functional ifferential equations · partial differential equations constitute a significant part of...

FUNCTIONAL IFFERENTIAL EQUATIONS

Guest Editor A.L. SKUBACHEVSKII

VOLUME 12, 2005 No. 1-2

DEDICATED TO A.D. MYSHKIS ON THE OCCASION OF HIS

85th BIRTHDAY

~ THE COLLEGE OF JUDEA & SAMARIA

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TABLE OF CONTENTS

Anatoly Dmitrievich Myshkis 3

J. A. D. Appleby. Exponential asymptotic stability of non-linear Ito-Volterra equations with damped stochastic perturbations. 7

J.A.D. Appleby and A. Rodkina. Asymptotic stability of polynomial stochastic delay differential equations with damped perturbations. 35

A. Ashyralyev andY. Ozdemir. A note on the nonlocal boundary value problem for hyperbolic-parabolic differential equations. 67

G. Belitski and V. Tkachenko. Linear functional equations on manifolds. 83

L. Berezansky and E. Braverman. On persistence of a delay differential equation with positive and negative coefficients. 95

A. S. Brat us and A. S. Novozhilov. Stabilizing effect of nonlinear damping for a system with follower force. 109

A. G. Chentsov. Some questions of asymptotic analysis: approximate solutions and extension constructions. 119

G. Derfel and A.D. Myshkis. On Verduyn Lunel's conjecture about small solutions. 149

R. Finn. Floating and partly immersed balls in a weightless environment. 167

M. Gil'. The Aizerman-Myshkis problem for functional differential equations with causal nonlinearities. 175

I. Gyori and M. Pituk. Asymptotically ordinary delay differential equations. 187

R. Kadiev and A. Ponosov. Relations between stability and admissibility for stochastic linear functional differential equations. 209

1

ANATOLY DMITRIEVICH MYSHKIS (ON THE 85-TH ANNIVERSARY)

April 13, 2004 marks the 85th anniversary of the renown mathematician, Professor Anatoly Dmitrievich Myshkis. A.D. Myshkis may be regarded by right as the founder of a number of scientific schools. Let us review some of the scientific achievements of A.D. Myshkis (partially in co-authorship). His first results, obtained in 1945, were devoted to conditions for a function uniformly continuously differentiable inside a domain that is differentiable on the boundary of the domain. Later on, he established conditions under which a function, having all kth uniformly continuous derivatives in a domain, could be extended to Rn whilepreserving smoothness. A.D. Myshkis initiated the theory of functional-differential equations in a series of papers published in 1949-1951, including [1]. He was the first to denote retardedtype equations. In formulating an initial-value problem, he proved theorems on the solvability of this problem for such equations and the corresponding functional-differential inclusions as welL Importantly, he obtained sharp estimates about the oscillatory character and asymptotic behavior of solutions to linear equations. Later on, a number of theorems were also obtained on the solvability of and properties of solutions to boundary-value problems with several leading terms. It turned out that distributions were a natural way of defining a solution. The applicability of the difference scheme to boundary- value problems for functional-differential equations was also shown. In addition to many interesting and unexpected properties of its solutions, the theory of functional-differential equations has many important applications to physics, control theory, biology, and economics. Later on, the theory of functional-differential equations was further developed in the works of many noted mathematicians such as N.N. Krasovskii, Yu.S. Osipov, Yu.A. Mitropolskii, L.E. Elsgolts, R. Bellman, K. Cooke, J. Hale, T. Kato and others. Hundreds of papers have been published, and international conferences are being organized in this field every year. Partial differential equations constitute a significant part of the research of A.D. Myshkis. For systems of partial differential equations, he investigated the existence of solutions to the Cauchy problem, obtained characteristics of the uniqueness sets, provided examples of unique continuation by means of a characteristic, along with examples of nonuniqueness when posing data on a characteristic. In his work on this topic, A.D. Myshkis was the first to introduce the notion of a generalized solution for a differential equation with a multi-valued

3

4 ANATOLY DMITRIEVICH MYSHKIS

discontinuous right- hand side. For this series of papers, he was awarded a prize by the Moscow Mathematical Society.

A.D. Myshkis also considered a variant of the Dirichlet problem for solving the Laplace equation where the solution is given, up to a constant on each connected component of the boundary but the gradient flow through any closed surface must equal zero. Solvability theorems were obtained for domains of a general form. He also considered one-dimensional systems of hyperbolic equations For the first-order semilinear systems, he introduced different notions of a generalized solution, proved solvability theorems for mixed boundary-value problems, and indicated the maximum set of existence of solutions. These results were extended to those systems with boundary conditions on intermediate lines, to quasilinear equations, and then to degenerate systems having a small parameter of the derivative with respect to time. Some of A.D. Myshkis results are devoted to multi-valued mappings .and differential inclusions. The well-known PoincareBendixon theorem, concerned with a stationary point inside a closed path, was extended to differential inc! usions and semi dynamical systems. This required some generalization of the Kakutani fixed point theorem. Some related topics are the analysis of accessibility sets for a differential inclusion and the construction of the family of integral curves in a given domain of the phase space. Some results in this domain can be found in [1]. Anatoly Dmitrievichs talent as a scientist, an organizer, and a teacher became apparent upon the formation of the Mathematical Division in the Technical Institute for Low Temperature Physics and Engineering of the Academy of Sciences of Ukrainian SSR. The. mainstream of research at the division at that time was the theory .of differential equations and applications. As for applications, they were concrete: in 1963 A.D. Myshkis and his pupils began studying the mechanics of weightless fluid. This was a very real problem in view of the progress of space technology in those years. In a short time A.D. Myshkis organized a collaboration with other institutes, recruited a group of young colleagues, and promoted their scientific growth. Soon enough, the following three directions of research became clear: (a) problems of hydrostatics (nonlinear problems of equilibrium surfaces of capillary liquid, the stability of equilibrium states, their bifurcation, and stability margin; (b) small oscillations of fluid approaching equilibrium with a free surface crooked by capillary forces; and (c) convection of weightless fluid caused by self-gravitation and the thermo-capillary effect. These investigations resulted in the monograph [8], which was the first of its kind to appear in the mathematical literature. Its revised version was published in English in 1987 (see [12]) and in Russian in 1990 (see [13]). The

ANATOLY DMITRIEVICH MYSHKIS 5

achievements mentioned in the present brief review represent just a part of the scientific work of A.D. Myshkis. It is also worth pointing out his contribution to approximate and numerical methods, difference equations and inequalities, turbulent systems, impulse impact systems, spectral problems with variable boundary, and his analysis of the influence of velocity forces on oscillatory stability. The perennial educational work of A.D. Myshkis prompted him to write several textbooks on mathematics. These textbooks were republished in different languages. A.D. Myshkis pays much attention to the methodology of applied mathematics. His view on this subject is set forth in [27, 9,10]. His particular way of thinking and a specific logic used in applied mathematics were clearly formulated in the literature for the first time. In these works, one can also find his original views about how engineers and other specialists should be taught mathematics. For many years, A.D. Myshkis remained a member of the presidium of the scientific methodological board in mathematics of the Ministry for Higher and Specialized Secondary Education and the chairman of the Section of Technical Universities of the Moscow Mathematical Society. At present, he is a member of editorial boards of well-known international journals such as Nonlinear Analysis. Theory, Methods, and Applications, Functional Differential Equations, and Journal of Difference Equations. In recent years, A.D. Myshkis has actively studied a new area in the theory of functional-differential equations devoted to so-called mixed functional- differential equations. These investigations are reflected in his monograph [14]. As Anatoly Dmitrievich celebrates his 85-th anniversary, he is still full of creative forces. We wish him health and every success in the future.

E. Litsyn, A. L. Skubachevskii

The list of books by A.D. Myshkis

1. Myshkis A.D. Linear differential equations with retarded argument. M.: Gostehizdat, 1951 [in Russian]; German trans!.: Veb. Deutsch. Veri. Der Wiss., 1955. 2. Myshkis A.D. Lectures on higher mathematics. M.: Nauka, 1964 [in Russian]; English trans!.: Mir Publishing House, 1972. 3. Myshkis A.D., Rabinovich I.M. Mathematician Pirs Bol from Riga. Riga: Zinatne, 1965 [in Russian]. 4. Zeldovich Ja.B., Myshkis A.D. Elements of applied mathematics. M.: Nauka, 1965 [in Russian]; English trans!.: Mir Publishing House, 1976.

'" ··~···

6 ANATOLY DMITRlEVICH MYSHKIS

5. Myshkis A.D. Mathematics for students of higher technical institutions: special courses. M.: Nauka, 1971 [in Russian]; English trans!.: Mir Publishing House, 1975. 6. Zeldovich Ja.B., Myshkis A.D. Elements of mathematical physics. Medium consisting of non-interacting particles. M.: Nauka, 1973 [in Russian). 7. Blekhman I.I., Myshkis A.D., Panovko Ja.G. Applied mathematics: object, logic, and details of approaches. Kiev: Naukova Dumka, 1976 [in Russian); German trans!.: Veb. Deutsch. Veri. Der Wiss., 1984. 8. Babskii V.G., Kopachevskii N.D., Myshkis A.D., Slobozhanin L.A., Tyuptsov A.D. Hydromechanics in zero gravity. M.: Nauka, 1976 [in Russian). 9. Blekhman I.I., Myshkis A.D., Panovko Ja.G. Mechanics and applied mathematics. M.: Nauka, 1983 [in Russian]. 10. Myshkis A.D., Muszinskii J. Ordinary differential equations. Warszawa: PWN, 1984 [in Polish). 11. Borisovich Yu.G., Myshkis A.D., Obukhovskii B.B., Gelman B.D. Introduction to the theory of multi-valued mappings. Voronezh: University Press, 1986 [in Russian]. 12. Myshkis A.D., Babskii V.G., Kopachevskii N.D., Slobozhanin L.A., Tyuptsov A.D. Low gravity fluid mechanics. Mathematical theory of capillary phenomena. Berlin: Springer-Verlag, 1987. 13. Babskii V.G., Zhukov M.Yu., Kopachevskii N.D., Myshkis A.D., Slobozhanin L.A. Tyuptsov A.D. Methods of solving the problems of hydromechanics for zero gravity. Kiev: Naukova Dumka, 1990 [in Russian). 14. Myshkis A.D. Mixed functional-differential equations. M.: MAl Press, in Contemporary mathematics. Fundamental trends, 4(2003), 5 120 [in Russian); English trans!.: to be published in Journal of Mathematical Sciences, 2005.

J

•

FUNCTIONAL DIFFERENTIAL EQUATIONS

VOLUME 12 2005, NO 1-2 PP. 7- 34

EXPONENTIAL ASYMPTOTIC STABILITY OF NONLINEAR ITO-VOLTERRA EQUATIONS WITH DAMPED

STOCHASTIC PERTURBATIONS

J. A. D. APPLEBY '

Abstract. This paper studies the convergence rate of solutions of the nonlinear It6 -Volterra equation

(1) dX(t) = (t(X(t)) + l K(t- s)g(X(s)) ds) dt + E(t) dW(t)

where K and E are continuous matrix-valued functions defined on R +, and the functions f and g are globally linearly bounded and satisfy Lipschitz conditions. (W(t))t>O is a finite-dimensional standard Brownian motion. It is shown that when the entries of K are all of one sign on R+, that (i) the almost sure exponential convergence of the solution to zero (ii) the p-th mean exponential convergence of the solution to zero (for some p;:: 1), and (a) the exponential integrability of the entries of the kernel K, (b) the exponential square integrability of the entries of noise term E are closely related. We show that under some additional technical conditions that (a), (b) imply (ii); (a), (b), (ii) imply (i); (i), (a) implies (b); (i), (b) imply (a); (ii) implies (a); and (ii), (a) imply (b). Hence we have an equivalence between p-th mean exponential convergence of solutions and (a), (b) for a particular class of problems, and that any two of (a), (b), (i) implies the third. The paper extends results obtained in the linear case by the above authors.

Key Words. Exponential asymptotic stability; Ito-Volterra equation; Volterra equation; Liapunov exponent; almost sure exponential asymptotic stability.

AMS(MOS) subject classification. 60Hl0, 60H20, 34K20, 45D05.

1. Introduction. The subject of this paper is the exponential asymptotic convergence of solutions of perturbed nonlinear Volterra equations, in

' Centre for Modelling with Differential Equations (CMDE), School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland

7

·--··

8 J.A.D.APPLEBY

which the perturbation is a random contribution which decays over time, or is "damped". We consider perturbations of Ito type (i.e., the solution of the equation is a semimartingale driven by Brownian motion). Sufficient conditions under which solutions of stochastic differential equations or stochastic delay differential equations are exponentially asymptotically stable in either a mean square (or p-th mean) or almost sure sense have been extensively studied. We refer the reader to our earlier work on linear equations [2] for a selection of references on this topic, and also to work on stochastic equations with unbounded delay.

For deterministic linear autonomous delay differential equations with bounded delay, it is known that uniform asymptotic stability and exponential asymptotic stability are equivalent. However, for deterministic linear equations with unbounded delay - for instance, for convolution Volterra equations - it is known that the zero solution can be asymptotically stable, or even uniformly asymptotically stable, and yet not be exponentially asymptotically stable. Necessary and sufficient conditions for the exponential asymptotic stability of the linear Volterra equation

(2) x'(t) = Ax(t) + l K(t- s)x(s) ds, t;::: 0

have been established by Murakami [17], [18], under a sign condition on the entries of the kernel K. Specifically, he establishes that when the zero solution of (2) is uniformly asymptotically stable, and the entries of K(t) do not change sign on R +, the zero solution is exponentially asymptotically stable if and only if

(3) {

00

IIK(s)lle75 ds < oo, for some 1 > 0.

The results of [17], [18] have been extended by the authors in [2] to deal with the linear It6-Volterra equation

(4) dX(t) = ( AX(t) + l K(t- s)X(s) ds) dt + I::(t) dW(t), t;::: 0

where W(t) = (W1 (t), ... , wr(t)) is an r-dimensional standard Brownian motion. The principal result of that paper establishes the equivalence of the following statements, when the entries of K(t) do not change sign on R+: (i). The zero solution of (2) is uniformly asymptotically stable, and

(5) foi!K(s)lle7JSds < oo, f" III::(s)ll2e27'

5 ds < oo for /1>/2 > 0.

""'"

EXAS IN ITO-VOLTERRA EQUATIONS

( ii). There exists X > 0 such that for every p > 0 there exists Mp = Mp(Xo) > 0 such that

(6) E[IIX(t)IIPJ :<::: Mp(Xo)e->!pt, t 2 o,

for all solutions of ( 4). (iii). There exists (30 > 0 such that

(7) ' 1 hmsup -log IIX(t)ll :<::: -f3o a.s.

t--;oo t

for all solutions of (4).

9

Conditions under which exponential stability is not assured, and in which the precise rate of non-exponential decay can be established for the scalar version of ( 4), were presented in this journal in [ 1].

In the present paper, we aim to establish necessary and sufficient conditions for p-th mean and almost sure exponential convergence of solutions of nonlinear convolution Ito-Volterra equations with damped stochastic perturbations of the form

(8) dX(t) = (!(X(t)) + f K(t- s)g(X(s)) ds) + E(t) dW(t), t 2 0,

The results we obtain in this note are often very similar to those obtained in the linear case, and go through using similar arguments. In other cases, the fact that the solution of (8) cannot be represented in terms of primitive deterministic functions (as is the case for the linear problem (4)), and the non-Gaussianity of the process mean that weaker results are proven, and to obtain these results, one must revisit the problem in the spirit of Murakami's original analysis to proceed.

To prove our results, we make some standard restrictions on the functions appearing in (8). We suppose j, g : Rd --+ Rd are continuous functions satisfying global Lipschitz conditions, that E is ad x r continuous and square integrable matrix function, and that K is a d x d continuous and integrable matrix function. As above W(t) = (W1(t), ... , Wr(t)) is an r-dimensional standard Brownian motion. In addition, we suppose that f(O) = 0, g(O) = 0. Therefore the unperturbed deterministic equation

(9) x'(t) = f(x(t)) + f K(t- s)g(x(s)) ds, t 2 0

has a unique continuous solution on R+; in particular, if x(O) = 0, x(t) = 0 is the unique solution, called the zero solution. However, the process X(t) = 0

••-c•c•

10 J.A.D.APPLEBY

for all t ::0:: 0 is not a solution of (8). By viewing the random contribution in (8) as a perturbation, we may ask whether the equilibrium solution of the unperturbed problem (9) is asymptotically stable in the presence of this perturbation, and determine the conditions under which the solutions are exponentially convergent to the equilibrium solution of the unperturbed problem.

We seek connections between the following four statements concerning (8):

(a) There exists an open ball of initial conditions I C Rd such that Xo E I implies

(10) lim sup~ log IIX(t)ll:::; -f3o, t-+oo t a.s.

for some (30 > 0, which is independent of X 0•

(b) There exists p ::0:: 1, (Jp > 0 and an open ball of initial conditions I~ Rd, such that X 0 E I implies

(11) E[IIX(t)IIPJ:::; Mp(Xo)e-!3pt, t 2': 0,

where Mp = Mp(Xo) > 0. (c)

(12) hoo IIK(s)lle"'s ds < oo, for some 11 > 0.

(d)

(13) hoo IIE(s)ll2e2"'s ds < oo, for some 12 > 0.

We establish relationships between (a)-( d). These are of three types; first, conditions which, together with (c), (d), are sufficient to imply (a), (b); second, showing that (d) holds when (c) and either of (a), (b) are true; and third, under a sign condition on the entries of K, we show that (b) implies (c), and (a), (d) imply (c).

As in [2], our results sharpen slightly the pointwise exponential decay requested on the kernel K and noise perturbation E that were presented by Mao in [14] as part of a subset of sufficient conditions which guarantee the exponential convergence of solutions. We do not consider this, in itself, to be a major improvement on earlier results. However, as may be seen from the list of results mentioned above, it is our intention to suggest that the weaker hypotheses (12), (13) on the "memory" and "noise" are synonymous

.,

EXAS IN ITO-VOLTERRA EQUATIONS 11

with the exponential decay of solutions of the stochastic integrodifferential equation.

If f is negative definite, in the sense that there exists a > 0 such that (x, f(x)) :S: -allxl\ 2 for all x E Rd, and a is sufficiently large, we show in Theorem 1 that (c), (d) imply (b). The conditions employed are by no means optimal for the problem; for example, for the scalar problem with f(x) = 0, and negative feedback from the integral term, Liapunov functional techniques provide conditions under which (b) is satisfied (for a related example, see [9]).

Our aim here, however, is to exhibit the equivalence between groups of the phenomena (a)-( d) for (at least) a subclass of equations of the form (8). Nonetheless, the proof introduces an alternative line of reasoning using the deterministic comparison principle as opposed to the Liapunov function/functional techniques (see [9]), or Razumikhin techniques (see [12]), so we include it here. Independently, Mao and Riedle [15] have worked on Ito-Volterra equations in weighted spaces, and have given similar sufficient conditions for exponential stability.

In Theorem 2, we show that (b), (c), (d) together imply (a). The proof can be achieved in several ways. One way of proceeding is to follow the model of Theorem 4.3.1 in Mao [11], and related sequels in [13]; a different proof presented here uses the idea of Lemma 1.3.1 for deterministic equations in Burton '[4], in which the integrability of solutions implies their asymptotic convergence to zero.

The second set of results (Theorems 3 and 4) follow the arguments of Theorems 4.1, 4.2 in [2], and show that the exponential integrability of L; follows from either (a), (b) together with (c).

In the final set of results (Theorems 5 and 6), the main technique used to obtain (c) is a Tauberian-type argument, modelled closely on the ideas developed in Murakami [17], [18], which exploit the analyticity and existence of the Laplace transform of a function H in the negative real half plane, which agrees with the Laplace transform of K in the positive real half-plane. This enables one to extend the domain of definition of K into the negative half plane, thereby establishing (c). The existence of a such a function H relies on either (a), (b), and in the almost sure case, also relies upon the almost sure exponential convergence to zero of the random variables

{" L;(s) dW(s), t---+ oo,

where L;(t) satisfies (d). The results of Theorems 1-6 are arranged together in some equivalences

in the last section of the paper. Theorem 7 shows that under some restrictions

•+•··

12 J. A. D. APPLEBY

introduced in earlier theorems that (b) is equivalent to (c) and (d), taken together. Theorem 8 gives conditions under which (b) implies (a), (c), and (d). Theorem 9 shows, under some technical restrictions, that any two of (a), (c), (d) implies the third.

The organisation of the note is as follows: definitions, a precise statement of the problem, and supporting theory is contained in Section 2. Theorems 1 and 2 are the topic of Section 3. The converse results relating to the square integrability of I; are contained in Section 4; those connected with the integrability of K are exposed in Section 5. Section 6 contains the equivalences and summaries collected in Theorems 7-9.

2. Background Material. We first fix some standard notation. Denote by R+ the set [O,oo). As usual, let x 1\ y denote the minimum of x, y E R.

Denote by C(/; J) the space of continuous functions taking the finite dimensional Banach space I onto the finite dimensional Banach space J. Let d be a positive integer. Let Md,d(R) denote the space of all d x d matrices with real entries, and C(R+; Md,d(R)) stand for all continuous dxd matrix-valued functions with domain R+. The corresponding d x r matrices and matrix functions lie in the spaces Md,r(R) and C(R+; Md,r(R)), respectively. We say that the function f : R + --+ Mn,m (R) is in £ 1 (R+) if each of its entries is a scalar Lebesgue integrable function, and in £2(R+) if each of its entries is a scalar square integrable function. The convolution of the function f with g is denoted by f *g. The transpose of any matrix A is denoted AT; the trace of a square matrix A is denoted by tr(A), and its determinant by det(A). Further denote by Id the identity matrix in Md,d(R). Let e; denote the i-th standard basis vector in Rd, say. Let llxll stand for the Euclidean norm of x E Rd, and llxllt be the sum of the absolute values of the components of x. If A= (Aij) E Md,r(R), A has operator norm denoted by IIAII, and given by

IIAII = sup{IIAxll : x ERr, llxll = 1}.

It further has Frobenius norm, denoted by IIAIIF, and defined as follows: if A = ( ai,j), is an d x r matrix, then

(

d r ) 1/2 IIAIIF = t; j; lai,jl2 = tr(AAT)~.

Since Md,r(R) is a finite dimensional Banach space, 11·11, II·IIF are equivalent, so there exist positive universal constants c1(d, r) ~ c2(d, r) such that

(14) c1(d,r)IIAII ~ IIAIIF ~ c2(d,r)IIAII, A E Md,r(R).

We will use one other matrix norm in this paper: for A= (a;j) E Md,d(R), we define IIAIIN = L,f;J L.J;J laiJI·

We revisit some of the important properties of Laplace transforms. We denote by C the set of complex numbers, and the real part of s E C by ~s. Iff : R+ -r Md,d(R), we can define the Laplace transform off at s E C to be

](s) = f' f(t)e-'1 dt.

If a E Rand j 000 llf(t)lle-"1 dt < oo, then ](A) exists and is continuous in A

for ~A 2 a, and analytic on ~A > -a. See for example, Churchill, p.171 [6], or Widder [19].

Standard definitions of stability for scalar linear Volterra equations will be referred to in this paper. Consider the scalar equation

(15) x'(t) = ax(t) + l k(t- s)x(s) ds, t 2 0

with resolvent defined by z(O) = 1, and satisfying

(16) z'(t) = az(t) + l k(t- s)z(s) ds

For any t0 2 0 and ¢ E C([O, t0]; R), there is a unique real-valued function x(t), which satisfies (15) on [t0 ,oo) and for which x(t) = ¢(t) fortE [0, t0].

We denote such a solution by x(t; t 0 , ¢). We recall the various standard notions of stability of the zero solution

of (15) required for our analysis; the reader may refer further to Miller [16]. For¢ E C([O, to]; Rd) we define I<Pito by 1</Jito = SUPo:::s:Sto ll¢(s)ll·

The zero solution of (15) is said to be uniformly stable (US), if, for every c: > 0, there exists 8(c:) > 0 such that t0 E R+ and¢ E C([O, t0]; Rd) with I <PI to < 8(c:) implies llx(t; to, ¢)1h < c: for all t 2 to. The zero solution is said to be uniformly asymptotically stable (U AS) if it is US and there exists 8 > 0 with the following property: for each c: > 0 there exists a T(c:) > 0 such that to E R+ and¢ E C([O, to]; Rd) with I <Pita < 8 implies llx(t, to, ¢)11! <£for all t 2 to +T(c:).

The properties of the resolvent z are deeply related to the stability of the zero solution of (15). It is shown in [16] that the zero solution of (15) is UAS if and only if z E U(R+). If a> 0, and k(t) 2 0 in (15) above, with a> J0

00 k(s) ds, Burton and Mahfoud [5] have shown that the zero solution is UAS. Murakami has proved in [18] that if the zero solution of (15) is uniformly asymptotically stable, and there exists 'f > 0 such that j 0

00 lk(s)le7 ' ds < oo,

,..,. "~··

14 J.A.D.APPLEBY

then there exists C > 0 and A > 0 such that the resolvent of ( 15) satisfies lz(t)l :::; Ge-M for all t ;::>: 0.

In this paper, we study general finite-dimensional nonlinear stochastic integro-differential equations (or Ito-Volterra equations) with stochastic perturbations. The equations studied are of the form

(17) dX(t) = (t(X(t)) + l K(t- s)g(X(s))ds) + L:(t)dW(t), t ;::>: 0

where (W(t))t>o is an r-dimensional Brownian motion on a complete filtered probability sp;ce (0, .r, (F(t))t?o, P), where the filtration is the natural one F(t) = a-(W(s) : 0:::; s:::; t}. X has deterministic initial condition X 0 = x. Therefore, (17) is the usual shorthand notation for the evolution of a process obeying

X(t) = x

+ l (t(X(s)) +los K(s- u)g(X(u)) du) ds + l L:(s) dW(s), t ;::>: 0.

The functions J, g are in C(Rd;Rd). Moreover, we assume they satisfy a global Lipschitz condition and f(O) = g(O) = 0. Therefore, there exists At > 0, A9 > 0 such that

(18) llf(x)ll :::; Atllxll, for all x E Rd,

and

(19) llg(x)ll :::; A9 llxll, for all x E Rd.

In addition, suppose

(20) L: E C(R+; Md,r(R)) n L 2 (R+),

and

(21) K E C(R+; Md,d(R)) n L 1 (R+)

Under the above hypotheses, there exists a unique continuous solution of (17). See [3], or [7], for instance.

In this paper, E[Z] denotes the expectation of an F-or F(t)-measurable random variable Z viz.,

E[Z] = { Z(w) dP(w). lwEfl

..


This immediately enables us to state some regularity properties of the moments of the solution of this equation. The functions

t r-7 E[ sup IIX(s)il 2], and t r-7 E[IIX(t)il 2

], os;ss;t

are bounded on compacts, and continuous on R +, respectively. A proof of this can be found in Problem 5.3.15 in [8], for example.

We reiterate our definitions of p-th mean and a.s. exponential convergence of solutions of (17) (see [2]).

DEFINITION 1. The Rd-valued stochastic process (X(t))t2:0 is p-th mean exponentially convergent, for p > 0, if there exists /3p > 0 such that

lim sup~ log(E[IIX(t)IIP]) ~ -/3p. t-+oo t

The definition of a.s. exponential convergence has a similar form: DEFINITION 2. The Rd-valued stochastic process (X(t))t2:0 is almost

surely exponentially convergent, if there exists (30 > 0 such that

lim sup~ log IIX(t)il ~ -f3o, t-H)O t

a.s.

This definition is, in turn, equivalent to the following: for every e > 0, there exists n, c n, with P[n,] = 1, and an a.s. finite random variable C(e) > 0 such that for all w E It, we have

IIX(t)(w)ll ~ C(e)(w)e-(ilo-c)t, t ~ 0.

For other technical points of stochastic analysis, we direct the reader to [8] or [11].

3. Sufficient conditions for exponential convergence of solutions. In this section, we present sufficient conditions under which solutions of (17) converge exponentially fast to zero, in a p-th mean sense (more specifically, in a mean-square sense), or an almost sure sense.

THEOREM 1. Let f, g, K, E satisfy (18}-(21) above. Suppose (12), ( 13) hold. If there exists a > 0 such that

(22) (x, f(x)) :S -allxll 2, X E Rd

'

and

(23) a> 1\.9 f" IIK(s)il ds

16 J. A. D. APPLEBY

then there exists X > 0, and M = M(Xo) > 0 such that

(24) E[IIX(t)ll 2]:::; M(Xo)e-2

-"1

, t 2 0.

'•+•··

Proof Denoting Xi(t) = (X(t), ei), and using Ito's rule, we obtain

diiX(t)ll 2 = ( 2(X(t), j(X(t))) + 2(X(t), l K(t- s)g(X(s)) ds)

(25) +III:(t)ll}) dt + ~ 2Xi(t) (~ L:;,1(t) dWi(t)) .

Since t H E[sup0:Ss::;t IIX ( s) 11 2] is bounded on compact sets, we have

(26) E [t+h ~ 2Xi(s) (~ L:;,1(s) dWi(s))] = O,

for any t, t + h 2 0. We now seek a bound on the second term in the drift on the right-hand side of (25). To this end, note that by successively using the linearity of the inner product, the Cauchy-Schwarz inequality, the Banach algebra property of the norm, (19), the inequality 2lxyl ~ x2 + y2

, and the integrability of K, we obtain the sequence of inequalities:

\·2(X(t), l K(t- s)g(X(s)) ds)j

- \l2(X(t), K(t- s)g(X(s))) dsj

< l2I(X(t), K(t- s)g(X(s)))l ds

:::; l211X(t)IIIIK(t- s)g(X(s))ll ds

< A9 l2IIX(t)IIIIK(t- s)IIIIX(s))llds

< Ag liiK(t- s)II(IIX(t)ll 2 + IIX(s))ll 2) ds

(27) :::; A9 f" IIK(s)lldsiiX(t)ll 2 +A9 liiK(t- s)IIIIX(s))ll2 ds.

Using (22), (25), (27), we have for t, t + h 2 0

IIX(t + h)ll2 -IIX(t)ll 2

rt+h - it (2(X(s), f(X(s))) + 2(X(s), (K *(go X))(s)) + III:(s)ll}) ds

t+h d r

+ [ I; 2Xi(s) I; L:;,1(s) dWi(s). t i=l j=l

Thus

IIX(t + h)ll2 -IIX(t)ll 2

< l+h (- (2a- A9 fo"" IIK(u)ll du) IIX(s)ll 2

+A9 fo' IIK(s- u)JIIIX(u)JI2 du + IJE(s)ll}) ds

rt+h d r

+ }, L 2Xi(s) L E;,1 (s) dW1(s). t i=.l j=.l

Set m(t) = E[IJX(t)ll2]. Noting that t >-+ m(t) is bounded on compact sets, by taking expectations across the last inequality, and using (26), we get

rt+h( r"" m(t +h)- m(t):::; lt - (2a- A9 lo IIK(u)ll du)m(s)

+A9 fo' IIK(s- u)llm(u) du + IIE(s)ll}) ds.

Since t >-+ m(t) is continuous, we obtain

(28)

D_m(t):::; -(2a- A9 fo"" IIK(u)Jl du)m(t)

+A9 lllK(t- s)llm(s) ds + IIE(t)IJ}.

Note that m(O) > 0. Define x(t) fort 2: 0 by x(O) = m(O), and

x'(t) = -(2a- A9 f' IIK(u)ll du)x(t)

(29) +A9 lllK(t- s)JJx(s) ds + IIE(t)ll}.

By the comparison principle (see Theorem 1.4.1 in [10], for example), we have

(30) m(t) :::; x(t) for all t 2: 0.

Next, define k(t) = IIK(t)ll fort 2: 0 and define z(t) fort 2: 0 by z(O) = 1 and

(31) z'(t) = -az(t) + l k(t- s)z(s) ds, t 2: 0,

where k E C(R+; R+) n U(R+), and a = 2a- A9 f0"" k(s) ds. By (23), we have a > f0"" k(s) ds. Hence by Theorem 1 in Burton and Mahfoud [5], we

18 J.A.D.APPLEBY

have that the zero solution of (31) is uniformly asymptotically stable. By (12), we have that J0

00 k(s)e718 ds < oo, and so, by Theorem 1 of [18], there exists ,\ > 0, and C > 0 such that

(32) z(t) :s:: ce-2>.t, t 2: 0.

By (29), (31), and (32), we have

x(t) = z(t)x(O) + l z(t- s)IIL:(s)ll~ds

(33) :S:: Ce-2>.tx(O) + Ce-2>.t l e2>.siiL:(s)ll~ ds.

Define X = ,\A f'2 , where /'2 > 0 is given by (13). By Lemma 3.1 of [2], we have

{t ' roo e-2.\t lo e2>."11L:(s)ll~ds :S:: e-2.\t lo e27zsiiL:(s)ii~ds.

Inserting this bound into (33), and using (30) gives (24), where we identify M = M(Xo) by

M(Xo) = C (11X(O)II2 + fnoo e27'.IIL:(s)il~ds). 0

To establish almost sure exponential convergence of solutions of (17), it is sufficient to know that (11), (12), (13) are true.

THEOREM 2. Let j, g, K, L: satisfy {18}-{21} above. Suppose {11}, (12) and {13) hold. Then {10} holds.

Proof. Choose initial conditions X 0 E J, so that (11) holds for some p 2: 1. Then, by Liapunov's inequality, there exists fh = (Jp/ p > 0, and M1 (X0 ) = Mp(X0 )

11P > 0 such that

E[IIX(t)IIJ :S:: Ml(Xo)e-fht, t 2: o.

Let fJo < (J1 A"f1 /\f'2 , where/'!, "!2 > 0 are defined by (12), (13). Define Y(t) = ef3ot X(t), K(t) = ef3ot K(t), i';(t) = ef3otL;(t). By construction, E[IIYIIJ E

U(R+). Therefore Y E U(R+), a.s .. From (12), (13), k E L1(R+), and f; E L2 (R+). Using (stochastic) integration by parts, the definition of Y and (17), we get

Y(t) = X(O) + fJo l Y(s) ds + l ef3os f(e-13° 8Y(s)) ds

(34) + l (los K(s- u)ef3oug(e-f3ouY(u)) du) ds + l t(s) dW(s).


