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HANDBOOK OF METRIC FIXED POINT THEORY
Handbook of Metric Fixed Point Theory
Edited by
William A. Kirk Department of Mathematics, The University of Iowa, Iowa City, lA, U.S.A.
and
Brailey Sims School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, Australia
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-90-481-5733-4 ISBN 978-94-017-1748-9 (eBook) DOI 10.1007/978-94-017-1748-9
or
and
Printed on acid-free paper
All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover 1 st edition 200 1 No part of the material protected by this copyright notice may be reproduced utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage retrieval system, without written permission from the copyright owner.
Contents
Preface
Contraction mappings and extensions
W. A. Kirk
1.1 Introduction
1.2 The contraction mapping principle
1.3 Further extensions of Banach's principle
1.4 Caristi's theorem
1.5 Set-valued contractions
1.6 Generalized contractions
1. 7 Probabilistic metrics and fuzzy sets
1.8 Converses to the contraction principle
1.9 Notes and remarks
2
Examples of fixed point free mappings
B. Sims
2.1 Introduction
2.2 Examples on closed bounded convex sets
2.3 Examples on weak' compact convex sets
2.4 Examples on weak compact convex sets
2.5 Notes and remarks
3
Classical theory of nonexpansive mappings
K. Goebel and W. A. Kirk
3.1 Introduction
3.2
3.3
3.4
3.5
Classical existence results
Properties of the fixed point set
Approximation
Set-valued nonexpansive mappings
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3.6 Abstract theory
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Geometrical background of metric fixed point theory
S. Pros
5
4.1 Introduction
4.2 Strict convexity and smoothness
4.3 Finite dimensional uniform convexity and smoothness
4.4
4.5
4.6
Infinite dimensional geometrical properties
Normal structure
Bibliographic notes
Some moduli and constants related to metric fixed point theory
E. L. Fuster
6
5.1 Introduction
5.2
5.3
Moduli and related properties
List of coefficients
Ultra-methods in metric fixed point theory
M. A. Khamsi and B. Sims
7
6.1 Introduction
6.2
6.3
6.4
6.5
6.6
Ultrapowers of Banach spaces
Fixed point theory
Maurey's fundamental theorems
Lin's results
Notes and remarks
Stability of the fixed point property for nonexpansive mappings
J. Garcia-Falset, A. Jimenez-Me/ado and E. Llorens-Fuster
7.1 Introduction
7.2 Stability of normal structure
7.3 Stability for weakly orthogonal Banach lattices
7.4 Stability of the property M(X) > 1
7.5 Stability for Hilbert spaces. Lin's theorem
7.6 Stability for the T-FPP
7.7
7.8
Further remarks
Summary
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Contents
8
Metric fixed point results concerning measures of noncompactness
T. Dominguez, M. A. Japon and G. Lopez
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8.1 Preface
8.2
8.3
8.4
8.5
8.6
8.7
8.8
Kuratowski and Hausdorff measures of noncompactness
¢-minimal sets and the separation measure of noncompactness
Moduli of noncompact convexity
Fixed point theorems derived from normal structure
Fixed points in NUS spaces
Asymptotically regular mappings
Comments and further results in this chapter
Renormings of £1 and Co and fixed point properties
P. N. Dowling, C. J. Lennard and B. Turett
9.1 Preliminaries
9.2 Renormings of £1 and Co and fixed point properties
9.3 Notes and remarks
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Nonexpansive mappings: boundary/inwardness conditions and local theory
w. A. Kirk and C. H. Morales
10.1 Inwardness conditions
10.2 Boundary conditions
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10.3 Locally nonexpansive mappings
10.4 Locally pseudocontractive mappings
10.5 Remarks
Rotative mappings and mappings with constant displacement
W. Kaczor and M. Koter-Morgowska
11.1 Introduction
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11.2 Rotative mappings
11.3 Firmly Iipschitzian mappings
11.4 Mappings with constant displacement
11.5 Notes and remarks
Geometric properties related to fixed point theory in some Banach function lattices
S. Chen, Y. Cui, H. Hudzik and B. Sims
12.1 Introduction
VII
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12.2 Normal structure, weak normal structure, weak sum property, sum property and uniform normal structure 343
12.3 Uniform rotundity in every direction 356
12.4 B-convexity and uniform monotonicity 358
12.5 Nearly uniform convexity and nearly uniform smoothness 362
12.6 WORTH and uniform nonsquareness 367
12.7 Opial property and uniform opial property in modular sequence spaces 368
12.8 Garcia-Falset coefficient 377
12.9 Cesaro sequence spaces 378
12.10 WCSC, uniform opial property, k-NUC and UNS for cesp 380
13 391
Introduction to hyperconvex spaces
R. Espinola and M. A. Khamsi
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13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
Preface
Introduction and basic definitions
Some basic properties of hyperconvex spaces
Hyperconvexity, injectivity and retraction
More on hyperconvex spaces
Fixed point property and hyperconvexity
Topological fixed point theorems and hyper convexity
Isbell's hyperconvex hull
