t) > 0 for t > 0 if k ^ 0and m^^4. This solution is unique up to constant factors.

Corollary 2.2 has the following generalization in which

where i,j = 1,... ,k, denotes the Wronskian determinant of the functionsI*!,..., «*.

Corollary 2.4. Let m be fixed, 0 < m d. In (2.7), let the coefficients becontinuous for 0 t < at (^ oo) fl/w/ /rave the properties that

a/w/ fAa/ f/ie mf/i order differential equation

/KW a ref of solutions u^t),.. ., um(t) such that

Then (2.7) has a solution satisfying (2.9) for n = 0, 1,. . ., d — m onQ<t<co.

Exercise 2.7. Prove Corollary 2.4.Exercise 2.8. Let/ = (f1,... ,/d) and y = (y1,..., y^). Assume that

f(t, y) is continuous for t 0, y 0; that/(r, 0) = 0; and that/(f, y) >0. Let c be any nonnegative number. Show that y' = —/(/, y) has atleast one solution y(t) for / 0 satisfying ||y(0)|| = c and y(f) > 0,y'(0 0 for / ^ 0.

Exercise 2.9. Let y, / be real-valued, (a) Assume that f(t, y, y') iscontinuous for t > 0, y 0, y' < 0; th'at/(r, 0,0) = 0; that/(f, y, y') ^0; and that solutions of y" =/(?, y, y') are uniquely determined by initialconditions. Show that there exists a c0,0 < c0 oo, such that if 0 < c <c0, then y" =f(t, y, y') has at least one solution y(t) for t 0 satisfyingy(0) = c, y(/) 0 and y'(0 0 for / 0. This is not contained inExercise 2,8, where the corresponding initial condition is

(b) Show that it is not always possible to take c0 = oo in (a), (c)Let /(/, y, y') be continuous for t^ 0, y > 0, y' ^ 0; /(r, 0, 0) = 0;/(', y, 0) ^ 0 for t > 0, y ^ 0; for every /? > 0, let there exist a

r<*>positive continuous function <p(z) = 9* (2) for z ^ 0 such that I u duj

<p(-u) = oo and \f(t, y, z)| 9^(2) for 0 < f ^ /?, 0 y < /?, 2 0.Let c > 0. Show that y" = /(r, y, y') has a solution on / 0 satisfyingy(0) = c and y(f) ^ 0, y'(0 = 0- (This is a special case of Theorem XII5.2 and Exercise XII 5.3.)

3. Second Order Linear Equations

This and the next section will be concerned principally with solutions ofoscillatory equations (cf. § XI 6) of the form

where ^(f) is a monotone function of /.Theorem 3.10[ooj. Let q(t) > 0 be continuous for 0 / < o> (^ oo)

and monotone; i.e.,

Then, for any solution u(t), the functions u* + u'2jq and qu* -f u'2 aremonotone, in fact,

Ifu(t) ^ 0 has a (finite or infinite) set of zeros (0 ) t0 < ^ < . . . , then

(3.50[oo]) *n ~ *»-i is nondecreasing [or nonincreasing] with n.

Furthermore, if

as f —»• co, then (3.1) possesses linearly independent solutions M0(0> Mi(0satisfying, as t -*• co,

Figure 1

If (3.1) is oscillatory at / = co, then ^ > 0 implies that the graph ofu = |w(/)| in the (f, w)-plane consists of a sequence of convex arches. Theassertion (3.3) implies that the successive "amplitudes" (i.e., maxima of|w|), which occur at the points where u — 0, are monotone. Correspond-ingly, the successive maxima of |u'|, which occur at the points whereu" — 0 or, equivalently, where u = 0, are monotone by (3.4). See Figure 1.The Sturm comparison theorems imply (3.5) and even more:

Exercise 3.1. Let q(t) ^ 0 be continuous and nonincreasing forT! t TS and let (3.1) have a solution u(i) with exactly three zeros' = Ti» Ts» Ts> where rt < ra < TS. (a) Show that u'(t) has exactly twozeros t = T2, T4 satisfying ^ < r2 < r3 < r4 < TS and that rm — T, ^r^+2 — TJ+I for j = 1,2,3. (b) After a reflection across a vertical line/ = rj+1 for j = 1,2 or 3, the graph of u = \u(t)\ for a quarter-waveTj+i = t = ri+z lies °ver tne graph of M = |w(r)| for the preceding quarter-wave rj ^ / Tm; i.e., |w(rm - /)| |M(Ty+1 + /)| for 0 / <; Ty+1 - T,,

The assertions (3.3), (3.4), (3.7), and (3.8) are consequences of Theorem1.2, but slightly different arguments for their proofs will be indicated.

Except for the assertion (3.5), either of the two cases of the theorem,corresponding to (3.2) and (3.2^), is a consequence of the other. This canbe seen from the following lemma which can be interpreted as a "dualityprinciple" between equations (3.1) and (3.11) in which u,u',q,dt arereplaced by u, —(sgn^)w, \jq, \q(t)\ dt, respectively:

Lemma 3.1. Let q(t) 0 be continuous for 0 t < co. Introduce thenew dependent variable v and independent variable s defined by

Then

(3.1) and the equation

r°>and 0 s < I dtj\q(t)\ ^ oo, are equivalent by virtue of (3.9); finally,

Jo

Proof. This lemma is trivial for, by (3.10), dv\ds + (sgn q)u — 0.Differentiating this relation with respect to t and dividing by |^(f)| gives(3.11). The relations (3.12) follow from (3.9) and (3.10).

Exercise 3.2. Find the analogue of Lemma 3.1 if (3.1) is replaced by(/?(/)«')' + q(t)u = 0, where XO > 0, q(t) j* 0.

Proof of Theorem 3.1. Note that if q is monotone, the functionsw2 + u'*/q, quz + u'z are clearly of bounded variation on any interval[0, a] c [0, w). The relations (3.3), (3.4) follow from (3.1).

In the case (3.20[oo]), the existence of a solution wx [or MO] in (3.80)[or (3.7^)] implies the existence of a solution w0 [or wj in (3.70) [or (3.8^)].This can be seen as follows: Let w0, MX be linearly independent solutions of(3.1). Then their Wronskian is a nonzero constant

Since this Wronskian is

it follows from Schwarz's inequality that

Hence (3.7^) implies (3.8^) [and interchanging MO and ult it is seen that(3.80) implies (3.70)].

Thus in view of Lemma 3.1, it only remains to verify (3.50[oo]) and theexistence of a solution MO satisfying (3.7^) in the case (3.2^):

Exercise 3.3. Assuming (3.2^), verify the existence of a solutionM0 ^ 0 satisfying (3.7^). Apply either Theorem 1.2^ or apply argumentssimilar to those used in the proof of Theorem 1.1 directly to the quantityM2 + u'*/q (instead of ||y||2).

Exercise 3.4. State and prove the analogue of Theorem 3.10[00] when(3.1) is replaced by (p(t)u')' + q(t)u — 0.

Corollary 3.1. Let q(i) > 0 be continuous and nondecreasing for 0 t < w (<; oo) and (3.1) oscillatory at t = <w. Let M0(f) be a solution o/(3.1)satisfying M0(f) —»• 0 as t -*• a>. Then

(possibly conditionally).Exercise 3.5. Verify Corollary 3.1.Exercise 3.6. Let J^t) be the Bessel function of order //. There exists

a constant c such that

See Lorch and Szego [SI] (or Hartman and Wilcox [l,p.239]).Note that if the conditions (3.20[oo]) of Theorem 3.1 hold for a contin-

uous q(t) > 0 and (3.60[oo]) does not hold, then q((o) = lim q(t) as t -*• o>satisfies 0 < q(ai) < oo. If <t> = c» and 0 < ^(oo) < oo, then (6.1) has apair of solutions MO, MJ satisfying

(3.15)

as / -* oo; cf. Exercise X 17.4(fl) or XI 8.4( ). In particular.

This will be used in the next section.

converges

When q(t) tends monotonously to oo [or 0], then there exists at least onesolution u0(t) & 0 which tends to 0 [or is unbounded]. When co = ooand q(t) is of sufficiently smooth growth, then all solutions tend to 0 [orare unbounded] as can be seen from the asymptotic formula supplied, e.g.,by Exercise XI 8.3 or XI 8.5.

A similar statement, without involving asymptotic integration, is givenby the following exercise:

Exercise 3.7. A monotone function H(t) on a t < oo satisfyingH(t) ~> oo as / ->• oo will be said to be of "irregular growth" if, for everyc > 0, there is an unbounded sequence of /-values a = t0 < ^ < . . .such that the open sets B = (t0, t£ U (tz, t3) u ... and C(n) = (tlt f2)

u

(/3, *4) U • • • U (tzn-i, /2n) have the properties

If H(i) is not of "irregular growth," then it will be said to be of "regulargrowth." Show that if q(t) is continuous and satisfies (3.20[o0]) and (3.60^),and if |log^(r)| is of "regular growth" on a t < oo for large a, then allnontrivial solutions of (3.1) satisfy (3.70[oo]) and (3.80[oo]). See Hartman[23].

Exercise 3.8. Letw ^ 0. Let (0 possess n + 1 continuous derivativesfor t 0 satisfying (— l)y+1>(/) 0 fory = 0,-..., n and 0 < q(ao) <oo. Let/(f) haven + 1 continuous derivatives for / ^ 0 and (— 1)J/O)(0 0 for j — 0 , . . . ,«+ 1 and /(oo) = 0. Then v" + q(t)v =/(0 has aunique solution v(t) such that (—1)V^(/) 0 for j = 0, . . . , n andv(i}(t) -»• 0 as / -* oo for j = 0 , . . . , n + 2. Prove this for n = 0,1, 2.(The cases n gZ 2 are more complicated; see Hartman [22].)

Exercise 3.9. Let 0 / < oo and assume that (3.1) is nonoscillatoryat t = oo. For a solution u =£ 0, put r = w'/M and £ = </w2 -f n'2; let «0

denote a principal solution (Theorem XI 6.4), r0 = MO'/WO and £0 =/•OO

?«o2 + «o'2- (fl) Let ?(0 °- Show that r' ^ 0 and ^(j) ds ^ r(0

!/(/ — /„) for large /. (b) Under the additional assumptions of Theorem3.10 with co = oo, E -> 0 as / -> oo for every solution u(t) of (3.1) if and

/•OO

only if tq(t) dt = oo. (c) Let q(t) < 0, dq 0 [or dq ^ 0] and (oo) = 0

[or ^(oo) = — oo]. Then, for the principal solution, r0 < 0, r0' > 0,£0 < 0, dE0 ^ 0 [or r0 < 0, r0' < 0, £0 > 0, dE0 < 0] for 0 / < ooand r0(oo) = 0, EO(<X)) = 0 [or r0(oo) = — oo, E0(ao) = 0]; and, for anonprincipal solution, r > 0, r' < 0, E > 0, dE > 0 [or r > 0, r' > 0,E < 0, </£ 0] for large / and r(oo) = 0 [or r(oo) = oo, £(oo) = - oo].In the case ^(oo) = 0, £(oo) = oo for all solutions if and only if

- tq(t)dt = oo. (d) Let (0 < 0, dq 0 [or dq ^ 0] and q(ao) =

—A2, where A > 0. Then, for the principal solutions r0 < —A, /•„' > 0,E0 < 0, dE0 > 0 [or r0 > -A, r0' < 0, £0 > 0, dE0 ^ 0] for 0 f < ooand r0(oo) = — A, E0(cd) — 0 in both cases. On the other hand,

f°° f*rt I (exp 2 I [—^(j)]l/i <&} dq(t) = oo is necessary and sufficient in order

that the nonprincipal solutions satisfy £(oo) = oo [or £(oo) = — oo]; inthis case, r > A, r' < 0, E > 0, dE > 0 [or r < A, r' > 0, E < 0, rf£ S 0for large /.

4. Second Order Linear Equations (Continuation)

This section concerns the function r = [t/02(0 + «i2(01H 0. where

M0(0 and u^t) are certain solutions of

when ^r(f) is monotone. The desired result will be deduced from resultsanalogous to those concerning (3.15), (3.16) and the following simplelemma.

Lemma 4.1. Suppose that q(t) > 0 is continuous onQ < t <GD (< oo),satisfies

an</ w monotone,

7%en /Aere exist real-valued solutions w0(f), M!(/) of (4.1) swc/i rAa/ /Aecomplex-valued solution z(i) = u9(t) + iu-Jif) satisfies

as t —> CD ; in this case,

or

Remark 1. It will be convenient to note that in (4.5+), the inequality|z|2 > 1 holds for 0 < / < o> if the conditions q > 0, </<7 jg; 0 are relaxed to9 0 for 0 < r r0 and q > 0, <ty 0 for t0 < t < (o for some fixed /0,0 < /0 < <o; cf. the end of the proof of the lemma.

Proof. If a* = oo, the existence of solutions w0, ut satisfying (4.4)follows from Exercise X 17.4(a) or XI 8.4(6); cf. (3.15)-(3.16). If w < oo,then q(t) can be defined at / = <w so as to be continuous there and theexistence of «0, ul is then trivial.

Lemma 3.1 shows that the assertions (4.5±) follow when (4.5+) isproved in the case (4.3+). For a fixed <p, consider the solution of (4.1)given by

Note that u' = Re {e"i9z'(t)} = -Im {e~i<fz(t)} + o(l) as / -> a>, by (4.4).Hence u2 + u'z = |z|2 + 0(1) = 1 + 0(1) as t -> to.

By Theorem 3.1^, assumption (4.3+) implies that (3.3.x,), (3.4^) hold.Since quz + u'2 -* 1 and w2 + t/'2/^ -*• 1 as / -» to by (4.2), it is seen that

In particular qu2 1. For fixed f, choose <p so that u(t) = \z(f)\. Thisgives the first of the four inequalities in (4.5+). Also, by (4.7), u\i) ^ 1if u(t) = 0. For / fixed, choose 9? so that u'(t) = 0. By Schwarz's in-equality, |w(f)| ^ 1401 and so |z(OI ^ 1. This is the second of the in-equalities in (4.5+). The last two are obtained similarly. This completesthe proof of Lemma 4.1".

As to Remark 1 following Lemma 4.1, note that the Wronskian ofi/o, M! is 1,

by (4.4). By the lemma, (4.5+) holds for / > t0, so that |z(/0)| S>,1,I2'('o)l = 1- Choose 0 and u"(i) ^ 0 for 0 < / ^ t0. Hence w'(0 w'(^o) = 0 and so"(0 «(^o) 1 for 0 < r ^ r0. As before, \z(t)\ ^ w(/) 1. This provesRemark 1.

Exercise 4.1. In Lemma 4.1, show that |z|2^ + |z'|2 1 + ^ in bothcases (4.3 ±).

Lemma 4.2. Lef ^(/) > 0 be of class C2 for t > t° with the propertiesthat

satisfies

and that either

or, for some t0 t°,

77ie« (4.1) possesses a pair of real-valued solutions «0(0, «i.(0 JMC/I thatz(f) = MO(/) + iujfa) satisfies, as t -*• oo,

and, for t > f°, «VA«r

according as (4.11 —) or (4.11 +) /zo/Js.Remark 2. Note that if q is of class C3, then Q has a continuous

derivative given by

Hence dQ 0 is implied by

Remark 3. If ^(/) satisfies the conditions of Lemma 4.2 and if q(t) ismonotone for large / with 0 < ^(oo) oo, then (4.10) is redundant. Forthe monotony of Q implies that 1 — Q(f) does not change signs for larget, so that q'lqy* is monotone by (4.9). Hence

tends to a limit as t -* oo. Since 0 < q(<x>) ^ oo and Q is monotone forlarge /, the relation (4.10) holds.

Proof. By the Liouville change of variables

(4.1) is transformed into

and Q is defined by (4.9); cf. § XI 2(xiii). The /-interval t° < t < oo istransformed into some s-interval (—00 <^) a < s < co (^ oo).

By Lemma 4.1 and the Remark following it, the differential equation(4.16) has a pair of real-valued solutions U0(s), U^s) such that Z(s) =U0 + /C/i satisfies, as s -*• co,

and the analogues of (4.5±) if z, z', q are replaced by Z, </Z/<&, (?. Inparticular |Z|2 :< 1 or |Z|2 1 according as (4.11—) or (4.11+) holds.By (4.15), the equation (4.1) has the solutions w0 = U0lql/i, ut = UJqlA,and z == w0 -f- «/!; thus ^rHz = Z at f = f(j). This gives Lemma 4.2.

Theorem 4.1. Let q(f) satisfy the conditions and u0, ul the assertions ofLemma 4.1 with a> = oo.

(i) Let (4.11 -) hold. Then

If, in addition, q(t) is continuous for t > 0, q(t) ^ 0 for 0 f < f °, a/w/<7(0 const. > 0/or large t, then

(ii) Lef (4.11 +) hold. Then

Proof. Let r = \z\ = (w02 + Uj2)1^ ^ 0. Then two differentiations of r

show that, by virtue of (4.1),

Since (4.13±) mean that 1 — qr* 0 or 0, the first and last inequalitiesin (4.17), (4.19) follow. Also, if q 0, then r" 0, so that the thirdinequality in (4.18) holds.

From (4.13—), it is seen that r(t) is bounded as t -> oo if</(/) ^ const. >0 for large t. Thus r' ^ 0 follows from r > 0, r" 0 in (4.18). Alsor ^ 0 follows from r > 0, r" 0 in (4.19). This proves the theorem.

Corollary 4.1. Let q(i) be continuous for t > 0 and of class C3 fort>t°^0. Let q < 0 for 0 < t < f° a/irf (4.14) hold for t > t°. Then(4.18)holds.

Exercise 4.2. Consider Bessel's equation

The variation of constants u = t^v transforms it into

so that a | 0 according as (0 ^) /j, | \. The real-valued solutionsv = Jfi), YJ(t) of (4.21) are such that, for some real number 0,

satisfies zr^e**, z' ~ ieu as /-*• oo. Use these facts in the following:(a) Show that

(b) Furthermore,

(c) The function r = /*(/u2 + Fu

2)^ > 0 satisfies (4.18) or (4.19) forr° = 0 according as /* > J^ or 0 /* < 1 . (af) Show that

Exercise 4.3. Let n ^ 0. Let (0 be continuous for t 0 and possessn + 2 continuous derivatives satisfying (— 1)J^O+1) ^0 for y = 0,.. . ,« + 1 and 0 < 9(00) <; QO. (a) Show that (4.1) has a pair of solutionsMo(0> "i(0 such that MO'M! — u0Ui = 1 and w = M0

2 4- «i2 satisfies(- 1)V(J> ^ 0 for j = 0 , . . . , « + 1 and w<J-) -»• 0 as t -+ oo for j = 1,. . . , » + 3, while w-*-! o rw-*Oas / -» -oo according as 9(00) < oo or9(00) = oo. See Hartman [22]. (b) Let u(i) & 0 be a real-valued solutionof (4.1) and let its zeros be (0 ) r0 < ^ < Let AJ+1/fc = A(AJffc);thus A1^ = Arfc = tk+l — tk, A2/fc = tk+z — 2tk+l + tk, Show that(- ly+'A'Vfc ^ 0 for; = 1 , . . . , » + 1.

PART H. A PROBLEM IN BOUNDARY LAYER THEORY

5. The Problem

This part deals with a generalization of a problem in boundary layertheory in fluid mechanics. The problem concerns existence and uniqueness

questions for a singular boundary value problem involving the autonomous,third order, nonlinear differential equation

and solutions on 0 fs / < oo satisfying the boundary conditions

where A, a, ft are constants. The problem will further be restricted to theconsideration of solutions of (5.1), (5.2) satisfying

in particular, it will be assumed that 0 ft < 1. [Questions of existenceand uniqueness without the restriction (5,.3) are not yet completelysettled.]

The cases A = 0 and A = Ji of (5.1) are often called the Blasius andHomann differential equations, respectively. As far as questions ofuniqueness are concerned, the cases A = 0, A > 0, A < 0 are quitedifferent. Although, all cases can be treated in a similar manner, a simpledifferent existence proof for the case A > 0 will be given. The existenceand uniqueness problems for the case A > 0 will be given in § 6, for A < 0in § 7, and for A = 0 in § 8. The asymptotic properties of the solutions forall cases will be given in § 9.

6. The Case X > 0

The existence theorem in the case A > 0 will be based on the followingsimple topological argument:

Lemma 6.1. Let y,fbe d-dimensional vectors andf(t, y) continuous on anopen (t, y)-set O such that solutions of initial value problems associated with

are unique. Let D° be an open subset of£l with the properties that all egresspoints from ii° are strict egress points and that the set Qe of egress points isnot connected. Let Q, denote the set of ingress points of Q° and S a con-nected subset of H° U Qe U Q; such that S n (Q° U Qt) contains twopoints (tlt «/x), (t2, y^ for which the solutions yfc) of (6.1) through (/,, yf)forj =1 ,2 leave Q0 with increasing t at points of different (connected) com-ponents of£le. Then there exists at least one point (t0, y0) e S (~\ (Q° u D,-)such that the solution y0(t) of (6.1) determined by y0(t0) = y0 remains inii° on its (open) right maximal interval of existence.

For the definition of egress, ingress, and strict egress point, see § III 8.Proof. If the lemma is false, there exists a continuous map TT : S->Qe

where, for (/„, t/0) e S, TT(/O, y0) is the first point (t, y), t t0, where thesolution through (t0, y0) meets Qe. The map TT is continuous since everyegress point of £i° is a strict egress point and solutions of (6.1) dependcontinuously on initial conditions (Theorem V 2.1). Consequently, theconnectedness of S implies that the image ir(S) <= Qe of S is connected.But this contradicts the assumption concerning the existence of (tlt yj,(>* y2)-

Theorem 6.1. Let X > 0, — oo <a < oo, 0 p < 1. Then there existsone and only one solution u(t) of (5.1), (5.2), (5.3). This solution also satisfies

In the application of Lemma 6.1, the following fact will be needed:the case /? = 1 of (5.1)-(5.2) has the trivial solution

This will imply that the set Qe below is not connected.The uniqueness proof will be given in this section for both A > 0 and

A = 0. It will be derived from Exercise III 4.1 and uses (6.2); cf. (8.4).Proof. Existence for X > 0. Rewrite the differential equation (5.1)

as a system of first order for a 3-dimensional vector y = (y1, y2, t/3),where y1 = u,yz = u't y3 = u",

Consider this equation on Q, the entire (t, y)-space. Introduce the open(f, 2/)-set

and the boundary sets

see Figure 2. It is readily verified that the set of egress points for Q° isQ1 U £12 and that all egress points are strict egress points. The set ofingress points is Qf. A solution y(t) through a point (/0»2/o)> wherey02 = yQ3 = o is not in Q° for small |/ - t0\, since «/2' = y3 = 0 and t/2" =ys' = -A < 0 at t = 0 imply that y\t) < 0 for small \t - t0\ * 0. Notethat the points of ii3 are neither ingress nor egress points since they arepoints on the trivial solutions (6.3).

Thus de = Q1 U Q? is not connected. Let S = {(t, y): t = 0, y =(a, ft, y) and y > 0 arbitrary}, where a, /? are fixed, 0 /S < 1. ThusS c: H° u Qt is connected. Let yy(0 be the solution of (6.4) through thepoint (/, y) = (0, a, £ y).