The first term on the right-hand side of (34) is constant, and therefore has almost sure limit as t-+ oo. So does the second, since Y E £l(R+), a.s. For the third, notice by (18) that

f lle;Jos f(e-;Josy(s))ll ds :'0 At lot IIY(s)ll ds,

so the almost sure integrability of Y ensures that the third term tends to a limit almost surely. The fourth term has an almost sure limit, using an identical argument. Since K * Y E L1(R+) almost surely, and (19) obtains, the bound

lot lifo' K(s- u)eiJoug(e-;JouY(u)) dull ds::; A9 f (K * Y)(s) ds,

gives the existence of an almost sure limit. Regarding the fifth, and final, term on the right-hand side of (34), note that f: E L2 (R+) implies that each of the entries f:i,J is a scalar square integrable function. By the martingale time change theorem, it follows that each of the scalar stochastic processes

f f:;,j(s) dWi(s)

has a limit almost surely, as t-+ oo, fori= 1, ... , d, j = 1, ... ,j. Therefore each of the processes

zi(t) = t f f:;,j(s) dWi(s) j=l 0

has an almost sure limit as t -+ oo, i = 1, ... , d. But zi(t) is just the ith

component in the vector f~ f:(s) dW(s), and so f~ f:(s) dW(s) must have an almost sure limit as t -+ oo.

Since every term on the right-hand side of (34) has an almost sure limit as t -+ oo, it follows that

Y* := lim Y(t) t-H:XJ

exists, a.s. But, as Y E L1(R+) a.s., it must be that Y* = 0, a.s., or else a contradiction is introduced. Therefore, for X 0 E I, we have

lim ei3at X(t) = 0, a.s .. t->oo

Therefore, (10) is established, with f3o = (31 II 'Yt II /'2· 0 The proof of this result can also be established along the lines of Theorem

3.2 in [2]. The above proof is inspired by Lemma 1.3.1 in [4].

,., ..

20 J. A. D. APPLEBY

4. Exponential square integrability of E. As mentioned earlier, the results in this section give sufficient conditions under which the noise is exponentially square integrable. Specifically, it is shown that if the solution of (17) is exponentially convergent in some sense, and the kernel K is exponentially integrable, then E must be exponentially square integrable.

The proof of first result follows that of Theorem 4.1 in [2] very closely. THEOREM 3. Let J, g, K, E satisfy (18}-(21} above. Suppose (10} and

(12} hold. Then {13} is true. Proof. Let (30 be defined by (10), 'Yl > 0 by (12). Let 'Y2 be a positive

constant satisfying 'Y2 < (30 1\ 'Yt, and define the process Y(t) = e72 tX(t). By (10), we have

(35) lim Y(t) = 0, Y E L1(R+) a.s. t-->oo

Define K(t) = e72tK(t). Then k E £l(R+), by (12). Integration by parts on the process Y, using (17) gives, on rearrangement,

l e728 E(s) dW(s) = Y(t)- X 0 - 'Y2 l Y(s) ds

rt rt r· _ (36) - lo e72' f(e- 728Y(s )) ds- lo lo K(s- u)e72"g(e-7'"Y(u)) duds.

Consider the limit as t -+ oo on the right hand side of (24). By (35), the first and third terms have almost sure limits as t -+ oo. The second term is constant, and hence has almost sure limit as t -+ oo.

Using the same argument as in Theorem 2 above, we note that

(37) llle7'' f(e-728Y(s))ll ds::; A1 liiY(s)ll ds,

and

(38) lllf K(s- u)e72"g(e-721'Y(u)) dull ds::; A9 l (K * Y)(s) ds.

Therefore, both the fourth and fifth terms on the right-hand side of (36) above have almost sure limits, as k and Y are integrable. As all terms on the right-hand side of (36) have an almost sure limit,

lim {t e7''E(s)dW(s) t-+oo lo

exists almost surely. The latter part of the argument of Theorem 4.1 in [2] now applies directly, and (13) is true. 0

The proof of the second result in this section follows that of Theorem 4.2 in [2] very closely.

THEOREM 4. Let f, g, K, 2:: satisfy (18)-(21) above. Suppose (11) and (12) hold. Then (13) is true.

Proof. By ( 11) and Liapunov's inequality, there exists M1 > 0, fJ1 > 0 such that E[ijX(t)IIJ::; M1e-/3't. Choose "12 > 0 such that "/2 < fJ1 11"(1, where "(1 > 0 is defined by (12). Define Y(t) as in Theorem 3 above. Then

lim E[ljY(t)IIJ = 0, E[IIYIIJ E L1(R+). t->oo

Defining f< as in Theorem 3 means that f< E U(R+). Using the triangle inequality on (36), employing the bounds obtained in (37), (38), and lastly taking expectations yields

E [Ill e~'·L:(s) dW(s)ll] :S: EIIY(t)ll + IIXoll + b2ll E[IIY(s)IIJ ds

(39) +At l E[ijY(s)IIJ ds + A9 l los IIK(s- u)IIE[jjY(u)ll] duds.

Every term on the right hand side of (39) is uniformly bounded on R+; the first term is bounded as t 1--t E[ljY(t)ll] is continuous on R+ and has zero limit at infinity; the second term is constant; the third is bounded as E[IIYIIJ E U(R+), while the fourth is bounded as IIKII * E[IIYIIJ E £l(R+). Thus, there exists D2 > 0 such that

E [ill e~28 Z::(s) dW(s)ll] ::; D2.

The remainder of the proof follows that of Theorem 4.2 of [2] verbatim. 0

5. Exponential Integrability of K. In this section, we establish conditions under which the kernel K is exponentially integrable, i.e., under which it satisfies (12). The proofs are of a similar flavour to the converse results established in Murakami [17], [18], and use Tauberian-type arguments. As in those papers, we impose the extra restriction that the entries of K do not change sign on R +.

Specifically, we show in Theorem 5 that if the solution of (17) is a.s. exponentially convergent, and 2:: is exponentially square integrable, then the kernel K is exponentially integrable, under an additional technical constraint (which does not seem very restrictive, but is, admittedly, difficult to check). In Theorem 6, we show that if the solution is exponentially convergent in p-th mean for some p 2: 1, K is exponentially integrable, under an additional constraint similar to that required to prove Theorem 5.

22 J. A. D. APPLEBY

Before proving our main results, we need to establish two important lemmata. The first abstracts the key ideas of Murakami's converse results, and is needed in both results. The second lemma is required for the proof of Theorem 5 only, and is used to show the a.s. exponential convergence to zero of

[xo E(s) dW(s) as t-+ oo,

when E satisfies (13). LEMMA 1. Suppose that K E C(R+; Md,d(R)) n L 1(R+) has entries

which do not change si11n on R +, and that there exists a complex-valued function H satisfying the following:

(Ml} There exists an open disk U containing the origin such that A f-t H(A) is analytic on U;

(M2} H(A) = K(A) for all ~A > 0; (M3} H(O) = K(O).

Then there exists 'Y > 0 such that

f" JJK(s)JJe~s ds < oo.

Proof Allowing for some additional, but minor, technical modifications, the proof of this result can be abstracted from the proofs of Theorem 2 in [18] (which gives a detailed proof in the scalar case) and Theorem 2 in [17] (which sketches the extension of the proof of Theorem 2 in [18] to the finite dimensional case). 0

The proof also requires an a.s. exponential estimate on the convergence rate of the family of random variables

{" CJ(s) dB(s) := f" CJ(s) dB(s) -l CJ(s) dB(s),

as t -+ oo, where CJ is a real square integrable function.

(40)

LEMMA 2. Suppose CJ E C(R+; R) n U(R+) which satisfies

roo e2~,sCJ(s)2ds < 00 lo '

for some "(2 > 0. Then

lim sup~ log I roo CJ(s) dB(s)l :": -"(2, a.s. t-too t lt

where (B(t)) 1>o is a one-dimensional standard Brownian motion. Proof. D~fine the process M = {M(t); t 2: 0} by M(t) = J~ a(s) dB(s),

t 2: 0. Then M is a continuous, square integrable martingale with square variation process (M) given by

(M)(t) = f a(s) 2 ds, t;::: 0

Since a E £2(R+), we have that E(M)(oo) < oo, and so there exists M(oo) such that

lim M(t) = M(oo), a.s. 1->oo Moreover,

E[M(oo) 2] = E(M)(oo) = f" a(s)2ds.

We write M(oo) = J000 a(s) dB(s). Therefore, as we understand the random

variable f100 a(s)dB(s) = M(oo)- M(t), we have

lim {oo a(s) dB(s) = 0, a.s. t-+oo lt

Since

(100

a(s) dB(s)) 2

= (M(oo)- M(t))2

- M(oo) 2- 2(M(oo)- M(t))M(t)- M(t) 2,

and M(oo)- M(t) and M(t) are independent random variables, taking expectations gives

E [ (1oo a(s) dB(s)r] = 100 a(s)2 ds t 2: 0.

We now can see that this expectation decays exponentially, for by ( 40), with C = J0

00 e27 28 a(s) 2 ds, we obtain

100 a(s)2 ds::; 100 e272(s-1la(s)2 ds::; e-2721 100 e2'Y2Sa(s)2 ds = ce-2721,

and therefore

( 41) E [ (1oo a(s) dB(s)r] ::; Ce-2721 , t;::: 0.

24 J. A. D. APPLEBY

Next, for every t E R+, there exists n = n(t) EN, n <:: 1 such that n- 1:::; t:::; n. Using the inequality (x + y) 2 :::; 2(x2 + y2), we have

(f' !7(s) dB(s) r :::; 2 ({1

!7(s) dB(s) r + 2 (L~~ !7(s) dB(s) r. Thus by Doob's martingale inequality, and using (41), we obtain

E [ sup (roo !l(s) dB(s))2

] n-l:St::Sn it

:::; 8 rn !7(s)2 ds + 2Ce-2~z(n-!) :::; 10Ce-2~'(n-l). ln-1

Thus for 0 < c < "'(2 , Markov's inequality yields

p [ sup I roo !7(s)dB(s)l 2:: e-hz-e)n]:::; 10Ce2~2 e-2en, n-1::St:Sn lt

Hence, for every c E (0, "'12), by the first Borel-Cantelli Lemma, there exists Q* <;;; Q with P[Q*] = 1, such that for each wE n•, there exists N(w) EN so that

sup I roo !7(s)dB(s)l < e-hz-e)n, n > N(w) + 1. n-l:St$n it

For each t E R+, there exists n = n(t) such that n <:: t <:: n- 1 > N(w), so

! Jog I roo !7(8) dB(s)l :::; ! Jog sup I roo !7(s) dB(s)l t it t n-19:0:n it

Thus, by taking limits,

n < -("'12- c)-- < -("'12- c). n-1

Jim sup! log I roo !7(s) dB(s)l < -("'12- c), a.s. t-+00 t it

Letting c + 0 though the rational numbers yields the required result. 0 We are now in a position to prove the main results of this section. THEOREM 5. Let f, g, K, I: satisfy (18}-(21) above. Suppose (10), (13)

hold, and that the entries of K do not change sign on R+. If there exist d solutions of (17) (Xj(t))t>o with initial conditions Xj(O) E I for j = 1, ... , d such that

(42) det (fooo g(X 1(s)):g(X2 (s)); .. · ;g(Xd(s))ds) ~ 0, a.s.,


then {12} is true. Proof. Let (XJ (t) ) 1~0 , j = 1, ... , d be d solutions of (17) arising from

the initial conditions Xi(O), j = 1, ... , d, and satisfying (10), (42). Define the d x d matrix function X on R+ by

(43) X(t) = [x1(t):X2(t): ... :xd(t)].

Further define F,G E C(Md,d(R);Md,d(R)) as follows: if X E Md,d(R) is

written in block form as X = [ X 1 : X 2 : ••• : xd]' then

so

F(X) = [!(X1):j(X2): ... :j(Xd)],

G(X) = [g(X1): g(X2

): .. • : g(Xd)] ,

(44) F(X(t)) = [!(X1(t)): j(X2(t)): .. · :j(Xd(t))l,

and

(45) G(X(t)) = [g(X 1(t)) :g(X2 (t)): ... :g(Xd(t))].

Also introduce the Md,d(R)-valued martingale

(46) M(t) = [fL:(s)dW(s): fL:(s)dW(s): ... : lE(s)dW(s)].

From (17), and (43)-(46) we get

(47) x(t)

= X(O) + f (nY(s)) + { K(s- u)G(X(u)) du) ds + M(t).

By (10), we have X(t) --> 0 as t --> oo, a.s., and X E V(R+) a.s .. By (18), (19), (44), (45), we have FoX E L1(R+) a.s., and Go X E V(R+) a.s .. Moreover, asK E L1(R+), we have K *(Go X) E L1(R+), a.s. Since L: E L2 (R+), we have that M(t) -t M(oo) a.s. as t -too (recall we proved the same result in Theorem 2). Therefore, with N(t) = M(oo)- M(t), we can write

(48) -X(t) = [" (F(X(s)) + f K(s- u)G(X(u)) du) ds + N(t).

... ,. """

26 J.A.D.APPLEBY

Fori= 1, ... , d, j = 1, ... , d, we have

(49) r

100 Ni,J(t) =I; L;i,z(s) dW1(s). l=! t

Since (13) implies

fo00

L;i,l(s) 2e2~"ds < oo,

by Lemma 2, we have

lim sup~ log 1100

L;i,z(s) dW1(s)l:::; -12 , a.s., t--too t t

for all i = 1, ... , d, l = 1, ... , d. Thus, for every c E (0, 12) there exist almost surely finite random variables Ci,z(c) > 0 such that

(50) 1100

L;i,z(s) dW1(s)l:::; Ci,t(c)e-(72 -•)<, t 2: 0 a.s.

Define Ci(c) = Lt=l Ci,z(c). Then, by (49), (50), we have,

IN(t)l < C·(c)e-(7,-£)t t > 0 a.s. t)J - t ' -

Setting C;(c) = dZ::f=1 Ci(c) > 0, we have

(51) IIN(t)IIN:::; c;(c)e-(~,-o)t, t 2:0 a.s.

From (10), (43), for every c E (0,/30/2), there exists an almost surely finite random variable C~ (c) > 0 such that

(52) IIX(t)IIN:::; C~(c)e-(fJo-•l<, t 2: 0 a.s.

Note by norm equivalence, that there exists c1 > 0 such that IIAIIN :::; c1IIAII for all A E Md,d(R). Using this and (52) together with (18) and (44) now imply

(53) IIF(X(t))IIN:::; c1IIF(X(t))ll:::; c!AfC~(c)e-(fJo-<)t, t 2:0 a.s.,

and, using (19), we have

(54) IIG(X(t))IIN:::; c1IIG(X(t)ll :::; A9c1 C~(c)e-(i3o-•)<, t 2: 0 a.s.

:j

,.

Now consider ,\ E C with lR,\ > 0. Then

1: \\e-'' 1,: Lo K(s- u)G(X(u)) duds\\ dt

< 1: e-rrv.t /,)IKII * IIG o X[[)(s) dsdt

= 1,:0 ([[KII * [[Go X[[)(s) 1: e-fR>-t dt ds

< ~,\ f" [[K[[(s) ds f" IIG(X(s))[[ ds < oo.

27

Denote the Laplace transform of X by x. (For any other function Y (say), denote its Laplace transform by Y.) Then by (48), (51), (52), (53), (54) for )R,\ > 0, we get

-x(>-) = f' e-A! {" F(X(s)) ds dt

(55) + f' e-A! {" (K *(Go X))(s) ds dt + fooo e->-t N(t) dt.

Noting that t r-+ F(X(t)) E U(R+) a.s., we have

fooo e-At 100

F(X(s)) dsdt

(56) = ~ (f" F(X(s)) ds- (FoX)(>-)), )R,\ > 0,

and since K E L1(R+), we obtain

fooo e-At 100

( K * ( G o X)) ( s) ds dt

(57) = ~ (fooo K(s) ds f" G(X(s)) ds- K(>.)(G:-X)(>-l), lR>. > o.

Inserting (55) and (56) into (57), we get, for f.r(>.) = f000 e->-t N(t) dt the

relation

-x(>-) = ~ (f' F(X(s)) ds- (F:-X)(>-l) +

(58) ~ (fooo K(s) ds fooo G(X(s)) ds- K(,\)(G :-x)(>-)) + N(>.),

for all lR,\ > 0. Note by (42) that det((G o X)(O)) i' 0, a.s. By (54),

,\ r-+ (Go X)(>.) is continuous for )R,\ ~ -(J', where (J' E (O,(Jo- E). Since the determinant of a matrix is a continuous function of its entries, ,\ r-+

... ,.

28 J. A. D. APPLEBY

det(G oX)(,\) is continuous for~,\:;::: -(3'. T~, there exists an open disc,

containing the origin (U', say), s~ that ( G o X)(,\) - 1 exists for ,\ E U'.

Then a~4) implies that,\ f-t (Go X)(,\) is analytic on U', it follows that

,\ f-t (Go X)(,\)-1 is analytic on U'. Note from (51) that,\ f-t N(,\) is continuous for~,\:;:::-"(', and analytic

on~,\> -"(1, where "(1 E (0,"(2 - c-). Now choose f.1. ="!'II (3' > 0. By the

above discussion, the function

H(A) = (Ax( A)+ f' (F o X)(s) ds- (F :-x)(A)

(59) + f' K(s)ds f' G(X)(s)ds+AN(A))(c-:-x)(At 1•

is well-defined (and analytic) on U = U' n { ,\ : ~,\ > - f.1.}, so H defined by (59) satisfies (Ml) in Lemma 1. Comparing (58), (59), we see that H also satisfies (M2), (M3) of Lemma 1. Therefore the conclusion of Lemma 1 holds, namely (12), which is the required result. D

The result of this Theorem requires an ancillary hypothesis which is unnecessary in the deterministic analysis. It can be shown in for the linear versioD: of (17) that ( 42) holds if the deterministic problem has a uniformly asymptotically stable zero solution. For the nonlinear problem, however, we have been unable to obtain sufficient conditions under which ( 42) holds in the finite-dimensional case. On the basis of the linear problem, however, it appears that ( 42) is not too restrictive.

In the scalar nonlinear case, we can present a condition on g which guarantees that ( 42) holds.

LEMMA 3. Consider the unique strong solution of the Ito- Volterra equa-tion

dX(t) = (f(X(t)) + l K(t- s)g(X(s)) ds) dt + E(t) dW(t)

where f, g E C(R; R) obey {18}, (1g), K E C(R+; R)n£l(R+), E E L2 (R+), E E C(R+; Rr), and W is r-dimensional standard Brownian motion. Suppose that

(60) g does not change sign on R, and any zeros of g are isolated,

and that E is not identically zero. Then

(61) f' g(X(s)) ds > 0, a.s.

h

1~ ,~


The proof of this result is presented at the end of this section. An example of a function which obeys (60), as well as a global linear bound and a Lipschitz continuity condition, is g(x) = I sin(x)l. In the deterministic case (when :S is identically zero) an examination of the proof of Lemma 3 reveals that the condition (60) along with the convergence of the solution to zero, guarantees (61), provided the initial condition is nontrivial. In the unperturbed case, this last stipulation simply guarantees that the solution is nontrivial.

We now turn to the second principal result of this section. THEOREM 6. Let f, g, K, E satisfy (18)-(21) above. Suppose (11)

holds, and the entries of K do not change sign on R +. If there exist d solutions of (17) (XJ(t)Jr:::o with initial conditions XJ(o) E I for j = 1, ... , d such that

(62) det (f" E[g(X1(s))J; E[g(X2 (s))]; ... ; E[g(Xd(s))J ds) f- 0,

then (12) is true. Proof The proof follows that of Theorem 5 very closely, so we sketch

the main points only. Using the same notation as Theorem 5, we have (47). By (11), we have E[IIXIIJ E L1(R+), and hence E[IIF o XII], E[IIG o XII] E U(R+). Therefore X, FoX, Go X E L1(R+) a.s. As in Theorem 5 above, M(t) -+ M(oo), a.s. Therefore, the right-hand side of (47) has an almost sure limit as t-+ oo, so X(t) -+ 0, as t-+ oo a.s .. Hence (48) obtains, as previously. Since E E L2 (R+), we have E[N(t)] = 0, for all t ~ 0. Therefore, taking expectations across ( 48) yields

(63) -E[X(t)] = {" ( E[F(X(s))] +fa' K(s- u)E[G(X(s))] du) ds.

For convenience, write

X1(t) = E[X(t)], F1(t) = E[F(X(t))], and G1(t) = E[G(X(t))].

By (11), there exists (31 > 0 such that

(64) IIXl(t)ll :S c;e-il't, IIFl(t)ll :S C~e-il't, IIGl(t)ll :S C~e-il't,

where C(, C~, q > 0, and (18), (19) are used. Since X1, F1, G1, K E U(R+), for~,\> 0, we have

- 1 ( roo - ) -X1(J\) = :\ lo F1(s) ds- F1(J\)

+~ (fooo K(s) ds fooo G1(s) ds- K(,\)Gl(>-)),

.,. •..

30 J.A.D.APPLEBY

using the argument of Theorem 5. The remainder of the proof runs exactly along the lines of Theorem 5,

and is therefore omitted. The function H needed for Lemma 1 is given by

H(>..) = (>..xt(>..)+ f" Ft(s)ds-F't(>..)+ fa'"" K(s)ds f" G 1(s)ds)G 1(>..)- 1

in this case. 0 We remark that in the scalar case the condition (60) on g, which ensures

that (42) holds, guarantees that (62) holds in the scalar case viz.,

f' E[g(X(s))] ds > 0.

Thus, in the scalar case, the conditions in Lemma 3 suffice to remove the technical condition (62) on the solution.

We close this section with the proof of Lemma 3, deferred from earlier. Proof (Lemma 3}. Define the set Z9 = {x E R : g(x) = 0}. Since

we wish zero to be a solution of the unperturbed solution it is known that 0 E Z9 , so Z9 is nonempty. By hypothesis, the members of Z9 are isolated. Introduce the set 10 ={wEn : X(O,w) E Z9 }. Hereinafter we assume without loss that g(x) :2': 0 for x E R.

As Z:: is nontrivial and deterministic, and the Brownian motions W 1, W2

,

... W" are independent, there exists a standard Brownian motion B and a nonzero function a such that

r {t t L lo Bj(s) dWj(s) =In a(s) dB(s). j=l 0 0

( ) 1/2

where a(t) = I:j=1 Z::j(t)2 and a E L2 (R+). Finally, this means that we may rewrite the Volterra equation as

(65) dX(t) = (t(X(t)) + l K(t- s)g(X(s)) ds) dt + a(t) dB(t).

To prove the lemma, we consider separately the cases where Io is a trivial or a nontrivial set. We consider first the case where P[Io] > 0.

Suppose that A = {f000 g(X ( s)) ds = 0} n /0 is a set of positive probability. If wE A then X(t,w) = X(O,w) E Z9 for all t :2': 0, as the members of Z9 are isolated, and tI-t X(t,w) is continuous. On A, as g(X(t)) = 0, and X(t) = X(O), (65) simplifies to

0 = f(X(O))t + l a(s)dB(s).

By the martingale convergence theorem, lim,_,00 JJ u(s) dB(s) exists and is a.s. finite. Therefore, for a.a. wE A we have that lim,_,00 j(X(O, w))t exists and is finite, so f(X(O, w)) = 0 for a.a. wE A. Thus on a nontrivial subset of A we have

f u(s) dB(s) = 0, t;::: 0.

which holds only iff~ u(s) 2 ds = 0 for all t :2: 0. But this is impossible, as u is continuous, and not identically equal to zero. Therefore if P[/0 ] > 0, P[A] =0.

In the case where P[/0] = 1, we therefore have P[f:' g(X(s)) ds =OJ= P[A] = 0, so (61) is true.

If, on the other hand, 0 < P[/0] < 1, consider the set

A'= {fooo g(X(s)) ds = 0} n 70 .

Thus wE A' implies X(O,w) rf. Z9 and f000 g(X(s,w))ds = 0. But, as the

members of Z9 are isolated, there exists a positive, random time T(w) = inf{t > 0 : X(t,w) E Z9 }, so that g(X(t,w)) > 0 for 0::; t < T(w). As g is nonnegative

{oo {T(w) lo g(X(s,w)) ds :2: lo g(X(s,w)) ds > 0.

Thus P[A'] = 0. Since {!000 g(X(s)) ds = 0} =AU A', and A, A' are trivial

sets, (61) holds if P[/0] > 0. Now we consider case where P[/0] = 0, or P[J0] = 1. Then

P[fo00

g(X(s)) ds] = P[A'].

In the argument above, however, we have shown P[J0] > 0 implies P[A'] = 0, so once again it follows that (61) holds. This secures the proof. D

6. Connections between (10)-(13). We now use Theorems 1-6 above to draw equivalences between the statements (10)-(13) for the problem (17). Where necessary, we use the technical restrictions (22), (23), (42), (62), and the sign condition on the kernel K required for the proof of Theorems 5, 6 (viz., that each entry of K does not change sign on R+).

The sharpest connection between the statements (a)-( d) echoes Theorem 5.2 in [2]. Under appropriate additional technical conditions, exponential convergence of solutions of (17) in p-th mean (for some p :2: 1) is equivalent

··~·

32 J.A.D.APPLEBY

to the exponential integrability of K and the exponential square integrability of I:.

THEOREM 7. Let f, g, K, I: satisfy {18)-{21) above. Suppose that {22), {23) hold, and that the entries of K do not change sign on R+. If there exist d solutions of ( 17) such that ( 62) holds, then the following are equivalent:

{i) {11) holds for all X 0 E Rd; {ii) {12), {13) are true. Proof We prove (i) implies (ii) first. If (i) is true, and the entries of K

do not change sign on R+, then (62) implies that (12) is true, by Theorem 6. But if (11) and (12) are true, then so is (13), by Theorem 4. Hence (i) implies (ii).

If (ii) is true, and (22), (23) hold, then we must have (11), by Theorem 1. Moreover, by Theorem 1, the result is true for all initial conditions Xo, so (i) is true. D

In this paper, we have been unable to show the equivalence between (10) and (12), (13), even in the presence of additional technical conditions, a result which is true in the linear case (see Theorem 5.1 in [2]). However, it is possible to use the results obtained here to show, under certain circumstances, that any two of (10), (12) and (13) implies the third.

THEOREM 8. Let f, g, K, E satisfy {18)-{21) above. Suppose that {22), {23) hold, and that the entries of K do not change sign on R +. If there exist d solutions of {17) such that {42) holds, then any two of the following implies the third:

{i) {10) holds for all X 0 E Rd; {ii) {12) is true;

{iii) {13) is true. Proof Suppose (i), (ii) are true: then Theorem 3 gives (iii) automatically. If (i), (iii) are true, then the fact that the entries of K do not change

sign on R+, together with (42), imply (ii), by Theorem 5. Finally, if (ii), (iii) are true, (22), (23) imply that (11) holds, by Theorem

1. But if (11), (12), (13) are all true, then (i) follows from Theorem 2. D In the final result, we observe that (11) is the strongest exponential phe

nomenon of the four mentioned at the outset, in some sense. For a particular class of problems, it transpires that (11) implies (10), (12) and (13).

THEOREM 9. Let f, g, K, E satisfy {18}-{21} above. Suppose that the entries of K do not change sign on R+. If there exist d solutions of {17) such that {62} holds, then {11) implies {10), {12) and {13).

Proof By Theorem 7, if the entries of K do not change sign on R+, and If there exist d solutions of (17) such that (62) holds, then (11) implies (12)


and (13). By Theorem 2, (ll), (12) and (13) together imply (10). 0 Acknowledgement

33

It is a pleasure to thank Prof. Salah Mohammed (Carbondale, Illinois) for motivating the inclusion of Lemma 3 in this paper. The author is also delighted to acknowledge David Reynolds and Alan Freeman for interesting discussions on the topics contained herein.

REFERENCES

[1] J. A. D. Appleby, Subexponential solutions of scalar linear Ito-Volterra equations with damped stochastic perturbations, Funct. Differ. Equ., 11(1-2) (2004), 5-10.

[2] J. A. D. Appleby and A. Freeman, Exponential asymptotic stability of linear ItoVolterra equations with damped stochastic perturbations, Electron. J. Probab., 8 Paper No. 22 (2003), 22pp.

[3] M. A. Berger and V. J. Mizel, Volterra equations with Ito integrals I, J. Integral Equations, 2(3) (1980), 187-245.

[4] T. A. Burton, Stability and Periodic solutions of Ordinary and Functional Differential equations, Academic Press, Orlando, Florida, 1985.

[5] T. A. Burton and W. E. Mahfoud, Stability criterion for Volterra equations, Trans. Amer. Math. Soc., 279 (1983), 143-174.

[6] R. V. Churchill, Operational Mathematics, McGraw-Hill, New York, 1958. [7] K. 'Ito and M. Nisio, On stationary solutions of a stochastic differential equation, J.

Math. Kyoto Univ., 4 (1964), 1-75. [8] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer,

New York, 1991. [9] V. Kolmanovskii and A. Myshkis, Introduction to the theory and applications of

Functional Differential Equations, Kluwer Academic, Dordrecht, 1999. [10] V. Lakshmikantham and M. R. M. Rao, Theory of Integrodifferential Equations,

Gordon and Breach Science, Lausanne, 1995. [11] X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker,

New York, 1994. [12] X. Mao, Razumikhin-type theorems on exponential stability of stochastic functional

differential equations, Stochastic Process. Appl., 65(2) (1996), 233-250. [13] X. Mao, Stochastic Differential Equations and Applications, Horwood, New York,

1997. [14] X. Mao, Stability of stochastic integra-differential equations, Stochastic Anal. Appl.,

18(6) (2000), 1005-1017. [15] X. Mao and M. Riedle, Mean square stability of stochastic Volterra integrodifferential

equations, System Control Lett., (2004), submitted. [16] R. K. Miller, Asymptotic stability properties of linear Volterra integrodifferentiaJ

equations, J. Differential Equations, 10 (1971), 485-506. [17] S. Murakami, Exponential stability for fundamental solutions of some linear func

tional differential equations, Proceedings of the international symposium: Functional differential equations, World Scientific, Singapore, 1990, 259-263.

[18] S. Murakami, Exponential asymptotic stability for scalar linear Volterra equations, Differentia/Integral Equations, 4(2) (1991), 519-525.

34 J.A.D.APPLEBY

[19] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, N.J., 1946.


VOLUME 12 2005, NO 1-2 PP. 35- 66

ASYMPTOTIC STABILITY OF POLYNOMIAL STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH DAMPED

PERTURBATIONS *

J.APPLEBY t AND A.RODKINA I

Abstract. The paper studies the almost sure asymptotic stability of a class of scalar nonlinear It6 stochastic delay-differential equation with polynomial nonlinearity in the drift, and deterministic and fading diffusion coefficient. We show, under conditions that guarantee the stability of the unperturbed deterministic equation, that the condition a 2 (t) log t --'> 0 as t --'> oo is sufficient for the almost sure asymptotic stability of solutions. If a is decreasing, this rate of decay is also necessary.

It is also possible to show that all solutions approach zero at a polynomial rate. If a decays sufficiently rapidly, we obtain the same upper bound on the rate of decay of the deterministic problem. Under some positivity assumptions, we can show that the result we obtain is optimal. When a decays more slowly, a weaker almost sure polynomial upper bound on the decay rate of solutions is established.

Key Words. Polynomial asymptotic stability, almost sure asymptotic stability, stochastic delay differential equation, diffusion process

AMS(MOS) subject classification. Primary: 60Hl0 Secondary: 34D05, 34F05, 60.160, 93E15

1. Introduction. The asymptotic theory of stochastic functional differential equations has been an area of interest to mathematicians for

* The research is supported under the Enterprise Ireland International Collaboration Programme (grant number IC/2004/003). The first author is also partially supported by an Albert College Fellowship, awarded by Dublin City University's Research Advisory Panel....

t School of Mathematical Sciences, Dublin City University, Dublin 9, Ireland l Department of Mathematics and Computer Science, University of the West Indies,

Kingston, Jamaica

35

36 JOHN APPLEBY AND ALEXANDRA RODKINA

some years, with important monographs appearing by Kolmanovskii and Myskis [6], Mohammed [13] and Mao [12]. Much of the effort in these works, and elsewhere, has concentrated upon equations whose spatial dependence does not depart too greatly from linearity. However, more recently, researchers have considered asymptotic results for stochastic functional differential equations when such global linear restrictions are violated, e.g., Rodkina [14], Rodkina and Nosov [15]. On the other hand, very precise results about the non-exponential rates of decay of highly nonlinear stochastic differential equations are also becoming available, e.g., Appleby and Mackey [3] and Appleby, Mao, and Rodkina [4]. In this paper, we attempt to employ some of the methods and conjectures generated in work on such non-delay equations to stochastic equations with delayed argument.

The paper studies the almost sure asymptotic stability of a class of scalar nonlinear Ito stochastic delay-differential equation

n

(1) dX(t) = (- aX(t)fi + L b1X(t- TJ)fi j=l

+ {70

bo(t- s)X(s)fi ds) dt + a(t) dB(t).

Here /3 > 1 is a quotient of odd integers. This super linear term makes global analysis more difficult. However, existence and uniqueness of solutions can be established under the similar sufficient conditions known in the deterministic case, namely

a> t lbJI + {0

bo(s) ds. j=l 0

Under this restriction, we show that when a(t) -+ 0 sufficiently quickly, then the solution is almost sure asymptotically stable. If fact, if a(t) 2 1ogt-+ 0 as t-+ oo implies limt-+oo X(t) = 0, a.s. Moreover, this rate of decay of a is the critical one; a slower decay rate would not ensure asymptotic stability. In fact, we show that if a is decreasing, and X(t) -+ 0, as t-+ oo a.s., then a(t) 2 logt-+ 0 as t-+ oo.