13.9 Set-valued mappings in hyperconvex spaces
13.10 The KKM theory in hyperconvex spaces
13.11 Lambda-hyperconvexity
Fixed points of holomorphic mappings: a metric approach
T. Kuczumow, S. Reich and D. Shoikhet
14.1
14.2
14.3
14.4
14.5
14.6
14.7
Introduction
Preliminaries
The Kobayashi distance on bounded convex domains
The Kobayashi distance on the Hilbert ball
Fixed points in Banach spaces
Fixed points in the Hilbert ball
Fixed points in finite powers of the Hilbert ball
14.8 Isometries on the Hilbert ball and its finite powers
14.9 The extension problem
14.10 Approximating sequences in the Hilbert ball
14.11 Fixed points in infinite powers of the Hilbert ball
14.12 The Denjoy-Wolff theorem in the Hilbert ball and its powers
14.13 The Denjoy-Wolff theorem in Banach spaces
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Contents
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14.14 Retractions onto fixed point sets
14.15 Fixed points of continuous semigroups
14.16 Final notes and remarks
Fixed point and non-linear ergodic theorems for semigroups of non-linear mappings
A. Lau and W. Takahashi
15.1 Introduction
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15.2 Some preliminaries
15.3 Submean and reversibility
15.4 Submean and normal structure
15.5 Fixed point theorem
15.6 Fixed point sets and left ideal orbits
15.7 Ergodic theorems
15.8 Related results
Generic aspects of metric fixed point theory
S. Reich and A. J. Zaslavski
16.1 Introduction
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16.2 Hyperbolic spaces
16.3 Successive approximations
16.4 Contractive mappings
16.5 Infinite products
16.6 (F}-attracting mappings
16.7 Contractive set-valued mappings
16.8 Nonexpansive set-valued mappings
16.9 Porosity
Metric environment of the topological fixed point theorems
K. Goebel
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17.1 Introduction
17.2 Schauder's theorem
17.3 Minimal displacement problem
17.4 Optimal retraction problem
17.5 The case of Hilbert space
17.6 Notes and remarks
Order-theoretic aspects of metric fixed point theory
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J. Jachymski
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18.1 Introduction
18.2 The Knaster-Tarski theorem
18.3 Zermelo's fixed point theorem
18.4 The Tarski-Kantorovitch theorem
Fixed point and related theorems for set-valued mappings
G. Yuan
19.1
19.2
19.3
19.4
19.5
19.6
19.7
Index
Introduction
Knaster-Kuratowski-Mazurkiewicz principle
Ky Fan minimax principle
Ky Fan minimax inequality-I
Ky Fan minimax inequality-II
Fan-Glicksberg fixed points in G-convex spaces
Nonlinear analysis of hyperconvex metric spaces
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Preface
The presence or absence of a fixed point is an intrinsic property of a map. However, many necessary or sufficient conditions for the existence of such points involve a mixture of algebraic, order theoretic, or topological properties of the mapping or its domain. Metric fixed point theory is a rather loose knit branch of fixed point theory concerning methods and results that involve properties of an essentially isometric nature. That is, the class of mappings and domains satisfying the properties need not be preserved under the move to an equivalent metric. It is this fragility that singles metric fixed point theory out from the more general topological theory, although, as many of the entries in this Handbook serve to illustrate, the divide between the two is often a vague one.
The origins of the theory, which date to the latter part of the nineteenth century, rest in the use of successive approximations to establish the existence and uniqueness of solutions, particularly to differential equations. This method is associated with the names of such celebrated mathematicians as Cauchy, Liouville, Lipschitz, Peano, Fredholm and, especially, Picard. In fact the precursors of a fixed point theoretic approach are explicit in the work of Picard. However, it is the Polish mathematician Stefan Banach who is credited with placing the underlying ideas into an abstract framework suitable for broad applications well beyond the scope of elementary differential and integral equations. Around 1922, Banach recognized the fundamental role of 'metric completeness'; a property shared by all of the spaces commonly exploited in analysis. For many years, activity in metric fixed point theory was limited to minor extensions of Banach's contraction mapping principal and its manifold applications. The theory gained new impetus largely as a result of the pioneering work of Felix Browder in the mid-nineteen sixties and the development of nonlinear functional analysis as an active and vital branch of mathematics. Pivotal in this development were the 1965 existence theorems of Browder, Gohde, and Kirk and the early metric results of Edelstein. By the end of the decade, a rich fixed point theory for nonexpansive mappings was clearly emerging and it was equally clear that such mappings played a fundamental role in many aspects of nonlinear functional analysis with links to variational inequalities and the theory of monotone and accretive operators.