If 0 < /? < 1, the point (t, y) — (0, a, ft, 0) e Q1 is a strict egress point ofii°. This makes it clear that if y > 0 is small, the arc (t, yy(0) leaves Q°

Figure 2. Projections of Q3, Q, on (y*, y^-plane.

for some t > 0 at a point of Q1. The same argument is valid for /S = 0and small y > 0.

It will be shown that if y > 0 is large, the solution arc (t, yY(t)) leavesQ° through a point of Q2, where y2 = I . Write the third equation of (6.4)as

Along the arc y = yy(t), with (t, yv(t)) e 0°, the component y2 is non-decreasing (for yv — y3 ^ 0). Hence

A quadrature gives, for / ^ 0,

Since 0 y2(0 1 so that a yl(0 = « + '»

Consequently, if y is sufficiently large, then y*(t) exceeds a given positiveconstant on a large /-interval [0, t] as long as (/, yv(t)) e fi°. Thus y8' — y3

implies that (t, yy(/)) leaves OP at a point where y2 = 1.

By Lemma 6.1, there is a y = y0 > 0 such that (t, yy(/)) e ii° U Qt onits right maximal interval of existence, which is necessarily 0 t < oo.On this solution, y2' = y3 >0 for / > 0, so that y2 > 0 for large /. Also,y3' < —A[l — (y2)2] < 0 for large t shows that lim y3^) exists as r -v oo.This limit is 0 since yz(/) < 1 for all t 0. Consequently, y2 -»• 1 ast -> oo (otherwise y3' < —const. < 0). This completes the existenceproof.

Uniqueness for X 0. The proof depends on the introduction of thefollowing new variables along a solution u = u(t) of (5.1) satisfyingu'(t) > 0: Let u be the new independent variable and z = u'z > 0 the newdependent variable; so that djdt — u' djdu s= z*d\du or d\du = z~^d\dt.Thus if a dot denotes differentiation with respect to u, then

The equation (5.1) is transformed into

the boundary conditions (5.2) into

and (5.3) into

Let z(u) be a solution of (6.8), (6.9) with 0 < z(u) < 1, z(u) > 0 on someinterval (a, «„]. Let u = £7(2) be the function inverse to z — z(u). Put

Then

while f/K/f/z = zdU/dz = z/V, so that (6.8) gives

Then (6.13), (6.14) constitute a system of differential equations for(-U, V\ in which the function on the right of (6.13) is an increasingfunction of F(> 0) and the function on the right of (6.14) is an increasingfunction of — U and a nondecreasing function of V for V > 0 and A 0.Hence if (C/^z), V^(z)\ (C/2(2), F2(z)) are any two solutions of (6.13)-{6.14)

such that U^Y = W) - «, Vi(F) > W*) > 0. Then £/2(z) >tfife), ^i(«) > ^*), and £/2(2) - ^(z), Kt(«) - K2(2) are increasingfunctions of z on any interval /?2 < z < 20 (< 1) on which the solutionsexist; cf. Exercise III 4.1(0)-(c).

Now suppose that (5.1)-(5.3) has a pair of distinct solutions u^t), w2(f)on / ^ 0 and suppose that u[(Q) > u"z(Q). By (6.2) or (8.4), u] > 0 for0 ^ t < oo and u] -»• 0 as / -»• oo. Let Z!(M), 22(w) be the correspondingsolutions of (6.8) defined by (6.7); 6^(2), £/2(z) the functions inverse to^(w), z2(w) and ^(z), F2(z) defined by (6.12). Thus t//z), K/z) > 0are defined for 02 < z < 1. Also Ujffi*) = UJP) = a and FjOT >VJJP). Then wj-*0 as r->oo implies that F! - F2-*0 as «-*•!,but Pj — F2 > 0 is increasing with z. This contradiction establishesuniqueness.

Exercised.}. Let w(f, A) be the solution supplied by Theorem 6.1.Modify the uniqueness proof, to show that if 0 < A < p, then u"(Q, A) <M"(0, p) and «(f, A) < w(r, /«), w'(/, A) < «'(/, ) for 0 < t < oo.

Exercise 6.2. If a ^ 0, the uniqueness assertion of Theorem 6.1follows by a variant of Exercise XII 4.6(ar) applied to (6.8) where u > a 0 for / > 0 [and the interval 0 / p of Exercise XII 4.6(a) is replaced by0 < / < oo].

Exercise 6.3. Let A > 0. Show that if — oo < a < oo and ft > 1,then (5.1), (5.2) has one and only one solution u(t) satisfying u'(t) > 1 for0 < / < oo. This solution also satisfies u" < 0 for 0 < t < oo andu" -*• 0 as / -*• oo.

Exercise 6.4. Give another proof of existence in Theorem 6.1 based onthe following: Put z = 1 — u'. Then (5.1) becomes

Define a sequence of successive approximations by letting z°(f) = 1,M°(0 s 0, and if z°,. . . , zn~\ w° , . . . , un~l have been defined, let zn(t) be asolution of

satisfying 2(0) = 1 - /?, z > 0, z' < 0 for / > 0; cf. Corollary XI 6.4.The function zn(t) is unique and satisfies zn(t) -*• 0 as / -> oo; see ExerciseXI 6.7. Let

Show that 1 = 2° 21 ... for t^ 0 and z1' > 22/ ... at f = 0;

cf. Corollary XI 6.5. In a similar way, define a sequence of successiveapproximations z0, zlt..., with

starting with z0(f) == 0. Show that 0 = z0 < zl < za < ... for / ^ 0 and«i'(0) ^ z2'(0) Also z,.(0 z*(0 for / ;> 0 and y, A: = 0, 1, . . .and z/(0) z*'(0) for j, k = 1,2, Show that the limits z(f) =lim zn(f) and z(f) = lim zB(f) and the last part of (6.15) define solutions of(5.1)-(5.3). (These are the same solution by uniqueness in Theorem 6.1.)

7. The Case X < 0

In case A < 0, the analogue of Theorem 6.1 becomesTheorem 7.1.* Let A, 0 be fixed, A < 0 am/0 £ < 1. Tte/i ffore exists

a number A = A(X, /3) and a continuous increasing function y(a) defined fora ^ A with the properties that y(A) = 0 and that if u(t) is a solution of

/Ae/t «'(») = 1 and (5.3) /toW if and only if v. ^ and 0 < w"(0) ^ y(a);in /Aw ca^e,

Thus for given A < 0 and ft on 0 £ < 1, the problem (5.1)-(5.3) hasone and only one solution if a = A. When a < A, there is no solution andwhen a > A, there is a family of solutions. In the case A < 0, the unique-ness proof of the last section breaks down, for the function on the right of(6.14) is not a nondecreasing function of V.

At one point, the proof will use the simple estimates for solutions of theWeber equation supplied by Exercise X 17.6; cf. Exercise XI 9.7. In theproof, the singular boundary value problem (6.8)-(6.11) will be considered.If z = Z(M) is a solution of this problem, a solution of problem (5.1)-(5.3)is obtained by inverting the quadrature

Note that the usual existence theorems apply to the differential equation(6.8) only if z > 0. Nevertheless, solutions of

"determined" by initial conditions

*A(A, /?) is continuous; Hartman [S4]. For applications, Hastings [SI],Hastings and Siegel [SI], Hartman [S4].

0 ^ ft < 1, y ±£ 0, will be considered (at least for small u — a > 0) even if/3 = 0. This is to be understood in the sense that u = u(t) is the solutionof (5.1) satisfying w(0) = a, w'(0) = ft ^ 0, u"(0) = £y ^ 0 and 2 = z(w)is determined by (6.7). In the critical case z(a) = 0, z(a) = 0 (i.e., ft =y = 0), (5.1) implies that u" = - A > 0, hence u" > 0 for small / > 0, andso u > 0 for small t > 0 and z > 0 for small u — a > 0.

The proof of Theorem 7.1 will be divided into steps (a}-(k).Proof, (a) A solution z(w) of (7.4), (7.5) with 0 < ft* < 1, y 0

satisfies Z(M) > 0 for u > a as long as 0 < Z(M) < 1.For if there is a point t/j > a, where z(u^ = 0, z(w) > 0 for a 0.

(b) Let /32 > 0 and y > 0. Then there exists an a° = a%*, ft, y) suchthat if a ^ a°, then the solution of (7.4), (7.5) exists on a w-interval[a*, a], Z(M) > 0 on [a*, a], and z(w) = 0 if u = a*.

Choose a° so large that a ^ a° implies that

Thus, by convexity, the statement concerning Z(M) is correct unless, indecreasing u from a, we encounter a first point where z(u^ = 0 beforeZ(M) vanishes. It will be shown that such a point u^ cannot exist if a°,hence a, is sufficiently large. If MJ does exist, then z(u^ > z(u) > z(a) = yfor M! < M < a. Hence

thus

From (7.4) and z(wx) = 0, u^z(u^ + 2A[1 — z(uj] = 0, so that MJ > 0 and

Consequently, the last two formula lines give

which is negative if a° (hence a ^ a°) is sufficiently large. This contra-diction proves the statement (b).

(c) Let z(w) be a solution of (7.4) for large u satisfying z(w) > 0 and0 < Z(M)< 1. Then z(u) -* 1 as u -> oo.

For the proof of this, use (5.1) rather than (7.4). A differentiation of(5.1) gives

By assumption, 0 < u' < 1 andw" > 0 for large/. At points where u'" = 0,it follows that «"" < 0. Hence u'"(t) can have at most one zero. Also,u'" < 0 for large / (for if u'" > 0, then u" > 0 implies that u is unbounded).

Suppose, if possible, that lim w'(0 < 1. Then (5.1) shows that uu" j>c > '0 for large t and some constant c. Hence u'u" ^ cu'/u, and so itfollows that Jtt'2 > c log u + const. -*• oo as t -*• oo. This contradictionproves (c).

Figure 4

(d) Let zv(u), z2(«) be two solutions of (7.4) satisfying either

(7.6) z2(a) = Zl(a) = /32, 0 < £ < 1, and 0 22(a) < 2x(a),

with a = ax = a2; or

cf. Figures 3 and 4. Let w = U}(z) be the function inverse to z = Z}.(M) andK, = z,(^(2)) f°r j ~ 1' 2- Then, as long as both solutions z = «,-(«)satisfy/32 < z < 1,

is increasing in z.

In particular, the arcs z = z^ii) and z = z2(w) in the (M, z)-plane do notintersect for u > a2 as long as /82 < z,2 < 1.

For a giveny, ((/,, V}) is a solution of (6.13)-(6.14). Since the right sidesof these equations are increasing functions in V, — U, respectively, theassertions (7.9), (7.10) follow from assertion (a) and from Exercise III4.1(tf)-(c) if J'lfo) > 0; i.e., zt(ai) > Oin (7.8). The case zfa) = ^(oc^= 0 follows from continuity considerations (in which we first obtain £/2— £/! > 0 is nondecreasing in place of (7.10)).

(e) Let Z^M), z2(w) be solutions of (7.4) satisfying (7.7) and zx(aj) ^ 0.Then there exists a positive c = e(al5 a2, /?) such that if

then the arcs z = z^w), z = z2(«) cannot intersect for u > a2 as long as£2 «i, 2a < 11 cf. Figure 4.

Let z3(«) be the solution of (7.4) with z3(oc2) = /?2, Zg(a2) = ^(aj).Let M = U3(z) be the inverse of z = z3(w) and ^3 = Zs(£/3(z)). Then L/j <«2 < t/3, 0 < ^3 < K! on some small z-interval /?2 z /S2 + d, inparticular, at z = /#2 + <5. By continuity, if c > 0 is sufficiently small in(7.11), then U^ < a2 < f/2 and 0 < Vi< Vl*iz = p* + d. Assertion (e)follows from (d) if a1} a2 are replaced by U^2 + 6), Uz(ft2 + <5), respec-tively.

(f) Suppose that (7.4) has a solution satisfying (6.9)-(6.11). Then thereexists a number y* = y*(a) with the property that the solution of (7.4)determined by (7.5) satisfies (6.9)-(6.11) if and only if 0 y ^ y*

It will first be shown that if y > 0 is sufficiently large, then the solutionof (7.4), (7.5) does not satisfy (6.9)-(6.11). Consider only a — uz ^ — (a + l)z. Divide by ZA and integrate over [a, u] toobtain z y — 2(<x + l)z'^ y — 2(1 + |a|). Integrating over a u a + 1 gives the contradiction z > 1 at u — a + 1 if y > 2(1 + |a|) + 1.

By assumption, there is a y ^ 0 such that the solution of (7.4), (7.5)satisfies (6.11). Let y* = sup y taken over all such y. It follows from (d)that solutions of (7.4), (7.5) with 0 y < y* satisfy (6.11) and solutions of(7.4), (7.5) with y > y* do not satisfy (6.11). The case y = y* followsfrom the hypothesis if y* = 0 or from continuity considerations if y* > 0.

(g) There exist a0, @0 (with 0 < /?0 < 1), y0 > 0 such that the solution of(7.4) determined by

exists for u a0 and

Consider a Riccati equation associated with (7.4) as follows; introducesuccessively

so that (7.4) becomes

Consider the Weber differential equation

If s denotes the logarithmic derivative of a nontrivial solution

the corresponding Riccati equation is

By Exercise X 17.6(a), (7.16) has a solution v = v(ii) such that s = v/v -^—M as « ->• oo. Let y(w) > 0 for large u and let <x0 be so large that

Define ft, > 0, y0 > 0 by

and let z(w) be the solution of (7.4) and (7.12). Thus

in particular r(a0) > j(a0).It will be verified that

for all M ^ a0 for which r(w) exists. On any interval <x0 u ax, wherer(u) ^ S(M) holds, a quadrature shows that w(u) ^ v(u) > 0 by virtue ofr = w/w, s = v/v, and (7.21). In this case, 0 < z(u) < 1 for a0 u ax

since z(u) = 1 — w(u) < 1.By (7.21), r > s holds at u = occ. Suppose, if possible, that there exists

a first u = ocx > a0, where (7.22) fails to hold, then j(at) r(at). But, by(7.18) and the last part of (7.19),

This contradiction proves that (7.22) and 0 < z(ii) < 1 hold for all u oc0.(h) There exists a number /40 = AQ(X, ft) with the property that if a ^ A&

then (7.4) has a solution satisfying (6.9)-(6.11).

Let 0 < 7 y0 and ax > max (a°, a0), where (a0, ft,, y0) is given by (g)and a° = a°(A, /S0, y) is given by (b). Consider the solution ZJ(M) of (7.4)determined by

Then, by (d), the solution zv(u) and the solution z0(w), determined by (7.12),cannot intersect. Since z^u) increases as long as 0 < z%(u) < 1 by assertion(a), it follows that zt(u) exists for u ax and /?0

2 < ZJ(M) < z0(w) < 1 foru > ax.

Applying (6), the solution Z^M) can be extended over an interval [a*, 04]such that ZJ(M) -> 0 as u -> a*. For a given /?, 0 @ < 1, there exists aunique w-value ^0 = y40(A, /3), a* 51 /40 < oo, satisfying z±(A^ = fP. PutXi = ^(^o). so tnat 7i > 0 by (a) or (A) according as /? > 0 or /S = 0.

As before, (d) implies that the solution z(u) of (7.4) satisfying z(a) = /S2

and z(a) = y, where a ^0 and 0 y y^ exists and 0 < z(w) < Zj(w) <1 for u > a. By (c), this proves (h).

(/) There exists a number A(X, /?) such that the solutions of (7.4) deter-mined by z(oc) = p*, z(a) = 0 satisfy (6.9)-(6.11) if a A; but if a < A,then no solution of (7.4) satisfies (6.9)-(6.11).

Let a* = — |23A|~^7r. It will first be shown that if z(w) is a solution of(7.4) and (7.5), where a < a,,,, then Z(M) assumes the value 1 for someu a — a* < 0. In view of (d), it is sufficient to consider the case y = 0.

Suppose, ff possible, that 0 < z(«) < 1 for a < 1. Let r = vv/w as in (7.14), so that (7.15)holds. Note that w = 1 — (P and w = —y — 0 at u = a, so that r = 0and, by (7.15), r = 2A//S < 0. As long as a < u 0, r 0 and 0 <w < 1, it is seen that f ^ — r2 + 2A. Under these circumstances, T(M) J?(M), where /?(«) = — |2A|H tan |2A|^(M — a) is the solution of k =-R2 + 2,*, jR(a) = 0 [for^ = -Rz + 2A is the Riccati equation belongingto v — Ikv = 0; cf. § XI 2(xiv)]. Hence

provided that a < « ^ a — a* <0 and r = wjw ^ 0. Clearly, theproviso r 0 is not needed, for 0 < |2A|^(w — a) < ir/2 on this M-range.Since tan t is not integrable over 0 t < \TT, it follows that w -+• 0 asu -> M0 for some «0 a — a*. This contradiction proves the assertion.

If a = aj has the property that the solution z(w) of (7.4) satisfyingz(a) = jff2, z(a) = 0 assumes the value 1 for some u > a1} then, by (rf),the same is true for all a < aj. Let /4 = sup aj. Then A < oo; in fact,a* < A ^ A0, where y40 is given by (h).

If a > A, then solutions of (7.4) and z(a) = /ft2, z(a) = 0 satisfy (6.9)-(6.11). By continuity, the same holds for a = A. It is clear from (d) thatif a < A, then no solution of (7.4), (7.5) satisfies (6.9)-(6.11).

(;') The number A(A, /?) in (/) is also given by A = inf AQ(h P) takenover the set of numbers /40(A, /?) with the property specified in (h).

In fact, A ^ inf A0(X, {$) is obvious from the inequality A ^ AQ(X, /?) inthe last proof. Also if ax < inf AQ(X, /?), then ax has the property specifiedin the last proof, so that the inequality a! < inf A0(h, P) implies thatQCJ < A; hence A ^ inf y40(A, /?).

(k) Proof of Theorem 7.1. By the characterization of A(X, p) in (i) and(j), equation (7.4) has a solution satisfying (6.9)-(6.11) if and only ifa ^ A(X, p). By (/), for a > A(X, p), there is a y* = y*(a) such that thesolutions of (7.4), (7.5) satisfy (6.9)-(6.11) if and only if 0 y < y*.

It is clear that (e) implies that y*(a) is an increasing function of a. Inparticular, y*(a) > 0 for a > A [since y*(A) ^ 0].

It will now be verified that y*(A) = 0. For suppose, if possible, thaty*(A) > 0. Consider solutions Z^M) and z2(«) of (7.4) determined byz,(A) = p*, z^A) = y*(A\ and zz(A) = pt zz(A) = \y*(A\ respectively.Let « = Ufa) be the functions inverse to z = Zj(u) and Vj(z) = z(f/3(z)).Then £/2(z) > ^(2) and ^(z) > K2(z) for £2 < z < 1. Let a < A and«(«) be the solution of (7.4) such that z(a) = p2, z(a) = Jy*(v4) and letU(z) be the inverse of z(u) and F(z) - z(U(z)\ Let 6 > 0 be fixed. Then,by continuity, V(P + $)> U^p* + 5), V^ + <5) > K(^2 + 5) for smallA — a > 0. Then, by (e), z(u) exists and z(u) < ZI(M) < 1 for large u.Consequently, z(ii) satisfies (6.9)-(6.11). Since a < A, this gives a contra-diction and proves y*(A) = 0.

The argument just completed can be used to show that y*(a — 0) =y*(a) for a > A. By considering solutions of (7.4) satisfying (7.5) andz(a) = p\ z(a) = y*(a) and applying a continuity argument, it is seen thatthe solution of (7.4) determined by z(a) = /32, z(a) = y*(a + 0) satisfies(6.9)-(6.11). Hence y*(a + 0) y*(a). This proves the continuity ofy*(a) for y.^ A. Thus Theorem 7.1 follows from the choice y(a) =2y*(a) since M" = \z.

8. The Case X = 0

When A = 0, the differential equation (5.1) reduces to

the boundary and side conditions are the same:

Theorem 8.1. // 0 < p < 1, then (8.1)-(8.3) has one and only onesolution for every a, — oo < a < ao. If p = Q, there exists a number

A ^ 0 such that (8. l)-(8.3) has a solution if and only if en ^ A; in this case,the solution is unique. In either case 0 < /? t>ut the proofs of somesteps will only be sketched because of their similarity to some of thearguments in the proofs of Theorem 6.1 and 7.1.

(a) If w(0 is a solution of (8.1), then either u(t) = 0 or u"(t) > 0 oru"(t) < 0 for all / for which u(i) exists. For (8.1) is a first order linearequation for «"; thus either u" = 0 or u jL 0.

(b) Let uy(t) denote the solution of (8.1) satisfying the initial conditions

Then uy(t) exists for / ^ 0, and

Since u'y > 0 for all t where uy(t) exists and t/y'(0) 0, it is clear thatthere exists a /„ such that uy(t) exists on 0 < t t0 and uy(t0) = 1. Inaddition, uy(t) ^ 1 for all / t0 for which uy(t) exists. Thus, fort ^ /„,

This makes the assertion clear.(c) The limit «/(<») is a continuous function of y > 0. It is clear that

t0 = tQ(y),mY"(t^ are continuous functions of y. Thus, by (8.8), uv'(t) ->My'(«xi) uniformly as / -> c» on closed bounded intervals of 0 < y < oo.Hence M/(OO) is a continuous function.

(d) The limit wy'(oo) is an increasing function of y > 0. This follows bythe arguments used in the proof of uniqueness in Theorem 6.1.

(e) The problem (8.1)-(8.3) has a (unique) solution for a = 0, /S = 0.For if u(t) is a solution of (8.1) and c > 0, then cu(ct) is also a solution of(8.1). (This follows from a direct verification.) Hence, for a = ft = 0,cu^ct) = WyO) for y — c3, thus wy'(co) = /^"/(oo). Since MI'(CO) 7* 0,there is a unique y > 0 such that wy'(a>) = 1 and uv(t) is the desired so-lution of (8.1)-(8.3) when a = /? == 0.

(/) The limit My'(°o) tends to oo as y -> oo. For a moment, denotewy(0 by Ma/}y(0 to show the dependence on a and /?, as well as y. It is clearfrom the arguments in step (d) in the proof of Theorem 7.1 that " (00)is a nonincreasing function of a and a nondecreasing function of $ and of

y. As in the proof of (e), wa0as(0 = aw101(af) for a > 0. Hence u'a0a»( oo) =a2wioi(°°) -" °° as a -> oo. This implies (/).

(#) If, for a fixed 0 on 0 0 < 1, the problem (8.1)-(8.3) has a solution«°(/) when a = a0, then it has a solution for a ^ a0.

The argument in the step (d) of the proof of Theorem 7.1 shows thatthe existence of a solution a = a0 implies the existence of a solutionwy(0 of (8.1) with M/(OO) < 1, wherey = «°"(0). Thus steps (c\ (d), (f) ofthis proof imply (g).

(/z) If j8 = 0 and a ^ —2, then (8.1)-(8.3) has no solution. Considerthe reduction of the problem (8.1)-(8.3) to (6.8H6.11) with A = 0. Thedifferential equation (6.8) with A = 0 is

and u is .on the range a 5s u < oo. Let a ^ —2 and a ^ M — 1, sothat z = —uzlzA ^ zfzlA. Hence z y + 2z* since /? = 0. In par-ticular, z 2zH and so ZA ^ M — a ^ M + 2. Consequently, z attainsthe value 1 on a < u 5! — 1 for all choices of y ^ 0.