It is also possible to show that all solutions approach zero at a polynomial rate as t -+ oo. If a decays sufficiently rapidly, we obtain the same upper bound as seen in the deterministic problem, namely that

1 .\::; limsuptP-1 IX(t)l::; A a.s.

t--+oo

1

where.\, A are given by.\ ={(a- ~J=l b1 - fr{0 b0(s) ds)(/3 -l)rP and 1

A= {(a- ~'J=1 IbJI- f070 lbo(s)l ds)(f]- 1)}-~- 1 . In the case when bJ ::0: 0,

ASYMPTOTIC STABILITY OF POLYNOMIAL SDDEs 37

b0(t) 2: 0 fortE [0, To], we have that 1

limsuptH IX(t)l = L, a.s. Hoo

where L is either 0 or .\. In the case when a decays more slowly, a weaker almost sure polynomial upper bound on the decay rate may be found.

The main idea used to prove the results in this paper is a variant of the following strategy: subtract from the solution X a diffusion process Y with the same diffusion coefficient; the process x = X - Y then obeys a perturbed delay-differential equation of the form

(2) x'(t) = -ax(t)fl + ~ bjx(t- Tj)fl + {ro b0(t- s)x(s)fl ds + f(t)

where f depends on x. The analysis revolves on bounding f independently of x (usually the behaviour of Y determines the behaviour of f). One then proves results about perturbed equations of the form (2), using the "known" asymptotic behaviour of f. These results can be applied pathwise to X once appropriate asymptotic information is available about Y.

The results on the rate of decay of solutions in this paper in part parallel those of [3] which is concerned with non-delay versions of (1); the results on the as~mptotic stability are motivated by work of Chan and Williams [5] which consider the asymptotic stability of diffusion equations under the condition a 2 (t) logt-+ 0 as t-+ oo. The importance of this condition in linear stochastic Volterra delay equations has been studied in Appleby [1]. Other papers which consider polynomial stability of stochastic dynamical systems include Liu [7], Liu and Mao [8, 9], Mao [10, 11], Zhang and Tsoi [16, 17]. The first paper and last pair of papers in this list are particularly germane here, as the authors consider strong nonlinear spatial dependence; in the other papers listed, the spatial dependence is close to linear, but there are many complicated non-autonomous features present. A summary of the literature on the non-exponential and polynomial stability of linear stochastic Volterra equations may be seen in Appleby [2].

2. Preliminaries. We first establish some standard notation. As usual, let x V y denote the maximum of x, y E R and x 1\ y the minimum. Denote by C (I; J) the space of continuous functions from I to J, and by C1 (I; J) the corresponding space of all functions with continuous derivatives. If f E C([O, oo ); R) is locally Lipschitz continuous, the following upper Dini derivative D + may be defined:

D+f(t) =lim sup f(t + h~ - f(t), t > 0 htO

If I is a compact interval, we denote the usual sup norm off E C(I; J) by

IIJIII = SUPtEIIf(t)l. Denote by L(R+) the space of all measurable real-valued functions which

are integrable on R+ and by L 2 (R+) all square integrable functions on R+. Let fJ > 1 be a quotient of odd integers, and a be a real number. Let

n E N, and ( Tj) j=o, ... ,n be a sequence of non-negative real numbers, with r = maxj=O, ... ,n Tj, and (bj)j=l, ... ,n be another sequence of real numbers. Suppose that 7/J E C([-r, OJ; R), b0 E C([O, r 0]; R), and CJ E C([O, oo); R).

Let (0, .F, (.F8 (t) )t2:o, P) be a complete filtered probability space and B= {B(t);.F8 (t);O :S t < oo} be a one-dimensional standard Brownian motion on it. The filtration (.F8 (t))t>o is the natural filtration for the standard Brownian motion B, viz., .F8 (t) = CJ{B(s) : 0 :5: s :5: t}.

Under these hypotheses, there exists a continuous adapted process X which is a strong solution, up to an explosion line Te > 0, of the Ito stochastic delay differential equation (1) relative to B, with initial function 7/J, viz. X obeys

(3) X(t)

(4) X(t)

7/J(O) + l {- aX(s)il + tbjX(s- Tj)il 0 j=l

+ J.~ro bo(s- u)X(u)13 du}ds+ l CJ(s)dB(s),

- 7/J(t), t E [-r,OJ.

Here, as is conventional, the explosion time Te is defined by

Te = lim Tm m-too

where Tm = inf{t > 0: IX(t)l = m}.

t E [0, Te),

In order to ensure that Te ( w) = oo for almost all sample paths w E n, we prove that it is suffices to assume

(5) a> f. lbil + {o lbo(s)l ds. j=l 0

We are interested in establishing the almost sure asymptotic stability of the solution of (3)-( 4), viz., we wish to prove

(6) lim X(t,w) = 0, !-too

wE Oo,

where P[Oo] = 1.

•


In proving results about asymptotic stability and decay rates of (3)-( 4), we invoke the following theorem due to Chan and Williams [5].

(7)

(8)

(9)

THEOREM 1. Let g be a locally Lipschitz continuous function with

g strictly increasing on R,

lim g(x) = oo, lim g(x) = -oo x......roo x-t-oo

g(O) = 0,

and suppose CJ is a continuous function on [0, oo). Then, there is a unique strong solution of the stochastic differential equation

dX(t) = -g(X(t)) dt + CJ(t) dB(t)

on [O,oo). (i) Suppose moreover that

(10)

Then X obeys (6). (ii) If CJ obeys

lim CJ(t) 2 log(t) = 0. t-+co

(11) CJ is decreasing on [0, oo), CJ(O) is finite and lim CJ(t) = 0 t-+co

and X obeys (6), then (10) is true.

3. Global existence of solution. Below we prove that solution of (3)-(4) exists on [0, oo).

THEOREM 2. Suppose CJ E C([O, oo); R), (3 > 1 is a quotient of odd integers, and condition ( 5) is fulfilled. Then there exists a unique continuous adapted process X which satisfies Eqn{3)-{4) on [0, oo) a.s.

Proof. Let

(12) V(t) = X 2 (t) + V1(t) + V2(t),

Vl(t) = /:1 ~ [bi[ tTi xtJ+l(s) ds,

2(3 TQ t

V2(t) = (3 + 1 j [bo(s)[ j Xil+l(u) duds. 0 t-s

It is easy to see that solution X exists and is unique on some interval [0, T). We have to show that T = oo. Let Tm = inf{t: [X(t)[:? m}.


Applying the Ito formula we have

(tiiTm ( n (13) X 2 (t 1\ Tm) = X(W + ln -2aX6+ 1(s) + 2 I: bjX(s)X(J(s- Tj)

0 j=l

+ 2X(s) fo b0 (u)XfJ(s- u) du + cr 2(s)) ds + M(t)

where

(tiiTm M(t) = lo 2X(s)cr(s)dB(s)

is a martingale. Using the inequality

labl:::; ~ + ~ r s , where 1 1 - +- = 1, r s

we get that

IX(t)XfJ(t- ri)l:::; XfJ+l(t) + f3XfJ+l(t- ri) (3+1 (3+1

and '

IX(t)XfJ(t- s)l:::; XfJ+l(t) + (3XfJ+l(t- s) (3+1 (3+1

Then

(14) lx(t) fo b0 (s)XfJ(t- s) dsi

ro (XfJ+l(t) (3XfJ+l(t- s)) ::;lo lbo(s)l !3+ 1 + (3+ 1 ds

XfJ+l(t) ro (3 ("'" = lo lbo(s)l ds + (3 + 1 lo lbo(s)IXfJ+l(t- s) ds.

Now we substitute (14) in (13) to give

(15) lo

t!ITm ( 1 { n X 2 (t 1\ Tm) :::; X(W + 2xfJ+~(s) -a+ -!3- L lbJI

0 + 1 j=l

(16) !oro }) 2(3 lot!ITm n . + lbo(u)Jdu ds+-(3- I:lbJIXfJ+l(s-ri)ds

0 + 1 0 J=!

(17) riiTm { 2(3 ro } + lo (3 + llo lbo(u)IXfJ+l(s- u) du + cr2 (s) ds + M(t).

We note that

(18) V{(t) = /:1 (~ lbiiXil+l(t)- ~ lb;IXil+l(t- Ti)),

v;(t) = /:1

( xil+~(t) fo lbo(s)l ds fo'o lbo(s)lxil+~(t- s) ds).

Applying the It6 formula to the functional V defined in (12), we get from (15) and (18) that

{t/\Tm ( 1 n V(t A Tm) :'0 V(O) + fo 2Xil+l(s) -a+ -(3

1 L lbjl

0 + J~l

1 InTo ) 2(3 lot/\Tm n + -(3 1

lbo(u)l du ds + -(3- L lbjlxil+l(s- Ti) + 0 + 1 0 j~l

InTo 2(3 lot/\Tm n + lbo(u)lxil+l(s-u)duds+-(3- L:lbiiXil+I(s)

0 + 1 0 i~l n 2(3 {t/\Tm ro L lbiiXil+l(s- Ti) ds + -(3- lo xil+l(s) lo lbo(u)l du i~l + 1 0 0

ro {t/\Tm - fo lbo( u)IXil+l(s- u) duds+ fo 0'

2(s) ds + M(t).

Thus

{t/\Tm ( n (19) V(t A Tm) :S V(O) + fo 2Xil+l(s) -a+ _f; lbjl

ro ) {t/\Tm (20) + fo lbo(u)ldu ds+ fo 0'

2 (s)ds+M(t).

By (5), upon taking the expectation of both sides of (19), we obtain

t' EV(t' A Tm) :S EV(O) + fo 0'

2 (s)ds =: K(t')

for some non-random number K = K(t') > 0. Then for every t' > 0

K ~ EV(t' A Tm) ~ P{t' > Tm} i>nf V(u), u m

and so

P{t' > T } < K(t') < K(t'). m - infu>m V(u) - m2

Hence limm_,ooP{t' > Tm} = 0. It means that Tm-+ oo a.s., that is T = oo a.s. 0

4. Asymptotic stability. Before we can establish the a.s. global asymptotic stability of solutions of (3)-(4), we must first prove a result concerning the solution of a related deterministic delay differential equation. Once this is done, we move on to prove the asymptotic stability of solutions, and converse results.

4.1. Uniform boundedness of a deterministic delay-differential equation. Let a > lbl > 0, r > 0 and consider the unique solution of the delay differential equation

(21) y'(t) = -ay(t) 13 + lbl(2 + y(t- r))13 +a, t > t 0 ,

y(t) = 'lj;(t), t E [to-r, to].

We will show in this section that there exists a c0 = c0 (a, b, !3) > 0 such that

(22) limsuply(t)l:::; co, t->oo

where c0 is independent of the initial function 7/J. The following result can be checked using elementary calculus. LEMMA 1. Let a > lbl > 0, and j3 2: 1 be a quotient of odd integers.

Then the function f E C(R +; R) given by

(23) f(x) = -ax13 + lbl(2 + x)f! +a

has a unique positive zero at c0 > 1. Moreover, f ( x) < 0 if and only if X> Co.

The following result can then be established for a particular real sequence.

LEMMA 2. Let c0 > 1 be the unique positive solution of f(x) = 0, where f is defined in (23) above, and let (bn)n:;:o be the real sequence defined by

(24) bo 2: co, (lbl )~ bn+l = -;;- (2 + bn)/3 + 1 , n 2: 0.

Then (a) If b0 >Co, then bn >co for all n 2: 0, and bn+l < bn· (b) If bo =Co, then bn =co for all n 2: 0.

Therefore

(25) lim bn =co. n->oo

This result is crucial in proving the following technical lemma, which enables us to establish the global almost sure stability of all solutions of the stochastic delay differential equation.

LEMMA 3. Let a > lbl > 0 and (3 > 1 be a quotient of odd integers. Let 1/J E C([-r, 0]; [0, oo)). Then the solution of (21) obeys (22), where Co> 1 is the unique solution of the equation f(x) = 0, and f is defined in (23).

Proof The first step in this proof is to show that the solution of (21) is globally bounded.

Let

(26) b0 = max y(t), tE[-r,O]

and c0 be as defined above. If b = b0 V c0 , then

(27) y(t) ::; b, t?: 0.

To see this, consider the constant function Yu(t) = c > b, fort ?: -r. Then Yu(t) > y(t) fortE [-r,O], and fort> 0, by Lemma 1, as c > c0 , we have

y~(t) + ayu(t) 13 - lbl(2 + Yu(t- r)) 13 - a

ac13 - lbl(2 + c) 13 - a= - f(c) > 0.

Therefore, a comparison argument gives y(t) < Yu(t) = c for t ?: -r. We can let c + b to complete the proof of (27).

We prove the result by showing that the solution of (21) can be bounded above by bn on some interval [Tn, oo), where (Tn)n?:O is an increasing sequence of numbers, regardless of its initial condition. For each n E N, we call the proof of this result STEP n.

To prove STEP 1, suppose first that c0 ?: b0 . In this case, ( 27) enables us to conclude.

Suppose now, to the contrary, that c0 < b0 . Then y(t) ::; b0 for all t?: 0, and therefore y satisfies the differential inequality

(28) y'(t) ::; -ay(t) 13 + lbl(2 + bo) 13 +a, t?: 0.

Let b1 be given by (24). Then, by Lemma 2, b1 < b0 , as b0 >co. In the case that y(O) ::; b1 < b0 , the comparison principle ensures that

we have

y(t) ::; b), t?: 0,

because y obeys (28), and y(O) ::; b1.

We now consider the other case, where c0 < b1 < y(O) :::; b0 •

Now, as y is bounded there either exists a finite and positive T1 = inf{t > 0 : y(t) = bi}, or y(t) > b1 for all t :2: 0. Suppose that this latter case holds. Then, as y obeys (28), y'(t) < 0 for all t :2: 0, as y(t)f3 > bt = lbl/a(2+b0 )f3+1. Therefore y is decreasing on (0, oo). Then y(t):::; b0 fortE [0, r] and y(r) < y(O) :::; bo. Therefore y(t) < b0 fort E [r, 2r]. Moreover, as y is decreasing, there is a 1 E (0, 1) such that

y(t):::; a1bo = y(r), t E [r,oo).

Hence

y'(t) :::; -ay(t)f3 + lbl(2 + a1bo)f3 +a, t :2: 2r.

Therefore, for t :2: 2r,

y'(t) < -abf + lbl(2 + albo)f3 +a

= -lbl { (2 +bo)f3- (2 + a1bo)f3} =: -(h < 0.

Hence, fort :2: 2r, y(t) < y(2r)- fJ1(t- 2r). As b1 < y(2r) < y(r) = a1bo, we may .define T{ = 2r + ( a 1 b0 - bi) f3! 1

• Then

b1 < y(T{) < y(2r)- fJ1(T{- 2r) < a1bo- fJ1(T{- 2r),

which yields the contradiction b1 < b1• Therefore, there is a finite T 1 > 0 such that y(T1) = b1. Thus, by virtue of (28), y satisfies the differential inequality

y'(t) :::; -ay(t)f3 + lbl(2 + bo)f3 +a, t :2: T1, y(TI) = b1.

The comparison principle now gives

(29) There exists T1 > 0 such that y(t) :::; b1 for all t :2: T1•

The proof of STEP 2 is slightly different from that of STEP 1. We present it here, however, as all subsequent STEPs 3, ... follow by an identical argument. Indeed by showing that STEP 2 follows from STEP 1, it is easy to see that the proof of STEP n + 1 follows from that of STEP n for a general n, thereby providing the necessary induction argument.

Let n1 = lTd rl Then

y(t):::; bh t E [n1r, (n1 + 1)r].

'

j;'j

Therefore, for t 2': ( n1 + 1 )T,

(30) y'(t) ::; -ay(t)fl + [b[(2 + b1)f3 +a, t 2': (n1 + l)T.

Let b2 be given by (24). Then, by Lemma 2, b2 < b1 , as b1 > c0 .

In the case that y((n1 + l)T)::; b2 < b1, the comparison principle ensures that we have

y(t) ::; bz, t 2': (n1 + l)T,

because y obeys (30), and y((n1 + l)T)::; b2 .

We now consider the other case, where c0 < b1 < y(O) ::; b0 •

Now, as y is bounded there either exists a finite and positive T2 = inf{t > (n 1 + l)T : y(t) = b2}, or y(t) > b2 for all t 2': (n1 + l)T. Suppose that this latter case holds. Then, as y obeys (30), y'(t) < 0 for all t 2': (n1 + l)T, as y(t)f3 > b~ = [b[/a(2 + bt)il + 1. Therefore y is decreasing on ((n1 + l)T, oo). Theny(t)::; b1 fortE [(n1 +l)T,(n1 +2)T] andy((n1 +2)T) < y((n1 +1)T)::; b1• Therefore y(t) < b1 for t E [(n1 + 2)T, (n1 + 3)T]. Moreover, as y is decreasing on ((n1 + l)T, oo), there is a 2 E (0, 1) such that

y(t)::; a2b1 = y((n1 + 2)T), t E [(n1 + 2)T, oo).

Hence

y'(t) ::; -ay(t)fl + [b[(2 + a 2b1)(J +a, t 2': (nt + 3)T.

Therefore, for t 2': ( n1 + 3) T,

y'(t) < -abg+[b[(2+a2bt)fl+a

= -[b[ { (2 + b1)f3- (2 + a 2b1)fl} =: -(32 < 0.

Hence, for t 2': (n1 + 3)T, y(t) < y((n1 + 3)T) - (32 (t (n1 + 3)T). As bz < y((n1 + 3)T) < y((n1 + 2)T) = a2b1 , we may define T~ = (n1 + 3)T + (a2b1 - bz)/321 > (n1 + 3)T. Then

b2 < y(T~) < y((n1 + 3)T)- (32(T~- (n1 + 3)T)

< a2b1 - f3z(T~- (n1 + 3)T),

which yields the contradiction b2 < b2 . Therefore, there is a finite T2 > (n1 + l)T such that y(T2) = b2 • Thus, by virtue of (30), y satisfies the differential inequality

y'(t) ::; -ay(t)fl + [b[(2 + b1)f3 +a, t 2': Tz, y(T2) = bz.


Since T1 ~ n 1r, and T2 > (n1 + 1)r, the comparison principle now implies that

(31) There exists T2 > T1 such that y(t) ~ b2 for all t :0:: T2.

The proof of STEPs 3, 4, ... , n follows exactly as that of STEP 2. We therefore can show that there exist an increasing sequence of finite times (Tn)n:;~ 1 such that

y(t) ~ bn, t :0:: Tn

where (bn)n:;-:o is defined in (24). Since Lemma 2 implies bn -t c0 as n -t oo, it follows that

limsupy(t) ~ c0 , t->oo

which completes the proof. 0

We now make an observation. Let c: > 0 be fixed, and y, be the unique continuous solution of the delay differential equation

(32) y;(t) = -ay,(t) 13 + lbl(2c: + y,(t- r,))13 + ac:13 , t > to,e y,(t) = 'lj;,(t), t E [to,e- T., to,eJ,

where r = r,c:/3-1, to = to,ec:/3-1, and

'lj;(t) = c:-1'1j;,(t/c:i3-l), t E [to-r, to].

If

y(t) = C 1y,(tjc:13- 1), t :0:: to,

then y solves the delay-differential equation (21). Lemma 3 now ensures that we have the following result.

LEMMA 4. Let a > lbl > 0 and j3 > 1 be a quotient of odd integers. Let c: > 0 be fixed. Then then the solution of (32) obeys

lim sup !y,(t)! ~ c:co, t->oo

where co > 1 is the number defined in Lemma 1. Instead of equation (21) and (32) we consider now

n

(33) y'(t) = -ay(t)f3 +I; lbjl(2 + y(t- Tj))f3 j=l

+ tro lbo(t- s)l(2 + y(s))13ds +a, t >to

y(t) = 'lj;(t), t E [to - r, to],

and


n

y~(t) = -aye(t) 13 + L lbJI(2t: + Ye(t- Tj)) 13

j=l

(34) + t70

lbo(t- s)i(2t: + Ye(s)) 13 ds + at:13 , t > to,e

Ye(t) = 1/Je(t), t E [to,e- Te, to,eJ,

where t: > 0 is fixed, T = T8 t:!3-I, t0 = to,et:/3- 1, and

·lj;(t) = t:- 11jJ,(tjt:13- 1), t E [to-T, to],

Let

(35) lbl =f. lbJI +fa lbo(s)l ds and a> lbl. j=1 0

In the same way as above we can prove Lemmas 5 and 6 which are analogous to Lemmas 3 and 4.

LEMMA 5. Let 1/J E C([-r, 0]; [0, oo)) and lbl be defined in {35). Let a > lbl > 0 and (3 > 1 be a quotient of odd integers. Then the solution of {33) obeys {22), where c0 > 1 is the unique solution of the equation f(x) = 0, and f is defined in {23).

LEMMA 6. Let a> lbl > 0 and (3 > 1 be a quotient of odd integers. Let t: > 0 be fixed. Then then the solution of {34) obeys

limsuply,(t)i S: t:co, t->oo

where c0 > 1 is the Bame number aB in Lemma 5.

4.2. Asymptotic stability of (3-4). The main theorem of this section is the following.

THEOREM 3. Let a > lbl ?: 0, r ?: 0 and (3 > 1 be the quotient of two odd integers. Let 1/J E C([-r, 0]; R) and a E C([O, oo); R) and suppose that X is the unique strong solution of

(36) dX(t) = ( -aX(t)!3 + bX(t- r) 13 ) dt + a(t) dB(t),

where X(t) = 1/J(t), t E [-r,O]. If

(37) lim a(t) 2 logt = 0 Hoo

Now, for each wE Oj, we define Z(t,w) = Z(t,w)- e:, t;::: T(e:,w). Then, using (40), we have

( 42)

where

z'(t,w) ::: -aZ(t,w)li + JbJ(2e: + IZ(t- T,w)l)li + ae:li,

t > T(e:,w) + r, Z(t) = '1/Je(t), t E [T(e:,w), T(e:,w) + r],

ll'I/Jelhr(e,w),T(e,w)+r] :':: 2e: + JJXJJ[T(e,w),T(e,w)+r] < oo,

and '1/Je E C([T(e:,w),T(e:,w) + r];R). Now, let P(t,w) = JZ(t,w)J, t ;::: T(e:,w). Then, by (42),

(43) D+P(t,w) :':: -aP(t,w)li + JbJ(2e: + P(t- r,w))li +ae:li, t>T(e,w)+r,

P(t, w) = 1'1/Je(t)J, t E [T(e:, w), T(e, w) + r].

where D+ signifies the appropriate Dini derivative. Now, let Pe(w) be the solution of

D+Pe(t, w) = -aPe(t, w)f' + JbJ(2e + Pe(t- r,w)).B +ae.B, t > T(e:, w) + r,

with P,(t) = 1 + P(t,w), t E [T(e, w), T(e, w) + r]. Then P(t, w) < P,(t, w), t;::: T(e, w). But by Lemma 4,

(44) lim sup Pe ( t, w) :':: eeo, t-+oo

where Co > 1 is the e:-independent number defined in Lemma 1. Therefore, fort;::: T(e, w)

IX(t,w)J :':: JX(t,w)- Y(t,w)l + JY(t,w)J

< JZ(t,w)J+e

- Je+Z(t,w)J+e < 2e + JZ(t, w)J - 2e+P(t,w)

< 2e + P,(t,w).

Therefore, by (44), for each fixed wE 03, and every e > 0, we have

limsupJX(t,w)J < e:(2+co). t-+oo


Thus for each w E 03 we may let c: + 0 to give

lim X(t,w) = 0. t->oo

But this is nothing other than (6), so the result is proven. D A result analogous to Theorem 3 can be established for the equation

(3-4). THEOREM 4. Let f3 > 1 be a quotient of two odd integers, (Tj)j=O, ... ,n be a

sequence of non-negative real numbers with T = maxj=O, ... ,n Tj, and (bj)j=l, ... ,n

be another sequence of real numbers. Suppose that 7/J E C([-T, 0]; R), b0 E C([O, To); R) and

n r· a> L lbil + ln lbo(s)l ds.

j=l 0

If X is the unique strong solution of (3-4} with X(t) = 7/J(t) fortE [-T, 0], and IJ E C([O, oo); R) obeys (37}, then

lim X(t) = 0, a.s. t->OO

To prove Theorem 4 we apply Lemma 5 and Lemma 6.

4.3. Converse result; necessary and sufficient conditions for asymptotic stability. The following converse result in Theorem 5 relies on the converse of Theorem 1. Once proven, we may state the main result of this section.

THEOREM 5. Let IIJI be a continuous and decreasing function. If the solution of (3-4) obeys (6}, then sigma obeys {37}.

Proof. Using integration by parts on (36), putting

(45) U(t) = e-t l e'II(s)dB(s),

and rearranging, we get

(46) U(t) = X(t)- X(O)e-t -l e-(t-s){ -aX(s)i3

+ ~ bjX(s- Tj)i3 + J.~ro b0(s- u)X(u)13 du- X(s)} ds.

Let e1 (t) = e-t. By hypothesis, the first term on the righthand side of (46) tends to zero as t -+ oo, almost surely, and the second term clearly has a

'

zero limit. Moreover, the third term has a zero limit as t -+ oo, almost surely, since the term in curly brackets has zero limit as t -+ oo, a.s., and e1 E L 1[0,oo). Hence, by (46),

( 47) lim U(t) = 0, a.s. t->oo

Also, U is the unique strong solution of the stochastic differential equation

( 48) dU(t) = -U(t) dt + a(t) dB(t).

Since lal is decreasing, and U obeys (47), (48), we may use the second part of Theorem 1 to allow us to conclude (37). 0

Combining the proofs of Theorems 5 and 3, we have the following result. THEOREM 6. Let fJ > 1 be a quotient of two odd integers, (Tj)j=O, ... ,n be a

sequence of non-negative real numbers with T = maxj=O, ... ,n Tj, and (bj)j=l, ... ,n

be another sequence of real numbers. Suppose that 'lj; E C([-T, 0]; R), bo E C([O, To); R) and

n ro a> I: lbjl + lo lbo(s)l ds > 0.

j=l 0

Let X be the unique strong solution of (3-4) with X(t) = 'lj;(t) fortE [-T, 0], and a E C([O, oo); R), where a 2 is a decreasing function on [0, oo).

Then the following statements are equivalent (i) limt-too a(t) 2 1ogt = 0,

(ii) 1imt-t00 X(t) = 0, a.s.

5. Polynomial asymptotic behaviour. In this section we concentrate on establishing the asymptotic behaviour of solutions of (3-4) in the case that the rate of decay of the noise perturbation is bounded above, in some sense, by a function which decays like a negative power oft, as t -+ oo. To do this, we seek to express the solutions in terms of a process Y (resp. a family of random functions, {Y(w) : wE !1}) whose asymptotic behaviour is in some way determined by the law of the iterated logarithm, and a process Z (resp. or family of random functions {Z(w) : wE !1}) which is the solution of a delay differential equation whose unique continuous solution is in C 1(0,oo).

5.1. Polynomial asymptotic behaviour of an auxiliary stochastic differential equation. Let Y = {Y(t);F8 (t);O ::'0 t < oo} be the solution of the stochastic differential equation

( 49) dY(t) = -Y(t) dt + a(t) dB(t)

with Y(O) = 0. Here u is the same function as in (3). Extend Y to [-r, OJ by letting Y(t) = 0 fortE [-r, 0].

We wish to consider polynomial decay in the solution of (3-4). To this end, we impose the following polynomial decay condition on u.

(50) There exists p 2: 1 and 'Yv > 0 such that

(51) 'Yv = inf{a > 0: {" (1 + s)P"' u(s) 2P ds = oo}.

In (51), in the case the set is empty, we define 'Yv = oo. This case arises, for example, if u(t) = e-t. We prefer to impose the hypothesis (51) to a stronger pointwise polynomial bound on u, as it is sufficient to establish the a.s. polynomial asymptotic stability of solutions of (3-4). Moreover, in earlier work of Appleby and Mackey [3] it was shown that an integral condition of the form (51) was also necessary if the solution is to be almost surely polynomially stable.

We now show that under (51), Y tends to zero at a polynomial rate. LEMMA 7. Suppose that u obeys (51) andY is the solution of (49). (i) If 'Yv > -/::r,

lim ti61Y(t)l = 0, t-+oo a.s.

(ii) lf'Yp '5. -/::r,

lim sup logiY(t)l < -'Y t-+OO log t - P>

a.s.

Proof Suppose that 'Yv < oo; the proof in the case when 'Yv = oo can be established by choosing f.L arbitrarily large in the proof below, and by taking the limit as f.L -+ oo through the integers.

Note also that the result holds if we can show that

(52) lim sup log IY(t)l < -'Y: t-+oo log t - P>

a.s.

whatever the value of fJ > 1. Define

I(t) = l u2 (s)e-2(t-s) ds, t 2: 0.

We consider the case where p > 1. Let 0 < J.l < 'Yp· Also define q > 1 such that p- 1 + q-1 = 1. Then by Holder's inequality, and the inequality 1 + t::; (1 + (t- s))(1 + s) for 0::; s::; t, we get

(1 + t) 2~' /(t) = (1 + t) 2~' l a2(s)e-(t-s)e-(t-s) ds

( r ) 1/p ( t ) 1/q < (1 + t)2~' Jo a2P(s)e-p(t-s) ds lo e-q(t-s) ds

( rt ) 1/p ::; kq Jo (1 + t)2~'-Pa2P(s)e-p(t-s) ds

( rt ) 1/p < kq lo (1 + s)2~'-Pa2P(s)(1 + (t- s))2~'-Pe-p(t-s) ds ,

where kq = (~f1q. In the case when p = 1, a similar estimate holds, but Holder's inequality is not employed.

Since J.l < 'Yp, if we define g(t) := (1 + t)2~'-Pa2P(t), then g E £ 1(0, oo). Also, if ep := e-Pt, then ep(t) -t 0 as t -too. Therefore

(53) for all J.l < 'Yp (1 + t)2~' I(t) ::; kq ((g * ep)(t)) 11P -t 0 as t -too.

Since I.(t) -t 0 as t -too, so there isH> 0 such that I(t) ::; H for all t 2:: 0. Hence

(54) loglog (l e28a2(s)ds) = loglog(e21/(t)) = log(2t+log/(t))

::; log(2t +H).

Suppose next that

By the representation

lim rt e2'a2(s) ds = 00. t-4oo Jo

e1Y(t) = Y(O) + l e'a(s) dB(s)

and by law of the iterated logarithm applied to J~ e•a(s) dB(s) we have

(55)

where

Y(t)2 = 1, a.s., lim sup e-2te2(t) t-+oo

e2(t) = 2 (l e2'a2(s)ds) log log (l e28a2(s)ds).

By (53), for any 0 < E: < "/p, (1 + t)27p-e I(t) --+ 0 as t--+ oo. Then, by this and (54), we have

(1 + t)27.-2•e-2tr/(t)

= 2(1 + t) 27P-e J(t)(1 + t)-e log log (l e25a 2 (s)ds)

:5: 2(1 + t)27p-e l(t) x (1 +We log(2t +H).

Therefore (1 + t)27p-2ee-2tr/(t) --+ 0 as t--+ oo, and so by letting E:--+ 0, we get

. loge-2t1/(t) hm sup

1 ::; -'Yp·

t-+oo og t

Combining this with (55) gives (52). In the case when

fo"" e25 a2 (s)ds < oo,

etY(t) tends to an a.s. finite limit as t--+ oo, a.s., and so (52) holds. 0

5.2. Structure of solutions of (3-4). The process Y defined by (49) in the last subsection enables us to write the solution of (3-4) in terms of a perturbed polynomial delay-differential equation.

Hereinafter, we take as given that X(t) --+ 0 as t --+ oo, a.s.; but by Theorem 4 this is assured once a 2(t) log t --+ 0 as t --+ oo. However, we choose not to impose any conditions on the data other than those necessary to determine a particular rate of decay of solutions.

With Y given by the solution of ( 49) and X the solution of (3-4), we define

x(t) = X(t)- Y(t), t ~ -r.

Then, it is easy to see that xis a process with C 1(0, oo) sample paths, and obeys the delay-differential equation

n {t (56) x'(t) = -ax(t)il + ~ bjx13 (t- rj) + lt-To b0(t- s)x13 (s) ds

+ f(t), t > 0,

where

(57) f(t) a(xil(t)- (x(t) + Y(t))il)

':1

•


n

+ LbJ{(x(t- Tj) + Y(t- Tj))fi- xfi(t- Tj)} j;J

+ {To bo(t- s)[(x(s) + Y(s))fi- xfi(s)] ds + Y(t), t 2: 0.

We now want to get an estimate on the rate of decay off which is independent of x, and depends only on the rate of decay of Y. To this end, by the mean value theorem, we have

(x(t) + Y(t))fi- xfi(t) = f3rl- 1(t)Y(t),

where lx(t)- ry(t)l::; IY(t)l. Therefore

n

lf(t)l :S lal/3177(t)lfi-JIY(t)l + L lbJif3177(t- Tj)lfi-IIY(t- TJ)I j;J

+ /3 {To bo(t s)lry(s)lfi-liY(s)l ds + IY(t)l.

Now, we notice that because the solution of (3-4) obeys X(t) -t 0 as t -too, a.s., and under condition (51) Y(t) -t 0 as t -t oo, a.s., it follows that x(t) -t 0 as t -t oo, a.s. Thus, ry(t) -t 0, a.s. and therefore, by (57), there exists an a.s. finite random variable C 2: 0 such that

lf(t)l::; c(IY(t)l + ~ IY(t- Tj)l + sE~~~o,t) IY(s)l)·

Hence, by Lemma 7, if 'Yp > (3((3 -1)-1, we have tl--riY(t)l-> 0 as t -too

a.s., and therefore tl--rlf(t)l -t 0 as t -t oo a.s. On the other hand, if "/p ::; (3((3- 1)-1

, by Lemma 7 we have

1. logiY(t)l 1m

1 ::; -"fp, a.s.

!-too og t

and so

lim sup log lf(t)l ::; -"fp, a.s. t-too log t

The analysis in this subsection may be summarised by a theorem. THEOREM 7. Suppose that a is a continuous function which obeys (51}.