Nonexpansive mappings represent the limiting case in the theory of contractions, where the Lipschitz constant is allowed to become one, and it was clear from the outset that the study of such mappings required techniques going far beyond purely metric arguments. The theory of nonexpansive mappings has involved an intertwining of geometrical and topological arguments. The original theorems of Browder and Giihde exploited special convexity properties of the norm in certain Banach spaces, while Kirk identified the underlying property of 'normal structure' and the role played by weak compactness. The early phases of the development centred around the identification of spaces whose bounded convex sets possessed normal structure, and it was soon
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discovered that certain weakenings and variants of normal structure also sufficed. By the mid-nineteen seventies it was apparent that normal structure was a substantially stronger condition than necessary. And, armed with the then newly discovered GoebelKarlovitz lemma the quest turned toward classifying those Banach spaces in which all nonexpansive self-mappings of a nonempty weakly compact convex subset have a fixed point. This has yielded many elegant results and led to numerous discoveries in Banach space geometry, although the question itself remains open. Asymptotic regularity of the averaged map was an important contribution of the late seventies, that has been exploited in many subsequent arguments. A turning was the discovery by Alspach in 1980 of a nonempty weakly compact convex subset of LIla, 1] which admitted a fixed point free nonexpansive mapping. This was quickly followed by a number of surprising results by Maurey. Maurey's ideas were original and set the stage for many of the more recent advances. The asymptotic embedding techniques recently initiated by Lennard and Dowling represent a novel development. For example, combined with one of Maurey's results their work shows that a subspace of L 1 [0, 1] is reflexive if and only if all of its nonempty closed bounded convex subsets have the fixed point property for nonexpansive mappings.
All of these developments are reflected in the contents of this Handbook. It is designed to provide an up-to-date primary resource for anyone interested in fixed point theory with a metric flavor. It should be of value for those wishing to find results that might apply to their own work and for those wishing to obtain a deeper understanding of the theory. The book should be of interest to a wide range of researchers in mathematical analysis as well as to those whose primary interest is the study of fixed point theory and the underlying spaces. The level of exposition is directed toward a wide audience, embracing students and established researchers.
The focus of the book is on the major developments of the theory. Some information is put forth in detail, making clear the underlying ideas and the threads that link them. At the same time many peripheral results and extensions leading to the current state of knowledge often appear without proof and the reader is directed to primary sources elsewhere for further information. Because so many authors have been involved in this project, no effort has been made to attain a wholly uniform style or to completely avoid duplication. At the same time an attempt has been made to keep duplication to a minimum, especially where major topics are concerned.
The book begins with an overview of metric contraction principles. Then, to help delineate the theory of nonexpansive mappings and alert the reader to its subtleties, this is followed by a short chapter devoted to examples of fixed point free mappings. A survey of the classical theory of nonexpansive mappings is taken up next. After this, various topics are discussed more or less randomly, including the underlying geometric foundations of the theory in Banach spaces, ultrapower methods, stability of the fixed point property, asymptotic renorming techniques, hyperconvex spaces, holomorphic mappings, generic properties of the theory, nonlinear ergodic theory, rotative mappings, pseudocontractive mappings and local theory, the non expansive theory in Banach function lattices, the topological theory in a metric environment, order-theoretic aspects of the theory and set-valued mappings.
We wish to thank all the contributors for enthusiastically supporting the project and for giving generously of their time and expertise. Without their contributions this
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Handbook would not have been possible. Special thanks are due to Mark Smith, who did almost all of the type-setting and formatting - an invaluable contribution - and to Tim Dalby. Both devoted many hours to proofreading the original manuscripts, and thanks to their efforts many oversights (including some mathematical ones) have been corrected. Lastly we must thank Liesbeth Mol, whose enthusiasm, understanding and gentle encouragement made our dealings with the publishers both a pleasant and an effective one.
ART KIRK AND BRAILEY SIMS, IOWA CITY AND NEWCASTLE, APRIL 2001