(i) If 0 < 0 < 1, then (8.1)-(8.3) has a solution.Consider the differential equation (8.9) and let zy(t) be the solution of the

equation corresponding to the solution «y(f) of (8.1). Also, let z°(r) be thesolution of (8.9) corresponding to the solution w°(0 of the problem(8.1)-(8.3) with a = £ = 0; cf. (e).

Let t — tp be the unique f-value where w°'(0 = /3 for 0 < /? < 1. Puta0 = M°(^) and let a < 0, i/j > a0; see Figure 5. Note that if y = 0, thenuy(t) s 0, wy'(0 = ft > 0, and so zy(u) s 02, zy(u) = 0. In particular, forsmall y > 0, zy(w1) < z\uj and zy(u) < z°(w) for a0 u «j. For such asmall y > 0, there is a HO, <x0 < «o < Mi» where zy(w0) = z°(w0)» zYM < 2°(«o)-Thus, by the arguments of (d) in the proof of Theorem 7.1, zy(u) < Z°(M) foru > MO. Consequently, zy(oo) 1.

The existence assertion (/) for the fixed a < 0 follows from (c), (d),

and (/). The assertion (i) for all a (for fixed /?, 0 < fi < 1) is a consequenceof (g). This proves (/) and completes the proof of Theorem 8.1.

9. Asymptotic Behavior

In this section, the asymptotic behavior, as t -*• oo, of solutions of(5.1)-(5.3) will be discussed. The results will be based on the asymptoticintegrations of second order, linear differential equations.

If u(t) is a solution of (5.1), put

Then h(t) satisfies the differential equation

Differentiating (9.2) gives

since h' = — u".In order to eliminate the middle term in (9.2), put

so that x satisfies

where

cf. (XI 1.9HXI 1.10). Thus

and, by (5.1),

Since 0 < u' < 1, u" > 0 and u' 1, M -^ f as ^ -> oo, there is a constantC such that for large /

/•oo

In addition, I M" fifr is (absolutely) convergent (since u'(t) -> 1 as / ->• oo),so that ^


provided that

It is easy to check (9.8), for an integration by parts (integrating u" anddifferentiating u"jt6) gives

by (5.1). The last integral is absolutely convergent and lim inf u"(t) = 0as t -* oo. Thus (9.8) holds.

Consequently, (9.7) holds, and thus (9.5) has a principal solutionx(t) satisfying, as / -»• oo,

where c 0 is a constant, while linearly independent solutions satisfy

cf. Exercise XI 9.6.From the last part of (9.6) and u ~ t,

hence

where c° is a constant. Thus (9.9), (9.10) become

In view of (9.4), the equation (9.2) has a principal solution satisfying

while linearly independent solutions satisfy

By treating (9.3) as a second order equation for ti in the same way that(9.2) was handled, it is seen that (9.3) has principal solutions satisfying

and that the linearly independent solutions satisfy

as / -> oo.If (9.1) satisfies (9.11), then, since u ~ t, it follows that th dt < oo;

thus

as / -^ oo. Substituting this into (9.11), (9.11') gives

as / -> oo, where c0 > 0, cl are constants.If (9.1) satisfies (9.12), then u ~ t implies that h = 1 - u' ~ c/2A+0(1) as

r -+ oo. Hence u(t) = f + O(tz*+1+f) as / -* oo for all e > 0. If this issubstituted into (9.12), (9.12') and if it is supposed that A < 0 (and 2A +e < 0), then

as / -> oo, where c0 > 0 is a constant.Theorem 9.1. Let X 0 am/ /<?/ M(/) te a solution o/(5.1)-(5.3). T/zew

r/iere exw^ constants c0 > 0, Cj JwcA that (9.13) A0/d!y as t-+ oo.Proof. For a given t/(f), it has to be decided whether h = 1 — u'

satisfies (9.11), (9.11') or (9.12), (9.12'). If A 0, (9.12) cannot hold, forotherwise h = 1 - i/->0, t-+ oo fails to hold. Thus (9.11), (9.11') arevalid and, as was seen, this gives (9.13).

Theorem 9.2. Let A < 0, 0 £ < 1, a (A, £), >v/iere ^(A, #), y(a)are given by Theorem 7.1. Lef u(t) be a solution o/(5.1)-(5.3). Then thereexist constants c0 > 0, Cj such that (9.13) holds if and only ifu"(0) = y(a);for other solutions u(t) o/(5.1)-(5.3), with a > ^(A, j8) a»i/ 0 < tt"(0) <y(a), /Ae asymptotic relations (9.14) Ao/</ (w///r a suitable constant c0 > 0).

Proof, (a) If t/*(0 is the solution of (7.1), (7.2) and w*"(0) = y(a), then(9.13) holds.

Using the notation of the proof of (g) in § 7, let z*(u) be related to«*(/) by (6.7) and let v(u) be a solution of Weber's equation (7.16) satisfyingv/v ~ —M as M-* oo and y(w) > 0 for large M. Letr*(w)= —z*/(l — 2*)and S(M) = z)/r; cf. (7.14), (7.17).

Then, for large u, r*(u) ^ s(u). For suppose that r*(u) > 5(w) for somelarge u = u0. In this case, r(u) > s(u) for M = MO if Z(M) = — z/(l — z)

belong to a solution of (7.1), (7.2) with w"(0) = y(a) + c for small |e|.But then, as in the proof of (g) in § 7, it follows that r(u) > s(ii) for allw ^ MO and that u(t) satisfies (5. l)-(5.3). This contradicts the main propertyof y(a).

Hence r*(w) s(u) for large u, and so 1 — z*(u) ^ c*y(w) for large uand some constant c* > 0. Since log v(ii) ~ — \uz as u -*• oo, it followsthat A = 1 — «*' cannot satisfy (9.14) and therefore satisfies (9.11).This gives (9.13).

(b) The problem (5.1)-(5.3) cannot have two distinct solutions satisfying(9.13).

Suppose, if possible, that there exist two solutions w^f), ut(t)of (5.1)-(5.3)satisfying (9.13) and, say «/(()) > w2"(0). Let z,(«) be the solution of (7.4)corresponding to w,(r) by virtue of (6.7) fory = 1, 2. Let t/,(z) be thefunction inverse to z = Z,(M) and ^(z) = z,(f/,(z)).

Then z^w), z2(«) satisfy (7.6) and, by (d) in § 7, the assertions (7.9),(7.10) hold. By (9.13), «/(/) —0 and u"(t) ~ t(l - u - } as t^ oo. Bvirtue of (6.7) and «,(/) ~ / as t -* oo, the latter relation implies thatz, — 2w(l - z/^) as u -» oo. Or, since 1 - z,^ = (1 - z,-)/(l + z/"*) ~K^ ~~ zj) as « —>• oo, we have z; -~ w(l — z,) as u —>• oo. Thus K7 '

>

t/j(l — z), also K, -* 0 as z -> 1 [since «/'(/) -»• 0 as r -> oo].The functions (/ = Ujy V = Vj satisfy the differential equation (6.14).

Hence

Consequently, as z ->• 1,

By (7.10), t/2(z) — f/^z) > 0 is increasing, and so there exists a constantc > 0 such that (72(z) — f/^z) §: c > 0 for z near 1. Also (7,(z) -^ oo asz -» 1. Therefore, d(F'1 — K2)/</z ^ ^c > 0 for z near 1, so that Fj(z) —F2(z) is increasing for z near 1. Since Fx(z) — K2(z) > 0 by C^-9)', thiscontradicts the fact that F,(z) -*• 0 as z -> 1 and proves (b) and Theorem9.2.

PART III. GLOBAL ASYMPTOTIC STABILITY

10. Global Asymptotic Stability

Consider a real autonomous system of differential equations

in which solutions are uniquely determined by initial conditions. Let


y0(0 be a solution for t 0. This solution is said to be globally asymp-totically stable when the system (10.1) has the property that if y(0 is asolution for small t 0, then y(t) exists for all t 0 and y(t) — y0(/) -»• 0as t -* oo.

This contrasts with the notion of asymptotic stability of § III 8 in thatit is not assumed here that the initial point y(Q) is near the initial pointy0(0) of y0(t). It will often be assumed that

and that y0(t) is the solution yQ(t) = 0, as in § III 8.Let the function/(y) have continuous first order partial derivatives and

let J(y) denote the Jacobian matrix (df/dy) = (df'/dy*), where j, k = 1,. . . , d. The criteria for global asymptotic stability to be obtained belowreduce in simple cases to conditions involving one of the two inequalities

or

where a dot denotes scalar multiplication. It is very curious that bothconditions (10.3) and (10.4), which in a certain sense are complementary,lead to stability.

The condition (10.3) which states that J(y)x - x ^ 0 whenever the vectorx is in the direction of ±/(y) can be replaced by the condition thatJ(y)f(y) • x ^ 0 whenever x = Gf(y), and G is a constant d X d, positivedefinite Hermitian matrix; i.e., by

Correspondingly, (10.4) can be replaced by

where G is the same as in (10.5). Actually, the conditions (10.5), (10.6)are not more general than (10.3), (10.4) in the following sense:

Exercise 10.1. In (10.1), let/(y) be a function of class C1 satisfying(10.5) [or (10.6)], where G = G* is positive definite. Let G^ be theself-adjoint, square root of G; cf. Exercise 1.2. Introduce the newdependent variable z = Gl/iy in (10.1) and show that the resulting systemfor 2 satisfies the analogue of (10.3) [or (10.4)].

The general criteria to be obtained will actually be generalizations of(10.3) or (10.4) involving nonconstant, positive definite Hermitian matricesG(y).

11. Lyapunov Functions

Recall that if V(y) is a real-valued function having continuous partialderivatives, then its trajectory derivative V(y) with respect to the system

is given by the scalar product

Lemma 11.1. Let f(y) be continuous on an open set E and such thatsolutions of (11.1) are uniquely determined by initial conditions. Let V(y) bea real-valued function on E with the following properties: (i) Ke C1 on E;(ii) V(y) and its trajectory derivative V(y) satisfy

on E. Let y(t) be a solution of (11.1) for t ^ 0. Then the w-limit points ofy(t), t 0, in E, if any, are contained in the set E0 = (y: V(y) — 0}.

Proof. Let tn < tn+1 -> oo, y(tn) -> y0 as n -» oo and y0 e E. ThenV(y(tJ) ~+ nVo) as n -* oo and V(y(t)) ^ K(y0) for f ^ 0. Suppose, ifpossible, that y0 $ E0, so that V(y0) < 0. Let y0(t) be the solution of (11.1)satisfying y0(G) = y0. Consider yQ(t) for 0 f ^ e, where e > 0 is small.Then V(y,(t)) < V(y0) for 0 < t£ e.

The continuous dependence of solutions on initial values (TheoremV 2.1) implies that \\y(t + ?„) — y0(t)\\ is small for 0 t e and large n.Hence | V(y(t + rn)) — K(i/0(r))| is small for 0 <; t and large n. Inparticular K(y(fn + e)) < V(y^ for large «. But this contradicts F(y(0) =v(y<i) for r ^ 0 and shows that y0 e £"0.

Corollary 11.1. Let f , V be as in Lemma 11.1, where E is the y-space,and let V(y)-> oo as \\y\\ —>• oo. Then all solutions y = y(t) of (11.1)starting at t = 0 exist for t ^ 0 and are bounded [in fact, y = y(0 w w /Aese/ (2/:l[/(2/) ^ V(y(®))} for t ^ 0]. //", in addition, there exists a uniquepoint y0, where V(y0) = 0 (i.e., ifE0 reduces to the point j/0), then \\f(y)\\ ^ 0according as \\y — yQ\\ ^ 0, and the solution y0(t) = y0 of (II.I) is globallyasymptotically stable.

Exercise 11.1. Verify Corollary 11.1.Corollary 11.2. Letf(y) 6 C1 for all y and let f(y0) = 0. Let G = G*

fe a refl/, constant, positive definite, Hermitian matrix and let the Jacobianmatrix J(y) = (df/dy) satisfy GJ(y)x • x < 0 for all y 7* y0 and all vectorsx ?± 0. Then the solution yQ(t) = yQ of (11.1) is globally asymptoticallystable [and, in particular, f(y) ^ Of or y y& yQ].

Proof. Put V(y) = G(y — y0) • (y — y0), so that V(y) ^ 0 according as\\y - yj > 0 and V(y) -> oo as ||y|| -* oo. Also %) = 2G(y - y0) -f(y).

It will be verified that P(y) :g 0 according as \\y —y0|| > 0. To thisend, we have

as can be seen by noting that/(y0) = 0 and that the derivative of

with respect to s is J(y0 + s(y — y0))(y — y0). Hence

where the. argument of / is the same as before. This shows that f%) =2G(y — yj -f(y) ^ 0 according as \\y — y0\\ >: 0 and proves the corollary.

Exercise 11.2. Let/(y) 6 C1 for all y and let GJ(y)x - x < 0 for all yand all vectors x, where G = G* is real, positive definite. Let y^t), y2(t) betwo distinct solutions of (11.1) starting at t = 0. Then y^(t\ y8(f) exist andG(yt(t) - &(*))' WO - 2/i(0) decreases for t 0.

Corollary 11.3. Let f(y] e C1 for all y and \\f(y)\\ -> oo as \\y\\ -* oo.Let G — G* be a real, positive definite matrix and let J(y) = (dffdy) satisfyGJ(y)f(y) -f(y) ^ 0 according as \\y - y0\\ ^ 0. Then f(y0) = 0 and thesolution y0(t) = y0of(\l.l) is globally asymptotically stable.

Exercise 11.3. Prove Corollary 11.3 by choosing V(y) = Gf(y) -f(y).The condition \\f(y)\\ -*• oo as ||y|| -»- oo in Corollary 11.3 can be

considerably weakened. Also, the constant matrix G can be replaced bysuitable matrix functions G(y) in Corollaries 11.2 and 11.3. This type ofresult will be considered in the §§ 12-13; cf. Corollary 12.1 and Theorem13.1.

12. Nonconstant G

Let E be a connected open y-set. Let G(y) — G*(y) be a (real) positivedefinite matrix and let G(y) be continuous on E. We can associate withthe matrix G(y), the Riemann element of arclength

if G = (gik(yy). By this is meant that if Cry = y(t\ a / ^ b, is an arcof class C1 in £, its Riemann arclength L(C) with respect to (12.1) isdefined to be

This is readily seen to be independent of any C1 parametrization of C.

We can also introduce a new metric r(ft, y^ on E by putting

taken over all arcs C:y — y(t), a t b, in E, of class C1 joining^ = y(d)and y2 = y(b). The function r(ft, y2) satisfies the usual conditions for ametric: r(ylt t/2) = r(y2, ft); Kft, ft) 0 according as lift - ft|| ^ 0;and the triangular inequality

Remark. Since G(y) is continuous and positive definite, it follows thatif EI is a compact subset of £, then there exist positive constants clt c2 suchthat Ci dy - dy ^ ds2 ^ c2 </y • */y. Thus if C is an arc of class C1 in £1?

then CiLe(C) ^ L(C) c2Le(Q, where L(C) is the Riemann and Le(C)the Euclidean arclength of C. In particular, if y° is an arbitrary point ofE and e > 0, there exists a d — d(y°, e) > 0 with the property that iflift - ftll ^ <5» lift — SbH = *» then> in determining Ky1} t/g) in (12.3), itsuffices to consider arcs C in ||y0 — y0|| ^ e. Hence if y° is an arbitrarypoint of £, then there exists a small d = <5(y°) > 0 and a pair of positiveconstants c10and c20, depending on y°, such that c10 ||ft — ya|| ^ r(ft, y2) ^20 lift - ftll if lift - ftll ^ <5 (or if Kft, y0) «5) fory » 1, 2.

The Riemann element of arclength ds will be called complete on they-set £ is it has the property that the convergence of the integral in (12.2)for a half-open arc C:y = y(t) of class C1 in E defined on a half-openinterval a t • b exists andis in £; i.e., d!s is complete if half-open arcs C of finite length (12.2) havean endpoint in E.

This concept of "complete" is equivalent to the usual notion that theset E considered as a metric space with the metric (12.3) be complete. Butthe fact will only be used in § 13. The following simple lemma will be usedsubsequently.

Lemma 12.1 Lei E be the y-space or the part of y-space \\y\\ 2* a > 0exterior to a ball. Let G(y) be of class C1 on E and G(y) = G*(y) positivedefinite. Then ds in (12.1) is complete on E if and only if every unboundedarc C:y(t), a / < i(^oo) of class C1 in E has an infinite Riemannarclength L(C).

Exercise 12.1. (a) Verify Lemma 12.1. (b) If, in Lemma 12.1, G(y) =/?%)/, where p(y) > 0 is a function of class C1, then a sufficient conditionfor (12.1) to be complete on £is that^y) c > 0, or that ||y|| p(y) ^c>0, or, more generally, that


Let/(y) be of class C1 on E and let y(t) be a solution of

on some /-interval. Let x(t) be a solution of the equations of variation of(12.7) along (12.3), i.e., a solution of the linear system

where J(y) is the Jacobian matrix J(y) — (dfjdy). Let G(y) e Cl on E andconsider the function

Its derivative with respect to / is easily seen to be given by

where B(y) = (bjk(y)) is the d x d matrix with elements

In particular,

(12.11) Y(y) = G(y)f(y)-f(y) implies that V(y) = 25(y)/(y) -f(y)

since y'(0 =/(KO) is a solution of (12.7).The matrix B has occurred in (V 7.11) and Lemma V 9.1 for a similar

purpose, where G = A*A. (For readers familiar with Riemann geometry,it can be mentioned that if f(y) is considered as a contravariant vectorfield; ft the components of its covariant derivative; and B°(y) = (bjk(y})is defined by bQ

jk = £ gjmf™> then B •*- B° is a skew-symmetric matrix.Thus (12.9) is not affected if B is replaced by £°.)

Note that if G(y) = G is a constant matrix, then B(y) = GJ(y).Theorem 12.1. Let f(y) be of class C1 on an open connected set E

containing y = 0. Let G(y) = G*(y) be of class C1 on E, positive definitefor each y, and such that ds in (12.1) is complete on E. Let q>(r) > 0 benonincreasing for r 0 and satisfy

Finally, let r(y) = r(y, 0) [cf. (12.3)] and

Then (i) every solution y(t) of y' —f(y) starting at t = 0 exists for t 0;(ii) t/(oo) = lim y(t) exists as t —> oo and is a stationary point, f(y) = 0aty = y(oo); (iii) ify(t) & y(oo), then

is a decreasing function for t ^ 0 and tends to 0 as t —>• oo; (iv) the set ofstationary points [i.e., zeros off(y)] is connected; hence (v) if the stationarypoints of f(y) are isolated (e.g., //del B(y) ^ 0 whenever f(y) = 0), thenf(y) has a unique stationary point y0 and the solution y0(t) s y0 is globallyasymptotically stable.

The proof will give a priori bounds for solutions y(t). Let

and let T(r) be the function inverse to 4>(r), then it will be seen that

In addition, r(y(0) = c implies that

and since

we have

If (12.12) does not hold but the initial point y(0) of a particular solutiony(t) is such that the definition of c in (12.16) is meaningful, then assertions(i)-(iii) are valid for this y(t).

Exercise 12.2. Using the example of the binary system where/(y) =(—y1 ,0)» G — !•> and E is the y-space, show that the additional assumptionin (v) concerning isolated stationary points cannot be omitted.

Proof. (i)-(Hi). Let y(i) be a solution oft/' =/starting at / = 0. Thenthe Riemannian length of the solution arc y = y(t) over [0, t] is the integralof v*(t), where v(t) is given by (12.14). Put r(t) = r(y(0) = r(y(t), 0) and

By the triangular inequality (12.4) and by (12.3), it is clear that r(t) < u(t).Since y(t) is a solution of y' =f, its derivative x = y'(t) — f(y(t)) is a

solution of the equations of variation (12.7). Thus (12.8)-(12.9) hold andso, by (12.13), v'(t) ^ -2<p(r(OXO- Consequently,

By (12.19), u = v* and M" = [y1^]' ^ -^(r(O)w'. Since <p(r) is non-increasing, r(0 M(/) implies that

Integrating over [0, /] gives

Since w(0) = r(0) and «' = vl/*, this can be written as

by (12.15). This inequality, ww 0, and r(f) ^ w(f) show that

Consequently, (12.16) holds on any interval [0, t] on which y(t) exists.Thus the monotony of 9? and (12.20) give

and so the inequalities in (12.17) hold on any interval [0,/] on whichy(t) exists. Consequently,

and if 0 / < ct>(<oo) is the right maximal interval of existence ofy(t), then the last integral converges as / -> <o. Since this integral is theRiemann arclength of the arc y — y(t\ 0 / < at, the completeness ofds implies that y((o) = lim y(t) exists as t -*• at and is in E. But then<a = oo by Lemma II 3.1. This proves (i), the existence of y(oo) in (ii),and (iii). The fact that/(y) = 0 at y = y(oo) follows from (iii) since theintegral in (12.21) is convergent as t -> oo.

Proof, (iv)-(v). Let EQ be the set of zeros off(y). In order to showthat EQ is connected, define a map P : E -> E0 of E onto E0 as follows:if y(t) is an arbitrary solution of y' =/for / ^ 0, put Py(Q) = y(°°)- Itis clear that the range of P is £"„. Since continuous maps send connectedsets into connected sets, it will follow that E0 is connected if it is verifiedthat P is continuous.

Let y° be an arbitrary point. It will be shown that P is continuous aty = i/0. If I?/ — y°\\ is small, there exist positive constants c10, c20 suchthat c10 \\y - y°\\ ^ r(y, y°) ^ c20 ||y - y°||; cf. the Remark following(12.4). Thus, in proving the continuity ofP : E -*• £"0 at y = y°, it can besupposed that E carries the metric defined by r(y^ y2) in (12.3).

Let y°(t) be the solution of y(t) satisfying y°(G) = y° and Md the Riemannsphere r(y°,y) ^ 6. Since c in (12.16) depends only on the initial pointy(G) of the solution y(t), it follows that the inequalities (12.17)-(12.18)hold with a constant c > 0 which can be chosen independent of y(G) e Ms.Hence if e > 0 is fixed, there exists a number te independent of y(0) e M6

such that r(y(t), y(aoj) < c if / ^ t . Let <5 = <5(«) > 0 be so small that

%(0,y°(0) < « for 0 t£ tf if r(y°, y(0)) < £(e). Consequently,r(«/°(oo), y(oo)) < 3e if r(y°, y(0)) < <5(e). This proves the continuity of Pat t/ = y° and completes the proof of (iv).

The main part of (v) follows from (iv). As to the parenthetical part of(v), note that if f(yj = 0, then B(Vo) = G(yJJ(y9) by (12.10), whereJ(y) = (dfjdy). This completes the proof of Theorem 12.1.

Corollary 12.1. Consider a map T of the y-space into itself given byT:yi =f(y), where f(y) is of class C1 for all y. Let the Jacobian matrixJ(y) = (dfldy) satisfy det J(y) * 0 and J(y)x • x ^ -?(||y||)|M|» for all xandy, where <p(r) > 0 is nonincreasing for r 0 and satisfies (12.12). ThenT is one-to-one and onto [i.e., T has a unique inverse T~l: y = /i(^i) defined

for all t/j]. In particular, there is a unique point y0 where f(y) = 0; further-more, the solution y0(t) = y0ofy' = f(y) is globally asymptotically stable.

Proof. Let £be the y-space and G — /in Theorem 12.1, and replacef(y) by/(y) - 2^ for a fixed y°.