Let (3 > 1 be a quotient of odd integers. Let X be the unique strong solution of (3-4} which obeys lim,_,00 X(t) = 0 a.s. Then X can be represented as

(58) X(t) = x(t) + Y(t), t > -T - ,

where Y is the solution of (49) and x is the unique continuous solution of the delay-differential equation

n t x'(t) = -ax(t)f' + ~bix!3(t- Tj) + [_

70 b0(t- s)xi3(s) ds

+ f(t), t > 0,

where x(t) = 1/l(t) for all t E [-r, 0], f E C(O, oo), and x(t)-> 0 as t-> oo. Moreover, if 'YP > 0 is the exponent in (51), then the following estimates

hold: (i) lhp "5. (3((3- 1)-1

, then

and

limsup logJY(t)/ < -'Y: t-+oo log t - P•

log /f(t)/ < -'"'(p, lim sup log t -

t-+oo

(ii) If 'Yp > (3((3- 1)-1, then

a.s.

a.s.

lim 0~/Y(t)/ = 0, a.s. t-+oo

and

lim t~/f(t)/ = 0, a.s. t-+oo

The rest of the paper is devoted to determining the rate of decay of solutions of (56) when the polynomial rate of decay off is given by either (i) or (ii) above. Once this rate of decay has been established, we may use the rate of decay of Y given in either (i) or (ii) above to determine the rate of decay of X, via (58).

5.3. Rate of decay of solutions of perturbed deterministic delay differential equations. The following two results show that the rate of decay observed for the corresponding unperturbed problem is recovered, once the size of the perturbation vanishes sufficiently quickly as t -1 oo.

THEOREM 8. Let a > /b/ :;:=: 0, r :;:=: 0 and (3 > 1 be a quotient of odd integers. Let 1/J E C([-r,O];R), and suppose that f E C([O,oo);R) satisfies

(59) lim t~ f(t) = 0. t-+oo

If x is the unique continuous solution of

(60) x'(t) = -ax(t)li + [b[x(t- 7)1i + f(t), t > 0, (61) x(t) = 'lj;(t), t E [-7,0],

which obeys limt->oo x(t) = 0, then

(62) ( 1 );&,

JimsuptP:_1[x(t)[ :'0 (a -[b[)(,6 1) · t->oo

Proof. Let Co > 0 be given by cg-1 = (a- [b[)-1/(li-1). Fix C > C0 ,

and let c: > 0 be so small that

(63) c

-,6 _ 1

+ aCii- c: -[b[Cii(1 +c) > 0.

Let

(64)

1

= ! ((1 + d'f - 1) P-1 1 C

x, 2 7 ( 1 + c) 11 li .

Then, as x(t) --1 0 as t --1 oo, there exists T0 (c) > 0 such that for all t:;:::: T0 (c:), [x(t)[ ::; x,. Next, let

(±c)/3-1 c [3-1

T1(c:)= ~x, -7>CxJ -7. Then by (64)

(65) T1(c:) > ( C )/3-t-2x, 7 = .. 7 "" ~' II'>

On account of (62), for every c: > 0, there exists T2(c:) > 0 such that

..JL tP-1[f(t)[ < c:, t > T2 (c).

Now, define T(c) = 1 + (T0(c) V T1(c) V n(c)). Thus, fort> T(c) + 7, as t > T2(c:), and T(c:) > T1 (c:), we have

(66) (t- T(c:) + T1(c:))ih[f(t)[::; tih[f(t)[ < c:.

The function g E C([r,oo);R) defined by g(t) = {(t- r)/t}-f3/((J-l) is decreasing on (7, oo), is positive fort> r, and obeys limt->oo g(t) = 1. Thus, fort> T(c:) + r, because g is decreasing and T1(c:) obeys (65), we get

g(t- T(c:) + T1(c:)) < g(T1(c:) + 7) < 9 (7 + (l+e)</ 1)/IJ 1)

= 1 +c.

This is equivalent to

(67) (t- T(c:) +T1(c:) 7 )-;/:-r

t-T(c:)+T1(c:) <1+c:, t > T(c:) + 7.

Now define

1

xu(t) = C(t- T(c:) + T1(c:)t$-l, t 2 T(c:).

Thus t >-+ xu(t) is decreasing on [T(c:,oo), so fortE [T(c:),T(c:) + 7], the definition of T1 (c:) gives

(68) 1 6

Xu(t) 2 C(t+Tl(c)t$-1 = 4Xe >X,.

Lett> T(c:) + r. Then

x~(t) + axu(t)(J lblxu(t- 7)(J- lf(t)l

= (t-T(c:)+T1(c:))-$-1 --+aC(J -L{ -C /3-1

-lbiC(J (t- T(c:) + T1 (c:) - 7 )-;/:-r - (t- T(c:) + T1 (c:) );1:-r lf(t)l }, t- T(c:) + T1(c:)

so, by first using (66), (67), and then using (63), we get

x~(t) + axu(t)(J -lblxu(t- r)(J -lf(t)l

> (t T(c:)+T!(c:))-;/:-r{/3-_:;1

+aC(J-IbiC(J(1+c:)-c:} >0.

Therefore, xu satisfies the delay-differential inequality

(69) x~(t) > -axu(t)(J + lblxu(t- 7)(J + lf(t)l, t > T(c:) + 7,

(70) xu(t) > x, 2 lx(t)l, t E [T(c:), T(c:) + 7].

.,

Since xis a solution of (60)-(61), with [x(t)[ ::; x, fort 2': T(e), it obeys the delay differential inequality

D+[x[(t) ::; -a[x(t)[il + [b[[x(t- r)[il + [f(t)[, t > T(e) + r,

[x(t)[ :<:: x., t E [T(e), T(e) + r].

Therefore, fort 2': T(e), [x(t)[ :<:: xu(t), and so the definition of Xu implies 1 l

limsuptP-1[x(t)[ :<:: limsuptll=lxu(t) =C. t--+oo t--+oo

Letting C +Co now gives (62). 0 THEOREM 9. Let a > [b[ 2': 0, r 2': 0 and f3 > 1 be a quotient of

odd integers. Let 1/J E C([-r,O];R), and suppose that f E C([O,oo);R) satisfies (59}. If x is the unique continuous solution of (60-61} which obeys limt->oo x(t) = 0, then either

1

(71) lim tH [x(t)[ = 0 t->oo

or

(72) 1 ) p:_1

JimsuptP:_'[x(t)[2': ((a_ b)(/3 1) t->oo

Moreover, if b 2': 0 then there is a constant L which assumes either the values 1

0 or ((a- b)(/3 -1))-P- 1 such that

(73) 1

limsuptP-1[x(t)[ = L. t->oo

1

Proof Define L : = ( (a - b) (/3 - 1))- H . According to ( 62) there exists 0 :<:: L 0 < oo such that

1

limsuptP-1[x(t)[ = L0 . t->oo

If £ 0 = 0 we have (71). Suppose now that L0 E (O,L). Then for every e E (0, L- L0 ), there is T1 (e) such that

l

tH [x(t)[ :<:: £ 0 (1 +e), t > T1(e).

By (59), it follows that ft00 lf(s)[ds is well defined for all t 2': 0 and moreover, as tilr[x(t)[l' :<:: Lg(1 + c:)il we have that

lxo [x(s)[il ds, loo [x(s- r)[il ds

.,.., .,..


exist for all t 2: 0. Since x(t) -t 0 as t -too, by integrating (60), we get

(74) x(t) =a t7

x(s)11 ds +(a- b) 1: x(s)P ds -[><> f(s) ds.

By (59), we have that

(75) 1 hoo lim t~- 1 f(s)ds = 0. t-too t

Fort> T1(c) + r

(76) ,t/-1 {' x(s)i1dsl:::; Lg(l +c)i1t~.'.~ {' s-1--r ds -t 0 lt-T lt-T

as t -too. Thus (74), (75), (76) imply

(77) 1 100 Lo:::; Ia- bllimsupU-1 lx(s)l11 ds. t-).oo t

Fort > T1 (c),

I roo I _/}_ t~- 1 }, lx(s)l 11 ds ::S: Lg (1 + c)i1tH J,oo s- ~- 1 ds

= Lg(l + c)i1(f3- 1).

Letting t -t oo and then c -1- 0 we get

1 100 limsupU-1 lx(s)ii1 ds:::; Lg(/3- 1). t-too t

Then (77) implies L0 :::; (a-b)Lg(/3-1), or L0 2: {(a-b)(/3-1)}-1/(11-1) = L, which contradicts the assumption that L0 < L. Thus Lo = L, and (73) holds. 0

The third result in this subsection concerns the rate of decay of a perturbed equation, when the size of the perturbation is bounded above by a polynomially decaying function, which decays to zero more slowly that t-Mi1-1).

THEOREM 10. Let a > lbl 2: 0, r 2: 0 and f3 > 1 be a quotient of odd integers. Let 'lj; E C([-r,O];R), and suppose that f E C([O,oo);R) satisfies

(78) lim log lf(t)l < -a Hoo logt -

where a E (0,/3(/3 -1)-1).

~

If x is the unique continuous solution of (60-61), which obeys x(t) --+ 0 as t --+ oo,

(79) I. loglx(t)l a 1msup < --. Hoo logt - f3

Proof. Since a:::; f3(f3 -1)-I, we can choose E E (0, a) such that a- E < f3/(f3 -1), or (a- c)(1- (3- 1

) -1 < 0. ForEE (0, a), as a> lbl > 0, we can define

2Tibjlf(a-e) Tibjlf(a-e) T (E) - > ----;--c,...--c;-'--..,.,-;:--;;--;-

1 - alf(a-e) - blf(a-e) alf(a-e) - lbjlf(a-e).

(The case b = 0 is covered by a result in Appleby and Mackey [3].) As f obeys (78), for every E E (0, a), there exists T2(c) > 0 such that

(80) lf(t)l < r<>+e/2, t > T2(c).

Let T0 (c) = sup{t > 1 + (T1(c) V T2(c)) : x(t) -:J 0}. If this set is empty, the result holds, as x(t) = 0 for all t ;::: T for some T > 0 implies (79). On the other hand, if the set is not empty, as x(t) --+ 0 as t --+ oo, there exists x, > 0 such that T(c) = sup{t > To(c) : lx(t)l = x,}. Then lx(t)l :'0: x, for all t;::: T(c). Notice, moreover, that T(c) > T 1(c) V T1(c) V 1.

Next, choose M1(c) > 0 such that

MJ(c)il ( ax/T(c)a-e -lblx/(T(c) + T)"-')

a - c ( ) c.-e +l --f3-M1 E x,T(c)T - 1 > 1,

and moreover suppose that

M ( ) (a- c)f3-1x,T(c)T+I

1 c > ~ . f3xe (aT(c)a-e -lbi(T(c) + T)<>-e)

Such an M1(c) exists, since f3 > 1, T(s) > T1(c), which implies that ax~T(c)a-e -lblx~(T(c) + T)a-e > 0. Moreover, forM> M1(c), we have

(81) M~ ( ax/T(c)"-e- lblx/(T(c) + T)"-')

a- c o:-e+l --(3-Mx,T(c)T - 1 > 1.

Next define

M2(c) = 2 (T(c) + T) a~' T(c) > 1.

Finally, choose M(t:) = M 1(t:) V M2 (t:), and Jet

0-'

xu(t) = M(t:)x, ( T~t:)) 13, t:;:: T(t:).

FortE [T(t:),T(t:) + T], as M(t:):;:: M2 (t:), the definition of M2 (t:) gives

( T(t:) )"fi'

Xu(t) :;:: M(t:)x, T(t:) + T :;:: 2x, > x,.

Therefore, fort> T(t:) + T, we have

x~(t) + axu(t)f3 -JbJxu(t- T)f3 -Jj(t)J

= t-(a-£){- apE M(t:)x,T(c) "fi'+lr"j"-l+(a-£) + aM(c)f3x~T(c)"-'

-JbJM(t:)f3x~T(c)"-' C ~ T) a-£- t"-'Jf(t)J }·

Thus, as t > T(c) + T > 1, t-'12 < 1, and t-(a-£)/!3-l+(a-£) < 1, by (80), we get

x~(t) + axu(t)f3 -Jblxu(t- T)f3- Jj(t)J

> rla-£) { - a p c M(c)x,T(c) "ii' +l + aM(t:)fix~T(t:)"-'

-JbJM(t:)f3x~(T(t:) + T)"-£- 1} > t-(a-£) > 0, where the penultimate inequality follows from (81), as M(t:):;:: M1(t:).

Therefore, xu satisfies the delay-differential inequality

(82) x~(t) > -axu(t)f3 + Jblxu(t- T)f3 + Jf(t)J, t > T(t:) + T,

(83) xu(t) > x,:;:: Jx(t)J, t E [T(t:), T(t:) + Tj.

Since xis a solution of (60)-(61), with Jx(t)J ::; x, fort;:: T(t:), it obeys the delay differential inequality (69). Therefore, fort:;:: T(t:), Jx(t)J::; xu(t), and so, the definition of Xu implies that

I. log Jx(t)J

1. Jog xu(t) a- E

1msup < lffiSUp = ---. t-too log t - t->oo log t f3

Letting c .j. 0 now gives (79). 0

'~

The results of this section can be generalized to cope with the equation

n ro (84) x'(t) = -ax(t)fl +I; bJx(t- TJ)fl + Jo b0 (s)x(t- s)fl ds

j=l 0

+f(t), t > 0,

(85) x(t) = 'lj;(t), t E [-T, 0],

where T = maxJ=O, ... ,n TJ· Exactly the same comparison functions xu are constructed, and we identify lbl := ~J=l lbJI + !;' lbo(s)l ds in the above proofs. We therefore have the following result.

THEOREM 11. Let fJ > 1 be a quotient of odd integers, (rJ)'J=o be a sequence of nonnegative real numbers, and (bJ)J=l be a sequence of real numbers. LetT = maxJ=O, ... ,nTJ· Suppose that 'ljJ E C([-r,O];R), f E C([O,oo);R), andb0 EC([O,r0];R), and

a> t lbJI + [' lbo(s)l ds. j=l 0

Let x be the unique continuous solution of {84-85) which obeys x(t) -t 0 as t -7 00.

{i) Iff satisfies

lim tl=-r f(t) = 0, t-too

1 1

then either A :<:; limsupt-too tP- 1 Ix(t)l :<:; A, or t~- 1 lx(t)l -t 0 as t -too, where

1

(86) A = f(a- tbJ ['b0 (s)ds) (fJ-1)}-P-1

,

l J=l 0 1

(87) A= {(a-~ibJI- ['lbo(sJidsl(fJ-1lrP-l

Moreover, if bj 2: 0, b0 (t) 2: 0 for all t E [0, To] there is a constant L which assumes either the values 0 or A such that

{ii) Iff satisfies

1

limsuptP- 1 Ix(t)l = L. t-too

lim log lf(t)l <-a t->oo log t - '

where a E (0, f3(f3- 1)-1), then

I. log Jx(t)J < a 1msup --. Hoo logt - f3

5.4. Rate of decay of solutions of (3-4). Now we may apply Theorem 7 and Theorem 11 to obtain the main result of the paper concerning the polynomial decay of solutions of (3-4).

THEOREM 12. Let f3 > 1 be a quotient of odd integers, (Ti)}=o be a sequence of nonnegative real numbers, and (bj )j=1 be a sequence of real numbers. Let T = maxi=O, ... ,n Tj. Let p ?: 1 and suppose there exists /p > 0 such that

lv = inf {a> 0: hoo (1 + s)2"'Pu2P(s)ds = oo}. Suppose that 7/J E C([-T,O];R), bo E C([O,To];R), and

a> t /bi/ + fo /bo(s)/ ds. j=l 0

Let X be the unique continuous solution of (3-4) which obeys X(t) -+ 0 as t -+ oo, a.s.

(i) If !v > -tr, 1

then either A :<:: limsupt->oo tP- 1 /X(t)/ :<:: A, a.s., or

1

limt->oo tP- 1 /X(t)/ = 0, a.s. where A, A are given by (86). Moreover, if bi ?: 0, b0 (t) ?: 0 for all t E [0, To] there is a random variable L which assumes either the values 0 or A such that

1

limsuptP-1 /X(t)/ = L, a.s. t->OO

(ii) If lv ::0: -lr. then

I. log /X(t)/ < lv 1m sup

1 _ --/3, a.s.

t->oo ogt

We close with a comment; the advantage of the condition (51) is that it enables us to show that there is always a polynomially decaying upper bound on the solution of (3-4), provided there is a polynomially decaying bound on 0'., viz., there is f.l > 0 such that

I. log /u(t)J 1msup

1 ::; -f.!,.

t->oo og t

ASYMPTOTIC STABILITY OF POLYNOMIAL SDDEs

In this case, one can show that It::; (3((3 -1)-1 implies

(88) lim log IX(t)l < _I!_ Hoo logt - (3'

a.s.

and if It> (3((3- 1)-1, there is L = 0 or L E [.\,A] such that

1

limsuptP-,IX(t)l = L, a.s. t-+oo

65

Such a result was not available in Appleby and Mackey [3] for all polynomial rates of decay of a.

The proof of ( 88) follows by choosing an integer p > p' > 1 so that It< (3((3- 1)-1 + p-1 and "/p =It- p- 1 > 0. Then by Theorem 12

lim sup log IX(t)l < Hoo logt -

It- p-1

(3 a.s. on flp

where OP is an almost sure event. Considering the behaviour on fl' -np>p.np proves (88).

REFERENCES

[1] J. A. D. Appleby. Almost sure stability of linear Ito-Volterra equations with damped stochastic perturbations. Electron. Commun. Probab., 7, Paper no. 22 (2002), 223-234.

[2] .J. A. D. Appleby. Almost sure subexponential decay rates of scalar Ito-Volterra equations. Electron. J. Qual. Theory Differ. Equ., Proc. 7th Call. QTDE, 1 (2004), 1-32.

[3] J. A. D. Appleby and D. Mackey. Almost sure polynomial asymptotic stability of scalar stochastic differential equations with damped stochastic perturbations. Electron. J. Qual. Theory Differ. Equ., Proc. 7th Call. QTDE, 3 (2004), 1-33.

[4] J. A. D. Appleby, X. Mao and A. Rodkina. Pathwise superexponential decay rates of solutions of autonomous stochastic differential equations, 2004, preprint.

[5] T. Chan and D. Williams. An 'excursion' approach to an annealing problem. Math. Proc. Camb. Phil. Soc., 105 (1989), 169-176.

[6] V. Kolmanovskii and A. Myshkis. Introduction to the Theory and Applications of Functional Differential Equations, Mathematics and Its Applications, Vol. 463. Kluwer Academic Publishers, Dordrecht, 1999.

[7] K. Liu. Some remarks on exponential stability of stochastic differential equations. Stochastic Anal. Appl., 19(1)(2001), 59-65.

[8] K. Liu, X. Mao. Exponential stability of non-linear stochastic evolution equations. Stochastic Process. Appl., 78 (1998), 173-193.

[9] K. Liu, X. Mao. Large time behaviour of dynamical equations with random perturbation features. Stochastic Anal. Appl., 19(2)(2001), 295-327.

[10] X. Mao. Almost sure polynomial stability for a class of stochastic differential equations. Quart. J. Math. Oxford Ser. {2}, 43(2)(1992), 339-348.


[11] X. Mao. Polynomial stability for perturbed stochastic differential equations with respect to semimartingales. Stochastics Process. Appl., 41 (1992), 101-116.

[12] X. Mao. Exponential Stability of Stochastic Differential Equations, Pure and Applied Mathematics, Vol. 182. Marcel Dekker, New York, 1994.

[13] S.-E. A. Mohammed. Stochastic FUnctional Differential Equations, Research Notes in Mathematics, Vol. 99. Pitman, London, 1984.

[14] A. Rodkina. On the asymptotic stability of nonlinear stochastic delay equations Gubo, A Mathematical Journal, 1(5) (2005), 23-42.

[15] A. Rodkina, V. Nosov. On stability of stochastic delay cubic equations. Dynam. Systems Appl., Special issue (2005), to appear.

[16] B. Zhang, A. H. Tsoi. Lyapunov functions in weak exponential stability and controlled stochastic systems. J. Ramanujan Math. Soc., 11(2)(1996), 85-102.

[17] B. Zhang, A. H. Tsoi. Weak exponential asymptotic stability of stochastic differential equations. Stochastic Anal. Appl., 15(4)(1997), 643-649.

~

VOLUME 12 2005, NO 1-2 PP. 67- 81

A NOTE ON THE NONLOCAL BOUNDARY VALUE PROBLEM FOR HYPERBOLIC-PARABOLIC

DIFFERENTIAL EQUATIONS

A. ASHYRALYEV' ANDY. OZDEMJR1

Abstract. The nonlocal boundary value problem

l d~~~tl + Au(t) = f(t)(O :0: t :0: 1),

d~('l + Au(t) = g(t)( -1 :0: t :0: 0),

u( -1) =au(!")+ ;3u' (.\) + <p, Ia!, 1/31 :0: 1, 0 < f", .\ :0: 1

for differential equation in a Hilbert space H with the self-adjoint positive definite operator A is considered. The stability estimates for the solution of this problem are established. In applications, the stability estimates for the solutions of the mixed type boundary value problems for hyperbolic-parabolic equations are obtained.

AMS(MOS) subject classification. 65N, 65J, 47D, 34G, 35M

1. Introduction. It is known that some problems in fluid mechanics (model of the motion of an ideal fluid filling, exhibiting both viscous and nonviscous phases) and other areas of physics and mathematical biology( taxisdiffusion-reaction model) lead to partial differential equations of the hyperbolic parabolic type. Methods of solutions of the nonlocal boundary value problems for hyperbolic-parabolic differential equations have been studied extensively by many researches (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12] and the references given therein).

' Department of Mathematics, Fatih University, Istanbul, Turkey and International Turkmen-Turkish University, Ashgabat, Turkmenistan

t Department of Mathematics, Fatih University, Istanbul, Turkey

67

68 A. ASHYRALYEV AND Y. OZDEMIR

In the present paper we consider the nonlocal boundary value problem

(1.1) d~~t) + Au(t) = g(t)(-1::; t::; 0), {

d~u~t) + Au(t) = f(t)(O :S: t::; 1),

u( -1) =au (J.t) + (Ju' (.>.) + <p, Ia I, l/31 ::; 1, 0 < J.t, .>.::; 1

for differential equations of mixed type in a Hilbert space H with self-adjoint positive definite operator A.

A function u(t) is called a solution of the problem (1.1) if the following conditions are satisfied:

i. u(t) is twice continuously differentiable on the interval (0,1] and continuously differentiable on the segment [-1,1]. The derivative at the endpoints of the segment are understood as the appropriate unilateral derivatives.

ii. The element u(t) belongs to D(A) for all t E [-1, 1], and the function Au(t) is continuous on the segment [-1,1].

iii. u(t) satisfies the equations and nonlocal boundary condition (1.1). In the paper [18] the following theorem on the stability was proved. THEOREM 1.1. Suppose that <p E D(A) and f(t) be continuously differ-

entiable on [0, 1) and g( t) be continuously differentiable on [ -1, OJ functions and (3 = 0 . Then there is a unique solution of the problem ( 1.1) and the stability. inequalities

max II u(t) IIH:S: M[ll <p IIH -199

+ max II g(t) IIH +max II A- 112 f(t) IIH], -19:00 09:01

max II A112u(t) IIH:S: M[ll A112<p IIH -199

0

+II g(O) IIH +j II g'(t) IIH dt+ max II f(t) IIH], 099

-1

du(t) d2u(t) -~~0 II dt IIH + ~~111 di2 IIH

+ max II Au(t) IIH< M[ll A<p IIH + II A112g(O) IIH -IStSI -

A NOTE ON THE NONLOCAL BOUNDARY VALUE PROBLEM ... 69

1

+ II f(O) IIH + -~~~0 II g'(t) IIH + J II f'(t) IIH dt] 0

hold, where M does not depend on j(t), g(t), and <p.

We are interested in studying the stability of solutions of the problem (1.1) for fJ # 0. We have not been able to obtain the same stability estimates for the solutions of the problem (1.1) for fJ # 0. Nevertheless, in the present paper the stability estimates for the solution of the problem (1.1) under a stronger assumption than f(t) be continuously differentiable on [0, 1] and g(t) be continuously differentiable on [-1, 0] functions are established. In applications, the stability estimates for the solutions of the of the mixed type boundary value problems for the hyperbolic-parabolic equations are obtained.

Finally note that the methods for numerical solutions of the nonlocal boundary value problem (1.1) in the case fJ = 0 have been studied extensively (see [14] - [17], [19] - [21]and the references therein).

2. The main theorem. First of all let us give some lemmas that will be needed below.

Lemma 2.1. The estimates hold:

(2.1) llc(t)IIH->H:; 1, IIA112s(tJIIH->H:; 1,

(2.2) IIA7 e-tA IIH->H:; Mr7 e-ot, t > 0, 0:; 1:; 1, o > 0, M > 0,

where itA 1/2 c(t) = e + e-itAl/2

A ' ( )

itAl/2 . s t = A-1/2e - e-•tAl/2

Lemma 2.2. The operator

I- a [c (Jl.)- As (Jl.)] e-A + fJ [s (>.) + c (>.)] Ae-A

has an inverse

T = (I- a [c (Jl.) -As (Jl.)] e-A + fJ [s (>.) + c (>.)] Ae-A) -1

and the estimate holds:

(2.3) IITIIH-tH:; M.

Proof. The proof of the estimate (2.3) is based on the estimate

11-a [c (J-L)- As (J-L)] e-A + ,B [s (..\) + c (..\)] Ae-AIIH->H < 1.

Using the definitions of c (J-L) and s (J-L) and positivity and self-adjointness property of A, we obtain

11-a [c (J-L) - As (J-L)] e-A + ,B [s (..\) + c (..\)] Ae-AIIH->H

::; sup 1-a [cos ( YPJ-L) - ypsin ( ypJ-L)] e-P 0Sp<oo

+,8 [ypsin (yp.\) + pcos ( yp..\)]1 e-P.

Since

cos (ypJ-L)- ypsin (ypJ-L) = JP+l cos (ypJ-L- J-Lo),

ypsin (yp..\) + pcos (yp..\) = ypJP+l cos (ypJ-L- J-Ld,

we have that

11-a [c (J-L) - As (J-L)] e-A + ,B [s (..\) + c (..\)] Ae-AII H->H

::; sup JP+l (1 + yp) e-P. OS:p<oo

It is easy to show that sup vPTI (1 + yp) e-P < 1. Thus Lemma 2.2 is o$p<oo

proved. Now, we will obtain the formula for solution of the problem (1.1). It is

known that for smooth data of the initial value problems

(2.4)

(2.5)

{ u" ( t) + Au ( t) = f ( t) , ( 0 ::; t ::; 1) ,

u (0) = u0 , u' (0) = uiJ,

{ u' (t) + Au(t) = g(t), (-1::; t::; 0),

u(-l)=u-1>

A

there are unique solutions of the problems (2.4), (2.5), and the following formulas hold:

(2.6) u (t) = c (t) u (0) + s (t) u' (0) + l s (t- y) f (y) dy, 0::; t::; 1,

and

(2.7) U (t) = e-(t+l)Au_1 + 1t e-(t-y)Ag (y) dy, - 1 :<;; t :<;; 0. -1

Using formulas (2.6), (2.7), and equation (1.1) we can write

(2.8) u (t) = [c (t)- As (t)] { e-Au_1 + 1: eYAg (y) dy}

+s(t)g(O) + l s(t y)f(y)dy.

Now, using the condition u( -1) =au (J.L) +flu'(.\)+ <p, we obtain the operator equation

(2.9) {I- a [c (J.L) As (J.L)] e-A + fJ [s (.\) + c (.\)] Ae-A} u_1

=a { c (J.L) { eYAg (y) dy + s (J.L) [g (0) -A { eYAg (y) dy]

+ [ s(J.L- y) f (y)dy} + fJ {-As(.\) [ eYAg(y)dy

+fJ { c(.\) [g(O)- A [ eYAg(y)dy] + [ c(.\- y) f(y)dy} +<p.

Since the operator

I- a [c (J.L)- As (J.L)] e-A + fJ [s (.\) + c (.\)] Ae-A

has an inverse

T = (I- a [c (J.L)- As (J.L)] e-A + fJ [s (.\) + c (.\)] Ae-Ar1,

72 A. ASHYRALYEV ANDY. OZDEMIR

for the solution of the operator equation (2.9) we have the formula

(2.10) u-1 = T [a{c(tt) I: eYAg(y)dy

+s (tt) [g (0)- A I: eYAg(y) dy] + { s (tt-y) I (y) dy}

+,8 {-As (.A) 1: eYAg (y) dy + c (.A) [g (0)- A I: eYAg (y) dy]

+ [ c (.A - y) I (y) dy} + cp] .

Hence, for the solution of the nonlocal boundary value problem (1.1) we have the formulas (2.7), (2.8) and (2.10).

Theorem 2.1. Suppose that cp E D(A), g(O) E D(A~), g'(O) E H, 1(0) E D(A~) and /'(0) E H. Let l(t) be twice continuously differentiable on [0, 1] and g(t) be twice continuously differentiable on [-1, 0] functions. Then there is a unique solution of the problem (1.1) and the stability inequalities hold:

(2.11) max llu(t)lln ~ M [ii'PIIn + max JJA-112g'(t)JJn -1$!$1 -1$!$0

+ IJA-1f2g(O) Jln + IJA-1/2 1(0) Jln + ~~s1JIA-l/2 l'(t) Jln] '

(2.12) -Ts~~~~~~~IL + -Ts~?~JIA1f2u(t)Jln ~ M [IJAif2cplJn

+ llg(O)IIn + max llg'(t)lln + III(O)IIn +max 11/'(t)lln], -I $1$0 0$1$1

(2.13) max llddull +max ~~~du2 11 + max IIAu(t)lln -1$1$0 t n 0$1$1 t n -1$1$1

s M [IIA'PIIH + IIA112

g(O)IIH + llg'(O)IIH + _rp~~~o llg"(t)IIH

+ IIA112

f(OliiH + llf'(O)IIH + ~~ llf"(t)IIH]'

where M does not depend on j(t), t E [0, 1], g(t), t E [-1,0] and <p. Proof. First, we obtain estimate (2.11). Using formula (2.10) and an

integration by parts, we obtain

(2.14) u_1 =T[a{c(J-L) [A-1 (g(O) -e-Ag(-1)

-I: eYAg'(y)dy)] +s(J-L) (e-Ag(-1)+ 1: eYAg'(y)dy)

+A-1 [1 (J-L)- c (J-L) j (0) -1~' c (J-L- y) f' (y) dy]} + ,B

X { -S (.\) (9 (0)- e-Ag ( -1) - [ eYAgt (y) dy)

+c(.\) (e-Ag(-1)+ ~o1 eYAg'(y)dy)

+s (.>.) f (0) + [' s (.>.- y) f' (y) dy} + <p] .

Using estimates (2.3), (2.1) and (2.2), we obtain

llu-1IIH S M [II'PIIH + -~~~0 IIA-11

V(tliiH

(2.15) + IIA-112g(O)IIH + IIA-112f(O)IIH +max IIA-112f'(tli1H] · 099

Using formulas (2.7), (2.8) and an integration by parts, we obtain

(2.16) u (t) = e-(t+l)Au_1 + A-1 (g (t)- e-Ag ( -1)

-~.-· '""'


-l1

eYAg' (y) dy, - 1 ::; t::; 0,)

(2.17) u (t) = [c (t)- As (t)] { e-Au_1

+K1 (g (0)- e-Ag ( -1)- I: eYAg' (y) dy)} + s (t) g (0)

+A-1 [1 (t)- c (t) f (0) -[ c (t- y) J' (y) dy, o::; t::; 1.) Using estimates (2.1) and (2.2), we obtain

llu (t)IIH ::> M [11u-1IIH + -Ifl~~o IIA-112g'(tJIIH + IIA-112g(OJIIH], -1::; t::; 0,

llu (t)IIH::; M [o~~~ IIA-112 f'(tJIIH + IIA-112 f(OJIIH

+ llu-1IIH + -Ifl~~o IIA-1129'(tJIIH + IIA-112g(OJIIH], 0 ::> t ::> 1. Then from (2.15) and the last two estimates, it follows (2.11).

Second, we obtain estimate (2.12). Applying A 112 to the formula (2.14) and using estimates (2.3), (2.1) and (2.2), we obtain

(2.18)

IIA112u-1IIH ::> M [11A112'PIIH + -Ifl~~o llg'(t)IIH

+ llg(O)IIH + llf(O)IIH + max llf'(t)IIH] · 099

Applying A 112 to the formulas (2.16), (2.17) and using estimates (2.1) and (2.2), we obtain

IIA 112u (t) II H ::; M [II A 112u-dl H + -Ifl~o llg'(t) II H + llg(O)IIH] , -1 ::> t ::> 0,

iiA 112u (t)iiH :S: M [~~lj_ llf'(t)IIH + llf(O)IIH

+ IIA112u-1IIH + _rn,~~o llg'(t)IIH + llg(O)IIH], 0 :S: t :S: 1.

Then from (2.18) and the last two estimates, it follows (2.12). Third, we obtain estimate (2.13). Using formula (2.14) and an integra

tion by parts, we obtain

u-1 = T [a { c (J.L) [A-1 (g (0)- e-Ag ( -1)

-A- 1 [g' (0)- e-Ag1 ( -1)- J: eYAg11 (y) dy])

+s(J.L) (e-Ag(-1)+A- 1 [g'(O)-e-Ag'(-1)- J: eYAg"(y)dy])

+A - 1 [ f (J.L) - c (J.L) j ( 0) - [ s (J.L) f' ( 0) + [ s (J.L - y) f" (y) dy]]} + (3

x { -s(.\) (g(O)- e-Ag(-1)- A-1 [g'(O)- e-Ag'(-1)- J: eYAg"(y)dy])

+c(.\) (e-Ag(-1) +A-1 [g'(O) -e-Ag'(-1)- [ eYAg"(y)dy])

+s (.\) f (0) +A - 1 [t (.\) - c (.\) f' (0) - [ c (.\ - y) !" (y) dy]} + <p] .