If x0 is fixed and the condition on J(y) is relaxed to J(y)(f(y) — z0) *(f(y) - x0) ^ -<p (\\y\\) \\f(y) - x0\\\ then it follows from Theorem 12.1that the equation/(y) = a?0 has at least one solution y.

Exercise 12.3. In Corollary 12.1, show that Tis one-to-one and ontoif "the assumption that f(y) is of class C1 and the condition on .%)" isrelaxed to the following: "/(t/) is continuous and satisfies

for all ylt yz in the sphere |]y|| ^ r." (This generalizes the first part ofCorollary 12.1; cf. the proof of Corollary 11.2.)

Exercise 12.4. (a) Let/X^eC1 for all y; <p(r) as in Theorem 12.1,p(y) > 0 of class C1 for all y and satisfying (12.5). Ifp(y) =/(y) • grad/>(^)and J(y) = (df(dy), assume that

Show that assertions (i)-(v) of Theorem 12.1 are valid with G(y) = pi*(y)I.(b) Verify that if ||/(y)|| ^ 0 according as \\y\\ 0 and p(y) satisfies allconditions of (a) except that p(G) = 0, that p(y) is merely continuous aty = 0, and that (12.22) holds only for y ^ 0, then the conclusions of (a)still hold, (c) What are the conditions on f(y) in order that part (b) beapplicable withXs/) = ll/(y)ll ?

13. On Corollary 11.2

In order to obtain an analogue of Corollary 11.2 in which G is replacedby a matrix function G(y), a property of complete Riemann elements ofarclength will be needed.

A set of points y on E will be said to be bounded with respect to themetric r(ylt y2) if for some (and/or every) fixed point y° E E, there is aconstant such that r(y, y°) ^ c for all y in the set.

Lemma 13.1. Let G(y) = G*(y) be continuous on a connected open y-setE and positive definite for each y&E. If ds in (12.1) is complete on E,then every subset ofE which is bounded with respect to the metric r(yly y^ hasat least one cluster point in E, hence a compact closure in E. In particular,every such subset ofE is bounded (with respect to the Euclidean metric on E).

The converse of this assertion is clear. Lemma 13.1 will be used onlyfor the proof of Theorem 13.1. Its use can be avoided, of course, bymaking the redundant assumption in Theorem 13.1 that ds has theproperty specified in Lemma 13.1, as well as being complete. In mostapplications, this fact will be clear. For a proof of Lemma 13.1, seeHopf and Rinow [1].

Theorem 13.1 Letf(y) e C1 on an open connected y-set E. Let G(y) =G*(y) be of class C1 on E, positive definite for fixed y, and such that ds in(12.1) is complete on E. Let the matrix B(y) defined by (12.10) satisfy

Then every solution y(t) of y' =f(y) starting at t = 0 exists for t 0;furthermore, ify^t), yz(t) are two distinct solutions for t 0, then

In particular, if there exists a stationary point y0, /(y0) — 0> then everysolution y(t) ^ yQ satisfies

(andf(y) ^Qfory^ 2/0)-The following proof could be simplified by using the known fact that if

y\-> Vz are two points of E, then there exists a geodesic arc C of class C2

joining them such that r(yl5 */2) is the Riemann arclength L(C) of C.Proof. Let y-^t) be a solution of y7 =/for 0 t T. Let yv = ^(0),

y& * Vi, and r0 = r(y» y2). The set of points ET: {y\r(y, y^i)) < r0 + 1for some t, 0 / ^ T} has a compact closure in E, by Lemma 13.1. Thus,by the Remark following (12.4) and a similar remark applied to the formB(y)x- x in (13.1), it follows that there is a constant c > 0 such that

Let C0:y = z(«), 0 u 1, be an arc of class C1 satisfying «(0) = yltZ0) = Vs and L(C0) is so near to r0 = r(ylt y^ that

Thus it follows that C0 c E?.

Let y(t, «) be the solution of y' =f satisfying y(Q, u) — z(«), so thaty(t, 0) = yj(/), and t/2(0 = y(l> 1) is the solution starting at yz for t = 0.Then y(t, u) is of class C1 on its domain of existence; Theorem V 3.1.

By Peano's existence theorem, there is an 5 > 0 independent of u suchthat the solution y — y(t, u) exists for 0 t ^ S for every fixed u,0 u 1. It will be shown that y(t, u) exists for 0 < t T. This is clearfor small u 0 by Theorem V 2.1. Suppose, if possible, there is a leastu-value e, 0 < e 1, such that if the right maximal interval of y(t, «) is0 < t < co, then co ^ T.

For fixed t, 0 / < co, let L(f) be the length L(C(0) of the arc C(t):y = y(t, w), 0 <^ w < e; i.e.,

where yu = dyjdu. Note that a; = yu(t, u) is a solution of the equations ofvariation (12.7) with y(r) = y(f, M). Hence the integrand in (13.6) isvl/i(t, u) where, for fixed u, v(t, u) is given by (12.8) with y(i) = y(t, u),aKO-y.CM'). By (12.9),

For small fixed / > 0, the arc y(t, u) is in ET since y(0, w) = z(u) is. Inthis case, (13.4) implies that

Consequently, by (13.5),

as long as y(/, u) e ET. The inequality (13.7) shows that as / increasesfrom 0 to co (^ T), y(/, w) cannot leave ET. Thus (13.7) is valid for0 / < o>.

It follows that the integral (13.6) with t = co is convergent. Thus, bythe completeness of ds in (12.1), lim y(t, u) exists as u —*• e and is in E fort = co (as well as for 0 t < co). This limit is y(t, t), so that this solutionexists for 0 / ^ o>. This contradicts the fact that 0 t < co is the rightmaximal interval of existence of y(t, e). Consequently, the solutiony(t, w) exists for 0 t T for every j/, 0 w 1.

In particular, y£t) = y(r, 1) exists for 0 / ^ T. Also, if * = 1 andt = Tin (13.6), and (13.7), then it follows from (12.3) that rfy^T), yz(T)) <e^TL(C0). Hence, by (13.5), r(yv(T\ y2(TJ) < r0 = f(yx(0)f y2(0)). Since71 can be replaced in this argument by any /-value 0 < / 2| T, it followsthat (13.2) holds on any interval on which y^t) exists.

It will now be shown that y^(t) exists for / ^ 0. If y^T) = ^(0), theny\(t) is periodic and exists for all T. If y^T) ^ y\($\ apply the argumentjust completed with y2 = y^T). Then yz(t) = yt(t + 7) exists for 0 / T; i.e., y±(t) exists for 0 t IT. Repetitions of this argument showthat ya(0 exists for / ;> 0.

If y0 is a stationary point and y(t) & y0 is a solution of y' =/, thenKy(0» 2/o) is decreasing. In particular,/(y) 0 for y ^ yQ and r(y(t), y0)is bounded for / ^ 0. Hence C+: y = y(t), f ^ 0, has a compact closure in£by Lemma 13.1.

Note that V(y) = G(y)f(y) -f(y) ^ 0 satisfies V(y) = 2£(y)/(y) -/(y) 0 by (12.11) and (13.1). Consequently, by Lemma 11.1, the co-limit pointsof C+ are zeros of V(y). But (13.1) shows that V(y) = 0 if and only iff(y) — 0. Hence y(t) ->• y0 as / -*• cx>. This proves Theorem 13.1.

Exercise 13.1. In Theorem 13.1, let assumption (13.1) be relaxed tojB(y)a: • a; < 0 for y e £" and for all x. Show that the following analogue ofthe first part of Theorem 13.1 is valid: (a) r(yi(f)» ^(0) 's nonincreasingand (b) ify(t) is a solution of y' —f(y) starting at t = 0, then y(t) existsfor / 0.

14. On "/GO* • Jt ^ 0 if jc -/CO = 0"

The last three sections have been concerned with the conditionJ(y)f(y) 'f(y) = 0 and its generalizations. In this section, the condition"J(y)x • x <: 0 if x -f(y) = 0" and generalizations will be considered.If/(y) ^

is of class C1 on a set £, and dsz = G(y) dy • dy& positive definite Riemannelement of arclength with C?( ) = G*(y) of class C1, the generalizedcondition is

where ,%) is defined in (12.10).In the first theorem, it will be supposed that

and that

(14.4) y = 0 is a locally asymptotically stable solution of (14.1);

cf. § III 8. By the domain of attraction of y = 0 is meant the set of pointsy0 of E such that solutions y(t) of (14.1) starting at y0 for / = 0 exist fort ^ 0 and y(t) -*- 0 as / -*• oo. If £" is open, the domain of attraction isopen.

Theorem 14.1. (i) Let f ( y ) be of class C1 on a connected open y-set Econtaining y = 0 such that (14.3) and (14.4) hold, (ii) Let «*> 0 be sosmall that \\y\\ e is in the domain of attraction ofy — 0 and let Ef be theset of points ytE satisfying ||y|| ^ c. (iii) On £e, let G(y) = G*(y) be ofclass C1, positive definite for each y and such that dsz = G(y) dy • dy iscomplete on Ef. (iv) Finally, let (14.2) hold for y e Ee and allx. Then y = 0is globally asymptotically stable.

Before proceeding to the proof, it will be of interest to formulate somecorollaries.

Corollary 14.1. Let (i), (ii) of Theorem 14.1 hold with E the y-space.Let there exist a function p(y) > 0 of class C1 on Ee: \\y\\ ^ e satisfying

and, ifp(y) =/(y) • grad p(y) andJ(y) - (dffdy), then

Then y — 0 is globally asymptotically stable.Note that if p s 1, (14.6) reduces to J(y)x • x < 0 when x -f(y) — 0.Exercise 14.1. Verify Corollary 14.1 by choosing G(y) — pz(y)I', cf.

Exercise 12.4.Exercise 14.2. Verify that Corollary 14.1 is applicable with p(y) =

il/(y)ll if

Corollary 14.2. Let (i) in Theorem 14.1 hold with E the y-space and let(14.7) hold. Let the eigenvalues of the Hermitian part Ja(y) = \(J + J*)ofJ(y) be ^(y) > • • • Ad(t/) and let

Then y = 0 is globally asymptotically stable. If d = 2, fAen (14.9) isequivalent to

Exercise 14.3. Verify this corollary by showing that (14.9) implies(14.8).

Theorem 14.1 will be deduced from the following result dealing with asolution y0(/), not necessarily y0(/) s 0.

Theorem 14.2. Let /(y) 0 be of class C1 on an open y-set E. LetG(y) = G*(y) be of class C1 on E and positive definite for yeE and let

(14.2) hold. Let y0(t) be a solution o/(14.1) on the right maximal interval0 ^ / < c o ( ^ o o ) with the property that there exists a number a > 0 suchthat r(y0(t), dE) > a > 0. Then there are positive constants d, Ksuch that,

for any solution y(t) of (4.1) with r(y0(0), t/(0)) < 6, there exists an increasingfunction s = s(t\ 0 / < a>, such that 5(0) = 0, 0 / < s(co) (<| oo) isthe right maximal interval of existence of y(t), and r(y(s(t)), y0(0) =Kr(y(G), y0(Q))for 0 < t < a>.

In this theorem r(yv, yz) is the metric associated with ds* = G(y) dy • dy.It is not assumed that ds is complete on £and the assumption r(y0(t), dE) >a means that if Cry = x(u), 0 u < 1, is a half-open arc of class C1

starting at the point y0(t), i.e., a;(0) = y0(0> and tne Riemann length L(C)is finite and L(C) a, then z(l) = lim x(u) exists as w ->• 1 and a;(w) e £".Roughly speaking, r(y0(t), dE) > a means that y0(?) is at least a distancea (in the /--metric) from the boundary dE of £ [i.e., the set {y:r(y, y(t)) ^ afor some /, 0 / < to} is in E}. Theorem 14.2 will be proved in the nextsection and Theorem 14.1 in § 16.

Exercise 14.4 Let f(y) ^ 0 be of class C1 on a bounded, connectedopen set E. Let y(t), t > 0, be a solution of (14.1) such that dist (y(t), dE) >a > 0, where "dist" refers to the Euclidean metric, (a) Let

satisfy y(y) ^ — c < 0 for some constant c > 0. Then the set of co-limitpoints of y(i), t ^ 0, is a periodic solution y0(t) of (14.1) which has d — 1characteristic exponents with negative real parts (and so is asymptoticallystable, in fact, Theorem IX 11.1 is applicable). See Borg [3]. (6) Showthat the condition y(y) ^ — c < 0 for y E Ecan be relaxed to the conditionT(0 — I» C — c(t — s) for 0 s < / < oo and a pair of constants

C, c > 0, where T(/) = y(y(u)) du. See Hartman and Olech [1].Jo


In this proof, notions of the length of a vector a; at a point y or orthog-onality of vectors xlt xz at y refer to the Riemann geometry; i.e.,(G(y)x - x)*A or G(y)x1 - x2 = 0. Similarly, arclength of an arc C refers toits Riemann arclength L(C); cf. (12.2).

Let yQ = y0(0) and IT the piece of the hyperplane

»:G(y«)/(y«)-(y-yt)-0through y0 orthogonal to f(y0) with a parametrization rr: y = z(p, u)for 0 ^ p ^ plt where u is any unit vector orthogonal to f(y0) at y0,z(p, u) = y0 -f /w. It is clear that all solutions of (14.1) with initial pointsnear y0 cross n.

For a fixed u, let y = y(t,p) be the solution of (14.1) determined by theinitial condition y(Q, p) = z(p, u). Let 0 < t < a)(p) <; oo be the rightmaximal interval of existence ofy(t,p). Thus y(t, 0) = y0(t) and eo(0) = o>.

For fixed u, consider the 2-dimensional surface S: y = y(t,p) definedon a (/, />)-set containing 0 / < o>(/j), 0 p ^ p^ On 5, consider thedifferential equation for the orthogonal trajectories to the parameter arcsp — const, [i.e., to the solution paths of (14.1) on S] determined by the

Figure 6

relations G(y)f(y) • dy\dp = 0, where y = y(t, p) and t — t(p). Let / =T(p, q) be the solution of this differential equation,

with initial condition

(so that the corresponding orthogonal trajectory starts at the pointy = y0(q}); see Figure 6. In (15.1) and later, subscripts/;, q denote partialdifferentiation.

Since the right side of (15.1) has a continuous partial derivative withrespect to the dependent variable t, the solution t — T(p,q) of (15.1),(15.2) is of class C1 and has a continuous second mixed derivative TM =TVQ; see Corollary V 3.1. Furthermore, as a function of p, TQ(p,q)satisfies a homogeneous linear differential equation by Theorem V 3.1,hence

since (15.2) implies T9(Q,q) = 1 > 0. The reparametrization of S givenby

will be used.Let D be the open subset of E which is the union of the "spheres"

K& y<»(0) < «/2 for 0 < t < w. Thus r(y, dE) £ a/2 if y e D. There isa constant ft > 0, independent of u, such that T(p, 0) exists for 0 :g /? ftfor every u and

Thus the orthogonal trajectory starting at y0 reaches, for every fixed «,the solution path of (14.1) through ar(0, ft) and has an arclength satisfying(15.5). Since r(y0,x(Q,p)) is less than or equal to the integral (15.5) for0 p ^ ft, it is seen that x(Q,p) e D for 0 < p ft.

The set of -values for which t — T(p, q) exists for 0 /?::£/? is open byTheorem V 2.1. Let q0, 0 < q0 < to be the least upper bound for this set.Put

for 0 5: or ±s T 5s ft and 0 :g ^ < q0. It will be shown that L(q, <?, T) isnonincreasing with respect to q for fixed a, r. Let v(q, p) denote the squareof the integrand. It will suffice to show that dv/dq ^ 0. To this end, notethat (15.4) implies that a;, = TJ(x) and, hence that xqv = TJ(x)xv + TQpf(x).In addition, G(x)f(x) • xp = 0 by the definition of T(p, q). Using thesefacts, (12.10), and

give 9y/a^ = 2TaB(x)xp - xv. Consequently, dvfiq ^ 0 by (14.2), (15.3).Thus L(q, 0, /8) L(0,0, /3) if 0 ^ < ^0. Since the integral in (15.5)

is £(0, 0, ft), it follows that L(q, 0, ft) < a/2 and so, x(q, p) e D for 0 ? <q0,0£p^p.

It will now be shown that

for Q p ^ ft, where 0 / < &>(/>) is the right maximal interval ofexistence ofy(t,p). Suppose, if possible, that (15.7) fails to hold for somep = o, 0 p° ft. Since the arguments to follow do not depend on theposition of/?0 on [0, ft], assume that/?0 = ft. Thus (15.7) fails to hold forp — ft. In particular, y° = lim y(q, ft) exists as q -> q& and j/° is in the

closure f> of D. There exists an orthogonal trajectory y = x(p) on S suchthat x(ft) as y, and x(p) is defined on some interval (0 ) a < /> < ft. Inparticular, the solution y = y(t,p) of (14.1) for a<p^ft crossesy ss z(p) with increasing f near T(^0, ft).

From the continuous dependence of solutions on initial conditions, itfollows that x(p) [and hence xv(p)] is the uniform limit of x(qtp) [andxv(q,p), respectively] zsq ->q0 on every closed interval (a <) T p ^ ft.Thus L(q, T, /?) is continuous at q = ^0 if L(qQ, T, /?) is defined by

and a < T < /?. By the monotone property of L, L(^0, T, /S) L(0,0, ft) <a/2. Thus the arc y = a:(/>), a a/2. Consequently, x(a) = lim x(p) existsas /> -> a + 0 and z(<r) e 5.

The limit relation x(q, p) -> ar(/?) as y ->• ^f0 holds unifofmly on the closedinterval a p ^ ^ if x(q,p) is equicontinuous with respect to /? oncr 2 —/>!-*• 0.It is easy to see that x = x(p) can be continued over the interval 0 p ^

/?. For if a > 0, the arguments above can be applied to p = a, instead ofp — ft, to obtain an extension to an interval a± ^ p ^ ft, where 0 < orx <a. Furthermore, the set of p = o^ which can be so reached is open andclosed relative to 0 p < /?, so that p = 0 can be reached.

This means that y — x(q,p) can be defined for 0 ^ q0, 0 /? ftand hence for 0 ^r ^ ^0 + c, Q p ^ ft for some > 0. But thiscontradicts the definition of q0. Thus the assumption that (15.7) fails tohold for some p = />° is untenable. In particular, q0 = eo.

By (15.8) with p± = 0, />2 =/>, and the definition of x(q,p) in (15.4),

By the continuity of G(y) at y = y0, there exist Constants /S > 0, C! > 0,c2 such that 1(0,Q,p) <c2pifQ ?£p ^ ft and r(z(/?,M),y0) _> Cj/); cf. theRemark following (12.4). Furthermore, ft, clt cz can be chosen independentof M.

Thus, if K = C2/CJ, then

for 0 ? < w and 0 ^/? < ft. Thus if y(/) is a solution of (14.1) with

initial point t/(0) = y(0, p) for some p (and u), Q^p^fi, then theassertions of Theorem 14.2, except for 5(6) = 0, follows with s(t) = T(p, t).On the other hand, if y(0) is near y0, then there exists a small j/J such thaty(t) crosses rr at / = /x near y0; i.e., y(t^ = y(Q,p) for some small/; > 0and some u. Also, it is clear that r(y(Q,p), yQ) is majorized by a constanttimes r(y(0), ?/0). Thus, if K is suitably altered,

for 0 / < to. Thus the assertions of Theorem 14.2, except for 5(0) = 0,hold with s(t) = ?x + T(p, r). In either of the two cases just considered,the modification of s(t) so as to satisfy s(0) = 0 is trivial. This provesTheorem 14.2.


Since the domain of attraction of y = 0 is an open set containing thesphere D(e):||?/|| ^ e, there exists an a > 0 such that £ (e -f- a):||y|| ^ 4- a is also in the domain of attraction. Let E* denote the open setE - S(e) obtained by deleting 2(e) from E. Then/(y) 0 on E* and themetric dsz = G(y) dy • dy satisfies (14.2) on £*.

Suppose, if possible, that Theorem 14.1 is false, then there exists a pointyQ E* on the boundary of the domain of attraction of y = 0. Lety = yQ(t) be the solution of (14.1) satisfying y0(0) = y0> hence

on the right maximal interval 0 t < co.Thus Theorem 14.2 is applicable with E replaced by E*. Hence all

solutions y(t) starting at points */(0) near yQ remain close to y — yQ(t) inthe sense of Theorem 14.2. In particular y(t) c E* on its right maximalinterval of existence. But this contradicts the fact that y0 is on the bound-ary of the domain of attraction of y = 0 and proves the theorem.

Notes

SECTION 1. Theorem 1.1 is due to Hartman [2], [23] and generalizes a result ofMilloux [1] on an equation of the second order; see last part of Theorem 3.1. Resultsof the type (1.9), (1.10) in the first part of Theorem 1.2 were given for the Bessel equationsby Watson [1] (cf. [3, pp. 488-489]) and for general scalar second order equations* byMilne [1] but, as the proof shows, these are consequences of older theorems on firstorder systems. (Szego [1] attributes the result to Sonine.) The last part of Theorem 1.2concerning (1.1!)-(!.13) and Exercise 1.3 are in Hartman [23].

SECTION 2. Most of the results of this section are due to Hartman and Wintner. ForTheorem 2.1, Corollaries 2.1 and 2.2, see [13]; for Corollaries 2.3 and 2.4, see [14];for Exercise 2.6 [except part (rf)J, see [14], [7]; and for Exercise 2.9, see [8]. ForExercise 2.6(</), see Wintner [19]. For a generalization of Theorem 2.1 to Banach

spaces, see Coffman [1]. Exercise 2.8 is a modification of a result of Hartman andWintner [16] and was suggested by Coffman [3J. For related results on a third order,linear equation, see Gregusf [1].

SECTION 3. The part of Theorem 3.1 concerning existence in (3.7oo) is due to Milloux[1]. The reduction of the proof of the theorem via Lemma 3.1 is in Hartman [23]; cf.Wintner [23]. For Corollary 3.1 and Exercise 3.5, see Hartman and Wintner [4]. Theresult of Exercise 3.7 was stated by Armellini; it was proved independently by Sansoneand Tonelli, see Sansone [1, pp. 61-67]; a simple proof is given by Hartman [23]. ForExercise 3.8, see Hartman [22]. For Exercise 3.9, see Hartman and Wintner [15].

SECTION 4. The main results of this section are in Hartman [22]. Parts of Exercise4.2 are given by Schafheitlin [1]. Exercise 4.3(a) is a particular case of a result onnonlinear differential equations of arbitrary order; see Hartman [22]. For Exercise4.30), see Lorch and Szego [1].

SECTION 5. Cf. Weyl [3] who proved an existence theorem for (5.1)-(5.3) for A 0and a = /3 = 0.

SECTION 6. Theorem 6.1 is due to Iglisch [1], [2]. The existence proof in the textand Exercises 6.1-6.3 are due to Coppel [1]; the uniqueness proof is that of Iglisch [2].

SECTION 7. Theorem 7.1 is a result of Iglisch and Kemnitz [1].SECTION 8. Cf. Grohne and Iglisch [1]. The arguments in the text are adapted from

Iglisch and Kemnitz [1].SECTION 9. For Theorem 9.1, see Coppel [1]; for Theorem 9.2, see Hartman [29].SECTION 11. For Lemma 11.1, see LaSalle [3]. Corollary 11.2 is in Hartman [24]

and is a generalization of a result of Krasovskil [1], [2], [3] (cf. [4] and Hahn [1, pp.31-32]). For related results on nonautonomous systems, see Wintner [8, pp. 557-559],Zubov [1], and Hartman [24, pp. 486-492].

SECTIONS 12-13. On the notion of completeness of ds, see Hopf and Rinow [1].Theorems 12.1 and 13.1 are due to Hartman [24]; see Markus and Yamabe [1] forweaker results. Corollary 12.1 is contained in Hadamard [4] (the proof in the text is inHartman [24]); the generalization in Exercise 12.3 is due to F. E. Browder [1] with aproof valid for Hilbert space. Theorem 13.1 is related to inequalities of Lewis [2], [3];cf. Opial [8].

SECTION 14. A condition of the type "J(y)x • x < 0 if x -f(y) = 0" was introduced byBorg [3]; cf. Exercise 14.4(a). The main results of this section are in Hartman andOlech [1]. They were suggested in part by the 2-dimensional case of Corollary 14.2in Olech [3].

Hints for Exercises

Chapter II

2.3. Consider the initial value problem y' —fj~l(y)£, y(Q) =0. Since\fvl(y)£\ = M\£\, Theorem 2.1 implies that this initial value problem has asolution y = Y(t, £) for |/| £ ft/A/|f|. By the implicit function theorem, thissolution is unique and satisfies/(K(/, f)) » /£. (Why?) In particular, if |f| = 1,then y - y(f, I) exists for |f| S A/A/. Replace I by cf, where c > 0, so thaty - F(/, c£) satisfies y' =fv~\y)c£, y(0) = 0 for |/| < ft/Me |||. By uniqueness,Y(t, c£) = Y(ct, |) for |/| < b\Mc |f|. Let |f| = 1 and / = 1 and rename c to/ obtaining r(l, /I) = K(r, I) for |£| - 1, |/| < b/M. Put y =g(x\ whereg(x) = 7(1, a;) for |a;| ^ ft/A/. Thus/(^(x)) s x and, by the implicit functiontheorem, g(x) is of class C1 for |a;| ^ bjM. The Jacobian matrix (dgjdx) is/71 (y) at y = (a:) and hence is nonsingular. Let />„ denote the open y-set whichis the image of |a;| < bjM under the map y = g(x), so that g(f(y)) = y for yon the closure D0 of />„. Then x =/(«/) gives a one-to-one map of D0 onto|x| ^ ft/M (for if there is an »0, |a?0| ^ ft/A/, such that/(j/) = x0 has two solutionsy - 2/i, 2/2e ^o» thence) is not single-valued; i.e.,^(x0) = yv and^o) = 2/z)-In order to show that /)0 contains D^: \y\ ^ ft/A/Mj, apply the result just provedfor x —f(y) on D:\y\ ^ ft to the map y —g(x) on |z| ^ ft/A/; this shows thatthere exists a domain £>l in |x| < ft/A/ such that y = g(x) is a continuous one-to-one map of the closure of Dl onto \y\ ^ ft/A/A/!.

4.1. Consider the continuation to the left of a solution y' =/(/, s/), y(c) =y0; see diagram.

4.2. Let <p(r) be a continuous function which is 0 for 0 ^ r ^ )^, 1 forY\ ^ r ^ [, 0 for r ^ %; in particular <p(l) = 1 and g<r) is bounded. Let

557

U(6) be periodic of period 2*, odd, and U(6) = 2QX(ir - 8)X £ 0 for 0 <0 ^ IT. Consider the differential equation for y = (yl, y2) given by y' —f(yl,y2),where /= (—y29<r)C/(0), yl<p(r)U(6)), y1 = rcos 6, y2 = rsin 0, and considerthe initial point (y1, y2) — (1, 0). Find the differential equations for r, 6. Notethat 8' = U(6), 0(0) = 0 has the maximal solution 0j(/) = -a sin2 / for0 ^ / ^ £*-, 0!(/) = TT for / ^ JTT and the minimal solution 0 = -0i(/). ThusSc, for c ^ £*-, is the circle (y1)2 + (y2)2 = 1.

4.3. (a) Show that the set of solutions y(t) of (4.1) is uniformly bounded for'o = ' = c m which case the proof of Theorem 4.1 is applicable.

Chapter III

4.1. (a) Consider z*(a) < y0k and DRzk(t) £fk(t, z(/)), * = ! , . . . ,</ . Let

y (/) be a solution of the system yk' =/*(/, y) + «, y(d) — y0 for smallf > 0. Suppose that c is such that a < c ^ b and that y,(/) exists for a gs r < cfor all small e. Suppose, if possible, that z*(/) ^ y/(/) does not hold fora ^ t ^c,k — I,... ,d. Then there is a largest tQ, a < tQ < c such that**(/) ^ y *(/)fora ^ / ^ /„,£ » 1, . . . , </but, for some/ z>(/) > yj(i) holds forsome /-values, / > /0 and / arbitrarily near to /„. In particular, zj(/0) = ye'(/0).But then Z)^^) ^f(t0, z(/0)) </J(/0, T/£(/O)) < y%t0), so that z^(r) < y^(t) forsmall / - /0 > 0. Contradiction. Thus zj(f) ^ ye

j(/) for a ^ / S c. Let c tendtoO.

4.1. (b) Let j/^f) be the solution of y' =/(/, y), y^a) = z(a). It suffices toshow that y/W < y0

k(t) for 0 < / ^ b, k = 1, . . . ,< / , since z* ^ yt* ^ y0k.

It is clear that yf < y.Qk or that (yQ

k — yf)' > 0 at r = o for every k ^ j\ hence2/i* < 2/0* f°r small / — a > 0 for k y*j and for k =j. If there is a first /-value/„ > a at which y^/,,) = y0*(/0) for some A:, then (y0* - y^)' ^ 0 at / = /0,but y^/o) 5^ y0(/0) by uniqueness, and the monotony of /* implies that(y0

fc - VikY > 0 at / = /0.4.5.Let U = 1 + r'(u), y(0= / , /o=w0= po- 0, and r= «4sin2 I / M .6.1. Let y(/) be defined for 0 < / ^ 1 and satisfy /2 < ?(/) < 4/2/3, '(0 is

continuous, g/(') ^ 2/ and lim <p'(f) does not exist as / -> 0. Put <o(/, u) =<p'(t)ul<p(t), so that o>(/, M) 3w/2/. Let y,/be scalars and put/(/, y) = 0,4y/3/,4/^/3 according as y < 0, 0 ^ y ^ fit or y > t% and / ^ 0.

6.2. Let/(/, y) be 0, (1 + e)y// or (1 + «)/« according as y ^ 0, 0 < y < /1+ ,or y > /1+ .

6.5. Put w0(/, M) = sup (/(/, yx) -/(/, y^lforlyj - y2l= u[or£ it}. tu0(/, w)is continuous for /o = ^ = 'o + fl ant^ ^ = u = 26 (why?); [^nondecreasing inu (for fixed /);]«>0(/, 0) » 0; |/(/, yx) -/(/, y2)| ^ o>0(/, |yx - ya|); o>0(/, u) £a>(t, u) for /„ < / ^ /0 + a, and 0 ^ u ^ 26. Suppose that u' = t«0(/, w),i/(/0) = 0 has a solution u°(t) & 0 on some interval [/„, /0 + «]. Then«°' = o>0(/, «°) ^ eo(/, «°). But the proof of Theorem 6.1 shows that this isimpossible.

6.6. (a) Suppose that there are two solutions on 0 ^ / ^ e and let u(t) denotetheir difference. Then « « « ' - • • • - u(d) - 0 at / = 0 and |«(d)(/)| ^ A(/),where

Successive integrations of u{d\t) give

Hints for Exercises 559

so that

Multiplying this relation by fk and adding gives

Note that A(/) > 0 is continuous for 0 < t < e and A(0) = 0. Suppose thatu(t) & 0, then A(/) ^ 0. Choose / in the last relation so that A(/) > 0 is themaximum value of A on [0, e]. Then replacing A(.s) by A(/) on the right gives theinequality

The factor of A(f) on the right is 2 efc = 1. Contradiction.6.7. (a) Suppose that there are two solutions y = y^t), yz(t) on 0 ^ t ^ e.

Put m(0 = ly^t) -yz(t)\. Then \DLm(t)\ £ o>//, m(/)). Suppose that thereexists a t = s,0 < s ^ , such that m(s) > a(s). Then the minimal (unique, byCorollary 6.3) solution of u = oj1(r, «), H(J) = m(j) satisfies «(/) ^ w(r) on itsleft maximal interval. Then «(/) can be extended over (0, s] and u( +0) = 0.This contradicts u(s) = m(s) > &(s). Consequently, m(i) ^ <x.(i) for 0 ^ t ^ e.Similarly, if w(e) > 0, then the minimal (— unique) solution of v' = o>2(f, t?),u(e) = w(e) exists on 0 < t ^ and satisfies 0 ^ i</) < m(/). Thus 0 ^lim y(/)//3(0 ^ lim m(/)//3(f) < lim a(/)/0(/) = 0 as / — 0. Thus v(0 = 0, butv( ) = /n( ).

6.8. Let <S(r) = ||2/2(0 - ^(011 and follow proof of Theorem 6.2.7.1. Consider the scalar initial value problem u' — 3u%, w^) = «j.7.2. Consider the differential equation u' — g(u), whereg(u) > 0 is continuous

for all u. The solutions are the level curves (/(/, u) — const, of U(t, u) =fu

t - dslg(s).Jo

8.1. It can be supposed that K(0) = 0 and that V has a strict minimum aty =• 0 for otherwise V(y) can be replaced by ±[^(3/) — F(0)]. Although it isnot assumed that V is of class C1, V has the trajectory derivative V = 0. Hencethe proof of Theorem 8.1 is applicable.

Chapter IV

2.2. (a) By the standard orthogonalization process, there exists a nonsingular,triangular Q(i) such that Z(f) = Y(i)Q(t) is unitary. This can be verified byfirst showing that if ylt..., yd are d linearly independent (Euclidean) vectors,then there exists a nonsingular, triangular d x d matrix R — (rik) with r^ & 0,rik =0 if k > j, such that the (column) vectors 8lt..., dd are orthonormal,where fy = rAyt + • • • + rjyy,. In fact, let = y j /Uy jH, so thatru = l/||yil|.In order to obtain <52, subtract the component of y2 along &i from y2

arjd normalizethe result to be a vector of length 1; i.e., dz = [y2 — (y2 • fi^J/Hyg — (y2 • d!) !!],where a dot denotes scalar multiplication. Thus S2 is a linear combinationrziYi + ''zaXz of Xi. ^2 w'tn ''aa 5^ 0- This process continues. If T is the matrixwith the columns ylt..., yA and A the matrix with columns dlt...t dd, then

A = RT is unitary. Hence A* = T*R* is unitary. This cari be applied toT = 7*(/) to give Q(t) = R* andZ(/) = A*. The construction shows that (?(/)is continuously differentiate. The desired result follows from Exercise 2.1for if C(t) = (cik) and Z(f) has the vectors 21(r),..., zd(/) as columns, then sinceCH = Z*AHZ and cik = 0 if j > kt it follows that c# = £z, • AHzj and z,fc =2;' AHzk ifj < k. [The construction of Q(t) shows incidentally that the diagonalelements of Q(t) are real-valued; also that Q(t) has only real entries if Y(t) hasonly real entries.]

2.2. (b) If i/^/),.... yd(r) are the columns of 7(/), let

in Exercise 2.1.8.2. (c) Induction on d— g. If < / — g = 0, then h(j) — I , the roots

A ( l ) , . . . , A(</) are distinct, and 7(0) is the Vandermonde matrix Wj)*"*"1) fory, A: = 1 d, so that del 7(0) = U WO - *0')] and (8.11) holds. Assume

i<i(8.11) for values of d — g less than a given d — g > 1. Assume that /r(l) > 1.Let A ?* A( l ) , . . . , A(^) and replace the first column of 7(0) by (A*-1, A*-2,.... 1).The resulting matrix, say K(A), corresponds to the case where the distinctroots of (8.9) are A, A( l ) , . . . , A(g) with corresponding multiplicities 1, A(l) — 1,/r(2),. . . . h(g). Thus by the induction hypothesis, det K(X) is obtained from theright side of (8.11) by replacing /r(l) by h(\) — 1 and multiplying by[A - A(l)]*"-1 IT [A - A(/)]*<>>. If the first column of K(X) is differentiated

j>ih(l) - 1 times with respect to A at A = A( 1), the resulting matrix is [h( 1) -1 ]! 7(0).Hence [A(l) - 1]! det 7(0) - ^(l}'1 det K(X)jB#M-1

at ^ = j^).8.3. (a) Determine «,(/) as a solution of (8.1) satisfying w3 = « / = • • • = =

w(d-j-i) ^ 0) M(d-i) „ i at / = a. Then W^t; uj = u^t) * 0 on [a, A). Assumethat 1 k <d and that H^(r) = Wj(t\ ult ..., ut) * 0 on (a, fr) for y —1,. . . , k. Consider the kth order linear differential equation with solutions11 » ML .. ., it* given by

This is an equation of the form u(k} + • • • = 0. Let a < t0 < 6; it will beshown that Wk+l(tQ) ^ 0. Since Wk(t0) ^ 0, it is possible to find constantsclt...,ck such that u0 = cûî) -f • • • + cûk(t) + uk+1(t) has a zero ofmultiplicity of at least £ at t = t0 and, of course, has a zero of multiplicityd — k — 1 at t = a. Hence, by assumption, M^(/O) ^ 0. It is clear thatW(t\ uk+l, ult..., Uk)l\Vk(t) - ^(/; </0, i » l f . . . , iiJ/WWl) = <*>(r) * 0 at f -/„. Hence W^+1(/0) 5^ 0 for a < t0 < b.

8.3. (b) Define /i(/) by h(i) — £[»]. Then (8.19) holds if t> is written in placeof u. Since a0u has */ 4- 1 zeros on (a, b), its derivative (a0v)', hence aâ^)', hasd zeros on (a, b). Thus [ai(aoZ>)T, hence ^[îCÔl't has d — 1 zeros on (fl, A).Continuing this argument, we find a zero / = B for A.

8.3. (c) Let w(r) 0 be a solution of (8.1) having d zeros on (a, 6). Thenu(t) = c1«1(/) + • • • -f cdud(t) for some constants Cj , . . . , cd. Suppose thatCk+i ^ 0 but Cfc+2 = • • • = cd = 0 for some k,0 ^k ^d — I . Let £*[«]denote the function on the left of (*) in the Hint to part (a). Then, by part (b),Lk[u] = 0 at some point t = 0, a < 0 < 6. This contradicts fKfc+1(0) ^ 0.

8.3. (d) Constants clt...,ca such that u — cû^t) + • • • + cdu<£t) has thedesired property satisfy a set of d linear, inhomogeneous equations. These

equations have a unique solution since the only solution for the correspondinghomogeneous system is cx = ... =» cd = 0 by part (c).

8.3. (e) Let u0(t) be any solution of L[u] = 1. By (d), there is a solutionw°(0 of (8.1) such that u(t) = «„(/) + «°(/) has the desired properties.

8.3. (/) Let a < t0 < b and suppose that t0^tlt...t tk. Then w0(/0) ^ 0by(6)sinceL[«0] = 1 •£ 0. Choose the number a such that u(/0) = w(/0) + <xw0(/0).Then w(f) s= v(0 - «(') - a«o(') nas at least ^ + * zeros on fr* $\ if fy» *1contains fc, f l t . . . , tk. Hence LE^KO) = 0 for some 6 on (y, <5) by (6). ButL[w](6) - L[i>P) - a.

9.1. (a) Let F(A, /) = det [M - R - G(/)], F(A) = det (A/ - /?) be thecharacteristic polynomials of R 4- G(t), R, respectively; so that F(A, t) -> F(A)as / -*• QO uniformly on any bounded A-set. In the complex A-plane, let the disc| A — A,-| <j e contain only one zero, A = A,, of F(A). It follows from the theoremof Rouche (in the theory of functions of a complex variable) that, for large /,F(A, /) has exactly one zero A//) in |A — A,| ^ e and A//) -* A3- as f -> QO. Byresidue calculus,

where FA = 3F/3A.and the line integral is taken over the circle |A — A^| = e.9.1. (b) Let RQ = Go^Go. G0(/) = Q^G(t)QQ, and consider the equation

for an eigenvector of R0 + G0(t) belonging, say, to A^r) and having its firscomponent z\i) = 1. Let /?x be the (d — 1) x (d — 1) matrix diag[A2 , . . . , Ad, £0] and similarly G^t) the matrix obtained from G0(r) by deletingthe first row and column. Then det [R0 + C0(r) - A^/)/] = 0, but det[R + Gt(t) — AÔ/d-il ^ 0 for large /. Hence, if zl is a. (d — l)-dimensionalvector, then z = (1, Zj) satisfies [R0 + G0(t) — At(r)]z = 0 if and only if(Rl + Gx(/) - ^(0/^K = -,?!(/), where g^t) =(<?210(0,. - - ,îo(0) andCfiioâio. • • • >gdio) is the first column of G0(t). Thus 2/j(/) = Qtff), whereKO = (1. «!(0) and Zl(0 - -[*! + C^/) - AiW/^J-^/).

9.1. (c) Let d(/) be the nonsingular matrix with columns yfi),..., yk(t\QoPk+i, • • • , QoPdl so that Qi*[R + G(/)]d has the form

where /?22(r) is a (d - k) x (d - k) matrix. Let /<(/) = Qll(R + C)Gi -Ax(0/ and let y, = A(t)et. Thus ux =0, y2 = [A2(0 - ^(t)]e2, ...,vk =[Afc(/) - Ai(OK. Note that /12(/)?/ = 0 implies A(t)y = 0, hence y = (const.) elt

for A = 0 is a simple eigenvalue of A(t), This implies that elt..., efc, y f e + 1 , . . . , vd

are linearly independent, for a linear combination of these vectors is of the formCj*?! + A(c?ez + • • • + CoPa). Let wr = A(t)vr(t) for r = 2 , . . . , d and writewr(t) as the linear combination wr(t) — Arle1 + • • • + Arfcefc + hrk+lvk+1 + • • • +hravd. Then hrl - 0 (otherwise, there is a vector i/ such that Ay = e^). Thus,with respect to the basis elt ez,.. ., ek, vk+l,... ,vd, the linear transformation

A(t) corresponds to the matrix

where H22(t) is a (d — k) ~x. (d — k) matrix. This means that if Q2(/) is thematrix with the columns (elt ez,..., ek, vk+l(t),..., t>d(/)) then Q2*A(t)Q2 --B(t\ i.e., (Q^tT^R + <7(r)K<2i(?i) = B(t) + ^(/)7. The desired conclusionfollows from this in the case k = 1. If A: > 1, repeat these operations on the(d - 1) x (</ - 1) matrix in the lower right corner of B(i) + ^(t)I.

11.1. What is the relation between Q(t) and Q0(t) in the proof of Theorem11.2?

11.3. The equation /V* + (3f — l)u' -f u = 0 has the formal solutionw = 1 + 1! / + 2! t* + • • • 4- k\ tk +

12.3. (a) If /n is not an integer, then there are two solutions J±fl(t) of the form

(12.12) with J^ = £ (-l)*(|02fc+^! T(A: + ^ + 1). If /i > 0 is an integer,*=o

there is the solution J (t) and a linearly independent solution obtained fromf« = J^(t) s~l J~z (s) ds which can be taken of the form

where the last power series can be determined by a substitution into the Besseldifferential equation.

13.2. (a) Apply Lemma 11.2 to X(t). Make the variation of constantsrj — Z(tjy and multiply the resulting system by P~\i) to obtain a systemtDrf = ... to which Theorem 13.1 applies.

Chapter V

5.1. Use Lemma 5.1 and its proof.5.2. Use Stokes' formula (5.3) when 5" is a rectangle a± ^ y1 ^ blta, yj ^ bj

and y* = const, for i 9* l,j. Differentiate the result with respect to biy then b{.6.1. Number the components of y so that (/\, . . . , id) becomes (0,.. . , 0,

(>+i» • • • » f d ) » where 0 ^j^d and (,+1,..., id are distinct integers on the range1 ^ ik ^ rf. Write t/' =/as a system of first order, say y' = z,-z' =f(t, y, z)or dy — z dt — 0, dz —fdt = 0. Choose the 2d x 2</ matrix A = A(tt yt z)so that

Id+j is the unit (d +j) x (d +j) matrix, P = diag [z*>+1,..., z»<«], and the last^ — y components of /4(c/«/ —zdt,dz —fdt) are z'wfe* — fkdy** for A: =y + ! , . . . ,< / .

6.2. (a) Write the differential equations (6.11) as a system of first orderequations for a 2</-dimensional vector (xt y) = (xl

t . . . ,«<*, yl,..., y*) in theform dxi - y*dt = 0, dyi + SZrjJfc2/ycdt = 0, where i = \,. ..,d. Choosethe Id x 2d matnx A = A(x, y) to be

where / is the unit d x d matrix and B — (bik) is the d x d matrix with theelements bjk = Z,r}

kmym. Apply Theorem 6.1.6.2. (c) Write y1 - x,y2 = t/,and<fc2 = h(y)(dx2 + dy\ where A = 1 + 9yX.

If x is used as the independent variable, the differential equations for geodesiesbecome dzyldx2 = £[1 + (dyldx)2]H(y), where #(y) = dlogh/dy. The initialvalue problem y(0) = 0, dy(0)jdx = 0 has the solutions 2/ == 0 and y = x3.

6.3. (a) Let o>, — pudy1 + p^dy* and </ft>, =qjdyl *dy2. Determineft>12 =/>i^2/1 + /72rf2/2 by the equations p1pn -pzpzl = ft, -/>!/>12 + /?2/>n = ?

6.3. (6) It can be supposed that pnpzz ^ 0 at y = 0, otherwise renumbera>j, co2. Let y1 = qKj/2, w1) be the solution of the initial value problem dylldyz ——pnlpu and y1 = ul at yz = 0. Thus y1 = nl(yz, ul) is of class C1 and^(0,0)1 dul 9* 0. Hence y1 — r)l(yz, ul) can be solved for ul = Ul(yl, yz) forsmall ||y||. In terms of (y2, w^-coordinates, coj is of the form toj = rx du1, where7j ^ 0 is a continuous function of (y2, u1), hence of (y1, yz). Similarly, letyz = rf(yl, uz) be the solution of dy*jdyl = —p^ilpzz and yz — «2 when y1 = 0and let «2 = t/%1, y2) be the inverse function. In terms of (y1, w2), o>2 has theform o>2 = 72 </M2 and T2 ?* 0. The transformation u> = U'(yl, y2) is of classC1 with a nonvanishing Jacobian at y = 0 and has an inverse y> — Y*(ul, uz).In (ul, w2>coordinates, o>, = Tj duj where T, is a continuous function of (ul, uz).Thus a =*2Tdul du2 and 7 = J^ 0. It can now be supposed thatT(ul, 0) s r(0, «2) = 1 for otherwise (w1, w2) are replaced by

where ± r(0, 0) > 0. Under the ^-transformation y -* M, the property ofhaving a continuous exterior derivative is not lost. Hence dTJSu2, dTJdu1 existand are continuous; Exercise 5.2. Let h(ul, uz) - (TJT2Y^ > 0. Thus =hT^du1, o>2 = (T^/h) du2, and hence,

Since d<alz = Kco^ A a>2» Stokes' relation is of the form (fceo^ = I KTdu1 du2.