Using estimates (2.3), (2.1) and (2.2), we obtain

IIAu-1IIH :S: M [IIA'PIIH + -T:s~~o llg''(t)IIH + llg'(O)IIH

(2.19) + IIA112g(O)IIH + IIA112 f(OliiH + llf'(O)IIH + 0~~lj_ llf"(t)IIH] ·

Using formulas (2.16), (2.17) and an integration by parts, we obtain

u (t) = e-(t+l)Au_1 + A-1 (g (t) - e-Ag ( -1)

-A-1 [g' (t)- e-Ag' ( -1) -l1

eYAg11 (y) dy]) , -1 S:: t S:: 0,

u (t) = [c (t) -As (t)] { e-Au_1 + A-1 (g (0) - e-Ag ( -1)

-A-1 [u' (0)- e-Ag' ( -1) - 1: eYAg" (y) dy])}

+s (t) g (0) + A-1 I (t)- c (t) I (0)

- [s(t)/'(0)+ [ s(t-y)f"(y)dy], 0 s; t s; 1.

Applying A to the last two formulas and using estimates (2.1) and (2.2), we obtain

II Au (t)IIH S:: M [11Au-1IIH + max llu"(t)IIH + \\A112g(O)\\H + llu'(O)IIH], -1:9$0

-1 s; t s; 0,

IIAu (t)IIH S:: M [max llf"(t)IIH + \\A112 1(0)\\H + 11/'(0)IIH 0:<;!$1

+ IIAu-1IIH + -~~0 llu"(t)IIH + \\A112u(O)\\H + llu'(O)IIH], o S:: t S:: 1.

Then from (2.19) and the last two estimates, it follows (2.13). Theorem 2.1

is proved.

., r.!

REMARK 1. Theorem 2.1 holds for the following multi-point nonlocal boundary value problem

d:~~t) + Au(t) = f(t)(O :S: t :S: 1),

d~\1) + Au(t) = g(t)(-1::; t :S: 0),

N L u( -1) = I; aiu (Mi) + I; f3iu' (>.i) + <p,

i::::::l i=l

N L I: Ia;!, I: lf3il :S: 1, i::::l i=l

0 < lli :S: 1, 1 :S: i :S: N, 0 < >.i :S: 1, 1 :S: i :S: L

in a Hilbert space H with self-adjoint positive definite operator A. REMARK 2. We cannot obtain the stability estimates for the solution

of the problem ( 1.1) in an arbitrary Banach space E with strongly positive operator A under the assumptions

(2.20) llc(t)IIE-+E :S: M, IIA11

2s(t)IIE->E :S: M.

Nevertheless, the nonlocal boundary value problem {1.1} generated by the following well-posed problem

(2.21) { u" (t) +Au (t) = f (t) (0::; t::; 1),

u' (t) +Au(t) = g(t) (-1::; t::; 0) ,u(-1) = .) = <p, Ia I, 1!31 :S: 1, 0 < fl, >. :S: 1,

u' (t) + Au(t) = g(t) (-1::; t::; 0)

for differential equations of mixed type in a Hilbert space H with self-adjoint positive definite operator A.

The stability estimates for the solution of the problem (2.21)in an arbitrary Banach space E with strongly positive operator A under the assumptions ( 2. 20) can be established.

3. Applications. First, we consider the mixed problem for hyperbolicparabolic equation

Vyy- (a(x)vx)x + ov = f(y, x), 0 < y < 1, 0 < x < 1,

Vy- (a(x)vx)x + ov = g(y, x), -1 < y < 0, 0 < x < 1, (3.1) { v( -1, x) = v (1, x) + vy(1, x) + <p(x), 0:::; x:::; 1,

v(y, 0) = v(y, 1), Vx(Y, 0) = vx(y, 1), -1:::; y:::; 1, v(O+, x) = v(O-, x), vy(O+, x) = vy(O-, x), 0:::; x:::; 1.

Problem (3.1) has a unique smooth solution v(y, x) for the smooth a(x) > O(x E (0, 1)), <p(x) (x E [0, 1]) and f(y, x)(y E [0, 1], x E [0, 1]), g(y, x)(y E [-1, OJ, x E [0, 1]) functions and o = const > 0. This allows us to reduce the mixed problem(3.1) to the nonlocal boundary value problem (1.1) in Hilbert space H with a self-adjoint positive definite operator A defined by (3.1). Let us give a number of corollaries of the abstract Theorem 2.1.

THEOREM 3.1. The solutions of the nonlocal boundary value problem (3.1} satisfy the stability estimates

-~~~ 1 llv(y)IIL,[o,!J:::; M [llf(O)IIL,[o,!J + oT~ llfv(Y)IIL,[o,!J

+ llg(O)IIL2[o,!J + -~~~0 ll9v(Y)IIL,[o,l) + ll<piiL,[o,l)],

-~~~1 llv(y)llwJ[o,!J:::; M [llf(O)IIL,[o,!J + 0T;{t llfv(Y)IIL,[O,!J

+ llg(O)IIL,[O,!J + -~~~0 ll9v(Y)IIL2[o,l) + ll<pllwi[o,l]]'

-T::a::?:1 llv(y)llw?ro,!J + _v:::a~o llvv(Y)IIL,[O,l) + 0~af1 llvw(Y)IIL,[O,!J _y_ _y_ _y_

:::; M [11<pllw:f[o,1J + llf(O)IIwJ[o,!J + llfv(O)IIL,[o,!J + oTt.f1 11fw(Y)IIL,[o,!J

+ llg(O)IIw:f[o,!J + ll9v(O)IIL,[o,!J + -~~~0 ll9vv(Y)IIL,[o,!J]

,.., ..

A NOTE ON THE NONLOCAL BOUNDARY VALUE PROBLEM... 79

where M does not depend on f(y, x) (y E [0, 1], x E [0, 1]), g(y, x) (y E [-1, OJ, x E [0, 1]) and <p(x) (x E [0, 1]).

The proof of this theorem is based on the abstract Theorem 2.1 and the symmetry properties of the space operator generated by the problem (3.1).

Second, let !1 be the unit open cube in the n-dimensional Euclidean space !Rn (0 < Xk < 1, 1 ::; k::; n) with boundary S, !1 = !1 US. In [0, 1] x !1 we consider the mixed boundary value problem for the multidimensional hyperbolic-parabolic equation

(3.2)

n Vyy- L (ar(x)vxclx, f(y, x), 0 :<_:; Y :<_:; 1,

r=l X= (x1, ... ,xn) E !1,

n Vy- I: (ar(x)vxclx, = g(y,x), -1::; y::; 0,

r=l X= (x1, ... ,Xn) E !1, v(-1,x) = v(1,x) +vy(1,x) + <p(x),x E !1, u(y,x) = O,x E S, -1::; y::; 1, v(O+, x) = v(O-, x), vy(O+, x) = vy(O-, x), x E !1.

We introduce the Hilbert spaces L2 (!1) of the all integrable functions defined on !1, equipped with the norm

{ )

1/2

llfiiL,(n) = r · ·11f(xWdxl .. · dxn

X Ell

Problem (3.2) has a unique smooth solution v(y, x) for the smooth ar(x) 2': 8 > 0, <p(x) (x E !1) and f(y, x) (y E (0, 1), x E !1), g(y, x) (y E ( -1, 0), x E !1) functions. This allows us to reduce the mixed problem (3.2) to the nonlocal boundary value problem (1.1) in Hilbert space H with a self- adjoint positive definite operator A defined by (3.2). Let us give a number of corollaries of the abstract Theorem 2.1.

THEOREM 3.2. The solutions of the nonlocal boundary value problem {3.2) satisfy the stability estimates

-~~~ 1 liv(y)IIL,(n) :<.:: M [ilf(O)IIL,(n) + o~:l-1 11fu(v)IIL,(n)

+ llg(O)IIL,(l'll + -~~~0 llgy(Y)IIL,(l'll + II'PIIL,(nl] ,

-~~~ 1 iiv(y)iiwl(n) :::; M [iif(O)iiL,(IT) + 0~:{! ilfv(Y)iiL,(IT)

+ llg(O)IIL,(IT) + -~~~0 ll9v(Y)IIL,(IT) + II'PIIw:}(IT)] '

-~~~ 1 iiv(y)iiw,'(IT) + -~~~0 llvv(Y)IIL,(IT) + 0~:{! llvvv(Y)iiL,(IT)

:::; Ml [ii'Pillw,'(n) + iif(O)iiwJ(IT) + llfv(D)IIL,(IT) + 0~:{! iifvv(Y)IIL,(IT)

+ iig(O)ilwj(IT) + ll9v(O)iiL,(IT) + -~~~0 li9vv(Y)IIL,(IT)]

where M does not depend on f(y, x) (y E [0, 1], x E !1), g(y, x) (y E [-1, 0], x E !1]) and <p(x) (x E !1).

The proof of this theorem is based on the abstract Theorem 2.1 and the symmetry properties of the space operator generated by the problem (3.2).

Acknowledgement The authors would like to thank the referee and Prof. E. Litsyn for their

helpful suggestions to the improvement of this paper.

REFERENCES

[lj KORZYUK,V.I. AND LEMESHEVSKY,S.V.,Problems on conjugation of polytypic equations, Mathematical Modelling and Analysis, 6(2001), no. 1, 106-116.

[2] VALLET,G., Weak entropic solution to a scalar hyperbolic-parabolic conservation law, Rev. R. Acad. Cien. Serie A. Mat., 97(2003), no. 1, 147-152.

[3] GLAZATOV,S.N., Nonlocal boundary value problems for linear and nonlinear equations of variable type, Sobolev Institue of Mathematics SB RAS, Preprint no. 46(1998), 26p.

[4] KARATOPRAKLIEV,G.D., On a nonlocal boundary value problem for hyperbolicparabolic equations, Differensial'nye Uravneniya,25(1989), no. 8,1355-1359( Russian).

[5] KARATOPRAKLIEVA,M.G., On a nonlocal boundary value problem for an equation of mixed type, Differensial'nye Uravneniya,27(1991), no. 1, 68-79( Russian).

[6] CHEN,G.Q. AND DIBENEDETTO, E., Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations.


[7] GERISH,A., KOTSCHOTE,M. AND ZACHER,R., Well-posed of a quasilinear hyperbolic-parabolic system arising in mathematical biology, Report on Analysis and Numerical Mathematics, Martin-Luther-Universitat HalleWittenberg, Germany, no. 04-24, August 31, 2004.

[8] SALAHATDINOV,M.S., Equations of Mixed-Composite Type. Tashkent: FAN, 1974, 156 p. (Russian)

[9] DJURAEV, T.D., Boundary Value Problems for Equations of Mixed and MixedComposite Types. Tashkent: FAN, 1979, 238 p. (Russian)

[10] BAZAROV,D. AND SOLTANOV,H., Some Local and Nonlocal Boundary Value Problems for Equations of Mixed and Mixed-Composite Types. Ashgabat: Ylym, 1995, 187 p. (Russian)

[11] VRAGOV, V.N., Boundary value problems for nonclassical equations of mathematical physics, Textbook for Universities, Novosibirsk: NGU, 1983. (Russian)

[12] NAKHUSHEV, A.M, Equations of Mathematical Biology, Textbook for Universities, Moscow: Vysshaya Shkola, 1995. (Russian)

[13] KREIN,S.G., Linear Differential Equations in a Banach Space. Moscow: Nauka, 1966. (Russian)

[14] ASHYRALYEV,A. AND MURADOV, I., On stability estimation of difference scheme of a first order of accuracy for hyperbolic-parabolic equations, Izv. Akad. Nauk Turkmenistan Ser. Fiz. -Tekhn. Khim. Geol. Nauk, no 1(1996), 35-39. (Russian).

[15] ASHYRALYEV,A. AND MURADOV, 1., On difference schemes second order of accuracy for hyperbolic-parabolic equations, in: Modeling the Processes in Exploration of Gas Deposits and Applied Problems of Theoretical Gas Hydrodynamics, Ilim, Ashgabat(1998), 127-138. (Russian).

[16] ASHYRALYEV,A. AND ORAZOV, M.B., Theory of operators and the methods of solutions of boundary value problems for equations of mathematical physics, 8 years Turkmenistan's. Independence, Ilim, Ashgabat(1999), 222-228. (Russian).

[17] ASHYRALYEV,A. AND ORAZOV, M.B., The theory of operators and the stability of difference schemes for partial differential equations mixed types, Firat University, Fen ve Muh. Bilimleri Dergisi 11(1999), no. 3, 249-252.

[18] ASHYRALYEV,A. AND YURTSEVER, A., On a nonlocal boundary-value problem for hyperbolic-parabolic equations, in: Application of Mathematics in Engineering and Economics 26, Heron Press and Technical University of Sofia(2001), 79-85.

[19] ASHYRALYEV,A. AND YURTSEVER, A., On difference schemes for hyperbolicparabolic equations, Functional Differential Equations 7(2000), no. 3-4, 189-203.

[20] ASHYRALYEV,A. AND YURTSEVER, A., On a nonlocal boundary value problem for semilinear hyperbolic-parabolic equations, Nonlinear Analysis. Theory. Methods and Applications 47 (2001), 3585-3592.

[21] ASHYRALYEV,A. AND YURTSEVER, A., A note on the second order of accuracy difference schemes for hyperbolic-parabolic equations in a Hilbert space, Proceedings of Dynamic Systems and Applications 4 (2004), 556-563.

VOLUME 12 2005, NO 1-2 PP. 83- 94

LINEAR FUNCTIONAL EQUATIONS ON MANIFOLDS '

G. BELITSKI AND V. TKACHENKOl

Abstract. Two general methods of investigation of linear functional equations on arbitrary manifolds are presented.

1. Introduction. The aim of the present paper is to demonstrate two methods for a study of linear operators

m

(1) L<p(x) = 'L: aJ(x)<p(FJ(x)), xEM, J~l

and related linear equations

(2) L<p(x) = 'Y(x), xEM,

where M is a smooth manifold, FJ : M -+ M are smooth mappings and aJ and 'Y are given smooth complex-valued functions on M.

There are many investigations dealing with a standard question: given a functional equation on M and a closed subset S C M, does the equation have a local solution in a neighborhood of S? Many essential results were obtained in this field (see [8, 9] for references). The next natural problem is following: assume a functional equation to be locally solvable at a neighborhood of every point of M; does it have a global solution on M?

Here we describe two methods which permit us to establish a connection between local and global solvability The first of them is based on a decomposition of the unit, connected with a joint dynamical behavior of the family {FJ}· We call it a decomposition method. It was used first in [1] for

• This research was partially supported by Israel Science Foundation, grant 186/01. t Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105,

Israel.

83

84 G. BELITSKI AND V. TKACHENKO

equations on llil.n with shifts and later for the same equations with shifts on Banach spaces [2]. In [3] we used the decomposition method to investigate one-dimensional equations with arbitrary mappings Fi. Here we apply it to equations with affine transformations in llil.n.

Our second method is applicable in a situation where "local solutions" of (2) are known on elements of some covering of M. An attempt to obtain a global solution by gluing these local solutions meets obstacles of a cohomological type, and Equation (2) has a solution if and only if the cohomological class corresponding to 1 is trivial. This is the reason for us to call such an approach a cohomological method.

2. Decomposition method. Given mappings Hi : M --+ M, a closed subset V C M is called an absorber for the family {Hi}~=! if V is Hrinvariant for all integers j = 1, ... , p, and for every point x E M there exists a neighborhood U and a number N such that

HiiHh ... Hj,(U) c IntV, s 2 N,

with }1, ... ,j8 being arbitrary integers from the set {1, ... ,p}. This notion of an absorber is very close to that of an at tractor used in the theory of dynamical systems. According to a common sense an at tractor attracts asymptotically points of M. As to an absorber, it absorbs every point of M with some its neighborhood in a finite number of steps. Such property is useful for constructing global solutions of functional equations.

THEOREM 1. Let Equation (2) be such that for some indices q1, ... , q., the mappings Fq,, ... , Fq, are Ck -diffeomorphisms and the functions

aq, (x), ... , aq,(x)

do not vanish on M. Assume Vi, ... , v; c M to be absorbers for the families

{Fq-, 1Fj}_,_, ... , {Fq- 1Fj}._,_, J-t-Ql 8 J-r-Qs

respectively. Then for every local Ck -solution 'Po of the equation at the set V =

Vi n ... n v; there exists a global Ck -solution coinciding with 'Po on the set W = Fq, (V1) n ... n Fq, (V,).

In particular, if the intersection is empty then a global Ck -solution does exist for every Ck -function 1·

LINEAR FUNCTIONAL EQUATIONS ON MANIFOLDS 85

Proof Let 'Po be a local Ck-solution of (2) in a neighborhood of V. This means that 'Po is a Ck-function on .MI satisfying (2) in a neighborhood of V. If we set <p = 7/J + 'Po, then ( 2) takes on the form

m

(3) L aj(x)?jJ(Fj(x)) = 1(x), X E .Ml, j=l

with m

1(x) = f'(x)- L ai(x)<p0 (Fi(x)), X E .MI. j=l

If 7/J is a solution of (3), then <p = 7/J +'Po is a solution of (2). Since the function 1 vanishes in a neighborhood of V, the function 7/J = 0 is a local solution of (3). Let us show that 1 may be decomposed into a sum

s

1 = :L /'i i=l

where /'i, i = 1, ... , s, are Ck-functions vanishing in neighborhoods V;, i = 1, ... , s, respectively.

For a proof let 1lu = 0 where U is a neighborhood of V. The subsets V;\U are closed and their intersection is empty. Hence the sets Ui = .MI\ (V; \ U), i = 1, ... , s, form an open covering of.MI and there exist Ck-functions Ti such that

' Ti(X) = 0, X If: Ui, L Ti(x) = 1.

i=l

The functions !'i(x) = 1(x)Ti(x) give us the required decomposition. To solve (3) now it is sufficient to solve the equations

rn

(4) L ai(x)?jJ(Fj(x)) = !'i(x), i = 1, ... 's, X E .MI. j=l

The substitution

(5) 7/J(x) = b(x),b(F_;;-1(x)), b(x) = a;;; 1 (F9~1 (x)),

converts Equation ( 4) with i = 1 into the equation

m

,b(x) + L iii(x),b(Fq~ 1 (Fj(x)) = 11(x), iii(x) = ai(x)bj(x), X E .MI. J#ql

(6)

The latter equation may be written in the form

-/iJ(x) = T-/iJ(x) + 'Yl(x)

with

m

T-/iJ(x) =- L aj(x)-/iJ(Fq~ 1 (Fj(x)). #ql

Since V1 is an absorber for the family Fq-;_ 1(Fj(x)) and the function -y1 vanishes in a neighborhood of V1, for every point of M there exists a neighborhood U such that the series

00

-/iJ(x) = L yn'Y!(x) n:::::O

contains a finite and the same for all x E U number of non-zero terms. Therefore the function .(iJ is a Ck-solution of (6) vanishing in a neighborhood of V1 and the function 1/J defined by (5) is a Ck-solution of ( 4) with i = 1 and vanishing in a neighborhood of Fq1 (V1).

Using the same arguments we obtain Ck-solution 1/Ji of (4) fori= 1, ... , s vanishing in neighborhoods of Fq,(V;), i = 1, ... , s, respectively. The sum of these solutions is a solution of (3) vanishing in a neighborhood of their intersection W. At last, the function <p = 'Po + 1/J is a Ck-solution of (2) coinciding with 'Po in the same neighborhood.

If the set V is empty, then an arbitrary function 'Po is a local solution and the above solution 1/J coincides with it in a neighborhood of W. 0

3. Equations with affine transformations. If M = JR.n and Fj(x) = x + ej with ej of ei for j of i, then there exists a linear functionall such that l(ej) of l(ei) for j of i, and the subsets

V+ = {x E !R.n : l(x) 2: 1}, V_ = {x E !R.n : l(x) S -1},

are non-intersecting absorbers for the families

FjFq~ 1 (x) = x + ej- eqn FjFq~ 1 (x) = x + ej- eq,

respectively, where q1 and q2 are such that

l(eq1 ) = minl(ej), J

l(eq,) = maxl(ej). J

:]

·":'!

•

It follows that if the coefficients aq1 and aq, do not degenerate anywhere then Equation (2) has a Ck-solution for every function 'Y E Ck, see [1].

If F1 are linear transformations in lR", then they have a common fixed point x = 0 and hence every absorber of the family contains that point. The following assertion states that under certain additional restrictions the local solvability of (2) at x = 0 implies the global one.

THEOREM 2. Let {F1(x)} = A1x,j = 1, ... ,m, be a family of invertible commuting linear mappings in JRn such that

(7) {IAI: A E specA;}n{IAI: A E specAj} = 0, i # j.

Aswme that the coefficients a1(x) do not vanish anywhere. Then the local Ck- solvability of Equation (2) at x = 0 implies its global Ck-solvability.

Proof. Since the operators A1 commute, there exists a direct decomposition

q

]Rn = EB LEe, a=l

such that every restriction A11Ea is an operator whose eigenvalues are of the same absolute value Pj,oo j = 1, ... , m, a= 1, ... , q.

Denote by p13 : JRn -+ E 13 the projection parallel to EB L: E". It follows a#

from (7) that for every fJ = 1, ... , q, there exists an index s = s(fJ) such that Ps,/3 > Pj,/3, j # s. We choose a norm in JRn such that

(8) IIP/3A;1Ajll < 1, .i # s,

and introduce the subsets

V13(E) = {x: IIP13xll :S t}.

It follows from (8) that for every E > 0 the set V13(E) is an absorber for the family A-; 1 11.1, j # s. Obviously, every neighborhood of x = 0 contains the intersection of the sets Vp(E) with sufficiently small E > 0. It follows from Theorem 1 that every local solution of (2) at x = 0 may be extended to a global one. D

For an illustration we consider the generalized Abel equation

(9) a(x)rp(x)- rp(Ax) = 1(x)

which is a particular case of (2) with m = 2. The conditions (7) are reduced to the form

A E spec A => IAI # 1.

Linear mappings with such a property are usually called hyperbolic. The following statement is an immediate consequence of Theorem 2.

CoROLLARY 1. Let an operator A be hyperbolic and let a(x) be a nonvanishing anywhere Ck-function. Then a local Ck- solvability of (9) at x = 0 implies its global Ck- solvability.

A local solvability of Equation (9) requires additional relations between eigenvalues of A and values of derivatives of a at x = 0. Namely, if cp is a Ck-solution of (9) then its formal Taylor decomposition

k

P(x) = L it+ ... +in=O

ix in Cit, ... ,in X1 · · · Xn '

1 [)it+ ... +incp(O) C· . = . . '

tt, ... ,tn q' 1 ,; t !l x'' "x'n vt . ... fin• u 1 · . · U n

satisfies the equality

a(x)P(x)- P(Ax) = f'(x) + o(lixW).

On the other hand, the latter relation may be considered as a linear algebraic system with respect to coefficients C;~, ... ,in. If it has a solution, then we say that (9) is formally Ck-solvable. In the case k = oo we say that (9) is formally solvable. A formal solvability of the general Equation (2) is defined in a similar way. It is well known and may be checked by an inspection that (9) is formally solvable for every I' E Ck if and only if

a<•l(o)- .X~' ... A;; # 0, Aa E spec A, i 1 + ... +in= s, s = 0, ... , k.

The corresponding relations for Equation (2) have the form

m

(10) l:aj•l (O).l.~j ... .l.!;j # 0, Aaj E spec A;, it + ... +in = s, s = 0, ... , k. j=l

Generally speaking, the above necessary conditions are not sufficient for a local Ck-solvability of (2). A relation between formal and local solvability was treated in [4, 5], where, in particular, it was proved that if k = oo and (7) holds then the formal solvability implies the local one. Combining the previous arguments we arrive at the following statement.

THEOREM 3. Let Fj(x) = Ajx,j = 1, ... ,m, be a family of invertible commuting linear mappings in JRn and let aj(x),j = 1, ... , m, be ceo_ functions non-vanishing anywhere. If (7) and (10) hold then Equation (2) has a global ceo -solution for every ceo -function I'·

It is easy to see that all conditions of Theorem 3 are essential for the operator in the right-hand side of (2) to be surjective.

4. Cohomological method. Let the mappings FJ : M -+ M be Ck-diffeomorphisms and let { U"' } be an open covering of M by sets which are invariant with respect to FJ,j = 1, ... , m. A Ck-function 'P: U U"' -+ C

"' is said to be a local Ck-solution of Equation (2) on U"' if

n

L aJ(x)cp(FJ(x)) = ')'(x), xE U"'. J~l

For operator L defined by (1) we denote by :F(k, { U"' }, L) the space of all functions 'Y E Ck(M, C) such that for every U"' there exists its local Ck-solution 'Pa on U"'.

Given 'Y E :F(k, { U"' }, L) and a collection { 'Pa} of local Ck-solutions of the latter equation, the differences

Paf3(X) = 'Pf3(x) - 'Pa(x), X E U"' n U13,

are Ck-functions satisfying the homogeneous equation

(11) LPaf3(X) = 0, X E u"' n Uf3.

If {Paf3} is a similar collection of solutions to the homogeneous equation corresponding to another set { <Pa} of local Ck-solutions of (2) on U"' with the same function 'Y, then

(12) Paf3(x) - Paf3(x) = 1/J13(x) - 1/Ja(x), X E U" nuf3,

where the functions

1/Ja(x) = 'Pa(x) - <Pa(x),

satisfy the homogeneous equation on u"'. More generally, a collection {Paf3(x)} of Ck-functions

Paf3 : U"' nuf3 -+ C

satisfying the homogeneous Equation (11) and the conditions

Paf3(x) + Pf3a(x) = 0, x E Ua n U13;

(13) Paf3(X) + Pf3,(X) + P,a(x) = 0, x E U"' n U13 n U,,

is called a cocycle. We denote by Z(k, {Ua}, L) the space of all cocycles corresponding to the covering { Ua } and the operator L. Cocycles p and p are said to be equivalent if (12) holds for some Ck-solutions 1/Ja of the homogeneous equation.

Denote by B(k, {Ua}, L) the space of all cocycles equivalent to zero and by

H 1(k, {Ua}, L) = Z(k, {Ua}, L)/B(k, {Ua}, L)

the quotient space of all equivalence classes. For every function 1 E F(k, {Ua}, L) we denote by [I] E H 1(k, {Ua}, L)

the corresponding equivalence class generated by local. solutions of (2). THEOREM 4. Given a function 1 E F (k, {Ua}, L), equation (2) has a

global Ck-solution if and only if[!] = 0. Given a cocycle c E H 1(k, {Ua}, L), there exists a function 1 E F ( k, { U"}, L) such that c = [!].

Loosely speaking, the space of cohomologies H 1(k, {Uet}, L) is a "measure of non-surjectivity" of operator L. Generally speaking, this space is infinitedimensional which means that in addition to conditions of a local solvability there exist non-trivial complementary conditions for the global solvability.

Proof. If <p is a global Ck-solution of Equation (2) and 'Pet is a local '' Ck-solution of its restriction to Ua then the difference 1/Ja = <p- 'Pet satisfies the homogeneous equation L1/Ja(x) = 0, x E Uet. Since 'Pf3 - 'Pet = ¢13 -1/Jet, we obtain [I] = 0.

Conversely, let [!] = 0. Then for every local Ck-solution 'Pet of (2) we have

'Pf3(x) - 'Pet(x) = 7/J13(x) - 1/Jet(x), X E Ua n U13,

where 1/Jet satisfy the same homogeneous equation on Ua and hence the function

cp(x) = 'Pet(x) - 1/Jet(x), X E Uet,

is a well - defined Ck-solution of (2). Let now an equivalence class c E H 1(k, {Uet}, L) be given and let

a cocycle {Petf3(x)} be a representative of c. In order to prove the second statement of Theorem 4 it is sufficient to find Ck-functions 'Pet(x), x E Uet, such that

(14) Paf3(X) = 'Pf3(x) - 'Pa(x), X E Uet n U13.


Equations (14) compose an one-dimensional version of the Cousin Problem. It is well known ( cf., [6]) that it has solutions in classes of smooth and analytic functions. If { 'Pa} are such solutions, then

f'(X) = Lrpa(x), X E Ua,

is a well-defined Ck-function belonging to C(k, {Ua}, L), and bJ = c. 0 To give a more detailed description of a connection between local and

global solvability we endow the space of cocycles

Z(k,{Ua},L) c II Ck(Ua n U/J;C) a,/3

with the topology induced by the direct product. THEOREM 5. Let the covering {Ua} be such that for every a the restric-

tion

L : Ck(Ua, C) --+ Ck(Ua, C)

is normally solvable. Then L is normally solvable in Ck (M, iC) if and only if the space B(k, {Ua}, L) is closed in Z(k, {Ua}, L).

Proof Let B(k, {Ua}, L) be closed, I'm E Im L and I'm --+ 'Y· Since 'Ymlua --+ 'Yiua and Lis normally solvable in Ck(Ua, C), there exists a sequence { 'Pa,m} E Ck(Ua, C) such that Lrpa,m(x) = 'Ym(x), x E Ua and 'Pa,m --+ 'Pa with Lrpa(x) = f'(x), x E Ua. If cocycles Pm = {Pa/3,m} and p = {Pa/3} are defined by

Pa/3,m(x) = 'P!3,m(x)- 'Pa,m(x), PaiJ(X) = 'P!3(x)- 'Pa(x), X E UanUiJ,

then Pm --+ p. According to Theorem 3 the assumption I'm E Im L implies that all cocycles Pm are equivalent to zero and hence belong to B(k, {Ua}, L). Since the latter space is closed, p E B(k, {U/3}, L) as well and again using Theorem 3 we conclude that 1 E Im L.

Conversely, let us assume Im L to be closed. Following [7] introduce the space of 0-cocycles

Z 0 (k, {Ua}) =II Ck(Ua, C), a

and the space of 1-cocycles

Z1 (k, {U,}) c II Ck(Ua n UiJ, q a,/3

formed by all collections of functions

P ={Pop}, Pof3 E Ck(Ua n Up, C),

satisfying the conditions (13) but not necessarily satisfying the homogeneous equation. It is known that the coboundary operator

8: Z 0 (k, {Ua}) --+ Z 1(k, {U,})

defined by

(8c) 0 f3(X) = 'Pf3(x)- cp0 (x), X E Ua n Ufj, C = {cp,},

is surjective [7]. Let now a sequence Pm = {Paf3,m} E B(k,{U,},L) converge top=

{Paf3} E Z(k, {U,}, L). Let us show that p E B(k, {U,}, L). Since 8 is a surjective operator, there exist functions cp, E Ck(U,, C) such

that p,p(x) = 'Pf3(X)- cp,(x), X E U, n Up. Similarly, PaP,m(x) = cpp,m(x) -'Pa,m(x), x E U0 nUp, where 'Pa,m E Ck(U", C). In addition, since 8 is normally solvable, we can assume that 'Pa,m converges to cp, in Ck(Uo, C) for every a.

Th~ local identities

'}'(X) = Lcp0 (x), 'Ym(x) = Lcp0 ,m(x), X E U0 ,

define the global functions 1' and 'I'm from the space Ck (M, C) and 'I'm --+ 1'· Since Pm E B(k, {Uo}, L), Theorem 3 implies 'I'm E Im L. According to our assumption Im L is a closed subspace and therefore 1' Elm L as well. Again applying Theorem 3 we obtain p =E B(k, {U0 }, L). 0

If a covering of M contains a finite number of maps U" and the spaces KerLiu. are finite-dimensional for all a's, then the space B(k,{Ua},L) is finite-dimensional as well. Hence it is closed and we arrive at the following conclusion.

CoROLLARY 2. If a covering {Ua} contains a finite number of elements and is such that the restricted operators

(15) L: Ck(U", C) --+ Ck(U", C)

are Fredholm for every a, then the operator (1) is normally solvable in Ck(M, C) and dim Ker L < oo.

However Coker£ may be an infinite-dimensional subspace in which case L is not Fredholm operator.

'

·~

~~

To illustrate the como!ogical method Jet us consider the equation m

(16) L aJ'p(FJ(x)) = 'Y(x), X E JR, J~l

with constant coefficients. We assume that the mappings FJ : lR -+ lR commute and have a common set of fixed points z1, ... , Zq which are all hyperbolic:

(17) Fj(z,) = AJ,<> f. 1, AJ,<> f. Ai,<>• j f. i.

Consider the open covering q

JR = U Ua a=l

with

ul = (-oo,z2); UQ = (Za-J,Za+Il, a= 2, ... ,q-1; Uq = (zq-J,OO).

THEOREM 6. Operator (16) is normally solvable in C""(JR, C) and dim Ker L < oo. In addition, dim H 1(oo, {Ua}, L) = oo.

Proof. First let us prove that the restrictions (15) are Fredholm operators. It is well known [8] that, because the hyperbolicity conditions (17), for every a there exists a C""-diffeomorphisms <l?" : Ua -+ lR such that . 1,,t. Hence they are linear mappings themselves, i.e., <PaFJ<P;;1(t) = AJ,at. The substitution x = <P;; 1(t) in (16) reduces the latter equation to the equivalent form

m

L aj"l/;(Ajat) = 'Y(t), t E JR. J~l

The Fredholm property of the restrictions now follows from Theorem 5.14 in [3] and the first statement of Theorem 6 follows from Corollary 2. To prove the remaining part of Theorem 6 we note that according to the same Fredholm property the space B(oo, {Uo}, L) is finite-dimensional. Therefore it is sufficient to check that dim Z 1(oo, {Ua}, L) = oo.

For a proof we note that the change of variable t =In <l? 1 (x), x E U1 n U2 ,

reduces the homogeneous equation (16) to a difference equation m

L ajB(t +In >.1t) = 0, t E JR. J~!


It is well known that the space of its smooth solutions is infinite-dimensional. If O(t) is a C 00-solution, then we define the cocycle p = {Pa/3} E Z 1(oo, {Ua}, L) by the relations

Pai3(x) = -P;3a(x) = { B(lnqh(x)), 0,

if a = 1 and ,8 = 2 otherwise.