Apply this to the rectangle with vertices (0, 0), (w1, 0), (u1, uz), (0, «2) to obtain

-log TV, w2) - I I " KTduldu2.Jo Jo

8.1. Cf. the proof of Theorem 8.1 and Exercise III 7.3.

Chapter VI

1.2. Let XOE E. Choose the enumeration of the components of x and ofYlt..., Ye such that Bn «• (60(a;0)), where /,_/ — 1,..., e, is nonsingular andlet 52i

= (^iXx))» where / = e + 1 ^ -I- rf and y" = 1,. . . , d. Write

* - (z1,..., z% y\ . . . , y«) and /%, z) - D^r)*^*)- Then the system (*)can be written as (dujdz)Bn + (duldy)Bzl = 0 or equivalently as (1.12). Hence(*) is complete on a neighborhood of x0 = (z0, y0) if and only if o> « —My,z) <fe is completely integrable at (y0, ZQ). Define the (</ + e) x (d + e)matrix B0 by

Let B0 = (ftfc(aO) and fl"1 = (<5,.fc(a;)), wherey, k = 1,..., d + e. Thus y*[«] =EAfcfc) duldxi for A: = 1,..., d + <?, let a, = Zkdik(x) dxk for / = 1,..., d + e.Thus 7j = • • • = Ye = 0 is complete at a;0 if and only if a^j,..., ad+e iscompletely integrable at x0. For any function w(x) of class C1, the total differ-ential du = S(3«/3a;') dx* can be written as

since (8ik) = (pik)~l. If u(x) is of class C2, a simple calculation of the exterior

derivative of du gives

There exist (unique) continuous functions eiik(x) — —eiik(x) such that

is the expression for */ak in terms of the base ctlt..., ad+e. Consequently, (1)and (2) give

for any function u of class C2. By Theorems 3.1 and 3.2, the system««+i, • • • , a«+d is completely integrable if and only if a^± = • • • = a^^ = 0implies that d<x.e+i — • - • = d«.^_a = 0; this is the case if and only if eiik = 0for /,_/ = 1,. . . , e and k = e + 1 e + d. In view of (4), the desired resultfollows with ciik — —eiik for i,j,k = !,... ,</.

3.2. Use Lemma V 5.1.3.4. See the proof of Lemma 3.1.3.5. If d(o exists and o = (o^,..., to,,), then d<ok is of the form dtok =

~'L'L(dfikildyi)dyi *<& + Wpkiidz* *dz*, where /7fcw - -pkji. What arethe conditions onpkii for (3.1)? Use Exercise V 5.1.

6.1. (b) See Exercise 3.5.6.2. Show that if such an A exists, the solution of (6.2) for fixed z0 and rj —y0

is of the form y = T(z)rj + y^z), where Y(z) is a nonsingular matrix of class C1

and yt(z) is a vector of class C1 for z near z0. Also, the map (z,»?) -> (z, y) givesdy — H dz = Y(z) dr). Since the form Y(z) dr\ in dq, dz has a continuous ex-terior derivative, the same is true of the form dy — H dz in dy, dz. Apply Lemma3.1.

8.1. (a) Let «0 = u(y0\ p0 = uv(y0). By Lemma V 3.1, there are continuousfunctions a, bk, ck of («lt ylf/»lf «2, y2,/?2) for small |«t- - «,(, |y< - y0l» I/'* - /'blfor j — 1, 2, such that

and that a = Fu,&* - dF/dy*, ck = 9F/9/ if (u^y^pj - (u2,y*p2). LetA > 0 be small. Apply (1) to (wi.yi,/^) = (u(y), y, uv(u)) and (uz,yztpz) =(w(y + Ay), y + Ay, «y(y -J- Ay)), where Ay is the vector with itsy'th componentequal to 0 or h according asy & m orj = m. It follows from (7.1) that

Let yh(t) be a solution of y' »» c, y(0) = y0, where c = (c1,..., cd) and theargument of c is («(y), y, MV(«), «(y + Ay), y + Ay, uv(y + Ay)) for fixed Ay.It is clear that if e > 0 is small and fixed and ht > hz > ... is a suitably chosensequence, then y(f) = lim yA(f), as h = /rn -*• 0, exists uniformly for \t\ ^ e andis a solution of y' - Fv(u(y), y, uv(y)), y(0) - y0. Note that if /V"(') «=[«(y*(0 + A2/) - «(yft(0)]M and 2/ is replaced by yh(t) in (2), then (2) can bewritten as ph

m'(i) = —aku\h - bm since (yft(f) + Ay)' = yfc'(0 •« c. Thus, ash - An -> 0, y?fc

w(/) -* Su(y(t))ldym and /JAm'(0 - -Fu3uldym - 8Fl3ym uni-

formly for |/| < e.8.1. (c) The condition Fv(u0, y0, p0) * 0 implies that (7.1) can be written in

the form (7.17), where y is a real variable if d - 2. Application of part (a) givesthe desired result; cf. Exercise 7.2 (a) for the corresponding equations of thecharacteristic strips.

8.2. (6) Let u(t), y(t),p(t) be a characteristic strip for F — 0, i.e., a solution of(7.9). Then the relation G(u(t), y(t\p(t}) — 0 can be differentiated with-respectto / to give Gu«' + Gv • y' + Gp • p' = 0. Hence H(u(t\ y(f),/KO) = 0 followsfrom (7.9).

9.1. Introduce the new variables (a;, x + y) instead of («, y).

Chapter VII

3.1. Use (3.1) and a circle 7.3.2. Consider the deformation (1 — j)/0(y) 4- sf(y) =/0(y) 4- J/i(y), 0 ^

s < 1 and small \\y\\.7.2. Make a real linear change of variables to bring A into a suitable normal

form.*10.2. Construct C as follows: Let a > 0 be large. C consists of the part of

the arc C«: \v* + G(u) = a for u ^ a, the line segments v — ±y for |w| ^ awith y > 0 and 4y8 4- G(a) = a and the part of the arc C/>: iua + G(«) =^for u £ -a with /? = ^y2 + C(-a).

11.1. Put z(/, y) =#x(M',,?«"%)), where y - (y1, y2) and z - (z1, z2). Thusz(/, y) is of class C1 and, for small |/| and |A|, z(t + h, y) = z(f, «(A, y)). ThusW, y)ldt - [rfz(/ + /r, y)/^]A_0 - ^ (3z(r, y)/9y')[<W, J/)/*=o-

11.2. (c) By (b), 9N is empty or dN = N. Correspondingly, N° = N or W°is empty. In the first case, N is open and closed, hence N = Af since M isconnected. In the second case, N° is empty implies that N is nowhere densesince N is closed.

13.1. Let « > 0 be arbitrarily fixed. Let i,j be such that 0 < yf — yt < e.

Then translates of the interval yi• < y < yt by m(j — /)a + h for h, m =0, ±1,... have endpoints of the form na + k and cover — oo < y < o>.

14.1. If the torus is cut along the circle F, it becomes a piece of cylindricalsurface. If a half-orbit remains in this piece of cylindrical surface for / J2: 0, itsset of co-limit points is a closed orbit; cf. the proof of Theorem 4.1.

14.2. There is an w0 such that T0 = C+(m0) is a Jordan curve by Theorem14.1. F0 cannot be contracted to a point; cf. the proof of Lemma 14.1. If thetorus is cut along F0, it becomes a piece of cylindrical surface on which thearguments in the proof of Theorem 4.1 become applicable.

Chapter VIII

2.2. Replace (2.1) by dy\dx - Y\X\ cf. the proof of Theorem HI 6.1.3.1. Introduce polar coordinates; cf. (3.14) where a = 0, F1 = xh(r),

F2 = yh(r).3.2. Let F(z)=(-yh(r),xh(r)\ r = (x* + yz)lA > 0 and h(r) a suitably

chosen continuous function for 0 ^ r ^ 1 with //(O) = 0.3.3. Let F(z) = (-A(r)*(0)t/, h(r)k(Q)x), where r - (xz + y2)* > 0, h(r) -

llogrl"1 or 0 according a s O < r ^ | o r r = 0 ; and for the respective cases(a), (b), or (c), let k(6) = 1, |cos 0|K or 0.

3.4. The only characteristic values (mod 2ir) are B — 0, IT. Apply Theorem2.1.

3.6. Cf. Exercise 2.2.4.2. Consider x' — —yx* — y5 — <p(x, y)y, y' = xy* — x3y2 + <p(x, y)x, where

<f(x, y) > 0 according as x2 + y* ^ 0.4.3. (b) Note that if y(r) = a(log l/r)-*/(fc-», then (4.36) has a positive

solution 0 = e(log 1/r)"1^*"11 provided that there exists an e > 0 satisfyingfl(k — 1) — cefc ^ a. This is the case if a > 0 does not exceed the maximumOf €/(k - 1) - C€fc.

4.4. (b) Let y(r) - a(log l/r)-*/<fc-i> and let 0 - «(log l/r)-1/^"" in (4.38).Then r(log \lr)du\dr = a - \u\(k - 1) - CM*] ^ const. > 0 if a exceeds themaximum of ul(k — 1) — c«*.

for a suitable constant y to be determined

Chapter IX

5.3 (A) Suppose that ^ » £| + F(f) has no solution ^. Let 0 < < r2 besuch that F({) - Oif ||f|| > rlf ill + E^F(£)|| < r.if ||f|| ^ r1>and Hf-^H < r2.Then I -+E~l(ES + F(£)) = I + E~1F($) maps the sphere ||£|| £ ra into itself,reduces to the identity on the boundary, and omits the point E~lr). This isimpossible (Brouwer).

5.4. (d) If not, ||zn* — zn|| > const, (c - 2S)n for large n [by part (c) appliedtoSM* instead of SJ.

5.4. («?) Let (yn, zn) - Tn(yQtz0), where z =*g(yj)\ so that ^«G/0), zn(t/0) arefunctions of yc. Let 2/n* = [yn(y0 + hej) - yn(y0)]//», zn* - My0 + **j) -*»<*)]/*. Then (y/,z/). -(^^ + y^_i + ^n-i.Cz^ +ZA»JLi J-^2n2n-i)» where the matrices 7 , Y£n, Zfn, Z^ tend to the Jacobian matrices

dv>z Yn, Zn evaluated at (yn_i, zn_i), as h -* 0. By (d), the functions ynh, zn

h of2/0 are bounded and equicontinuous (for fixed n) and satisfy \\zn

h\\ ^ Ilyn*ll f°r

n — 0 ,1, . . . . Hence, they have limits (un, vn), as A tends to 0 through suitablyselected subsequences; (un, vn) = T**(UO, v0\ u0 = eit \\vn\\ < ||ifj|. By unique-ness, selection is unnecessary.

11.2. If the assertion is false, apply Stokes' formula to § / *(£) dp -f1^) dp.

Chapter X

1.4. (d) In order to obtain the sufficiency of the first criterion, make the changeof variables $ = [I — G0(t)]rj for large / and apply part (c). The sufficiency ofthe second criterion follows from that of the first and from part (b) or can beobtained from (c) by the change of variables f = [/ + G0(/)]~

lj7.ft rT n

4.1. (a) For the inequality involving a(t), write =1 + I in (4.18) and applyJo Jo JT

integration by parts to the second integral.4.1. (b) Note that a(t) is a solution of the differential equation a' + (c — «)<r =

V and integrate this equation over the interval [$, /].4.2. (a) Multiply the differential equation a' + (c — f)a — y for a by 0*~l,

integrate over [0, t], and apply Holder's inequality to the integral on the right.4.6. (b) Without loss of generality, it can be supposed that (?(/) is continuously

differentiate, otherwise G(t) can be approximated arbitrarily closely by suchmatrices. By Exercise IV 9.1, for large f, there is a continuously differentiate,

/•oo

nonsingular Q(t) such that I lQ'(f)\\dt < <», lim Q(i) as t -*• oo is nonsingular;

(2-J[£ + G(t)]Q = diag [^(/),..., Afc(f), E^f)l where £22(/) is a (d - k) x(d — k) matrix and E0 — lim £^2(/) as /-*<*, is of the form E0 = diag (Elt E2);El is an m x m matrix with eigenvalues having positive real parts; £2 k ann x n matrix with eigenvalues having negative real parts. The change ofvariables £ = Q(t)r} gives a system

If rj = (y, x, z), where y is a A:-dimensional, a; an m-dimensional, and z an n-dimensional vector, then the system for r\ has the form

where G;fc is a rectangular matrix. There exist continuous functions y(0» Vo(')/•»

such that v(0 -> 0 as t -* oo and VoW^ < °° and ||G«(OII ^ V<0 + Vo(0

for /,; » 1, 2, 3 and, in addition, ||C,XOII ^ VoW for y = 1, 2, 3. It followsthat solutions r^t) = (y(t\ x(t\ z(t)) for which x(t\z(f) - 0(ll2/(OII) as / -^ ooare bounded.

4.6. (c) If r)(t) 5* 0 is a solution of the system in (b) such that x(t), z(t) =o(\\y(t)\\\ then I = Q(/)j?(r) has the desired form.

4.6. (d)lfy — (y\ ..., yk) in (6), introduce the new variables u = (ul wfc)f t

in place of y, where y1 — u1 exp I A,(j) dsforj**l,...,k. Apply Lemma 4.3.

4.7. Use Exercise IX 5.3 (a).4.8. (a) Let Kn = {(*. y, z): \\z\\ £ £0(||*|j + ||y||)}, *n0 = {(*, y, z): ||*|| <

0||y||}, and S-{(ar,y,z): a: = *0, y « y0, ||z|| < $fl(||*0|| -fc ||y0||)}. ApplyExercise 4.7. To verify that Kn n 7W(5) is not empty, use Exercise IX 5.3 (/>)and show that there exists a z(n) such that the z-coordinate of Tn(x0, y0, z(n)) is 0.

4.9. (a) Note that \\yn+l - yj ^ Vo(")0 + 26) ||i/n!|, hence \\yn+1\\ <llyJI [1 + Vo(«Xl + 26)]. Thus ||yj| 3 c0 ||t/0||, where c0 - O [1 + VoOOO +20)] is a convergent (infinite) product. Hence 2 HyB4.i — yn|| is majorized byc0 WO + 20)£v0(«).

4.9. (6) From the arguments in part (a) and 1 — v*oOOO + 20) ^ 1 —3vo(«) > 0> note that there exists a constant ct such that if the inequality in(4.44)holdsfor« = 0, 1,... .y.then \\yt\\ < Cl \\yt\\ and \\yk+l - Vi\\ <cx \\Vj\\ sk

for k - 0,1,... ,y - 1, where sfc - v0(*) + Vo(A: + 1) + V0(* + 2) + • • • .Show that there exists a G/O), a(J)) such that if Tn(a;0, yw), z(i]) = (a;n, yw, zn), theny> ^y^ and zj =0. As in Exercise 4.8, ||zn|| ^ %6(\\xn\\ + \\yn\\) for n -0,1,...,/ For large;, we have ||*'|| < 6 \\y^ = 6 ||yj|, hence ||«"|| ^ 6 \\yn\\for /i = 0,1,...,/ [For suppose, if possible, that ||ar'|| > 6 \\y>\\, then <$ <(1 - a)0/3 implies that \\xk\\ > 6 \\yk\\ for k - 0,1, . . . ._/ . But then llx^1!! <fi0 H^ II. where 60 = a + <5(2 + 0)(1 + 0)/2 < 1, and so ||a || < 00* ||x°||. Forlarge y, this contradicts ||z'|| > 6 \\yj\ and ||z0|| = ||*°|| < 0 \\yjl.] Thuslk*ll ^ 0 ||yfcll, l|zfc|| ^ 0 ||y*|| for ^ = 0,...,/ It follows that \\yk - yj\ <c\ llyooll Jfc-i for A: = 1,... ,y". Let (y0, z0) be a limit point of the sequence(3/(l)» ZU))» (y<2)» z<2))>

11.2. Use Exercise 4.2.17.1. Since R is not in a Jordan normal form if h < d, let y be a constant

matrix such that Y~1R Y = E — diag [/(I),. .., •/(£)] is a Jordan normal form.It can be supposed that the diagonal elements of 7(1) are 4(1) =0, so thath(l) = h. Since ad_h+l = • • • = ad — 0, Kcan be chosen of the form

where 7A is the unit h x h matrix and 0 is the rectangular (d — h) x h zeromatrix; cf. Exercise IV 8.2 where Y =• 7(0). The change of variables I = Yrfreplaces (17.4) by

Note that the only nonzero elements of G(i) Y are in the first row. The first helements of the first row of G Y are —p^^^ ..., —pA, and the other elementsof this row are linear combinations of plt..., pd with constant coefficients.It is then necessary to consider the factor G(t)YQ(i) of Q~\t)Y-lG(t)YQ(i),where Q(t) is described by (14.1H14.2). Actually, Q(f) is of the form

where Q^t) is an A x h matrix corresponding to the matrix Q(f) in (14.1) withd replaced by A; Q£f) — tl~?D2 where P = h — j + <n and D2 is a constantdiagonal matrix; and 01} Oa are rectangular zero matrices.

I


17.3. (a) Consider the differential equation for the function u'fu and integratethis equation over the interval [t, s].

17.4. (a) and (b) This can be obtained from Exercise 4.6. It is more easilyobtained directly by assuming that q(i) has a continuous derivative and replacingthe second order equation by a first order system for y = (y1, y2) where yl =u' +f*u, y2 = u' -/*«, and /(/) = 1 + q(t) in case (a) and /(/) = 1 - q(t)in case (b). Introduce a new independent variable s, ds =/1'* dt.

17.5. Write (17.23) as a linear system y' = ff(t)y for the binary vectory = («' +/^w, «' -/Kii). For parts (a) and (b), letds=gdt,z = (ylIW,yiW\ fX-g + ih, whereg(t\ h(f) are real-valued,^ > o, W = exp

'tihdt . There are solutions satisfying z2 = o(\zl\) and*1 =0(|z2|) by

Theorem 8.1 and Exercise 11.2. For the end of (A), suppose that / > 0and /He L1. Since/H« e La and fX*-Xf'(f 6 Lp, /'/f 6 L1. Hencelim /(/) > 0 exists as / -> oo, but /H e L1. For part (c), suppose /£ C2.Let y = Q(/)z = (z1 - z26/(a + c), - z16/(a + c) + **), Q~1HQ is diagonalwhere a =/'^, 6 =/'/4/, c = (a2 + 62)^. Introduce the new independent

variables = I \Ref^(r)\ dr. Note that the assumptions imply that bl(a -f c) ->0

as / -*• oo and |[ft/(a + c)]'| // < <x>.

17.6. Introduce the new dependent variable z defined by u = ze~'z/4; cf.(XI 1.9). Apply Exercise 17.5 (b) to the resulting equation for z.

Chapter XI

4.1. If u(t) is a solution of (4.U), (4.2), then Q(t) is a solution of (4.1 A), (4.2).Apply Green's relation (2.10) to/ = -lu(t),g — -Aw(r).

4.3. (a) Denote by (4.19n) the equation which results if u is replaced by unin (4.19). Multiply (4.19n+1) by MO, (4.190) by un+l, subtract, and obtain vn' —~(*n+i ~ ^)ro"n+i«o- Divide this relation by /v^2 and differentiate to get(4.20) with v = vn = /v/0

2(wn+1/M0)'. The last part follows from yn'/*V =

(*-n+i ~ -Wi ~ ^rl(.uu+ljuj) and L'Hopital's rule.4.3. (b) Note that vn = />oW0

2(««+i/«o)'. Write uoi - uXOfori - 0, . . . , k - 1and ult = v^t) for i = 1,. . , k - 2. If u^t) for i = 0, . . . ,k -j - 1, and/>, > 0, r} > 0 have been defined, put pi+1 = l/o«f0. Consider u =c0«o + • • • + cû^ or rather u = c0w00 + • • • + CÛQ,^. Then

Hence/jjttfoôM^M/Uoo)'/^^]' = câ, + • • • + cfc_1tti.fc_8. Continuing this proc-ess gives the desired result with a0 = I/MOO. fli = /'oWoo/Wio. fl2 — Pi"loluw>

4.4. (c) There is a constant c such that £/n(f) = cZ)(/, /0,. . . , tn_i), and£)(/, f 0 , . . . , /n_j) is positive or negative according as the number of inversionsin / , /„ , . . . , /„_! is even or odd.

5.1. Choose a=0 , A = 1 , 0 < 5 < 1 . It suffices to construct a non-negativecontinuous q(t) = q(t; m, s) such that (5.1) has a solution u(i) & 0 satisfying

f1

w(0) = w(l) =0 and m(')?(0 dt ^ [m(s) + «]/$(! - 5). To this end, letJo

0 < 6 < min (s, 1 — s), u(t) a function having a continuous, nonpositive secondderivative on 0 / ^ 1, such that u(i) - /andw(f) = (1 - t)(s - <5)/(l - s - S)on 0 ^ / ^s - d and s + d t £ 1, respectively. Let y(f) = 0, -w"/", or 0according a&Q<t£s —6,s — d£tga + d t o T S + 6£t£l. Then

Use w(r) ^ /w(5) + « if |/ - s\ < d and d > 0 is small, u(t) ^ s - d for\t - s\ < <5, and -«"(/) £ 0.

5.3. (a) It can be supposed that /=/+ ^ 0. (Why?) Put w = «(f) andh = —£#' — fu, a = 0 in (2.18) and differentiate (2.18) to obtain a formulafor «'(')• Let a = max \u'(t)\ and note that |i/(f)l ^ « min (/, 6 - t). The

f1 ffrfunction G(t) = I 51^| ds + \ (b — s) \g\ ds is nonincreasing or nondecreasing

Jo Jtaccording as 0 51 t ^ b{2 or b/2 ^ t ^ b.

5.3. (b) Multiply the differential equation by w; use uu" — (uu')' — u'2 andintegrate.

5.3. (d) It can be supposed that u - 0 at / = 0, b. Let 0 = at < • • • < ad = bbe d zeros of w. If a = max \u(d}(t)\, then \u(t)\ <:<x.U\t- ak\ld\ [This is aconsequence of Newton's interpolation formula but can be proved directly:there exists a B = 0(/0) such that «(/„) = M(d)(0)I3(/0 - afc)/rf! as can be seen byconsidering v(t) - u(t) - CU(t - ak)ld\, where C = dl u(t0)IU(t0 - ak). Hencev — 0 at / = t0, a l 5 . . . , a,;, so that its dth derivative vanishes at some / = 6].Since ^ = 0, aa - b and 0 t ^ Z>, it follows that II |f - a,| ^ /fc(6 - Od~*when afc ^ / ^ a*:+j. For 0 ^ t ^ b and A: = 1,.. ., d — 1, one hast*(b- f)d~*^ m»x W^ ~ O""1, tA~\b - t)} ^ b\d - I)"-1/*/" by differentialcalculus. Hence |«(/)| ^ <x(bdldl)[(d - l^fd*]. In addition, «' has at leastd - 1 zeros, say a^ ..., a'd_i- Thus |«'| ^ aH |r - a/|/(</ - 1)! < oJj^1](d - 1)1 Similarly, \u"\ < *b*-zl(d - 2)!,..., Iw^-"! S <tbjl I. Choose /* inthe differential equation so that |«(d)(/*)| = a.

5.5. (a) Use (2.43) where/KO = 1-5.5. (b) Use § 2(xiii) and part (a).5.8. Use v(t) — arctan«'/y' == arctan/7«'//w' and note that ^ ^ 0 implies

that the zeros of u and «' separate each other since u" fs 0 [or M" ^ 0] if u ^ 0[or « < 0].

6.1. Assume first that J: ^ £ t £ tz and let ult uz > 0 in (6.2). Next, ifJ isnot a closed bounded interval, let «x(/), K2(/) ^nearly indpendent solutions of(6.1), so that by the first step, for any [tlt t2]£J, there are constants clt c2 suchthat cj2 + c2

2 = 1 and the solution u = c^t) + c^(i) > 0 on [tlt tz]. Use alimit process involving [tlt t%] -* /.

6.2. Use the device (1.7) to reduce (6.1) to an equation of the form dzujdsz +q0(s)u =0 on an interval 0 ^ s < <x>. From q^s) ^ 0, it follows thatdzujds2 ^ 0 when u 0.

6.3. Suppose that the assertion is false. Then (6.1) has a solution u(f) & 0with two zeros a, b in /. Suppose that b is not an endpoint of /. Then «(f)changes sign at t = b. Let (6.1e) be the differential equation obtained byreplacing tj(t) by g(f) — «, e > 0, and 7 (»?, a, ft) the corresponding functional.

Then /e(^; a, /3) is positive definite for e > 0 on /42(a, /?) for all [a, /?] cJ. LetM (/) be the solution of (6.1 ) satisfying u (a) = 0, ue(a) — w'(a). Then wc(f) ^ 0for t i* a. But we(/) -»> u(t) as e -* 0 uniformly on compact intervals of J, and so,for small «, we(/) = 0 at a / near 6.

6.4. Show that if i/2(/) ?•* 0 on /x ^ / ^ /2 [i.e., if (3.12) is disconjugate on?! < r < r2], then (3.In) is disconjugate on t± ^ / ^ f2. Compare the functionalsbelonging to (3.1!) and (3.12).

6.5. (a) Let w(/) 5* 0 for T ^ f < to. It can be supposed that u(t) is a non-principal solution; cf. first part of Corollary 6.3. Then

cf. (6.14).6.5. (b) Consider J:0 ^ t < ir and the differential equation u" + u — 0. A

principal solution is u — sin /. There is no principal solution satisfying w(0) = 1.6.8. This is a consequence of Exercise 6.5(0), but can be obtained directly as

follows: Note that uT(t) > 0, wr'(0 ^ 0 for a < t < T. (Why?) Also, ifa < S < T < a>, then u^t) £ uT(t) for a ^ / ^ S since u^t) * uT(t) for/ > a and uT(t) ^ 0 for a ^ r < T. Hence uT'(G) ^ 0 is a nondecreasingfunction of T, so that lim wT'(0) exists as T -* to. This implies the existence ofw0(0 satisfying (6.19). Also «„(/) is a principal solution by (iii) of Theorem 6.4.

6.10. Make the change of variable u = vu^(f) in (6.302) and apply Corollary6.4.

6.11. For 0 ^ t ^ \j-n, put uiQ = 1 +y2 sin ///'; /?, s 1 and = — uiQ"fuio =(sin //y)/(l -f /2sin tlj). For ly'w < / < o>, extend the definitions of qit ujo sothat «jo becomes the principal solution of (6.33,), witTi/>, & 1. Thus <jj(t) -* 0,

_/ -»• oo, uniformly on bounded intervals of 0 ^ / < oo, but w,0(/) -* oo asj -* oo for every / > 0.

7.2. (6) Introduce the new dependent variable v = u exp — c I g(s) ds and note

that, in the resulting differential equation, v" + G(t)v' + F(t)v — 0, we haveF(0 < 0.

7.3. Let (/) = nit2 and Q(0 = nit.7.4. Let «(/) be the solution of (7.1) satisfying w(0) = 1, u'(0) = 0. Then

r = u'\u satisfies the Riccati equation (7.4) and r(0) = 0. Suppose u(t) > 0for 0 ^ / ^ b. A quadrature of (7.4) gives r = R — Q, where R(t) »

- I r2(5) ds and /?' - -r2. Since Q 0 for a ^ / < A, *' = -r2 =

-(R - Q)2 < -(/?2 4- e2) and /? ^ 0 for a £ t < b. If z & 0, its logarthmicderivative rt = z'/z satisfies rx' = — (i^2 4- Q2) and r^a) — 0. Also, rx(0 -»• — ooas t -* /0 if / = t0 is the first zero of z greater than a. By Theorem III 4.1,R < T! on the interval [a, f0).

7.9. Make the change of variables u — Wj(t)z in (7.35;), where w,(0 =

exp |c>QXj) ds, use Theorem 7.4 and Exercise 7.2(a).Jo"

8.1. Introduce the new dependent variable u — tl^v in Bessel's equation.8.2. Use (2.22), (2.28), and Exercise X 1.4(rf).8.3. (a) Use § 2(xiii).8.4. (a) Write (8.8,) as a first order system for («,/>,«')» and aPP^y a linear

f*change of variables. It can be supposed that/70(w0'y0 — u0iV) exp I (r0//?0) ds =

1; so that neither 2 |/v*o/IJol nor 2 l/Wo'l exceed £ |/»0(«0 + i>0X«0 + »o)'l +

£ l/>o(«o ~ "oX^o ~ *>b)'l + lexp - I (r0lp0) ds\.

8.4. (b) Assume that a(t) has a continuous derivative. For a fixed choice ofP±, note that 0 — exp ±i I 9^(5) ds satisfies the differential equation u" +

Jo[q T \iq~l/iq']u = 0. Apply (a), identifying this equation with (8.8^ and (8.1)with (8.80). (In order to show that the conditions on q, including the paren-thetical conditions, imply that a solution u of (8.1) and its derivative u' arebounded, let^r =g + ih, where g(t), h(t) are real-valued. If E — g |«|2 +1«'|2,then E' -g' |w|2 -I- ih(uu' - u'u), and so \E'\ ^ const, (l^'l + \h\)E for large f.This implies that E, hence u and u', are bounded.) For another proof when qis real-valued, see Exercise X 17.4(a).

8.5. Let 2 - M/X so that (8.12) becomes (f'^z'Y + (/*(%(')> - 0, whereg(t) . i +f/4fK - 5f*ll6fX; cf. (2.34H2.35). If w denotes either of thefunctions in (8.11), then w' = ±lfMhHWt where h(t) - 1 +/'2/16/3 Thus(f-Xw'Y - -fXhw ± /(V'O'M'. Using ±/w = w'lfXhX, it follows that H>satisfies (/-^w')' - (A^'/~K*-^M'/ +fXhw = 0. Identify the equations for wand z with (8.80) and (8.8}). Apply Exercise 8.4(a).

8.6. (a) and (b) Consider the case u(T)u'(T) ^ 0, say u'(T) > 0, u(T) ^ 0.Integrate (8.13) over [T, t] to get

Note that

Hence if T / ^ C/ ^ T -I- 0/C and «' ^ 0 on [T, t], t £ 5, then

Suppose that there is a (first) t = T*, T < T* £ U, where u' - 0, so that

at f = T. This is valid for T ^ t ^ r*. Integrate over [r, T*] to get anestimate for u(T*\ and hence by (**) with / - T, for «'(D > 0. If no T* exists,then (*) is valid for T ^ / ^ J7, and an integration over [T, U] gives the desiredresult.

8.6. (d) The proof of part (a) shows that if b - a = A < 0/c and a <> t £ b,then |«'(')! is majorized by either

or by one of the two quantities

Thus, in any case,

where |«(r*)| = max |«(T)| for a < r < b. Integrating this inequality over[a, T*] gives an estimate for \u(T*)\ which, together with the last inequality,gives the desired result.

8.7. Case 1: g(b) - g(d) > 0. Let a ^ c S b. Put/r(/) = P(t) - N(t) + h(c)where P(c) = N(c) — 0 and P(t\ N(t) are the positive and negative variation ofh [so that />(/), N(t) are nondecreasing and var h - P(b) + N(b) - P(d) - N(a)]Write

Integrating by parts and using P(d) < 0 ^ N(b) gives (821) with inf h replacedby A(c). Cose 2: ^(6) — (a) — 0. The arguments just used show that, in thiscase, (8.21) can be improved to

Case 3: ^(i) — g(d) < 0. It is sufficient to consider the case that g(t) is apolynomial, for otherwise g can be approximated by polynomials and a limitprocess applied to (8.21). For a polynomial g(t) with g(a) > g(b), the interval[a, b] can be divided in a finite number of subintervals, a — f0 < tl < • • • < fn =6, so that g(t) is either decreasing on [/,, ti+l] or g(ti+i) — g(t,) for each y. Iffv. 61 - ff*. /,,,!. then, bv Case 2.

8.8. Reduce (8.20) to the form (8.13) by introducing the new independent

variable s — s(t) = I exp I — I p(r) dr \ dr.

9.6. Apply § 2 (xiii) and Corollary 9.2 with A = 1.9.9. (a) Use Lemma 9.5 and Corollary 9.3.9.9. (b) Use Lemma 9.5 and Theorem X 17.5 [cf. (X 17.22)].10.1. Suppose that 7(»j; a, b) ^ 0 for all [a, b] c J and r\ e A^a, b) but that

(10.2) is not disconjugate on /. Let (z(/), y(t)) * 0 be a solution of (10.2)and x(t) — 0 at points t = a,b of J. Suppose b is not an endpoint of /, say/ - j8e/ and ft > b. Put %(/) = x(t) for a < / < 6, »?0(/) - 0 for A < / < ^;thus Tfo e >/i(«» ^)- For an arbitrary »? 6 A£at b\ put »? (0 = »j0(r) + «»?(/) andy(*) = /07 ; fl, )- Thus /(«) > 0,/(0) « /(%;«, 6) = D. Hence dj(^df - 0 ite — 0. Using (10.22) and integrations by parts, it is easy to see that (djldf)f=0 =2P(/>X(A) •»?(/»). If »K*) - a^W, it follows that x'(b) = 0, hence x(t) = 0.Contradiction.

10.2. If (Pi*1)' + QiX = 0 has a solution x(i) & 0 vanishing at two points

t = a, ft, where a ^ a < 0 ^ b then /(??, a, /?) = (/>2V • T?' - Q2i? •»?) dt is

not positive definite on At[a., ft]. Let ij(t) = x(t).10.4. Note that B = />-». Let BX denote the self-adjoint, positive definite

square root of B; cf. Exercise XIV 1.2. Since M is nonsingular and A/"1 =

-U~lBU*~l dt, it is seen that

From the relation ||c||2 - (U~lB^}Br\4Uc • c = BrXUc • BXU*~lc andSchwarz's inequality ||c||2 ^ ||£-H0c|| • \\B^U*~lc\\t it follows that

Finally, ||fl-^£/c||2 = B~lUc • Uc =Px-x.10.5. Cf. The proof of Theorem 10.4.10.7. Without loss of generality, it can be supposed that U0(d) = U(a) — I.

Since V(a)U~\a) = V*-\d)V*(a\ it follows that A = K0 in (10.12) Hence(10.21) holds with s « a and K^ - /. If K0 - A < 0, it follows that {...} in(10.21) is positive definite, hence nonsingular, and so del £/(/) 5* 0 for / ^ a.Conversely, suppose that KQc • c > 0 for some vector c ^ 0 and suppose, ifpossible, that det U(t) * 0 for t ^ a. Then Sa(t), denned by (10.34) withs = a and 7*(r) == /, satisfies (10.48) with s - a. But M = -K0 £ 0. This is acontradiction.

Chapter XII

3.2. Write (3.12) as a first order system for the 2</-dimensional vector(_Z' -F*z,z).

3.3. (a) Use the Sturm comparison theorems, e.g., Corollary XI 3.1.

3.3. (b) Repeat the proof of Theorem 3.3 with r = H/)||2 + I x(s) • K(s)x(s) ds.3.3. (c) User - K/)||. Jo4.1. (b) Introduce x — tx0fp as a new dependent variable; see Theorem 4.1

and Remark 1 following it.4.2. Introduce x — tx0lp as a new dependent variable in (4.1).4.3. Let /?! > 0 be arbitrary. Then ||/|| ^ M, where M ** aRf + b if

||a:'|| ^ RI. Show that if /?j - r*, then the two inequalities in (4.20) hold.4.4. If there exist two solutions, let *(/) denote their difference and r(t) =

||a</)||2. Show that r"(t) > -(200 + W)r(i)\ cf. the proof of Theorem 3.3.4.6. (a) If #!(/), xz(t) are two solutions, the difference x± — x2 cannot have a

positive maximum at a point t0,0 < /0 < p. Otherwise a?t' = xz', (jCj — xzy > 0at / = /0.

4.6. (b) The boundary value problem x" - -3(x')X, x(0) = x(2) = 0 has thesolutions x = 0 and x = [1 — (1 — /)4]/4.

4.6. (c) Suppose that there are two solutions ^(f), »2(0 an^ tnat *(') =

^i(') — (0 has a positive maximum at a point t0,0 < tQ <p. It can be supposedthat x(t) > 0 for 0 < / < / > , otherwise (0, />) is replaced by the largest interval

containing t = /0 on which x(t) > 0. Put

or <x(r) - 0 according as V(') — X2'(0 ?* Oor = Oand P(f) = f(t, x^t), xz'(t)) —f(t, xz(t), xz'(t)). Then x(t) satisfies a differential equation of the form x" «•a(fX + /?(/), where «(/) is bounded and measurable and 0(t) St 0. Hence, if

ny(t) = exp — I a($) <ft, then (X'K)' = KOft') ^ 0. Introduce the new

Joindependent variable 5, where ds = y(t) dt, so that dx\ds — yx' and dzx/ds* ^ 0.

4.7. (a) If there are two, say Xj(t) and x2(r), let x = xl — x2. Then **(/) >0 [ <0] at any / where x has a positive maximum [negative minimum].

4.7. (b) At a point where «(r) has a maximum, z'(0 = 0 and x"(t) ^ 0. At apoint where x'(t) has a maximum, x"(t) — 0.

4.7. (c) If A = 0, there is the solution x(t) = 0. Use Theorem 2.3 to show thatthe set of A is open on 0 ^ A < 1 and use part (&) to show that the set of A isclosed.

4.8. Let h(t) be periodic of period p and of class C1. Then

has a unique solution x(t) = T0[h] of period p by Exerise 4.7. Show the applica-bility of Schauder's fixed point theorem.

5.2. On 0 5| / ^ 1, consider the family of real-valued functions x —1 + «•/?*(* - l//t)4 or x = 1 according as 0 ^ / ^ l/« or 1/n < t ^ 1.

5.3. If (5.21) is assumed instead of (5.29), then r(t) has no maximum on0 < / < p. Hence if 0 ^ tQ ^ t ^ p, then KO ^ max (r(t0), r(p)) = r(t0); i.e.,r(t) is nonincreasing, so that (5.30) holds. When (5.29) holds, the proof ofTheorem 5.1 shows that r «• ||a;e(0ll2 satisfies (5.30). But the inequalities (5.30)are not lost in the limit process e = e(/i) -* 0.

5.4. Introduce the new dependent variable x — u — «o(/), then the boundaryvalue problem becomes x" =f(t, x, x')and*(0) = «0 — u^Q)tx(p) «• uv — «0(/?),where /(/, x, x') = h(t, x + MO(/), x' + «„'(/)) - h(t, uQ(t\ uQ'(f)\ Note that±/(/, ±/?, 0) 0 for R > 0 since h is nondecreasing in u. Also, |/| ^9<|:c'| + c) for a constant c ^ |MO'(')|- App^y Corollary 5.2.

5.5. (a) Let m = 1, 2, By Theorem 5.1, (5.26) has a solution on0 ^ / ^ m satisfying a;(0) = x0, x(m) = 0. For any p > 0, the sequences offunctions xm(t), xm'(t) for m ^ p are uniformly bounded and equicontinuous on0 ^ r ^ /?.

5.6. Note that r — \\x(t)\\2 satisfies r" > 0 for large /. Thus if the assertion isfalse, there is an m > 0 such that 0 < m2 ^ r(t) ^ Rz for large t. Define<j(t) - 2[x • /•(/, x, x') + ||z'||2]/||a;||2, where x = x(t). Hence r' - q(t)r - 0 and^(r) ^ 2h(t)lm2. Apply the last part of Corollary XI 9.1.

5.8. (b) Define a function a^fco) of x0 for ||x0|| < R as follows: a^ — yc(pt x0).This gives a map of the ball ||a;0|| < R into itself. Apply Brouwer's fixed pointtheorem.

8.2. Let£(f) be a bounded, measurable function for / ^ 0. Then

is a bounded solution of (6.1) satisfying P$(0) = y0. For x(/) a measurable

function on/ ^ 0 such that HO II ^ p,let^(/) =/(f,a</))andlett/(/) - TJtffj]be defined by the last formula line. Let SB = X> =» L°°( Y) and consider 70 as amap of Sp into D.

9.1. It can be supposed that y(f) & 0, otherwise introduce y — y^t) as a newdependent variable in (9.5). The problem is reduced to showing existence ofsolutions [for the transformed equation (9.5)] which exist for large / and tendto 0 as / -*• oo.

Chapter XIII

2.1. Suppose first that (2.26), hence (2.28), holds. This is equivalent to

max (||z(r)||, ||</(/)||) ^ K\\z(s) - y(s)\\ for 0 r < s < t

if y(0) G W0, z(0) 6 HV If 2(0) £ W* let z(0) - z°(0) + y(0), where y(0) e W0,2°(0) e ffj. Follow the proof of Lemma 2.2 to obtain analogue of (2.7). Thisgives the "if" part under the assumption (2.26). The proof under the assumption(2.27) follows by a modification of the proof of Lemma 2.5.

6.3. Let 0 < <tfjt) 6 B and

It suffices to show that 1, Vz e D» cf. the proof of Theorem 6.4. Let the integerrt > 0 be such that n < r < 11 + 1. Then

where y*(0 = 0 or yf(0 = v(^ — y) according as r <y or / ^ y. Thus Vi 6 D;in fact, \Yj\0 ^ S e~jK \v\0 by condition (</) for Def. Similarly, a use of thefirst part of (6.9) shows that v>2 6 D.

7.1. Use Exercise XI 8.8 and Theorems 5.3, 5.4.13.1. Use Theorem 12.1 and Exercise 7.1.13.2. Use Theorem 12.3.

Chapter XIV

1.1. If the assertion is false, then (1.30) holds for all solutions. Hence|det y(/)|2 has a finite limit as t -+ a>. This contradicts (l.Soo) and (1.6).

1.2. (a) Let U be a unitary matrix such that U*AU = D is diagonal,say D = diag [V, • • • » V3, where ^ > 0 for j - 1,..., d. Put D* -diag ( A l f . . . . Ad). Then D - D^D^saidA = UDU* - (UD*U*}(UDKV*\Let /4^ = VD*V*.

1.2 (6) It is sufficient to suppose that ^4 = D is diagonal, say D =diag [V, • • • » *«i2L (Why?) Let />'^ be any Hermitian, positive definite squareroot of D. Let the vector y and eigenvalue A of DM satisfy DXy =* Ay. Then

Dy — A2y, so that y is an eigenvector of D belonging to the eigenvalue A2. Itfollows that D* - diag [Ax AJ. (Why?)

1.2. (c) Consider only f near a fixed /0. Then there exists an e > 0 such thatA(t)y -y ^ f\\y ||2 for all vectors y. If A(f) is multiplied by a positive constantc > 0, its square root is multiplied by c1 . Hence it can be supposed that«l!j/!l2 ^ A(t}y • y < 0||y||8, where 0< c < fl < 1. Then

(1 - c)||yf £ [7 - A(t)Jy • V £ (1 - 9) W and ||7 - A(t)\\ ^ 1 - c < 1.

Show that AM(i) - B(t), where

Note that (fl(/))2 = /4(f) follows from [(1 - r)W]" = 1 - r since the powers of7 — v4(r) commute. Also B(t) is Hermitian and positive definite since an > 0,«i +«« + • • • = 1. (This gives another proof of the existence of /(**.) Finally,the series for B(i) — Al<*(i) can be differentiated term-by-term.

2.1. (c) Consider the binary system y1' «* — y2, yz' = — y^f2 for / ^ 1 andExercise XI 1.1 (c).

2.1. (d) For necessity, cf. (6); for sufficiency, cf. the proof of Theorem X 1.1.2.2. Cf. §IV 5.2.3. (a) Let y(t) be solution given by Theorem 2.1. Verify (-l)nt/(B)(0 ^ 0

by induction on n.2.4. Write the sum of the first two terms p<fc)uld} + pi(t)u(*-l} of (2.7)

as />o(')«~1('Xa(')«('l~~1))', where a(r) - exp I pj(s) dslp9(s) > 0. Divide the re-Jo

suiting equation by p0(t)l«(t) and consider the result as a first order system fory - («, -«', «*,. . . , (-l)rf-»«<d-«,(-l)d-1a(0«u-1)),i.e.,fory = (y\ . . . , y"),where y* - (-ly-V^1' for y - 1, . . . ,«/- 1 and yd - (-O'-X/W""Theorem 2.1 implies the existence of a solution satisfying (2.9) for n = 0,1,..., d -1. (Why « > 0?) From (2.7), it follows that (-l)"^"' +

Piu(d~l}) £ 0.2.5. Prove (2.9) by induction on n. To this end, first show that if d > 2 and

(2.9) holds for n = 1,.. . , d - 2, then it holds for n « d - 1. Note that

(-1VW' +Piu(*~l}} ^ 0, hence [(-I)1*-1*!"1-1) exp I (pjpjds]' ^0. ShowJo

that if (-l)<*-1a(d-1)(fl) < 0 for some a on 0 ^ a < oo, then

and ( — l)d~2«(<*~2)(/) is increasing for r j£ a. If d =£ 3, this is incompatiblewith ( -i)<*-2M<*-2) > 0 and the fact that lim M(d~3)(0 exists as / -* oo.

2.6. (a) Use analogue of Corollary 2.3 for 0 < t < « and the Remarkfollowing Theorem 2.1; for uniqueness, use Exercise XI 6.7.

2.7. There exist positive continuous functions a,^/),..., am(f) such that theexpression on the left of (2.12) can be written as

§IV8(ix). Thus (2.7) is

Write this as a first order system for y «• (y1,..., yd), where

and y*-"** = (-l)d~m+*~1a*_i{ajt_2[. - .(«o"((l~m))'.. -I'}' for A: - 1,..., w.Apply Theorem 2.1.

2.8. Suppose first that (*) if T > 0, then there exists a solution y^t) ofy' - -/ on 0 < t < T satisfying ||t/r(0)|| = c and yr(/) £ 0, t/r'(0 ^ 0 for0 ^ / ^ r. Choose a sequence of T-values rt < T2 < ... such that Tn -» ooand y(0) = lim t/r(0) exists as /i —• oo, where T — T(n). Then y(t) — lim yr(/),T — T(n) and w -*• QO, exists uniformly on bounded /-intervals (why?) and is asolution with the desired properties. Thus it only remains to prove (*) for everyfixed T. To this end, suppose first that solutions of y' — —f are uniquelydetermined by initial conditions and let y(t, t/0) be the solution satisfying y = y0at t — T. Since y ss 0 is a solution, it follows that y(t, y0) exists on 0 ^ / ^ Tif \\Vo\\ is sufficiently small; Theorem V 2.1. Since/ £ 0 and y0 ^ 0, it followsthat y(t, y0) > y0 on 0 < / < T and so \\y(0, y0)\\ > ||yj. Thus if c> 0 issufficiently small, \\y(0, yg)\\ = c holds for some small ||y0||. Let S be the set ofnumbers c0 such that, for every c on 0 < c < c0, there is a yQ with the propertythat \\y(0, yQ)\\ —c. It is clear that S is closed relative to the half-line 0 < c0 < o>(why?) and also open relative to 0 < c0 < oo (why?). Hence S contains allc0 > 0. This proves (*) in the case that the solutions of y' = —f are uniquelydetermined by initial conditions. In other cases, approximate / by smoothfunctions.