The space of all such cocycles is infinite-dimensional which completes the proof. 0

Let us note in conclusion that, as we saw in Section 3, the local solvability of Equation (16) at every fixed point, i.e., inclusion ry E :F (oo, {Ua},L), is equivalent to its formal solvability at every such point. It follows from Theorem 6 that infinitely many additional conditions are required for the global solvability.

We are thankful to L.Berezanskii for useful discussions and remarks.

REFERENCES

[1] Belitskii G., and Tkachenko, V. Solvability of linear difference equations in smooth and real-analytic vector functions of several variables, Integ. Eq. Oper. Theory, 18 (1994), 123-129.

[2] J.Morawiec, and J.Walorski. On the existence of smooth solutions of linear functional equations, Integr. Eq. Oper. Theory, 39 (2001), 222-228.

[3] Belitskii G., and Tkachenko, V. One-dimensional functional equations, Birkhiiser, (2003), 206 pp.

[4] Belitskii G., Functional equations and conjugacy of diffeomorphisms of a finitesmoothness class, Funct. Anal. and Appl., 7, 4, (1973), pp.17-28.

[5] Belitskii G., Functional equations and conjugacy of local diffeomorphisms, Math. Sbornik, 91, 4 (1973), 565-579.

[6] Hormander L. Introduction to complex analysis in several variables, D. Van Nostrand Comp., (1966), 208 pp.

[7] Godement R. Topologie algebrique et tMorie de fisceaux, Hermann, (1958). [8] Kuchma M. Functional equations in a single variable, Warszawa (1968), 385 pp. [9] Kuchma M., Choczewski B., and Ger R. Iterative functional equations, Cambridge

University Press, (1990), 552 pp.

"

VOLUME 12 2005, NO 1-2 PP. 95- 107

ON PERSISTENCE OF A DELAY DIFFERENTIAL EQUATION WITH POSITIVE AND NEGATIVE

COEFFICIENTS

L. BEREZANSKY ' AND E. BRAVERMAN t

Abstract. In this paper we consider the equation

x(t) = c(t)x(t) a(t)x(g(t)), t 2: 0,

where c(t) 2: a(t) 2: 0, g(t) ~ t. We prove that under some conditions every positive solution of this equation has the following property

0 < liminf x(t) < limsupx(t) < oo. t-+oo t-}oo

Impulsive and nonlinear equations with positive and negative coefficients

x(t) = c(t)x(t) f(t, x(g(t))), x(r/) = I;(x(r;)), lim r; = oo, J->00

are also considered.

Dedicated to A natolii Dmitrievich Myshkis on the occasion of his jubilee

Key Words. Delay equations, positive and negative coefficients, nonoscillation, persistent solutions, impulsive equations

AMS(MOS) subject classification. 34Kll, 34K25, 34K45

• Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel. The research was partially supported by the Israeli Ministry of Absorption

t Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, T2N 1N4, Canada. The research was partially supported by NSER.C and AIF Research Grants

95

96 L. BEREZANSKY AND E. BRAVERMAN

1. Introduction. One of the first monographs on delay differential equations [1] was written in 1951 by A.D. Myshkis. This book had a significant influence on all further development of the theory of functional differential equations. Together with other classes of equations, in [1] and the next monograph [2] A.D. Myshkis considered a class of so-called unstable type linear functional differential equations. In particular, this class includes a linear differential equation with several delays and positive coefficients:

m

(1) x(t) = L ak(t)x(gk(t)), k=!

where ak(t) :;:>: 0, gk(t) :::; t. One of the main results obtained in [1, 2] for equation (1) is the following.

If an initial function is positive on some interval, then for the solution x(t) of the equation we have:

(2) N = liminfx(t) > 0. t-+oo

The property (2) (together with the boundedness of solution x(t)) is called persistence; it is one of important properties for differential equations of Mathematical Biology. For instance, for equations of population dynamics it means guaranteed non-extinction of the population; moreover, the size of the population does not fall beyond the lower bound N.

If we assume in Eq. (1) a;(t) :::; 0 (a stable type equation), then [3] any positive solution under rather natural constrains tends to zero: lim x(t) = 0,

t-+oo which means that the zero solution is an attractor for all solutions of the equation.

We consider here the case when Eq. (1) contains both negative and positive coefficients, where the positive term "prevails" over the negative one. Naturally this extends a class of unstable type equations which was considered by A.D. Myshkis. In this paper we obtain persistence conditions for linear differential equations of unstable type with positive and negative coefficients.

Stable type equations with positive and negative coefficients were studied in [4, 5].

We also consider some nonlinear delay differential equations and delay differential equations with impulses.

It is to be noted that it was A.D. Myshkis and V.D. Milman who first introduced the notion of an impulsive differential equation in 1960 [6]. Since then several monographs and more than 1000 papers have been published on impulsive equations.

ON PERSISTENCE OF A DELAY EQUATION 97

2. Preliminaries. Consider the scalar linear delay differential equation

n

(3) x(t) + 'Lct(t)x(gt(t)) = f(t), t 2:0, 1=1

with the initial condition

(4) x(t) = rp(t), t < 0, x(O) = Xo,

under the following assumptions (al) c1(t) is a locally essentially bounded on [0, oo) function; (a2) g1(t) is a Lebesgue measurable function, g1(t) :'0 t,

limsupg1(t) = oo; I -too

(a3) <p: ( -oo, 0) --+ R is a Borel measurable bounded function.

DEFINITION 1. A solution X(t, s) of the problem

n

x(t) + 'Lc1(t)x(g1(t)) = o, t 2: s, 1=1

x(t) = 0, t < s, x(s) = 1,

is called a fundamental function of {3). DEFINITION 2. We say that a function is nonoscillatory if it is either

eventually positive or eventually negative. We say that a function x(t) is persistent if

(5) 0 < liminfx(t) < limsupx(t) < oo. t-too t--too

LEMMA 1. {see [1}) Suppose for {3) conditions {a1)-{a3) hold. Then for the solution of {3 ), {4) we have the following representation

{t n t (6) x(t) = X(t,O)x0 - lo X(t,s) t;c1(s)rp(g1(s))ds + fo X(t,s)f(s)ds,

where <p(t) = 0, t 2: 0. LEMMA 2. {see [3}} Suppose {a1)-{a3) hold, ct(t) 2: 0. If there exists a

nonnegative solution of the inequality

(7) u(t) 2: t c1(t) exp {l u(s)ds}, t 2: 0; u(t) = 0, t < 0, 1=1 91(1)

and

(8) 0:; cp(t) :; xo,

then the solution of initial value problem {3), (4), with f(t) = 0, is positive. Consider also the following linear delay equation with positive and neg

ative coefficients

(9) x(t) + a(t)x(g(t))- c(t)x(t) = 0, t 2': 0,

and the corresponding linear inequality

(10) i;(t) + a(t)y(g(t))- c(t)x(t):; 0, t 2': 0.

LEMMA 3. (see [4, 5]) Suppose {a1}-(a3} hold,

(11) a(t) 2': c(t) 2': 0, fo00

[a(s)- c(s)]ds = oo,

and

(12) limsupc(t)[t- g(t)] < 1. t-+oo

Then 1} lfy(t) is a positive solution of (10} fort 2': t0 2': 0, then

y(t) :; x(t), t 2': t0 2': 0, where x(t) is a solution of (9) and x(t) = y(t), t:; t0 .

2} For every nonoscillatory solution of (9} we have lim x(t) = 0. t-+OO

3) There exists a positive solution of (9). Consider now the following linear equation with one delay

(13) x(t) = a(t)[x(t)- x(g(t))], t:::: 0.

LEMMA 4. [8] Suppose for parameters of Eq. (13) hypotheses (a1)-(a3} hold, g ( t) is an increasing function,

a(t) :::: 0, limsup(t- g(t)) < oo, lim sup rt a(s)ds < 1. t-+oo t-+oo J g(t)

Then every solution of ( 13) has a finite limit. In particular, for the fundamental function X(t,s) of (13} we have

sup IX(t,s)i < oo. t?.:s?O

!i

3. Linear Equations. In this section we consider the following equation with one delay

(14) x(t) = c(t)x(t)- a(t)x(g(t)),

and a corresponding differential inequality

(15) y(t) :0: c(t)y(t)- a(t)y(g(t)),

where for parameters of (14) conditions (a1)-(a3) hold. THEOREM 1. 1) Suppose (a1)-(a3) hold, c(t) :0: a(t) 2 0. Then there

exists an eventually positive solution of Eq. (14) and for any positive solution x(t) of (14) and for the solution y(t) of (15), with y(t) = x(t), t ::; 0, we have y(t) 2 x(t) > 0. Suppose for initial conditions (4) inequality (8) holds. Then for the solution of (14), (4) we have x(t) :0: x0 > 0. In particular,

(16) lim inf x(t) :0: x(O) > 0. t->oo

2) Suppose conditions (11), (12) hold. Then (14) has no persistent solutions (every positive solution tends to zero}.

Proof. 1) In the space £ 00 [0, T] of all essentially bounded on [0, T] functions with a usual sup-norm consider the following operator equation

(17) w(t) = c(t)- a(t) exp {- ft w(s)ds}, t :0: 0; w(t) = 0, t < 0. }g(t)

Denote the sequence: wJ(t) = c(t) - a(t),

(18) Wn(t) = c(t)- a(t) exp {- {t Wn-l(s)ds}, Wn(t) = 0, t < 0. }g(t)

Inequality w0 (t) :0: 0 implies w1(t) :0: w0 (t). By induction we can prove Wn(t) :0: Wn-l(t) 2 wo(t) = c(t)- a(t) :0:0 and Wn(t)::; c(t).

There exists a pointwise limit of the nondecreasing nonnegative sequence Wn ( t). Let w ( t) = lim Wn ( t), then by the Lebesgue Convergence Theorem

n->oo u(t) is locally integrable and

lim (Fwn) (t) = (Fw)(t), n->oo

where operator F is denoted by the right-hand side of (17). Thus (18) implies that w is a nonnegative solution of Eq. (17).

Hence function x defined by the equality

(19) x(t) = ef;w(s)ds, x(t) = O,t < 0,

is a positive solution of equation (14). Since x(O) = 1 then x(t) = X(t, 0), where X(t, s) is the fundamental function of Eq. (14). We have X(t, 0) > 0. Similarly we can prove that X (t, s) > 0, t ;:::: s ;:::: 0.

For a solution y( t) of inequality (15) we have

y(t) = c(t)y(t)- a(t)y(g(t)) + f(t),

where f(t) ;:::: 0. Then by Lemma 1

y(t) = x(t) +fat X(t, s)j(s)ds,

where x(t) is a solution of (14) with x(t) = y(t), t ::; 0. If x(t) > 0, then y(t) ;:::: x(t) > 0.

Suppose now that x(t) is a solution of (14) and for initial conditions inequality (8) holds. By substituting

x(t) = exp {l c(s)ds} z(t), x(t) = z(t), t::; 0,

into Eq. (14) we obtain that z(t) is a solution of the linear equation with a nonnegative coefficient

i(t) + a(t) exp {- rt c(s)ds} z(g(t)) = 0, }g(t)

and with the same initial conditions ( 4) for which inequality (8) holds. Inequality (7) for this equation has the form

u(t);:::: a(t) exp {- r c(s)ds} exp { r u(s)ds}; u(t) = 0, t:::: o. }g(t) }g(t)

A nonnegative function u(t) = a(t) is a solution of this inequality. Lemma 2 implies z(t) > 0. Hence also x(t) > 0. Similarly to the above proof (which is given for the fundamental function X(t, s)), x(t) can be presented in the form

x(t) = x(O) exp {l w(s)ds}, t > 0,

where w(t) is a nonnegative solution of Eq. (17). Hence x(t) :::>: x(O), t > 0, and inequality (16) holds, which completes the proof.

The statement of 2) follows from Lemma 3. 0 Let us demonstrate the sharpness of conditions of Theorem 1. EXAMPLE 1. If the condition c(t) :::>: a(t) does not hold, then a solution

with initial conditions satisfying {8} can become negative. Let (!?(t) = 1, x0 = 1.1, g(t) = t- 1, a(t) = 2, c(t) = 1. Then the solution of the equation

x(t) = x(t)- 2x(t- 1) = x(t) - 2, 0::; t ::; 1.

is x(t) = -0.9et + 2 for t E [0, 1], so x(1) = -0.9e + 2 < 0. Hence the solution becomes negative at x = ln(2/0.9) f::; 0.7985.

EXAMPLE 2. Similarly, if c(t) :::>: a(t) holds but {8} is not satisfied, then the solution can become negative. For example, let (!?(t) = 10, x0 = 1, g(t) = t- 1, a(t) = 0.5, c(t) = 1. Then the solution of the equation

x(t) = x(t) - 0.5x(t- 1) = x(t) - 5, 0::; t::; 1,

with x(O) = 1, is x(t) = 5 - 4et, so x(1) = 5- 4e < 0 and the solution becomes negative at x = ln(l.25) f::; 0.223.

In addition to the positiveness of solutions Theorem 1 claims that as far as (8) is satisfied then the solution does not tend to zero (moreover, it is not less than the initial value). However if (8) does not hold then a positive solution can become less than the initial value and can tend to zero as Example 3 demonstrates.

t2 - 1 ExAMPLE 3. The equation x(t) = x(t)- -t-2 -x(t- 1) has a solution

1 x = t as can be easily checked. We can begin anywhere at t > 1 including

the previous part of the solution as prehistory. The solution tends to zero; {16) does not hold since {8) is not satisfied for any initial point.

THEOREM 2. Suppose {a1}-(a3} hold,

(20) g(t) is an increasing function, limsup(t- g(t)) < oo, t-+oo

(21) a(t):::: 0, lim sup rt a(s)ds < 1, {"' lc(s)- a(s)ids < oo. Hoo }g(t) Jo

Then every solution of {14) is bounded. Proof Rewrite Eq. (14) in the form:

x(t) = a(t)(x(t)- x(g(t))) + [c(t)- a(t)]x(t).

Denote by Y ( t, s) the fundamental function of the following equation

y(t) = a(t)[y(t)- y(g(t))].

Lemma 1 implies that for the solution of (14), (4) we have

x(t) = Y(t, O)x(O) + l Y(t, s)[c(s)- a(s)]x(s)ds

Conditions (20),(21) and Lemma 4 imply

Hence

Then

C = sup IY(t,s)l < oo. t2:s?:O

lx(t)l ::; Clx(O)I + C llc(s)- a(s)llx(s)lds.

lx(t)l ::; Clx(O)I exp { C lic(s)- a(s)ids}

< Clx(O)I exp { C fo'"" lc(s)- a(s)ids} < oo.

We have supt?:O lx(t)l < oo, which completes the proof. 0 REMARK 1. It is well known that if

f" la(s)lds < oo, f" lc(s)lds < oo,

then every solution of (14} has a finite limit. COROLLARY 1. Suppose the conditions of Theorem 2 hold and c(t) 2

a(t) 2 0. Then every solution of (14}, (4) with initial conditions satisfying ( 8), is persistent.

4. Impulsive and Nonlinear Equations. Now let us assume that the equation (14) is subject to linear impulsive perturbations

(22) x(rn = bjx(rj), j = 1, 2, .. 0 ,

and the following conditions hold: (a4) t0 < r 1 < r2 < ... < rk < ... satisfy )im ri = oo;

J->00

(a5) bi > O,j = 1,2, ....

3

• •

If we make a substitution [9, 10, 11, 12]

(23) z(t) = II btx(t), o::;-rj:St

then the function z(t) is an absolutely continuous solution of the equation

(24)

where

(25)

z(t) = c(t)z(t)- a1(t)z(g(t)),

at(t)=a(t) II bt. g(t)5,r;::;t

We also consider the following impulsive inequalities

(26) X(Tfj ~ bjX(Tj), j = 1,2, ... ,

and

(27) X(Tfj 2': bjX(Tj), j = 1, 2, ... ,

Thus, Theorem 1 and Theorem 2 applied to the continuous solution z(t), as well as the representation

x(t) = II bjz(t), O:STJ:St

imply the following results. THEOREM 3. Suppose (a1}-(a5) hold,

n

c(t) > a1(t), liminf II b1 > 0. - n-too

j:l

Then there exists an eventually positive solution of (14), (22} and for any positive solution x(t) of (14},(22) and for the solution y(t) of (15),(27), with y(t) = x(t), t ~ 0, we have y(t) :::>: x(t) > 0. Suppose for initial conditions (4) inequality (8) holds. Then for each solution of (14), (4),(22) we have

lim inf x(t) > 0. !--too

Suppose conditions (11}, (12} hold, where a(t) is replaced by a1 (t), and n

lim sup II b1 < oo. Then ( 14), ( 22) has no persistent solutions (every positive n-too j=l

solution tends to zero).

THEOREM 4. Suppose (a1)-(a5), (20) and (21} hold, where a(t) is re-n

placed by a1(t), and lim sup II bi < oo. Then every solution of impulsive n-too j=l

equation (14), (22) is bounded. CoROLLARY 2. Suppose {a1}-(a5), (20) and (21) hold, where a(t) is

replaced by a 1 ( t),

n n c(t) 2: a1(t), lim inf II bi > 0,

n-->oo lim sup II bi < oo.

n-too j=l j=!

Then every solution of impulsive equation (14), (22}, (4), with initial conditions satisfying (8}, is persistent.

Further, let us assume that the negative term in (14) is nonlinear

(28) x(t) = c(t)x(t)- f(t, x(g(t))),

where the following condition holds: (a6) f(t, u) satisfies Caratheodory conditions: it is Lebesgue measurable in the first argument and continuous in the second one.

We will.assume that the initial value problem (28)-(4) has a unique global solution x(t), t 2: 0.

THEOREM 5. Suppose (a1}-(a3}, (a6} hold. 1} If

(29) f(t, u) 2: a(t)u, u > 0,

and (11),(12} are satisfied then (28} has no persistent solutions (every positive solution tends to zero). 2) If

(30) f(t, u) ::::; c(t)u, u > 0,

and c(t) 2: 0 then any solution of (28), (4), with the initial conditions satisfying {8), has property {16}.

Proof 1) Due to (29) the solution of (28) is also a solution of inequality (10). Thus by Lemma 3 any positive solution tends to zero. 2) Similarly, under (30) the solution of (28) is also a solution of inequality (15), with a(t) = c(t). Thus by Theorem 1 every solution with the initial conditions satisfying (8) has property (16). 0

ON PERSISTENCE OF A DELAY EQUATION

EXAMPLE 4. The equation

x(t) = x(t) - 2x(t- 0.2)(x(t- 0.2) + 1) 1 + Jx(t- 0.2)

has no persistent solutions. As can be easily checked,

2u( u + 1) > 1.6u ;::: u, u 2:': 0, 1+fo -

105

thus hypotheses of Theorem 5, part 1}, are satisfied and any positive solution of this equation tends to zero.

EXAMPLE 5. Any solution of the equation

. x(t-2) x(t) = x(t) - 1 + 0.5x2(t- 2)

with the initial conditions satisfying {8} has property {16}, since the the hypotheses of Theorem 5, part 2}, are satisfied due to the inequality

u -< 1 + u2 '- u, u ;::: 0.

THEOREM 6. Suppose {a1}-{a3}, {a6}, {20}, {21} and {29} hold. Then every positive solution of {28} is bounded.

Proof. Suppose x(t) is a solution of (28). Inequality (29) implies

x(t) = c(t)x(t)- a(t)x(g(t))- h(t),

where h(t) 2: 0. This equation can be rewritten in the form

x(t) = a(t)(x(t)- x(g(t))) + [c(t)- a(t)]x(t) h(t).

Denote by Y ( t, s) the fundamental function of the following equation

y(t) = a(t)[y(t)- y(g(t))].

By Lemma 1 for the solution of (14), (4) we have

x(t) = Y(t,O)x(O) + tY(t,s)[c(s)- a(s)]x(s)ds- tY(t,s)h(s)ds.

The proof of Theorem 1 implies Y(t, s) > 0. Thus

0 < x(t) ::; Y(t, O)x(O) + t Y(t, s)[c(s)- a(s)]x(s)ds.

Similar to the proof of Theorem 2 we have

0 < x(t)::; Cx(O) exp { C lfc(s)- a(s)fds}

::; Cx(O) exp { C fooo fc(s)- a(s)fds} < oo,

where

C =sup Y(t,s) < oo. t,s~O

Finally, sup x(t) < oo, which completes the proof. 0 t>O

COROLLARY 3. Suppose (a1)-(a3}, (a6), (20} and (21) hold,

0 ::; a(t)u::; f(t, u) ::; c(t)u, u > 0.

Then every solution of {28}, (4), with initial conditions satisfying (8}, is persistent.

We also introduce nonlinear impulsive conditions for (28)

(31) x(rf) = Ij(x(rj)), j = 1, 2, ... ,

and will assume that the initial value problem (28), (31), (4) has a unique global solution x(t), t ~ 0.

THEOREM 7. 1} Suppose there exist such a(t), bj that (a1)-(a6) hold and

(32) f(t, u) ~ a(t)u, Ij(u) ~ bju, u ~ 0.

If (11} and(12) hold, where a(t) is replaced by a1 (t) (which is denoted by n

(25)), lim sup II bj < oo, then (28), {31} has no persistent solutions (every n-too j=l

positive solution tends to zero). 2} Suppose there exists a sequence bj, such that (a1}-(a6} hold,

n

c(t) ~ a1 (t) ~ 0, ~~~~f II bj > 0 and j=l

(33) f(t, u) ::; c(t)u, Ij(u) ::; bju, u ~ 0.

Then for any solution of (28}, (31), (4), with the initial conditions satisfying (8), we have lim inf x(t) > 0. t->oo

~;


REFERENCES

[1] A. D. Myshkis, Linear Differential Equations with Retarded Argument. Gosudarstv. Izdat. Tehn.-Tear. Lit., Moscow-Leningrad, 1951. 255 pp.

[2] A. D. Myshkis, Linear Differential Equations with Retarded Argument. Second edition, Nauka, Moscow, 1972. 352 pp.

[3] I. Gyi.iri and G. Ladas, Oscillation Theory of Delay Differential Equations. 1991, Clarendon Press, Oxford, 1991.

[4] L. Berezansky, Y. Domshlak and E. Braverman, On oscillation of a delay differential equation with positive and negative coefficients, J. Math. Anal. Appl. 274 (2002), 81-101.

[5] L. Berezansky and E. Braverman, On Oscillation of equations with positive and negative coefficients and distributed delay II: Applications, Electron. J. Diff. Eqns. 2003 (2003), No. 47, 1-25.

[6] V. D. Milman and A. D. Myshkis, On the stability of motion in the presence of impulses (Russian), Sibirsk. Mat. Z. 1 (1960) 233-237.

(7] N. Azbelev, V. Maksimov and L. Rakhmatullina, Introduction to the Theory of Linear Functional Differential Equations. Advanced Series in Mathematical Science and Engineering, 3. World Federation Publishers Company, Atlanta, GA, 1995.

[8] J. Dibllk, A criterion for convergence of solutions of homogeneous delay linear differential equations, Annales Polonici Mathematici, LXXII.2 (1999), 115-130.

[9] L. Berezansky and E. Braverman, Oscillation of a Linear Delay Impulsive Differential Equation, Comm. on Appl. Nonlin. Anal., 3 (1996), No.1, 61-77.

[10] J. Yan and C. Kou, Oscillation of solutions of impulsive delay differential equations, J. Math. Anal. Appl. 254 (2001), No. 2, 358-370.

[11 J J. Yan and A. Zhao, Oscillation and stability of linear impulsive delay differential equations, J. Math. Anal. Appl. 227 (1998), No. 1, 187-194.

[12] L. Berezansky and E. Braverman, Oscillation and other properties of linear impulsive and nonimpulsive delay equations, Applied Mathematics Letters 16 (2003), 1025-1030.


VOLUME 12

2005, NO 1-2 PP. 109-118

STABILIZING EFFECT OF NONLINEAR DAMPING FOR A SYSTEM WITH FOLLOWER FORCE

A. S. BRATUS' AND A. S. NOVOZHILOVI

Abstract. Addition of nonlinear damping for a system with "follower" force is shown to eliminate "jump" in critical value of the force magnitude between the system without damping and that with vanishingly small linear damping. Increasing the force magnitude leads first to "soft" self-excitation (it is shown analytically), so that a stable limit cycle appears. As numerical studies show further force increase leads to a second transformation whereby "sharp" self-excitation makes the system's trajectories go to infinity. The maximal range of the force magnitude which corresponds to the existence of a stable limit cycle in the vicinity of the origin is close to the magnitude of jump in the critical force magnitude of the linear system which appears with addition of the linear viscoelastic damping.

Key Words. Hopf bifurcation, Follower force

AMS(MOS) subject classification. 37, 70

Introduction. The theory of stability of elastic systems mostly deals with conservative systems, i.e. with the systems for which the theorem of energy conservation holds. However there are problems of practical importance where nonconservative elastic systems should be considered. These problems are connected, e.g., with the torsion of shafts, pipes conducting media, panels exposed to the flow of fluids or gases, etc. Nicolai [1] and Ziegler [2, 3] were the first to attract the attention of scientists and engineers to the nontrivial behaviour of such systems. A significant contribution to their study

' Russia, 127994, Moscow, Obraztsova, 15, Moscow State University of Communications Means, Department of" Applied Mathematics-!"

I National Institutes of Health, National Center for Biotechnology Information, 8600 Rockville Pike, Bethesda, MD, 20894, Bldg. 38A, Rm 8N811H

109

110 A. S. BRATUS AND A. S. NOVOZHILOV

was made by Bolotin [4], Ziegler [5] and Leipholz [6, 7]. These books are the most well known in this field. On example of sufficiently simple mechanical system Ziegler discovered [3] that any small viscous damping can give rise to instability in nonconservative elastic system where the magnitude of critical "follower" force change by "jump" (Ziegler's paradox). Various sides of this phenomenon were studied in [8]-[15]. All these investigations are devoted to linear nonconservative systems with linear damping forces. Mathematical aspects of Ziegler's paradox were studied in [16], [17], [18], [19]. In this investigation an important part is played by adjoint problems (see also [20, 21]) and spectrum perturbation theory for nonselfadjoint operators and matrices [22]. It should be noted that all the interesting cases are connected with appearance of multiple eigenvalues (frequencies). The main goal of this paper is to show on the example of the original Ziegler system that addition of nonlinear damping can eliminate "jump" of the critical value of the force magnitude between the system without damping and that with vanishingly small linear damping. Increase of the force magnitude leads first to "soft" self-excitation (it is shown analytically), so that a stable limit cycle appears. As numerical studies show the further force increase leads to a second transformation whereby "sharp" self-excitation makes trajectories of the system go to infinity. The maximal range of the force magnitude that corresponds to the existence of a stable limit cycle in the vicinity of the origin is close to the magnitude of jump in the critical force magnitude of the linear system which appears with addition of the linear viscoelastic damping.

1. Ziegler's Paradox. In 1952 Ziegler [3] considered the following twodegree-of-freedom system: two-bar linkage with each weightless rigid bar being of length I and attached masses m and 2m (see Figure 1). The lower mass is loaded by a follower force P. The torsional springs in the hinges exhibit viscoelastic properties providing the restoring moments -c<p1 - bciJI and -c(<p2- 'PI)- b(cp2- <P1).

(1)

The governing set of equations in the absence of gravity is then

3ml2<f>l + mi2<P2 + b(2cpl- <P2) + (2c- Pl)<p1 +(PI- c)<p2 = 0;

mz2<h + mi2'P2 + b( <P2 - rpJ) + c( '1'2 - 'PI) = 0

STABILIZING EFFECT

FIG. 1. Mechanical system with two-degree of freedom considered by Ziegler {3}

or equivalently

(2)

where

±1 = X3, ±2 = X4j

X3 = (p 3)xl + (2- p)x2- c(3x3- 2x4), ±4 = (5- p)x1 + (p- 4)x2- c(4x4- 5x3)

x1(r) = 'Pl(t), x2(r) = 'P2(t),x3 = :i1, X4 = x2,

-t ~ _ b _ Pl T- v 2ml 2 ' c- C1 p- C·

111

The eigenvalues of the matrix of the system (2) are the roots of the characteristic equation

(3) ,\4 + 7c>.3 + (2c2 - 2p + 7),\2 + 4c>. + 2 = 0.

The system (2) has a single equilibrium state- the origin. Condition Re A; < 0 is necessary and sufficient for its asymptotic stability. Using the RouthHurvitz criterion this condition may be reduced to the following inequalities for the parameters: 7c > 0, p < i~ + c2 ,p < ~~ + c2

, 2 > 0. Combining the presented inequalities we have the only condition p < ~~ + c2

. Therefore, for vanishingly small values of the damping parameter the limiting value of the critical force is if, = ~~ ~ 1.464.

On the other hand, critical force in the system (2) with c = 0 can be shown to be p, = ~ - v'2 ~ 2.086. Thus, if, # p,, and this is actually the


Ziegler's paradox [3]. This phenomenon is observed only in the nonconservative systems.

Systems of the kind x + cG(p):i: + A(p)x = 0 had been studied in [16], where x = x(t) is a vector of dimension n 2: 2 ,A(p), G(p) - real analytical matrix functions of p, c > 0 - small parameter. It had been established, that if for the case c = 0 there exists value p0 with purely imaginary double eigenvalues and only a single corresponding eigenvector (the case of a simple Jordan cell) then the set of matrices G(p) for which lim p, = p0 for c--+ 0 is a set of null measure in the space of all matrices. To be more precise the set of these matrices is described by the following equality

(4) (G(Po)uo, vo) = 0,

where u0 is an eigenvector which corresponds to double eigenvalue Ao, i.e. A(p0 )u0 = A0u0 , v0 - an eigenvector of adjoint problem AT(Po)v0 = ..\0v0 . For the Ziegler system (2) the matrix A(p) has the form

A(p)=(3-p p-2) p-5 4-p

For p0 = 7-~0 the corresponding eigenvalue ..\0 = J2 is double with the only eigenvector u0 = (3- 2VZ, 1). The eigenvector of the adjoint problem is vo = ( -3- 2VZ, 1). If the set of damping matrices

G(p) = ( 9u 912 ) 921 922

is considered the equality ( 4) is possible if and only if

922 - 9n + (3- 2vz)921 - (3 + 2vz)912 = o. It is easy to verify that in the case of the Ziegler system (1) this condition is not valid. In the space R4 of elements of matrix G(p) the last equality determines a hyperplane of dimension 3. Therefore this set has a null measure in R4. In other words the general case in this situation is destabilizing effect of damping.

2. System with nonlinear damping. Consider now the following system

X1 = Xa, X2 = X4j

(5) ±a= (p- 3)x1 + (2- p)x2- c(3xa(1- x¥)- 2x4(1- x~)), :i:4 = (5- p)x1 + (p- 4)x2 - c(4x4(1- x¥)- 5xa(1- xm

STABILIZING EFFECT 113

as obtained from (2) by adding nonlinear terms, which describe influence of the response amplitudes on damping properties of the viscoelastic hinges. (It may be noted that similar addition of nonlinearity in case of a dynamically unstable single-degree-of-freedom system leads to the well-known van der Pol equation).

The system (5) will be studied locally, in the vicinity of its origin which is its only fixed point. Jacobian of the system ( 4) at the equilibrium state precisely coincides with the matrix of the system (2). Therefore the eigenvalues of the Jacobi matrix of the system (5) should be the roots of the characteristic polynomial (3). The latter may be written, for the case of the critical force p = p, = ~~ + c;2 as

57 (1 1) (6) ,\4+7c:.\3+14,\2+4c:,\+2 = (7,\2+4) 7,\2+c:.\+2 =0

Thus, at p = p, two purely imaginary conjugated eigenvalues appear as ,\1,2 = ±(}r)i which do not depend on c:. It can be easily seen that two other eigenvalues have negative real part.

In such a case the Andronov-Hopf bifurcation may appear in the set of nonlinear differential equations [23, 24] implying birth of a limit cycle. As two eigenvalues have negative real part, qualitative behaviour of the trajectories of the system according to the Reduction Principal [23] is completely governed by the restriction of the dynamical system (5) on a central manifold. The central manifold corresponds to the neutraily stable equilibrium state and is attracting in the case involved. To invoke the Andronov-Hopf bifurcation theorem it has to be shown that the eigenvalues cross the imaginary axis with non-zero speed, i.e. d~e>.J i= 0. Using implicit differentiation

p p=p.

of the characteristic function one can get the following expression:

d,\1 Re-dp p=p.

484c: 5488c2 + 1681 > o.

Thus, in the considered case it is sufficient to know just the first Lyapunov coefficient (if it is nonzero) of the normal form oft he dynamical system (5) on a central manifold.

It should be noted that we are scanning the system ( 4) through values of c:, whereas the critical parameter is p(c:). Therefore the first Lyapunov coefficient is a function of c:. It may be calculated [25] as

(7) 1

11 = -Re(p,C(q,q,q)). 2wo

114 A. S. BRATUS AND A. 8. NOVOZHILOV

Here q E C4 is the eigenvector, corresponding to the purely imaginary eigenvalue and p E C 4 is the eigenvector of the transposed Jacobi matrix, corresponding to the conjugate purely imaginary eigenvalue, both vectors being normalized according to the condition (p, q) = 1. The angular brackets denote here the common dot product, w0 is the imaginary part of the (purely imaginary) eigenvalue, C - multilinear vector polynomials in expansions of the nonlinear terms in (5):

4 ff3Fi(~,O)l xjykul; Ci(X, y, u) = El o~iJ~kO~t e~o

J, '

i = 1, ... , 4.

Here F;(~, 0) are the RHSs of the system (5), ~ is a state vector, zero value of argument indicates the critical value of the parameter. The last expression is obtained by projection onto the central manifold with the use of the well-known formula for the first Lyapunov coefficient of two-dimensional dynamical system [24), [25], [26].

Furthermore, if xi, x§ in the system (5) are replaced by Kxf, Kx§ (K > 0) respectively, the result would not change since the factor K doesn't influence the sign of 11 according to (7).

3. Main results and conclusions. Using the formula (7) the following exact analytical expression for 11 (c) may be obtained as

(8) 0'(1568c4 + 1820c2

- 1845)c 11 (c) =

14 (5488c2) + 1681)(28c2 + 81)

For stability analysis only the sign of the Lyapunov coefficient is important (and not its magnitude) as long as it controls either birth of a stable limit cycle (if 11 < 0) or disappearance of the unstable limit cycle (if 11 > 0). These are the cases of so-called "soft" and "sharp" selfexcitation respectively. Analysis of the solution of (8) indicates that if

0 < c < c * = xhlfs;:s-445 ~ 0.806, then 11 (c) < 0, and a stable limit cycle appears, its amplitude being proportional to the square root of the excess of the parameter over its critical value ("soft" self-excitation). Thus, for the "soft" case whilst the linear approximation predicts instability of the equilibrium of the system (5), its trajectories starting in the vicinity of the origin do not go to infinity after the critical force is exceeded, rather self-excited oscillations appear. In case c > c* the self-excitation is of a "sharp" type.

Bifurcation diagram of the system ( 5) is presented in Figure 2. Bifurcations of co-dimension 1 correspond to the curve p.(c) = ~~ + c2

. The upper


boundary of domain D (curve 2 in Figure 2) is obtained numerically. Having taken a fixed point in the space of parameters the technique for searching a stable limit cycle was used [25]. To be more precise the values of c: were fixed and the values of p were changing with small step from small to bigger numbers until the stable limit cycle can be detected. Due to numerical investigation trajectories of the system (5) starting in a small neighbourhood of the origin are localized in the vicinity of the origin ("soft" self-excitation) for values of parameters within the domain and go to infinity otherwise.

It can be seen from Figure 2 that the first Lyapunov coefficient changes its sign at the point with p - which is almost at the critical value of force for the linear system without viscoelastic damping. Besides as numerical experiments show the range of the existence of the stable limit cycle with small c: is close to the magnitude of jump of the critical force in the linear system (2). Increase of the magnitude of the follower force from p, ,:::; 1.46 to fl. ,:::; p,(c:,) is accompanied with stable limit-cycle oscillations. The latter disappears when the upper boundary of is exceeded. On the other hand, if only linear damping terms are retained, the whole range of critical forces between and - which corresponds to the "soft" self-excitation - disappears, and critical force for c: -+ 0 exhibits a jump from p, to p,.

It should be noted that the nonlinear terms almost do not influence instability of the equilibrium for the values of follower force exceeding the critical force for the undamped system (1).

Graphical illustration is presented in the following of the projections of the four-dimensional state space onto subspace x1, x2, x4 . Numerical studies show that the system ( 4) may possess not only a stable limit cycle but - for certain range of parameters- an unstable limit cycle as well. Figure 3 illustrates two numerically generated limit cycles for the case p = 1.9, c: = 0.12. The inner one is stable; whereas the outer is unstable (it had been generated by integrating the system ( 5) in backward time).

Increasing magnitude of the follower force leads to increasing amplitude of the stable limit cycle, and at the value of the force on the upper boundary of the domain D (see Figure 2) the orbit of the stable limit cycle "collides" with that of the unstable one. The attracting manifold in the state space disappears accordingly implying global reconstruction of the phase-space pattern. The trajectory leaving the origin goes to infinity. It should be noted that the assertion in the last paragraph did not been proved analytically and is confirmed only by numerous numerical investigations.

It may be noted, in conclusion, that the system (5) may be regarded as a model for a variety of structures loaded by follower forces and for aeroelastic


FIG. 2. Bifurcation diagram of the system (5). Curve 1 - p = % + E:2

; Curve 2 - the curve obtained numerically that shows the maximal value of p( e) for which a stable limit cycle exists in the vicinity of the origin

p

2.2

i [ j ll --------~---------t---------~--------~-

p,(e,)(2.114 ' r t 1

--------~--------- --------- -------p- (2 .086) _______ .. _________ t_________ --- --------· • I I I

2~--------1--------- ---------~!____ --U----------1 f I II

' ' I I' ' 'D I I I I II I I I II ' [ II

t.e~--------i---------,.------- r--------1+----------! ! I II I t I II

I \ I 11 , , I .I I II

1.sr--------1------- r---------r--------p---------· I l I .I

' I II . I I' ' ' I I t I II

p-,(1.48) i -14~--------~---------L---------~--------~--------. I I I II

I I I .I I I I II I I I '1',(0.708) lle,(O 806) I I J1 k ' ' I II

1.2 0~2 0~4 e o:s o:a l

FIG. 3. Two limit cycles of the system (5). The inner one is stable, the outer one is unstable. The values of the parameters are p = 1.9, E: = 0.12

x,

x;

/


systems prone to flutter.

Thus we have shown (but only partly analytically) that addition of nonlinear damping in the original Ziegler system eliminates "jump" of critical force in the linear system (1).

REFERENCES

[1 J E. Nicolai, Uber die Stabilitat des Gedriickten und Gedrillten Geraden Stabes, Vordrang vor dem Mathemaitsche, April-May, 1927, pp. 31-47.

[2] H. Ziegler, Stabilitatsprobleme bei Geraden Staben und Wellen, Zeitschrift Angew. Mathematische Physik, 2, 1951, pp. 265-289.

[3] H. Ziegler, Die stabilitatskriterien der Elastomechanik, Ingeneer Archive, 20, 1952, pp. 49-56.

[4] V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Oxford, Pergamon Press, 1963.

[5] H. Ziegler, Principles of Structural Stability, Massachusetts, Waltham, 1968. [6] H. Leipholz, Stability of Elastic Systems, Amsterdam, Sithhoff and Noordhoff, 1980. [7] H. Leipholz, Analysis of Nonconservative, Nonholomonic Systems, New-York,

Springer, 1980. [8] G. Herrmann, and I. Jong, On the Destabilizing Effect of Damping in Nonconser

vative Elastic Systems, Journal of Applied Mechanical Transaction, ASME, 32, 1965, pp. 592-597.

[9] S. Nemat-Nasser, and G. Herrman, Some General Considerations Concerning the Destabilizing Effect in Nonconservative Systems, ZAMP, 13, pp. 305-313.

[10] V. Bolotin, and N. Zinzher, Effect on Stability of Elastic System Subjected to Nonconservative Forces, International Journal of Solid Structures, 5(9), 1969, pp.965-989.

[11 J I. Andreichikov, and V. Yudovich, On Stability of Elastic Systems with Small Internal Damping, Mechanical Solids, 2, 1974, pp. 78-87 (in Russian).

[12] G. Denisov, and V. Novikov, On Stability of Elastic Systems with Small Internal Damping, Mechanical Solids, 3, 1978, pp. 41-47 (in Russian).

[13] R. Scheid!, H. Trager, and K. Zeman, Coupled Flutter and Divergence Bifurcation of a Double Pendulum, International Journal of Nonlinear Mechanics, 19, 1983, pp.163-176.

[14] D. Chu, and F.C. Moon, Dynamic Instabilities in Magnetically Levitated Models, Journal of Applied Physics, 54(3), 1983, pp. 1619-1625.

[15] A. Miloslavsky, Stabilizing Influence of Small Damping on Abstract Nonconservative Systems, Advancements of Mathematical Sciences, 41(1), 1986, pp.199-200 (in Russian).

[16] N. Banichuk, A. Bratus, and A. Myshkis, Dynamic Stability of Nonconservative Mechanickal Systems with Small Damping, Applied Mathematics and Mechanics, 53(2), 1989, pp. 206-214 (in Russian).

[17] N. Banichuk, and A. Brat us, On Dynamic Stability of Elastic Systems with Small Dissipative Forces, Mechanical Solids, 5, 1990, pp. 166-174 (in Russian).

[18] N. Banichuk, and A. Bratus, On Stability of Nonconservative Systems with Small Damping that Allows Divergencal Solutions, Mechanical Solids, 1, 1992, pp. 134-143 (in Russian).


[19] A. Bratus, On Various Cases of Instability for Elastic Nonconservative Systems with Damping, International Journal of Solid Structures, 30, 1993, pp. 3431-3441.

[20] S. Prasad, and G. Herrmann, The Usefulness of Adjoint Systems in Solving Nonconservative Stability Problems, International Journal of Solid Structures, 5, 1969, pp. 727-735.

[21] S. Prasad, and G. Herrmann, Adjoint Variational Methods in Nonconservative Stability Problems, International Journal of Solid Structures, 8, 1972, pp. 29-40.

[22] M. Vishik, and L. Lyusternic, Solution of Certain Perturbation Problems in the Case of Matrices and Selfadjoint and Nonselfadjoint Differential Equations, Advancements of Mathematical Sciences, 15(3), 1960, pp. 3-80 (in Russian).

[23] V.I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, New-York, Springer, 1983.

[24] J.E. Marsden, and M. McCraken, The Hopf bifurcation and its Applications, NewYork, Springer, 1976.

[25] Yu. uznetsov, Elements of Applied Bifurcation Theory, New-York, Springer, 1995. [26] B.D. Hassard, N.D. Kazarinoff, and Y.-H. Wan, Theory and Applications of Hopf

bifurcation, Cambrige, Univ. Press, 1981.

•


VOLUME 12 2005, NO 1-2 PP. 119-148

SOME QUESTIONS OF ASYMPTOTIC ANALYSIS: APPROXIMATE SOLUTIONS AND EXTENSION

CONSTRUCTIONS*

A. G. CHENTSOVl

Abstract. The problem of attainability under constraints of an asymptotic character is considered. The attainability effect is realized in the class of approximate solutions defined as filters and ultrafilters in the space of usual solutions. A more concrete representation of this asymptotic effect is realized (usually) by extension procedures. These procedures are realized as compactifications very often. It is advisable to consider the connection of such compactifications and compactifications in general topology; in this investigation, a connection with the Stone-Gech compactification is considered. In addition, we use the zero-dimensional Stone compact space elements of which are ultrafilters. We establish the possibility to identify approximate solutions and generalized elements for a very general problem of asymptotic attainability. As a result, we obtain a direct universal procedure of extension.

Key Words. Ultrafilter, compactification, zero-dimensional space.

1. Introduction. In many applied problems, the consideration of settings with constraints of an asymptotic character is required. We can note problems of control, calculus of variations, and mathematical programming. In this connection, see, for example, the investigations [1]-[4]. The abovementioned case in connected (in particular) with problems unstable with respect to standard constraints; we keep in mind different variants of the weakening of these constraints. We can not fix a variant of the corresponding weakening. Therefore, we must realize an "enumeration" of weakened (standard) constraints. Usually, we realize the "enumeration" of sets of a given family. It is possible to suppose that this family forms the corresponding

' Supported by the Russian Foundation for Basic Research, projects 03-01-00415 t Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sci.,

16 Kovalevskaya St., 620219 Ekaterinburg Russia

119

120 A.G. CHENTSOV

constraint of asymptotic character. We can introduce approximate solutions using the above-mentioned family similarly to [1, ch.III] in idea. We note that, in this part, we use not only sequential approximate solutions (as in [1, ch.III]). We use nets, filters and, in particular, ultrafilters of the space of usual solutions; see [5]-[7]. Moreover, we can use two-valued finitely additive probabilities for approximate solutions.

So, we consider nonsequential approximate solutions. In this part, we are disposed to use filters and, in particular, ultrafilters. This is connected with the following singularity. If a set is fixed, then we consider the totality of all nets in this set; but, it is difficult to consider the above-mentioned totality as a set (we are oriented to the Zermelo formalism). Indeed, for operators used in nets, the corresponding domains can be arbitrary sets. As a corollary, it is difficult to realize constructing the set of required nets by the axiom of the selection for propositional function; see [8, §II.2]. Of course, it is possible to consider the class of all nets in a fixed set, using constructions of the Godel-Bernays-Neumann axiomatics; see [7],[9]. But, we use another possibility: instead of nets, we employ filters. Namely, we consider filters as approximate solutions. Of course, we can consider ultrafilters in this capacity. This step leads to extension constructions. The Stone-Cech compactification can be· noted. Of course, we keep in mind the realization in the class of ultrafilters or (0,1)-measures.

We consider the problem of attainability in a topological space. The employment of ultrafilters as approximate solutions generates the natural problem of the representation of filter limits; see, for example, the conclusion of Chapter 1 of [10].

We use some special notion of a compactificator for the investigation of problems of asymptotic analysis in the above-mentioned sense. Our compactificators realize a more suitable representation of attraction sets arising in a topological space of estimates under the validity of" asymptotic constraints" in the space of solutions. Of course, we fix an operator from the solution space into the estimate space; so, we have the goal operator. In addition, we realize the representation of the basic attraction set in the estimate space as a continuous image of an auxiliary attraction set in the space of generalized elements. This representation can be consider as an extension of our problem. In this investigation, direct procedures of extension are considered. For these procedures, the identification of approximate solutions and generalized elements is possible. Very often, under this identification, we obtain a compact space. Then, the corresponding compactification of our problem is realized.

'

SOME QUESTIONS OF ASYMPTOTIC ANALYSIS .. 121

In [1],[2] and [11]-[14], compactifications of the space of usual solutions are used too. These extension procedures can be considered as indirect: the used generalized elements (measures or measure-valued functions) differ from approximate solutions. Such a property is characteristic for extension constructions of [15]-[21] too. Indirect extensions permit us to obtain highly concrete results; in this connection, we note the property of an asymptotic nonsensitivity (with respect to a part of constraints) in [15]-[21]. But, in this investigation, we consider questions of the structure of attraction sets and approximate solutions. Therefore, we give the basic attention to direct versions of extension for which generalized elements are near to approximate solutions. The profound sense of the next consideration consists in the following (see [22, p.407-409]).

Fix a nonempty set E (the space of usual solutions) with a nonempty family E of subsets of E. Consider E as the constraint of asymptotic character. For brevity, we call E the asymptotic constraint. The intersection Eo of all sets of the family£ has the sense of the set of precise solutions in [1, ch.III]. Fix a topological space (H, e) and a mapping h from E into H. We call (H, e) the estimate space. Then h( e), e E Eo, are the precise estimates. It is natural to supplement {h( e) : e E Eo} by attraction elements arising as the limits of the following type. Namely, we consider nets (ea) in E with the following two properties:

1) under U E £ the inclusion ea E U holds starting from a certain index; 2)the net (h(ea)) is convergent (we keep in mind the Moore-Smith con

vergence). Of course, the simplest variant of such a net is a sequence (e;)~1 in E

with properties 1) and 2). In the class of all nets (ea) in E with the last two properties, we realize all limits of nets of the type (h( ea)). Then, we obtain the set of all attraction elements or an attraction set.

In this constructing, instead of nets, we can use the corresponding dual objects realized by filters :F in E. It is known [6],[9] that nets and filters generate equivalent (in essence) theories. We use the convergence of a filter base in H instead of 2); see [5, ch.I]. This base can be realized (in particular) as the image of a filter :F in E. The corresponding analog of the property 1) (for a filter F) is the requirement £ c :F. By this analogs of 1) and 2) we can introduce the set of approximate solutions-filters. Here, we have the set in the strict sense (see [8]).

2. General definitions and designations. We use quantifiers (Y, :3) and sheafs ( &, V, ==;., =) in their natural interpretation; :3! replaces

122 A.G. CHENTSOV

the expression "there exists ... and unique". We use the symbol ~ for the equality by definition. We take the axiom of choice. We call a family any set all elements of which are sets too. If x is an object, then by { x} we denote the singleton containing x. We use the traditional set-theoretic notions. Now, we recall only some of required definitions.

By P(X) (by P'(X)) we denote the family of all (all nonempty) subsets

of a set X. Let IR be the real line and N ~ {1; 2; ... }, N c R By Fin(S) we denote the family of all nonempty finite subsets of a set S; of course, a family can be used instead of S.

If A and B are sets, then by BA we denote (see [8, ch.IIJ) the set of all functions from A into B. Of course, we use the terms an "operator" and a "mapping" too. If A and Bare sets and f E BA, then: 1) forCE P(A), we

use f 1(C) ~ {f(x) : x E C} E P(B) and the restriction (JIC) E B 0 off 6

to the set C, (JIC)(x) = f(x) for x E C; 2) forD E P(B), we use the usual

inverse image f-1(D) ~ {x E A I f(x) ED} of the set D. In this connection, see [8, ch.II]. We use images and inverse images of families (see, for example, [23]): if X andY are sets and, moreover, f E yx, then

(2.1) f 1[X] ~ {l(U): U EX} E P(P(Y)) \fX E P(P(X))

(the image of a family) and

(2.2) r 1[Y] ~ {f-1 (V) : V E Y} E P(P(X)) \fY E P(P(Y))

(the inverse image of a family). The image and inverse image of a nonempty family (see (2.1) and (2.2)) are nonempty too. If X is a family andY is a set, then

Xly ~{X n Y: X EX} E P(P(Y)).

Consider some special families. If S is a set, then by iJ[S] we denote the set of all families S E P'(P(S)) for which

(2.3) \fA E S \f B E S 30 E S : C c An B.

We call (2.3) a semimultiplicativity of S; of course, (2.3) means that the family S is directed by the direction dual with respect to the inclusion. Then, we have the property: for a set S, the set j30 [S] of all families B E iJ[S] such that 0 f/c B is the set of all filter bases in S; of course,

J30[S] ={BE P'(P'(S)) I VB1 E B \fB2 E B 3B3 E B: B3 c B1 n B2}. (2.4)


We employ families of the set (2.4) for constructing filters. Here, we use the traditional constructions of [5]. If X is a set, then

(2.5) J[X] ~{.FE P'(P'(X)) I (An BE .F \lA E .F \IB E .F)& &({HE P(X) IF c H} c .r \IF E .F)}

is the set of all filters in X; of course, by (2.4) and (2.5),

(2.6) J[X] c flo[X].

Each filter base generates a filter: see [5, ch.I]. If X is a set and BE flo[X], then

(2.7) (X- fi)[B] ~ {L E P(X) I3B E B: B c L} E J[X]

is the filter generated by the B; of course, by (2.6) we can use (2.7) for 13 = .FE J[X]. In addition, for a set X and a filter .F, the equality (X -fi)[.F] = .F is realized. We use maximal filters or ultrafilters very often. If X is a set, then

(2.8) Ju[X] ~{.FE J[F]IVQ E J[X] ((.F c Q) ==;. (.F = Q))}

is the set of all ultrafilters in the set X; see [5]-[7]. In the following, the image of a filter base is used very often. By (2.1)

and (2.4) the following known [5, ch.I] property is realized: if X and Y are sets, BE flo[X], and f E yx, then

(2.9) f 1 [13] = {t(B): BE B} E flo[Y].

Of course, in (2.9), we can use a filter instead of B. Other properties of filters and bases are given, for example, in [5, ch.I] and [7, ch.II]. If X is a set and X E P(P(X)), then

(2.10) Jo[XIX] ~{.FE J[X]I X c .F},

(2.11) ~[XIX] ~ {.FE Ju[X]I X c .F}.

We use the standard notions of general topology; see [5]-[9]. If (X, T) is a topological space ( T is a topology of the set X) and x E X, then (see (2.4))

N~(x) ~{GET I x E G} E flo[X]

124 A.G. CHENTSOV

and, moreover, by analogy with [5]

(2.12) Nr(x) ~ (X- fi)[N~(x)] E ~o[XIN~(x)];

see (2.10). So, in (2.12) the filter of all neighborhoods of a point is considered. We use ultrafilters of the set (2.11) in constructions of asymptotic analysis.

If (X, r) is a topological space and A E P(X), then (by definition) ci(A, r) is the closure of A in the space (X, r) and riA is the topology of A induced from (X, r); see [5]-[9]. If (X, r1) and (Y, r2) are topological spaces, then

C(X,TJ,Y,Tz) ~ {! E yx I r 1h] c 71}

is the set of all continuous operators from (X, rt) into (Y, r2 ). We use the traditional notion of the convergence of a filter base; see [5, ch.l]. Namely, if (X, r) is a topological space, BE ,B0 [X], and x EX, then by definition

(2.13) (B ~ x) {=} (Nr(x) c (X- fi)[B]).

Of course, in (2.13), we can take a filter instead of B (see (2.6)); if :FE ~[X] (and x EX), then

(2.14) (:F ~ x) {=} (Nr(x) C :F).

We use the natural connection of filters and nets. Recall some known [9, ch.2] properties of the Moore-Smith convergence. In this part, we follow the symbolics of [15]-[22]. We define a net in a set X as any triplet (D, :j, f), where (D, :j) is a nonempty directed set (D is a nonempty set and :j is a direction in D) and f E X 0 ;

(2.15) 6

(X- ass)[D; :j; f] = {S E P(X) j:Jd1 ED \:ldz ED ((d1 :j dz) =* (f(dz) E S))} E ~[X]

is the filter associated with the net (D, :j, f). In terms of (2.14) and (2.15), we can introduce the known Moore-Smith convergence: if (X, r) is a topological space, (D, :j, f) is a net in X, and x E X, then by definition

(2.16) ((D, :j, f) ~ x) {=} ((X- ass)[D; :j;f] ~ x).

So, the convergence (2.13),(2.14) is basic for us. But, in idea, nets are similar to sequences. Therefore, we note (2.16). Recall the following known [6, §1.6]


property: if X is a nonempty set and :FE ~[X], then, for some net (D, :::5, f) in X, the equality

(2.17) :F = (X - ass)[D; :::5; f]

is fulfilled. Taking account of (2.16) and (2.17), we conclude that filters and nets in a fixed set are "identified". We note a simple property: if X is a set and A E P(X), then ,80 [A] C ,80[X]; hence, ~[A] C ,80[X] too.

If (X, r) is a topological space, then by definition (r- comp)[X] is the family of all compact [6, §3.1] (in (X,r)) subsets of X and

(2.18) (r- comp)0[X] ~ {S E P(X) I3K E (r- comp)[X]: S C K};

if (X, r) is a Hausdorff space (see [5]-[9]), then.

(2.19) (r- comp)0[X] = {S E P(X) I ci(S,r) E (r- comp)[X]}.

In (2.18) and (2.19), we consider the family of all precompact sets in the topological space (X, r). Recall the known [5, ch.I] property of the ultrafilters convergence in a compact topological space. For us, the following modification is essential: if (X, r) is a topological space and A E (r- comp)[X], then

(2.20) V:F E ~u[A] 3a E A : :F :b;. a.

We use (2.20) in constructions of attraction sets for compactifiable case of our basic problem. Note the important property [5, ch.l]: if X and Y are sets, f3 E ,80 [X], and hE yx, then

((X- fi)[f3] E ~u[X]) => ((Y- fi)W[B]] E ~u[Y]).

For us, the following particular case is important: if X and Y are sets, :FE ~u[X], and hE yx, then (see (2.6) and (2.7))

(2.21) (Y- fi)(h1(:F]] E ~u(Y].

We use (2.21) in compactification constructions; this property has the very simple sense: the ultrafilter image is an ultrafilter base.

Note one useful representation: if X is a set, (Y, r) is a topological space, :FE ~[X], f E yx, andy E Y, then

(2.22) (f 1[F] :b;. y) ¢=;. (f-1[Nr(Y)] C :F).

'""''- , .. ._,~

126 A.G. CHENTSOV

3. Approximate solutions and attraction sets. We use E, H, 0, and h in correspondence with Section 1: E is a nonempty set, (H, 0) is a topological space, and h E HE. In addition, we fix E, H, 0, and h in the following. We consider nonempty families of subsets of E as constraints of asymptotic character; in Section 1 we used E E P'(P(E)) for this.

Following [15]-[18], forE E P' (P( E)), we introduce the set (AS)[ E) of all z E H such that there exists a net (D, ::S, f) in the set E with the properties

(3.1) 0 (E C (E- ass)[D; ::S; f])&((D, ::S, h of) -+ z).

In idea, (3.1) corresponds to the scheme of [1, ch.III] concerned with constructing approximate solutions. But, we consider a nonsequential variant. Using (2.15)-(2.17),(2.22), and (3.1), we obtain that

(3.2) (AS)[E] = {z E H 13.7'" E ~o[EIEJ : h1 [.r] b z} = = {z E HI ~o[EIE U h-1[No(z)]] f 0}.

In connection with (3.2), it is worth noting that

(3.3) ~0 [EIE U h- 1[No(z)]] = {.7'" E ~o[EIEJ I h1 [.7'"] b z} Vz E H.

By (3.1) and (3.2) we realize the set of all results attainable in asymptotic sense. It is reasonable to consider certain filters in E as operations realizing these results. In other words, we can consider (see (3.2),(3.3)) such filters in E as approximate solutions in the attainability problem with the "asymptotic constraint" E. In this connection, we obtain that VE E P' (P(E))

(3.4) (~- sol)[E] ~ U ~o[EIE U h-1 [No(z)]] =

zEH

= {.7'" E ~o[EIE] l3z E H: h1 [.7'"] b z}.

Elements of the set (3.4) are used as approximate solutions in the abovementioned sense. In addition,

(3.5) (AS)[E] = {z E H 13.7'" E (~- sol)[E]: h 1[.7'"] b z} = = {z E H 13.7'" E ~0 [EIE]: h 1[.7'"] b z} VEE P'(P(E)).

We can supplement {3.5) by the following obvious corollary of suppositions of [5, ch.I]: if (H, 0) is a Hausdorff space, then

(3.6) VEE P'(P(E)) VJ'" E (~- sol)[E] 3!z E H: h1 [.7'"] b z.

r.:!

Of course, (3.6) permits us to define a surjection from the set (3.4) onto the attraction set (3.5). This surjection characterizes the rule realizing all attraction elements.

Let Z be the set of all families Z E P'(P(E)) for which

n U-/= 0 VK E Fin(Z). UEIC

Then, by (2.5) and (3.2) we obviously have the representations: V£ E P'(P(E))

(3.7) (AS)[£]= {z E HI£ U h-1[Ne(z)] E Z} =

= {z E HI£ U h- 1[N3(z)] E Z}.

In connection with (3.7), we note the property [17, p.39,40,51]:

(3.8) (AS)[£]= n cl(h1(U),B) V£ E ,B[E]. UEe

PROPOSITION 3.1. (AS)[£]= {z E H 1 U n h-1(V) -1= 0 VUE£ '<IV E Ne(z)} '</£ E ,B[E].

The proof follows from (3.7) and (3.8) directly. We also note one simple property:

(3.9) Er ~ { n U: K E Fin(£)} E ,B[E] V£ E P'(P(E)). UEIC

We add the following obvious property to (3.9):

(3.10) (AS)[£]= (AS)[£r] V£ E P'(P(E)).

As a result, from Proposition 3.1, (3.9), and (3.10), we obtain the general representation

(AS)[£]= { z E HI Unh- 1(V)-/= 0 VUE Er WE N8(z)} '</£ E P'(P(E)). (3.11) We note that, for concrete problems, the requirement £ E ,B[E] is usually fulfilled. Therefore, in the following, (3.11) is used very rarely, since the representation of Proposition 3.1 is simpler.

128 A.G. CHENTSOV

4. Ultrafilters in constructions of approximate solutions: general notions. In this section, we consider a hypothetical setting of the problem of attainability under constraints of asymptotic character. Now, we use only improved approximate solutions defined as ul trafil ters in E. Of course, usually, such approach is combined with an extension. But, now, we simply consider the corresponding "part" of the set (3.4) of all approximate solutions-filters. It is known that (see [5, ch.I])

(4.1) V:F1 E ~[E]3:F2 E ~u[E]: :F1 C h

From (2.10),(2.11), and (4.1), we obtain that

(4.2) '18 E P(P(E)) 'if:F1 E ~o[EI8] 3:F2 E ;fu[EI8] : :F1 C h

By (4.2) we have the following property: '18 E P(P(E))

(4.3) (~o[EI8] f. 0) ¢=? (;fu[EI8] f. 0).

From (2.8),(3.2), and (4.3), we obtain that

(4.4) (AS)[e] = {z E H 1 ;fu[E 1 e u h-1 [No(z)]] f. 0} VeE P'(P(E)).

By (3.3) we have the obvious representation: VeE P'(P(E)) Vz E H

(4.5) ;fu[Eie u h-1[N0 (z)]] ={:FE ;fu[Eie]l h1[:F] ~ z}.

Then, we introduce the set of all improved approximate solutions: if e E P'(P(E)), then

(~u- sol)[e] = U ~[Eie u h-1[N0(z)]] ={:FE ~[Eie]l3z E H: (4.6) zEH

h1[:F] ~ z}.

PROPOSITION 4.1. For any e E P'(P(E)), the following chain of equalities takes place

(AS)[e] = {z E H I3:F E ~[Eie]: h1 [:F] ~ z} =

= {z E H I3:F E (~u- sol)[e]: h 1[:F] ~ z}.

The proof is the obvious combination of (3.5) and (4.4)-(4.6). Of course, by (3.6) we have the property: if (H, B) is a Hausdorff space, then

VeE P'(P(E)) V:F E (~u- sol)[e]3!z E H: h 1[:F] ~ z. In this connection, we note that by (2.8),(2.10)-(2.11), (3.4), and (4.6)

(4.7) (~u- sol)[e] c (~- sol)[e] VeE P'(P(E)).

From (4.7), we obtain the possibility to essentially reduce the used class of approximate solutions.


5. Ultrafilters and procedures of a compactification in the problem of asymptotic attainability. We use the natural scheme of constructing generalized problems. This scheme is oriented to the known Stone-Cech compactification. In addition, ultrafilters in E are used for generalized elements. It is possible to call this scheme a direct extension procedure, since improved approximate solutions and generalized elements are identified. We note that indirect procedures of extension are considered in [15]-[21]; in addition, several useful concrete results were obtained. Now, we investigate questions of the structure of the above-mentioned problems of asymptotic analysis.

DEFINITION 5.1. If (K,r) is a compact [6, ch.3} topological space, p E KE, q E C(K,r,H,B), andh = qop, then the pmcession (K,r,p,q) is called a compactificator.

PROPOSITION 5.1. Let £ E ,B[E], (K, r, p, q) be a compactificator and (H, B) be a Hausdorff topological space. Then,

Ko ~ n cl(p1(U),r) E (r- comp)[K] UE£

and (AS)[£]= q1(K0) E (B- comp)[H]. The corresponding proof follows from Theorem 2.5.2 of [15]. In Propo

sition 5.1, several corollaries of compactification are realized. DEFINITION 5.2. The procession (E, H, B, h) is called compactifiable if

there exists some compactificator. REMARK 5.1. If a procession (E,H,B,h) is compactifiable and (H,B)

is a Hausdorff space, then, for any£ E ,B[E], the attraction set is realized by

means of neighborhoods. For any S E P(H), we introduce ~[S] ~ { G E 81 S c G}; moreover, suppose that

.Ne[S] ~{HE P(H) I :JG E ~[S]: G c H}.

Then (see [17, ch.3} and [24}), the following property is fulfilled: if (H, B) is a Hausdorff space and the procession (E, H, B, h) is compactifiable, then V£ E ,B[E] VM E .Ne[(AS)[£]] :JP E £ VQ E £

(5.1) (Q C P) =;.(ME .N0[cl(h1(Q),B)]).

By (5.1) we obtain that the attraction set is realized in terms of the sets h1(U), U E £, with "any precision". Therefore, the fact of compactificability of (E, H, B, h) itself is very important.

130 A.G. CHENTSOV

PROPOSITION 5.2. The following two conditions are equivalent: 1) h 1 (E) E (0- comp)0[H]; 2) the procession (E,H,O,h) is compactifiable.

The proof follows (in fact) from definitions (in particular, see (2.18) and known properties of continuous functions). We omit this proof. Note that the connection of the conditions 1) and 2) was noted by E.G. Pytkeev.

. 6 0 Introduce (h- LIM)[.F] = {z E HI h 1[.F] ==? z} V.F E ~u[E]. PROPOSITION 5.3. If h 1(E) E (0- comp)0[H], then

(h- LIM)[.F] E P'(cl(h1(E),O)) V.F E ~u[E].

Proof Let h 1(E) E (O- comp)0 [H]. Using (2.18), we choose K E (0-comp)[H] for which h 1 (E) C K. Then, hE KE. Fix .FE ~u[E]. By (2.11)

(5.2) K ~ (K- fi)[h1[.F]] E ~u[K].

From (2.20) and (5.2), we obtain the convergence K ~ y for some y E K. Then, No(y) C (H- fi)[K] (recall that K E ,60[H]; see Section 2). Introduce

1{ ~ (H- fi)[h1[.F]] E ~[H].

Then, (H- fi)[K] c 1{ and, as a corollary, No(y) c 1{ or h 1[.F] ~ y. Then, (h- LIM)[.F] E P'(H). But, h E h 1(E)E and h 1[.F] E ,60[h1(E)]. Then, (h- LIM)[.F] c cl(h1 (E),O); see [5, ch.I]. 0

CoROLLARY 5.1. If (H, 0) is a Hausdorff space and h 1 (E) E (0 -comp)0[E], then

V.F E ~u[E]3!z E H: (h- LIM)[.F] = {z}.

The corresponding proof follows from (2.5) and (2.13); see [5, ch.I]. In the following, unless otherwise stated, we suppose that (H, 0) is a

Hausdorff space and h1 (E) E (0- comp)0[H]. By Proposition 5.3 and Corollary 5.1 we define the operator

(5.3) SJ: ~u[E]-t cl(h1(E),O)

by the following rule: (h- LIM)[.F] = {SJ(.F)} V.F E ~u[E]. From (5.3), we obviously have

Sj : ~u[E] ---7 H.


Note the obvious property [5, ch.I]: if x E E, then

F~ ~{FE P(E) I X E F} E Ju[E].

Then, we introduce the operator

(5.4) X ---7 F~ : E ---7 Ju[E]

defined by m. The operators S) (5.3) and m (5.4) are connected by the following obvious equality:

(5.5) h = S) om.

In fact, by (5.5) we obtain the natural extension of h; as a result, we have the operator (5.3). This extension is realized by the immersion (5.4) of E in Ju[E]; of course, m 1(E) C Ju[E].

Now, we give very briefly one simple procedure of a compactification. Recall that by (2.19), for

t ~ Blcl(h 1 (E),O),

the topological space

(5.6) (cl(h1 (E),B),t)

is a nonempty com pactum; of course, (5.6) is the compact subspace of (H, B). PROPOSITION 5.4. The operatorS) (5.3) is a surjection from Ju[E] onto

cl(h1 (E),B):

cl(h1(E), B)= SJ1(Ju[E]).