2.9. (a) Let y0(t, a.) be the solution of y" = / satisfying y0 — 0, y0' = a ^ 0at/ = 1. This solution exists for 0 ^ / ^ l i f — a ^ 0 is sufficiently small. Also2/o > 0,2/o' ^ 0. W ^ 0 on 0 ^ / £ 1, since y" =/ ^ 0. Let c = t/0(0, a) > 0for some small — a > 0 and y(t, ft) be the solution of y" =/satisfying y(G) = c,2/'(0) = /? <; 0. For c fixed, let S be the set of ft < 0 for which there exists a'o = 'o(0) > 0 such that ?/(/, ft) exists on 0 ^ / < /0,«/(/, /?) > 0 for 0 ^ ft ^ /0>2/(/o» 0) = 0. Then y'(t, P) ^ 0, y"(t, ft) > 0 for 0 ^ / ^ /0. Show that 5 is notempty and is open. Let /?„ = sup ft for /? 6 S and show that y(t, ft0) is thedesired solution.

2.9. (6) Consider y" = 3(1 + y'2)1+V/2A, where 0 < A < 1. This has nosolution satisfying y(Q) > 1 and y(t) > 0, y'(/) ^ 0 for all / ^ 0; cf. Hartmanand Wintner [8, pp. 396-397].

3.1. Use Theorem XI 3.2; cf. diagram for Exercise 3.5.3.2. Introduce the new independent variable where ds — dtlp(t) and s(0) = 0.3.5. Use Sturm's second comparison Theorem XI 3.2 to show that if the arch

of the graph of u — |w0(/)| on /„ ^ / ^ /n+1 is reflected across the line / = /n+1,then it lies "over" the arch of u = |w0(/)| on /n+1 ^ / ^ /n+2; see diagram. Use"alternating series" argument to prove (3.14).

3.8. Let 0 ^ s < oo and let G(s, t) be the solution of M* + <j(t)u = 0 satisfyingthe initial conditions u = 0, u' = 1 at / = s [e.g., if q(t) = 1, then G(s, /) =

f°°sin (/—s)]. Show that v(s) = I G(s, /)/(/) dt is uniformly convergent on boundedJs

^-intervals and is a solution of v" + qv —f. Note that when q(i) > 0, then\G(s, /)|2^ \Gt(s, s)\*jq(t) ^ \lq(t) and \G,(s, Ol2 ^ \Gjs, s)\* = 1 for / £ * byTheorem 3.1 oo- The proof of v ^ 0 in the case n — 0 depends on argumentssimilar to those in the proof of Corollary 3.1. When n = 1, write dv/ds »

apply integration by parts twice. For « = 2, use the fact that w — v"lq is asolution of w" + qw — (flq)", for w + v =flq.

3.9. (a) Cf. Exercises XI 7.5 and XI 7.6. Use r' + r* + q - 0, hencer' + q < 0, r' + r2 < 0.

3.9. (6) Cf. Corollary XI 9.1 for "only if." For the "if" part, use dE -/•oo

«2<fy 2s 0 and that 1 tq(t) dt = oo implies that «'(°°) - 0 for all solutions.

Suppose that £"(<») > 0 for some nonprincipal u, so that r' + Efu* — 0 gives—r' ^ c/M2, c > 0. Integrate over [/, oo), multiply by u to obtain «'(') ^c«0(0 = const. > 0.

3.9. (c) First prove the assertions concerning rc(oo), r(oo) by using CorollariesXI 6.4 and 6.5.

3.9. (d) For the first part, cf. the hint for part (c). For the second part, useExercise X 17.4 (b) and dE = u*dq.

4.1. Let u be as in (4.6), then u2q + u'2 is a quadratic form in (cos q>, sin <p) forfixed /. In the case (4.3 +), (4.7) holds and shows that the eigenvalues A of thisform satisfy A < 1. Using (4.8), it is seen that the equation for the eigenvalues isA2 - A(|z|2 q + |z'|2) + q « 0. The desired inequality merely expresses the factthat the largest root A of this equation satisfies A 1. The case (4.3 —) followsfrom (4.3 +); cf. Lemma 3.1.

4.2. (d) Make the change of independent variables t = e* in Bessel's equation(4.21); then the change of dependent and independent variables V — (t2 — f^Y^v,da . (/a - WA ds for / > p.

4.3. (b) Take a number a such that, up to a constant factor, u = u0(t) cos a +Ui(t)sinct: It can be supposed that u = MO, for otherwise replace u0 and uvby «0 cos a + MJ sin a and — u0 sin a + ur cos a, respectively. Define a contin-uous 0(/) by 6(t) ~ arc tan w0(0/«i(/) and 0(f0) = 0, so that B(tn) = nv. Then0'(0 = l/(«p

2 + i/!2) > 0. Let / = /(0) be the inverse function of 0 = 0(f).Then (-l)'+1</>>(0)/</0J > 0 for j - 1 , . . . , /i + 2 and /„ = *(**) for n « 0,1 Note that a mean value theorem of Holder implies that if a function/(0) has / ^ 1 continuous derivatives, then AJ/(/ITT) = ^(d'tldB*)^ where0 = y is some number satisfying nn ^ y ^ (/i +y)fr. This fact is easily verified

from A>>(/!7r) = f* [A*-1 dt(6 + /t7r)/rf0] f/0 = ! " • • • ([</'*(»«• + 0r + • • • + 0,-)/Jo Jo Jo

d&] det dez... dbj.6.3. The existence proof is similar to that of Theorem 6.1 where fl° is chosen

to be {(t, y): /, yl arbitrary, t/2 > 1, y3 < 0}. For uniqueness, use a variant ofExercise III 4.1(rf).

12.1 (a) Cf. the Remark following (12.4).12.1 (b) Introducing polar coordinates with r = \\y\\, it is seen that ds >

p(y) dr ^ /?0(r) dr if p0(r) = min p(y) for ||y|| = r. Use Lemma I 4.1.12.4. (a) The ds is complete by Exercise 12.1(6). If y(t), 0 ^ r ^ 1, is any C1


arc joining y — 0 and y, then its Riemann length is not less than p0(u) du

if p0(u) = min p(y) for \\y\\ — w; cf. the proof of Exercise 12.1(A). Hencer(0, y) is not less than this integral.

14.3. If a(t/) = Aj + Ag, verify that, for fixed y and arbitrary vectors x, z,

(Jx • a:)»2||a + 11*11* (Jz -z)-(x- z)[Jx -z+x-Jz}£ a(||*||2 ||z||« - (x • zf).

To this end, note that the inequality is not affected if/ is replaced by JH and x,z are subject to an orthogonal transformation; thus it can be supposed thatJH =* diag [Alt Ad]. The left side of the desired inequality is

or, equivalently,

This gives the stated inequality. Let z =f(y) and x -f(y) =• 0.

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SUPPLEMENT

P. B. Bailey, L.F. Shampine and P.E. Waltman[SI] Nonlinear two point boundary value problems, Academic Press,

New York (1968).

W.A. Coppel[SI] Stability and asymptotic behavior of differential equations, Heath & Co.,

Boston (1965) (X17).[S2] Disconjugacy, Lecture Notes in Math. No.220, Springer, Berlin

(1971) (XI 5 and Notes, end).

J.W. Evans and J.A. Feroe[SI] Successive approximations and the general uniqueness theorem,

Amer. J. Math., 96X19'74.) 5015-510 (III 9).

P. Hartman[SI] Principal solutions of disconjugate n-th order linear differential

equations, Amer. J. Math. 91(1969) 306-362 and 93(1971) 439-451(XI Notes, end).

[S2] On disconjugacy criteria, Proc. Amer. Math. Soc. 24(1970) 374-381(XI 5).

[S3] The stable manifold of a point of a hyperbolic map of a Banach space,J. Differential Equations 9(1971) 360-379 (IX, Appendix).

[S4] On the existence of similar solutions of some boundary value problems,SIAM J. Math. Anal. 3(1972) 120-147 (XIV 7).

S.P. Hastings[SI] An existence theorem for a class of nonlinear boundary value problems

including that of Falkner and Skan, J. Differential Equations 9(1971)580-590 (XIV 7).

S.P. Hastings and S. Siegel[SI] On some solutions of the Falkner-Skan equation, Mathematika 19(1972)

76-83 (XIV 7).

M.W. Hirsch, C.C. Pugh and M.Shub[SI] Invariant manifolds, Bull. Amer. Math. Soc. 76(1970) 1015-1019

(IX Appendix).

A.Yu. Levin[SI] Nonoscillation of solutions of the equation x^ + pjWx*""1* +...+

Pn(t)x = 0, Uspechi Mat. Nauk 24(1969) No.2(146) 43-96; RussianMath. Surveys 24(1969) 43-99 (XI Notes, end).

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71(1969) 229-230 (XI 8).

Index

Adjoint, dth order equation, 66linear system, 62system of total differential equations,

119see also Associate operator

Admissible, 438,462,481Angular distance, 451Arzela theorems, 4Ascoli theorems, 4Associate operator, 486Asymptotic integration, boundary layer

theory, 534difference equation, 241 (Ex. 5.4),

289(Ex. 4.8), 290(Ex. 4.9)dth order equation, 314, 447(Ex.

9.2(c))perturbed linear system, 212, 259,

273,445,447(Ex.9.3)second order equation, 319(Ex. 17.4,

17.5), 320(Ex. 17.6), 369, 375,446,447(Ex. 9.2)

Asymptotic lines, 107(Ex. 6.3 (c))Asymptotic phase, 254Asymptotic stability, see StabilityAttractor, 160,213

domain of attraction, 548Autonomous system, 38, 144,202

Banach spaces, admissible, 438, 462,481

associate, 484class J, Jt, 452, 484lean at w, 439Lp, L ,L0 ,M,436,453open mapping theorem, 405, 437,

439,464quasi-full, 453, 467, 471, 474, 475Schauder fixed point theorem, 405,

414, 425stronger than L, 437] 453

Bessel equation, 87(Ex. 12.3 (a) )asymptotic behavior of solutions,

371 (Ex. 8.1)integral of Ju,513 (Ex. 3.6)Jf + Kf, 518(Ex. 4.2), 519(Ex.

4.3)zeros of solutions, 336(Ex. 3.2(</)),

519(Ex. 4.3)Blasius differential equation, 520Bohl theorem, 199Boundary layer theory, 519Boundary value problems, 337, 407,

418,519adjoint, 410linear, first order, 407linear, second order, 418nonlinear, second order, 422, 433,

434(Ex. 5.4)periodic, nonlinear, 412, 435 (Ex.

5.9(6))singular, third order, 519Sturm-Liouville, 337

Bounds for solutions, 30autonomous system, 543derivatives, 428equation of variation, 110second order equation, 373(Ex. 8.6),

374(Ex. 8.8)see also Differential inequalities

Brouwer fixed point theorem, 278

Cantor selection theorem, 3Cauchy, characteristic strips, 133, 135

initial value problem, 131, 137, 140Center, 159, 215, 216(Ex. 3.1)Characteristic direction, 209,220Characteristic equation (polynomial),

65Characteristic roots and exponents, 61,

252,253

607

608 Index

Characteristic strips, 133, 135Comparison theorem, principal solu-

tions, 358second order equations, 333second order systems, 391 (Ex. 10.2)Sturm, 333

Complete integrability, 118, 123, 128Complete Riemann metric, 541, 546Complete system of linear partial dif-

ferential equations, 119, 120(Ex.1.2), 124

Conjugate, points (Jacobi's theorem),391 (Ex. 10.1)

solutions, 385,386, 399Constant coefficients, dth order equa-

tion, 65linear system, 57second order equation, 324(Ex. 1.1)

Continuum, 16Curvature, quadratic form, 106(Ex. 6.2,

6.3(6))lines of, 107(Ex. 6.3 (<0)Riemann tensor, 106(Ex. 6.2)

Cycles, see Limit cycles

Derivate, 26Dichotomies, adjoint systems, 484

definitions, 453first order systems, 474Green's functions, 461, 476, 477higher order systems, 478second order equations, 483 (Ex. 7.1),

496(Ex. 13.1, 13.2)Difference equations, 241 (Ex. 5.4), 289

(Ex.4.8),290(Ex.4.9)Differential forms, 101,120Differential inequalities, 24

equations of variation, 110linear systems, 54partial, 140, 141<Ex. 10.1), 142(Ex.

10.2)Disconjugate equation of second order,

351, 362positive solutions, 351, 352(Ex. 6.2)variational principles, 352see also Nonoscillatory

Disconjugate system of second order,384

Disconjugate system of second order,criterion, 388, 390, 391 (Ex. 10.1,10.2), 420, 421 (Ex. 3.3)

Domain of attraction, 548

Egress point, 37, 175, 202, 278, 281,520

Eigenfunction, 338, 342interpolation, 344(Ex. 4.4)L2-approximation (completeness),

338Equations of variation, 96, 110

bounds for solutions, 110Equicontinuity, 3Equivalence, differential equations, 258,

271 (Ex. 14.1)maps, 258

Ergodic, 193, 194, 198, 199Euler differential equation, 85Existence in large, 29

for linear systems, 31,45Existence theorems, boundary value

problems, 337, 407,418, 519Cauchy problem, 137invariant manifolds, 234, 242, 296linear system of partial differential

equations, 124maximal solution, 25monotone solution, 357, 506, 514

(Ex. 3.8)PD-solution, 497Peano, 10Picard-Lindelof, 8see also Periodic solution, Solutions

tending to 0Extension theorem, 12Exterior, derivatives, 102

forms, 101,120

Fixed point theorems, see Brouwer,Schauder, Tychonov

Floquet theory, 60, 66, 71, 302(Ex.11.3)

Focus, 160, 215, 216(Ex. 3.1)Formal power series, 78, 79, 261Frechet space, 405, 436Frobenius, factorization, 67

Perron-Frobenius theorem, 507(Ex.2.2)

total differential systems, 117, 120

Index 609

Fuchs, theorem, 85type of differential equation, 86(Ex.

12.2)Fundamental matrix or solution, 47

adjoint system, 62analytic system, 70characteristic exponents and roots, 61Floquet theory, 60, 71, 302(Ex.

11.3)system with constant coefficients, 57unitary, 62(Ex. 7.1)

Geodesies, 106(Ex. 6.2)Global asymptotic stability, 537Green's formula, 62, 67, 327, 385Green's functions, 328, 328 (Ex. 2.1,

2.2), 338, 409(Ex. 1.2), 439, 441,461, 476, 477

Gronwall inequality, 24generalized, 29systems, 29(Ex. 4.6)

Haar's lemma, 139Half-trajectory, 202Hermitian part of a matrix, 55, 420Homann differential equation, 520Hypergeometric equation, 509(Ex.

2.6(6))Kurnmer's confluent form, 509 (Ex.

2.6(c))Whittaker's confluent form, 509(Ex.

2.6(d))

Implicit function theorem, 5, 11 (Ex.2.3)

Index, see Jordan curve, Stationarypoint

Indicial equation, 85Inhomogeneous equations, see Varia-

tion of constantsIntegrability conditions, 118, 119, 123,

128Integral, first, 114,124

(m + l)st, 478Invariant manifold, 228, 234, 242, 296Invariant set, 184, 184(Ex. 11.2, 11.3)

Jacobi, system of partial differentialequations, 120

Jacobi, theorem on conjugate points,391 (Ex. 10.1)

Jordan curve, flow on, 190index and vector field, 149,173theorem, 146

Jordan normal form, 58, 68

Kamke uniqueness theorem, 31van Kampen uniqueness theorem, 35

application, 113Kneser, H., theorem, 15Kronecker theorem, 194

Lagrange identity, 62, 67, 327Legendre equation, 87(Ex. 12.3(6))

associated equation, 87(Ex. 12.3 (c)),508(Ex.2.6(a))

Lettenmeyer theorem, 87, 91 (Ex. 13.2)Lienard equation, 179Limit cycle, 145, 151, 152, 156, 178,

181(Ex. 10.4, 10.5), 190,253Limit points, «- and a-, 145,184

set of w-limit points, 145, 154, 155,158, 184(Ex. 11.3, 11.4), 190, 193

Linearizations, differential equations,244, 257(Ex. 12.1), 258

maps, 194, 245, 257Linearly independent solutions, 46, 64,

326Liouville, formula, 46

Sturm-Liouville problems, 337substitution, 331volume preserving map, 96(Ex. 3.1)

Lipschitz, continuity, 35- and L-Lipschitz continuity, 107

Lyapunov, asymptotic stability, 38, 40,539

function, 38, 40, 539order number, 56, 294, 301, 303stability, 38, 40theorem on second order equation,

346uniform stability, 40

Manifold, definition of 2-manifold, 182flow on, 183invariant, 228, 234, 242, 296P(B,D) -manifold, 484stable and unstable, 238, 244,255

610 Index

Maps, associated with general solution,94,96,231

associated with periodic solution, 251equivalence, 258invariant manifolds, 234linearization, 194, 245, 257volume preserving, 96(Ex. 3.1)

Matrix, exponential of, 57factorization of analytic, 75Jordan normal form, 58, 68logarithm of, 61norm, 54trace (tr),46see also Fundamental matrix

Maximal interval of existence, 12Maximal solution, 25Minimal set, 184, 184(Ex. 11.2, 11.4),

185, 190,193Monotony, 500

functions of solutions, 500, 510, 518,519(Ex. 4.3)

solutions, 357, 506

Nagumo theorems, 32, 428Node, 160, 216, 216(Ex. 3.2, 3.3), 219Nonoscillatory equations of second or-

der, 350, 362criteria, 362necessary conditions, 367, 368see also Disconjugate

Norm, matrix, 54on R*. 3

Open mapping theorem, 405, 437, 439,464

Operator, associate, 486closed, 405graph, 405null space, 462

Order number, 56, 294, 301, 303Oscillatory, 333, 351, 369, 510

see also NonoscillatoryOsgood uniqueness theorem, 33

Peano, differentiation theorem, 95existence theorem, 10

Periodic solution, associated map ontransversal, 251

characteristic roots, 61,252

Periodic solution, existence, 151, 178,179(Ex. 10.1(6)), 179, 181(Ex.10.4), 198, 407, 412, 415, 416,435(Ex. 5.9(A))

Floquent theory, 60, 66, 71, 302(Ex.11.3)

see also Limit cycle, StabilityPerron, singular differential equation,

91(Ex. 13.1)theorem of Perron-Frobenius, 507

(Ex. 2.2)Perturbed linear systems, 212, 259, 273

invariant manifolds, 242, 296linearization, 244, 257(Ex. 12.1)see also Asymptotic integration

Picard-Lindelof theorem, 8Poincare-Bendixson theorem, 151

generalized, 185van der Pol equation, 181(Ex. 10.3)Polya's mean value theorem, 67(Ex.

8.3)Principal solution of second order

equation, 350, 355comparison, 358continuity, 360definition and existence, 355, 357

(Ex. 6.5)monotone, 357

Principal solution of second order sys-tem, 392, 398 (Ex. 11.4)

Priifer transformation, 332

Quasi-full Banach space, 453, 467, 471,474, 475

Reduction of order, 49second order equation, 64, 327

Regular growth, 514(Ex. 3.7)Retract, 278

quasi-isotopic deformation, 280(Ex.2.1)

Riccati, equations, 331, 364differential inequality, 362, 364, 368generalized equation, 226(Ex. 4.6)

Riesz, F., theorem, 387Rotation number, 191,198Rotation point, 158, 173 (Ex. 9.1)

see also Center

Index 611

Saddle-point, 161,216, 218Sauvage theorem, 73

partial converse, 74Schauder fixed point theorem, 405, 414,

425Schwartz, A. J., theorem, 185Second order, linear equation, 322

linear system, 384, 418nonlinear, 174,422see also Asymptotic integration,

Boundary value problems, Bounds,Comparison theorem, Dichotomies,Disconjugate, Liouville, Monot-ony, Nonoscillatory, Principal so-lution, Priifer, Riccati, Solutionstending to 0, Sturm, Variation ofconstants, Zeros

Sectors (elliptic, parabolic, hyperbolic),161

index of stationary point, 166level curves, 173(Ex. 9.1)

Self-adjoint, operator, 342dth order equation, 398

Singular point, simple, 73, 78(Ex.11.2), 84, 86(Ex. 12.2)

regular, 73, 78(Ex. 11.1), 85, 86(Ex. 12.2)

Small at QO, 486Solution, definition, 1, 46(Ex. 1.2)

continuity with respect to initial con-ditions or parameters, 94

differentiability with respect to initialconditions or parameters, 95, 100,104, 115

D-solution, 437PD-solution, 462

Solutions tending to 0, binary systems,161, 208, 209, 211, 220

linear systems, 500perturbed linear systems, 259, 294,

300, 304, 445second order equations, 510, 514,

(Ex. 3.7)see also Stability

Spirals, 151, 159,190, 211, 216, 220see also Focus

Square root, non-negative Hermitianmatrix, 503 (Ex. 1.2)

Square root, differential operators, 354,392

Stability, asymptotic, 38, 40, 537global, 537Lyapunov, 38, 40orbital, 157, 254periodic solutions, 158, 178, 179,

253, 302(Ex. 11.3)uniform, 40

Stationary point, 144,183, 209, 212, 220index, 149, 166, l73(Ex. 9.1)see also Center, Focus, Node, Saddle-

pointSturm, comparison theorems, 333, 362

majorant, 334separation theorem, 335Sturm-Liouville problems, 337

Successive approximations, 8, 40, 45(Ex. 1.1), 57, 236, 247, 296(Ex.8.1)

bracketing, 42(Ex. 9.2), 43(Ex. 9.3,9.4)

general theorem, 404Superposition principle, 46, 63, 326

Topological arguments, 203, 278, 520Toroidal function, 508(Ex. 2.6(a))Torus, 185

flow on, 195Total differential equations, 117, 120

adjoint, 119complete integrability, 118, 123,128

Trace (tr),46Transversal, 152, 184, 196Tychonov fixed point theorem, 405, 414,

425, 444

Umlaufsatz, 147Uniqueness theorems, 31, 109

dth order equation, 33(Ex. 6.6)Kamke's general theorem, 31, 33(Ex.

6.5)van Kampen, 35L-Lipschitz continuity, 109Nagumo, 32null solutions, 211, 212(Ex. 2.3, 2.4)one-sided, 34, 110

612 Index

Uniqueness theorems, Osgood, 33second order boundary value prob-

lems, 420, 421 (Ex. 3.3), 423, 425,427(Ex. 4.6)

U,v)-subset, 281,291, 293

Variation of constants, 48, 64second order equations, 328, 329

Variational principles, 352, 390, 399

Wazewski theorem, 280

Weber equation, 320(Ex. 17.6), 382(Ex. 9.7) ,529

Wintner theorem on existence in large,29

Wirtinger inequality, 346(Ex. 5.3(6))Wronskian determinant, 63, 326

Zeros of solutions of second order equa-tion, monotony, 519(Ex. 4.3)

number of, 344see Disconjugate, Nonoscillatory,

Oscillatory, Sturm

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