Proof. Use (5.3). Fix z E cl(h1(E), B). Then, h 1(E)nY =f. 0 VY E Ne(z). Therefore [5, ch.I],

(5.7) h- 1[Ne(z)] E ,6o[E]

and, for some ultrafilter U E Ju[E], we have the inclusion h-1[No(z)] C U. As a corollary, for h1[U] E ,60 [H], we obtain that (see (5.7))

Ne(z) c (H- fi)[h1[h- 1 [N0 (z)]]] c (H- fi)[h1[U]].

132 A.G. CHENTSOV

So, h 1[U) =4 z; therefore, z = SJ(U) E SJ 1 [~u[E)). We established the inclusion

cl(h1(E), IJ) c SJ 1 [~u[EJ].

0

CoROLLARY 5.2. The topological space (~u[E], S)-1 [t)) is compact. The proof follows from the definition of compactness with the employ

ment of (2.2) and Proposition 5.4. REMARK 5.2. We note that (~u[E], S)-1[t)) can not be a T1-space.

Therefore, in terminology of [5], in Corollary 5.2, we have a so called quasicompact topological space.

By (2.2) we have the following property:

(5.8) S) E C(~u[E), S)-1[t), cl(h1(E), IJ), t).

From the definition oft and (5.8), we obtain the obvious corollary:

(5.9) S) E C(~u[E), S)-1 [t], H, IJ).

THEOREM 5.1. The procession (~u[E], S)-1[t], m, SJ), where m is defined by ( 5. 4), is a compactijicator for which

(5.10) ~u[E) = cl(m1(E), S)-1 [t)).

Proof. From (5.4),(5.9), and Corollary 5.2, we have the property of a compactificator for the considered procession. So, it is sufficient (see (5.4)) to establish the inclusion

(5.11) ~u[E) C cl(m1 (E), S)-1 [t)).

Fix :FE ~u[E) and G E NZ-•[t](:F). We choose T E t such that G = S)-1 (T); see (2.2). Then, T E Nr(SJ(:F)). By Proposition 5.4, SJ(:F) E cl(h1(E), IJ). Using the definition oft, we have a neighborhood r E N3(SJ(:F)) for which

(5.12) T = r n cl(h1(E),IJ).

But, r n h 1(E) of 0. Let u E E be a point such that h(u) E r. Of course, by (5.12) h(u) E T and SJ(m(u)) E T (by (5.5)). Then, m(u) E G. So, G n m 1(E) of 0. Since the choice of G was arbitrary, the property :F E cl(m1(E), S)- 1[t]) is established. But, the choice of :F was arbitrary too. Therefore, (5.11) is established. 0


6. The employment of the Stone-Cech compactification,l. In this and the following sections we consider a natural possibility connected with constructing a compactificator on the base of the approach for which the idea concerned with constructions of the Stone-Cech compactification is used. We strive to equip Ju[E] with a topology of the very traditional zerodimensional compactum; see [5]-[7] and [9]. For the realization of our goal, an "extended" consideration is useful.

If A E P'(E), then suppose that

(6.1) ~[A]~ {j E HA I t(A) E (0- comp)0 [H]}.

In ( 6.1), we have analogs of h of the previous section. By analogy with Corollary 5.1 we obtain that

VA E P'(E) Vf E ~[A] VF E Ju[A]3!z E H: {z E HI t[F] ~ z} = {z} (6.2) (recall that we consider the case of a Hausdorff space (H, 0) ). Taking account of (6.2), we introduce the following definition. Namely, for A E P'(E) and f E ~[A], the operator

(6.3) S'l~[f] : Ju[A] -+ H

is defined by the following rule: if F E Ju [A], then S)~ [!](F) E H has the property

(6.4) ! 1[F] ~ S'l~[j](F).

Now we note that hE ~ [E]. Therefore, by (6.2),(6.4), and the definition of S) we have the property:

(6.5) S'l(F) = S'l~[h](F) VF E Ju[E].

So, by (6.5) we obtain some extension of the setting of Section 5. It is useful to note that by (2.18) and (6.1)

(6.6) (hiA) E ~[A] VA E P'(E).

Therefore, for A E P'(E) and FE Ju[A], we have the point

S'l~[(hiA)](F) E H.

From general properties of ultrafilters (see [5, ch.I]), the following property takes place:

(6.7) F ={A E P(E) IAn F /- 0 VF E F} VF E Ju[E].

134 A.G. CHENTSOV

We recall that :F C P'(E) 'r/:F E ~[E]. In (6.7), we have an important property of ultrafilters in E. We note that V:F E ~[E] 'r/ A E :F

(6.8) :FIA ={An F: FE :F} ={FE :F 1 F c A} E ~[A].

The following particular case is very important: if :F E ~u[E] and A E :F, then [5, ch.I]

(6.9) :FiA E ~u[A].

We combine (6.6) and (6.9). In addition, we have the property: if :FE ~u[E] and A E :F, then

fJ~[(hiA)](:FIA) E H.

By analogy with (5.3) we obtain that 'r/:F E ~u[E] VA E :F

(6.10) fJ~[(hiA)](:FIA) E cl(h1 (A), 0).

In connection with (6.10), we note the property: if :FE ~u[E] and A E :F, then

(6.11) (hiA)1[:FIA] = h1[:FIA] E ,Bo[H].

PROPOSITION 6.1. If :FE ~u[E] and A E :F, then

SJ(:F) = fJ~[(hiA)](:FiAl·

Proof Fix :F and A in the correspondence with the conditions of our

proposition. We have the base h 1[:F] E ,80[H]. For u ~ SJ(:F) E H, we have the equality

(6.12) (h- LIM)[:F] = {u};

see the definition of the operator (5.3). Then, by (6.12)

(6.13) (h1 [:F] =b. u)&('rfz E H ((h1 [:F] =b. z) =* (z = u))).

For :FIA E ~u[A] and v = .lj~[(hiA)](:FiA), we obtain (by (6.6) and (6.11)) the convergence

h1 [:FiA] =b. v.

"'

SOME QUESTIONS OF ASYMPTOTIC ANALYSIS.. 135

Then, Na(v) C (H- fi)[h1[FIA]]. In addition, FIA C F; see (6.8). Therefore, h 1[FIA] c h 1[F] and

(H- fi)[h1[FIAlJ c (H- fi)[h1[F]];

then N8(v) c (H- fi)[h1[F]] and h 1[F] =b. v. By (6.13) u = v. 0 PROPOSITION 6.2. IfF E Ju[E], fiE ~[E], and hE ~[E], then

(3A E F: (/!lA) = (hiA)) =? (SJ~[/I](F) = SJ~[h](F)).

The proof is an obvious analog of the proof of Proposition 6.1. This corollary corresponds to the conclusion of [10, ch.1]. By Proposition 6.2 we can introduce the natural factorization of ~[E]. Namely, for FE Ju[E], we can introduce the equivalence relation ~=~F by the rule: for f E ~[E] and g E ~[E]

(!~g)<==? (:JA E F: (!lA) = (giA));

in our case, by Proposition 6.2 we obtain that \f h E ~ [E] \f h E ~ [E]

(/!~h)= (SJ~[h](F) = SJ~[h](F)).

In this direction, some generalizations are possible (in particular, under constructing ~ [E] and, for a fixed F E Ju[E], the equivalence relation ~, the generalization on the case of non-Hausdorff space (H, e) is realized). But, we do not consider these obvious generalizations.

PROPOSITION 6.3. IfF E Ju[E] and A E F, then SJ(F) E cl(h1(A), e). The proof is the direct combination of (6.10) and Proposition 6.1. THEOREM 6.1. IfF E Ju[E], then the following equality takes place:

ncl(h1 (A),e) = {SJ(F)}. AEF

Proof. Fix FE Ju[E]. Recall that, by definition of S) (5.3), \fy E H

(6.14) (h1[F] =b. y) =? (y = SJ(F)).

Moreover, by analogy with ( 6. 7)

(6.15) (H- fi)[h1[F]] ={A E P(H) IAn F =J 0 VF E

E (H- fi)[h1[F]]} E Ju[H];

136 A.G. CHENTSOV

of course, we use (2.21). Choose arbitrarily

(6.16) q E n ci(h1 (A), IJ). AEJ'

Then, H n h 1(A) f. 0 \fA E :F \fH E Ne(q). Let A E Ne(q). Then (see (6.16)),

An F f. 0 \IF E (H- fi)[h 1[:FJ].

By (6.15) we obtain that A E (H- fi)[h1[:FJ]. We establish that

Ne(q) c (H- fi)[h1[:F]].

So, h 1[:F] ~ q. By (6.14) q = f)(:F). So, since the choice of (6.16) was arbitrary, we have the inclusion

n cl(h1(A),IJ) c {f:l(:F)}. AEJ'

Using Proposition 6.3, we obtain the required equality. 0

7. The employment of the Stone-Cech compactification,2. In this section, we construct a compactificator using the traditional variant of the space of generalized elements. Namely, we use the set of all ultrafilters in E. This set is converted in a zero-dimensional [6, ch.6] compactum. Here, we use the construction of the Stone representation in [23, ch.I] for the traditional (in topology) case of algebra of all subsets of E. We note that another variant of similar constructions is realized (in fact) in general topology; for example, see [6, ch.3] (we keep in mind the Stone-Cech compactification and the Wallman extension). But, we use somewhat another language.

First, we introduce the operator

(7.1) Ua : P(E) ---+ P(;yu[E]);

namely, we suppose that \fA E P(E)

(7.2) Ufl(A) ~{:FE ;yu[Ej I A E :F}.

We consider the image of P(E) for the operator (7.1),(7.2): in the following

(7.3) E ~ U/t(P(E)) = {UH(A): A E P(E)} E P'(P(;yu[E])).

'"


Of course, (7.1) is a mapping from P(E) onto E. It is known [23, ch.I] that E is an algebra of subsets of ~u[E]. We note that Ufl is a bijection from P(E) onto E having several useful properties:

Ufi(A1 n A2) = Ut~(Al) n UH(A2) \fA1 E P(E) \fA2 E P(E);

UH(E \A)= ~u[E] \ UH(A) \fA E P(E);

UH(0) = 0. In addition, U11 ({x}) = {Fn \ix E E. It is obvious that Eisa topological base (the base of some topology).

By Tfi we denote the topology of the set ~u[E] generated by the base E :

6 (7.4) Tfi = { G E P(~u[E]) I \fQ E G :JB E E: (Q E B)&(B c G)}.

We use reasoning similar to the brief discussion in [23, ch.I] and the conclusion of [25, ch.IV]. Then,

(7.5) (~u[E], Tfl)

is a zero-dimensional compactum, that is a zero-dimensional compact Hausdorff space. For the completeness of our account, we consider a scheme of the corresponding proof in Supplement. In addition, E is the family of all subsets of ~u[E] open-closed in the space (7.5).

We note that, for any ultrafilter FE ~u[E], the family

(7.6) {Ut~(A): A E .F}

is a local base of the space (7.5) at the point F. In other words, (7.6) is a fundamental system of neighborhoods ofF in the space (7.5).

LEMMA 7.1. .lj E C(~u[E], Tft, cl(h1(E), 11), t). Proof Recall that (5.6) is a compactum and, in particular, a regular

topological space. Therefore, any point y E cl(h1(E), 11) has a fundamental system of neighborhoods closed in the compactum (5.6). So, at any point of this compactum, we have a local base of closed sets.

Fix U E ~u[E]. Consider the filter base h1 [U] and the corresponding filter (H- fi)[h1[U]] E ~[H]. Then,

h1 [U] b SJ(U)

by definition of .lj (5.3). So,

(7. 7) Ne(SJ(U)) c (H- fi)[h1[U]].

138 A.G. CHENTSOV

We recall that h 1(A) c cl(h1(E), e) VA E U. In addition, by definition oft we have

(7.8) cl(h1(A), t) = cl(h1(A),e) VA E U;

indeed, (5.6) is a closed subspace of (H,e). Let

N E N,(SJ(U)).

Using the regularity of the space (5.6), we choose a neighborhood F E

N,(SJ(U)) closed in the sense of (5.6) and having the property

FeN.

So, we use the closed neighborhood F of the point SJ(U). Let FE N0(SJ(U)) be a neighborhood for which

F = cl(h1(E), e) n F;

see [17, p.36]. By (2.7) and (7.7)

3B E h1[U]: B c F.

Fix B E h 1[U] for which B C F. Let iP E U have the property B = h 1(iP). Then,

(7.9) h1 (ii>) c F.

Recall that Ufl ( iP) E Nr6 (U); we use the basic property of the family (7.6). Let V E U8 (iP). Then, by (7.2) iP E V. Of course, h 1(iP) E h 1[V], where h 1 (iP) c cl(h1(E),e). By Proposition 6.3 and (7.8)

(7.10) SJ(V) E cl(h1(iP), t).

From (7.9), we have the inclusion

h 1(iP) c F n cl(h1(E), e).

Therefore, h 1 (iP) c F and, as a corollary, cl(h1(iP), t) C F. By (7.10) SJ(V) E

F. By the choice ofF we obtain that SJ(V) EN. So,

(7.11) UH(iP) E Nr6 (U): SJ(.F) EN V.F E UH(iP).

SOME QUESTIONS OF ASYMPTOTIC ANALYSIS.. 139

Since the choice of N was arbitrary, we have the continuity of Sj at the point U. But, the choice of U was arbitrary too. Therefore, Sj is continuous as the mapping from (Ju[E], Tfi) into the compactum (5.6). 0

THEOREM 7.1. S)EC(Ju[E],Ta,H,O). The corresponding proof is realized by the direct combination of Lemma

7.1 and the definition of the topology t (see (5.6)). We use the operator m (5.4).

PROPOSITION 7.1. The procession (Ju[E], Tfi, m, SJ) is a compactificator.

The proof follows from the compactness of the space (7.5),(5.4), (5.5), and Theorem 7.1. The following statement corresponds to the standard property of compactifications in general topology (see [6, §3.5]); it is given for the completeness.

PROPOSITION 7.2. Ju[E] = cl(m1(E), Tfi)· Proof. Recall that m 1(E) = {F~: x E E} E P(Ju[E]). Then

cl(m1(E), Tfi) C Ju[E].

Let FE Ju[E]. Then, we can use (7.6) as a local base of the space (7.5) at the point F. Let A E F. Then, by (2.5) A E P'(E). Let a EA. For F~ E m 1(E), we obtain that A E F1 (see Section 5). Therefore, by (7.2) F1 E Utl(A) and

m 1(E) n Ufi(A) # 0.

Since the choice of A was arbitrary, we have (by the basic property of the family (7.6)) the inclusion FE cl(m1(E), Tfi). So,

Ju[E] C cl(m1 (E), Tfi).

0 PROPOSITION 7.3. If£ E P'(P(E)), then the set ~[EI£] E P(Ju[E])

is closed in the space (1. 5). Proof Let FE cl(~[EI£],Tfi). Then, FE Ju[E]. In addition,£ c F.

Indeed, suppose the contrary: £ \ F of 0. Let A E £\F. By the known [5, ch.I] property of ultrafilters we have the inclusion E \A E F. Consider

(7.12) Uti(E\A) E NTfi(F).

By the choice ofF and (7.12) we have the property

J~[EI£] n Ull(E \A)# 0.

140 A.G. CHENTSOV

Choose U E ~[EIE] n Uft(E \A). Then, U E tY'u[E], E c U, and E \A E U (see (7.2)). Since A E E, for the filter U E lY'[E], we have the following two inclusions

(7.13) (A E U)&(E \A E U).

But, by (2.5) the relation (7.13) is impossible. So, E \ :F = 0. Therefore, E c :F and :FE ~[EIE]; see (2.11). Then,

c!(~[EIEJ, 7ft) c ~[EIE].

0

Introduce the following auxiliary attraction sets: if E E P'(P(E)), then by definition (AS)[E] is the set of all ultrafilters :F E tY'u[E] such that there exists a net (D, j, f) in the set E for which

(7.14) (E C (E- ass)[D; j; f])&((D, j, m of) ...2"!4 :F).

Of course, from the constructions of [15, p.39,40] and (7.14), we obtain that

(7.15) (AS)[E] = n cl(m1(U), 7ft) YE E ,B[E]. UEl

In connection with (7.15), we note the following corollary of Propositions 5.1 and 7.1:

(7.16) (AS)[E] = Sj1((AS)[E]) YE E ,B[E].

In this scheme, tY'u[E] is used as the space of generalized elements and (7.15) is the corresponding admissible set. So, we realize the extension of the initial attainability problem using the Stone-Cech compactification.

THEOREM 7.2. (AS)[Ej = ~[EIEJ YE E P'(P(E)). Proof. Fix E E P'(P(E)). Let :F E (AS)[E]. Then :F E tY'u[E] and,

for some net (D, j, f) in E, we have (7.14). Fix A E E. By (2.15) and (7.14), for an index o E D, we obtain f(d) E A for d E D, o j d. Let

Do~ {dEDI o j d}; o E D0• If dE D0, then A E F}(d)" Now, we use the reasoning similar to the proof of Proposition 7.3.

Indeed, let A E E\:F. Then, E\A E :F; we use the corresponding analog of (7.13). Therefore,

Uft(E \A) E NT.(:F).

" :r,


By (7.14) we have an index r;, E D for which Yd ED

(r;, ~d)=* (F}(dl E Ua(E \A)).

We use the definition of the operator (5.4). Let 'Y E D have the property (o ~ 'Y)&(r;, ~ "'!). Then,

F}hl E Ua(E\ A).

So, E \A E F}(~l· But, 'Y E D,. Therefore, A E F}(~)· Then, by axioms of a filter

(E \A) n A E F}bl;

but (E \A) n A = 0. We obtain an obvious contradiction; see (2.5). So, A ~ & \F. Then, A E F. Since the choice of A was arbitrary, & c F. By (2.11) we have the inclusion FE J?.[E\&]. We obtain the inclusion

(7.17) (AS)[&] c J?.[E\&].

Let U E J?.[E\&]. Then, by (2.11) U E ~u[E] and & CU. Using Proposition 7. 2 and the axiom of choice, it is possible to construct a net (llli, !:;;, g) in the set E for which

(7.18) (llli, !:;;, m o g) ..!.!!.., U.

Let Q E &. Then, Q E U and

(7.19) Ua (Q) E Nra (U)

(see properties of the family (7.6)). By (7.18) and (7.19) we have an index ,\ E lili for which Yd E lili

(,\I:;; d)=* (F!(dl E Uo(O)).

This means that n E F!(d) for d E llli, ,\ I:;; d. In other words (see Section 5), g(d) E Q for all d E lili such that ,\ I:;; d. Then (see (2.15))

Q E (E- ass)[llli; !:;;; g].

Since the choice of n was arbitrary, the inclusion

& c (E- ass)[llli; !:;;; g]

142 A.G. CHENTSOV

is established. Now, from (7.18), we have (see (7.14)) the inclusion U E

(AS)[f]. So,

(7.20) J?.[Eif] c (AS)[f].

From (7.17) and (7.20), we have the required equality (AS)[f] =~[Elf]. D PROPOSITION 7.4. The following equality holds:

(~u- sol)[f] =~[Elf] Vf E P'(P(E)).

Proof. By (4.6) we have the inclusion (~u - sol)[f] C ~[Elf]. Let :F E ~[Elf]. Then :FE ~u[E] and f C :F; see (2.11). By the definition of SJ (5.3) we havethe property: SJ(:F) E H satisfies the condition

(7.21) h 1[:F] b SJ(:F).

By (4.6) and (7.21) we obtain that :FE (~u- sol)[f]. So,

\f,;[Eif] C (~u- sol)[f].

As a corollary, the equality (~u- sol)[f] =~[Elf] holds. D CoROLLARY 7.1. The set (4.6) is compact in the space (7.5):

(~u- sol)[£]= (AS)[f] =\f.;[ Elf] E (Tfl- comp)[~u[E]j Vf E P'(P(E)).

The corresponding proof follows from Proposition 7.3 and the compactness of the space (7.5); moreover, see Theorem 7.2.

From Corollary 7.1, we have (in particular) the following important property. For the considered case of the Hausdorff space (H, e), under the validity of the condition h 1(E) E (e- comp)0 [E], (~u- sol)[f] = (AS)[f] Vf E

P'(P(E)). This conclusion means that, in the given exhausting case (from the point of view of possibilities of a compactification; see Proposition 5.3), we can identify approximate solutions and admissible generalized elements in the extension scheme realized by the Stone-Cech compactification. In both variants, ~u[E] is the set from which the corresponding concrete choice of elements is realized. We have a highly perfect direct scheme of an extension of the problem of asymptotic attainability. This scheme is realized by the application of the classical construction of general topology: we keep in mind the Stone-Cech compactification.

'


8. Supplement. Here, we consider the scheme of the proof of the compactness property of the space (7.5).

Proof. For brevity, now, we suppose that (in the following) '{! ~ Ua. Then (see Section 7) '{! is a bijection from P(E) onto E. In addition, for A E P(E) and B E P(E)

(8.1) '{J(A n B)= 'P(A) n'P(B), 'P(A u B)= '{J(A) u 'P(B);

moreover, '{J(E\L) = Ju[E]\'P(L) 'IL E P(E). Finally, 'P(0) = 0. Of course, (8.1) is extended to the case of any finite intersections and unions.

Let 1/J be the bijection inverse of '{!;

1/J: E-----+ P(E).

And what is more, 1/J has the properties similar to (8.1 ): if lK E E and lL E E, then

(8.2) (1/>(IK u JL) = 1/J(IK) u 1/>(lL))&('l/J(IK n JL) = 1/>(IK) n 1/J(lL));

(8.3) 1/J(Ju[E] \ §) = E \ 1/J(§) 'cf§ E E.

The validity of (8.2) and (8.3) follows from the corresponding properties of 'P (moreover, we use the fact that 'Pis a bijection). Moreover, 1/J(0) = 0. We recall (7.4); of course, E C ra.

Let K E P'(E). Consider the family X~ 1/J1[K]; X C P(E). If X E X, then, for some K E K, the equality X = 1/>(K) holds; as a corollary,

(8.4) 'P(X) = 1.(!(1/J(K)) = K E K.

Suppose that

(8.5) Ju[E] = U JL. ~E~

In this proof, we use the following stipulation: if n E N and ( is a family of

sets, then (n is used instead of ( 1•n, where 1, n ~ { i E N I i ~ n}. Then,

n

(8.6) 3n EN 3(J4)iEl,n E Kn: Ju[E] = U14. i:::::l

Indeed, let us assume the contrary: n

(8.7) {(lLi)iEl,n E Kn I Ju[E] = U14} = 0 'In EN. i:;:::: 1

144 A.G. CHENTSOV

As a corollary, we have the property: 'linEN '1/(JL;);El,n E Kn

n

(8.8) n(tru[E] \ 14) # 0. i=l

Recall (see Section 7) that t'ru[E] \ lL E E '1/lL E K. By (8.3), for lL E r;, we have the representation

(8.9) 1/J(~u[E] \ JL) = E \ 1/;(JL) E { E \X : X E X}.

On the other hand, by definition of X we obtain that

'II X E X :JJL E r;: X= 1/J(lL).

Therefore, by (8.3) we have the following property; namely, '\IX EX :JJL E r;:

(8.10) E \X= 1/J(t'ru[E] \ JL).

Introduce the family Y ~ {1/J(t'ru[E] \lL): lL E r;}. Then, by (8.9) and (8.10)

(8.11) Y={E\X: XEX}.

If lL E r;, then 1/;(JL) E X and, as a corollary, <p( 1/;(JL)) = JL; we obtain that :JX EX: lL = <p(X). Then,

u lL = u <p(X); ILEK XEX

see (8.4). From (8.5), we obtain that

(8.12) tru[E] = U <p(X). XEX

By (8.8) and the definition of Y we have the property: 'lin E N 'I!(Yi);EJ,n E yn

n

(8.13) nY; o~ 0. i=l

Indeed, fix q EN and (r;);EI,q E yq. We have the finite procession of subsets of E; as a corollary,

<p(rj) E E '1/j E 1, q.

Of course, 1/l(<,o(rk)) = rk fork E 1,q. Let r E 1,q. Consider rr E y. By the definition of Y we have the equality

rr = 1/I(Ju[E] \ 11),

where n E K. Of course, by properties of 'P

<,o(rr) = Ju[E] \st. - 6

From (8.11), we obtain that X = E \ r r E X and

<,o(X) = Ju[E] \ <,o(q = n E K.

Since the choice of r was arbitrary, we have the property

(8.14) Ju[E] \ <,o(ri) E K ViE l,q.

By (8.8) and (8.14) we obtain that

q q

(8.15) n<,o(ri) = n(Ju[E] \ (Ju[E] \ <,o(ri))) of 0. i:::::l i=l

In addition (see (8.1),(8.15)), we have the property

q q

(8.16) <,o(nri) = n<,o(ri) of 0. i=-1 i:::::l

But, by the definition of <p, from (8.16), we obtain that

q

nri of0 i=l

(recall that <,o(0) = Ufl(0) = {.F E Ju(E] I 0 E .F} = 0). So, (8.13) is established. From (8.13), we obtain that

(8.17) n Y of 0 VK E Fin(Y). YEIC

By (2.4),(3.9), and (8.17) we have the property Yr E ~o[E] (recall that K of 0, X of 0, andY of 0 by (8.11)). Then,

(E - fi)[Yr] E J[E].

146 A.G. CHENTSOV

In addition, Y c Yr C (E- fi)[Yr]. Recall that, for some

WE ~u[E],

we have the inclusion (E- fi)[Yr] c W. As a corollary, Y c Yr C W. Then, in particular,

(8.18) E\XEW VXEX.

In addition, by (8.5) 3JL E "' : W E lL. Let E E "' be a set such that

W E E; ~ ~ 1/;(E) E P(E) and <p(~) = E. In particular W E <p(() and ~ E X; by (7.2) and the definition of <p we have the inclusion~ E W. But, by (8.18) E \ ~ E W. So,

(8.19) (~ E W)&(E \ ~ E W).

But, W is a filter and, as a corollary (see (2.5) ),

(8.20) (0 fi W)&(A n BE W VA E W VB E W).

From (8.19) and (8.20), we obtain the obvious contradiction (indeed, ~n (E\ 0 = 0 and, moreover, ~ n (E \ ~) E W by (8.20)). This contradiction means that (8.7) is impossible; therefore, (8.6) is fulfilled. So, (8.5)=?(8.6). Since the choice of"' was arbitrary, the following property is fulfilled: V1 E P'(E)

(8.21) (~u[E] = UJL) ==? (3/C E Fin(!) : ~u[E] = U JL). ILE7 ILEIC

Since E is a base of the space (7.5), from (8.21), we have the compactness property of this space. 0 We make brief remarks connected with the property of zero-dimensionality of the space (7.5). Let M E P(E); we preserve the previous stipulation with respect to the definition of <p. Choose an ultrafilter

U E ~u[E] \ <p(M).

Then, by (7.2) M 'i U and, as a corollary, E \ME U. Then, U E <p(E \ M), where <p(E \ M) E E. In addition,

(8.22) <p(E \ M) = ~u[E] \ <p(M) E E.

From (8.22), we have the property: ~u[E] \ <p(M) Era; as a corollary, <p(M) is closed. Since the choice of M was arbitrary, E is a family of open-closed subsets of ~u[E] in the required sense.


REFERENCES

[1] J. Warga, Optimal control of differential and functional equations., Academic Press, New York, 1972.

[2] L.C. Young, Lectures on the calculus of variations and optimal control theory., Saunders, Philadelphia, Pa, 1969.

[3] Duffin, R.J. Linear inequalities and related systems, Ann. of Math. Studies, 38 (1956), 157-170.

[4] Gol'stein, E.G. Duality theory in mathematical programming and its applications. Nauka, Moscow, 1971 (Russian).

[5] N.Bourbaki, General topology. Nauka, Moscow, 1968 (Russian). [6] R.Engelking, General topology, PWN, Warszava, 1977. [7] E. Cecb, Topological spaces, Publishing House of the Czechoslovak Academy of Sci

ences, Prague, 1966. [8] K. Kuratowski, and A. Mostowski, Set theory, PWN -Polish Scientific Publishers,

Warszawa, 1967. [9] J.L.Kelley, General topology. Van Nostrand, Princeton, N.J., 1955.

[10] R.E. Edwards, Functional analysis, Holt, Rinehart and Winston, New York- Chicago - San Francisco - Toronto - London, 1965.

[11] Gamkrelidze, R.V. Foundations of optimal control theory. lzdat. Tbil. Univ., Tbilissi, 1977 (Russian).

[12] N.N.Krasovskii and A.I.Subbotin, Game-theoretical control problems., SpringerVerlag, 1988.

[13] N.N.Krasovskii, Control to dynamic system. Problem about minimum of guaranteed results., Nauka, Moscow, 1985 (Russian).

[14] A.I.Subbotin and A.G.Chentsov, Optimization of guarantee in control problems., Nauka, Moscow, 1981 (Russian).

[15] A.G.Chentsov, Finitely additive measures and relaxations of extremal problems., Plenum Publishing Corporation, New York, 1996.

[16] A.G.Chentsov, Asymptotic attainability., Kluwer Academic Publishers, DordrechtBoston - London, 1997.

[17] A.G.Chentsov and S.I.Morina, Extension and relaxations., Kluwer Academic Publishers, Dordrecht - Boston - London, 2002.

[18] A.G.Chentsov, Well posed extensions of unstable control problems with integral constraints, lzvestiya: Mathematics, 63:3 (1999), 599-630.

[19] A.G.Chentsov, To the question about correct extension of a problem about the choice of the probability density under restrictions on a system of mathematical expectations, Uspechi math. nauk, T.50, No 5(305), pp.224-242, 1991 (Russian).

[20] A.G.Chentsov, Universal Properties of Generalized Integral Constraints in the Class of Finitely Additive Measures, Functional Differential Equations, 1-2 (1998), 69-105.

[21] A.G. Chentsov, Finitely additive measures and problems of asymptotic analysis, Non-smooth and discontinuos problems of control and optimization. Proc. volume from the IFAC Workshop, Chelyabinsk, Russia, 17-20 June 1998, Pergamon, 1-12.

[22] A.G. Chentsov, Two-valued measures and extension constructions, Functional Differential Equations, 3-4 (2003), 407-439.

[23] J .Neveu, Bases rnatM.matiques du calcul des prbabilites, Mason, Paris, 1964. [24] A.G.Chentsov, Topological constructions of extensions and representations of attrac-

148 A.G. CHENTSOV

tion sets, Proceedings of the Steklow Institute of Mathematics, Suppl. Issue, 1 (2000), 35-60.

[25] N.Dunford, and J.T.Schwartz, Linear operators, Vol. no. 1. Interscience, New York, 1958.

VOLUME 12

2005, NO 1-2

PP. 149-166

ON VERDUYN LUNEL'S CONJECTURE ABOUT SMALL SOLUTIONS

G. DERFEL' AND A. D. MYSHKIS t

Abstract. The paper discusses the asymptotic behaviuor of the solutions of nonautonomous delay equation

(0.1) y'(t) = a(t)y(t- 1), 0 :=; t < oo,

where a(t) E C"" . Assuming that a(t) # 0 for all t E R.r (though a(t) possibly tends to 0, as t-+ oo), we prove that under some additional conditions on a(t), every solution y(t) of (0.1), decreasing fast enough as t -+ oo, vanishes identically. We consider the case, when a(t) is an 1-periodic function with simple isolated zeros, as well. In this case, due to Verduyn Lunel, equation (0.1) has small (super-exponentially decreasing) solution if and only if a(t) has a sign change. How small can such a small solution be? Under some additional conditions on a(t), we prove, that if

iy(t)l $ Cexp{-'Ytlnt}, t > T > 0,

for some C > 0 and 'Y > 'Yo, then y(t) vanishes identically. Examples demonstrate that our results are sharp up to the constants.

Key words:. nonautonomous delay equation, small solution, the Shilov S spases.

Key Words. Nonautonomous delay equation, small solution, the Shilov S spaces.

1. Introduction. Consider the linear nonautonomous delay equation

(1.1) y'(t) = a(t)y(t- 1), 0:::; t < oo,

' Partially supported by a grant from Israeli Academy of Science and Humanities. Dept. of Mathematics, Ben-Gurion University, P.O. Box 653, 84105, Beer-Sheva, Israel

t Partially supported by grants of Russian foundation for Basic Research and Foundation for Basic Research of ministry of Railways of RF. Moscow State University of Communications, 127994 Moscow, Russia

149

150 G. DERFEL AND A. D. MYSHKIS

where a(t) is a real-valued or complex-valued continuous function. To define a solution of (1.1), one has supply it with an initial condition

(1.2) y(t) = Yo(t), -1 ::; t < 0.

A small solution of (1.1) is a a solution, which decays more rapidly than any exponential [6].

The question whether linear delay equation can possess nontrivial small solution has been studied by D. Henry [7], S. Verduyn Lunel [9, 10, 11], Y. Cao [1], K. Cooke & S. Verduyn Lunel [4], K. Cooke & G. Derfel [2], G. Derfel [3] and others.

The following principal question due to Verduyn Lunel [10], however, is still open.

Problem. ([10]). Suppose that the coefficient a(t) in {1.1} is a realvalued continuous function and that there exist constants m and M such that 0 < m < la(t)l < M. Can equation {1.1) have a nontrivial small solution?

In the case, when a(t) is a real analytic function bounded away from 0 and oo the nonexistence of smal solutions has been proven by K. Cooke and S. Verduyn Lunel [4]. More precisely, they proved the following

Theorem. ([4]). Let a(t) be a real analytic n x n-matrix valued function and y(t), y0 (t) E Rn. If a(t) is bounded throughout R with I det a(t)l > 0, then {1.1)-{1.2} has no nontrivial small solution.

Another important result related to Verduyn Lunel's conjecture is due toY. Cao [1]

Theorem. ([1].) If a(t) be a bounded scalar function such that la(t)l > m > 0, then a small solution of {1.1) must have an infinite number of zeros in any unit interval.

In this paper we deal with (1.1), under various assumptions similar to those in Verduyn Lunel's problem and present some new results related to this problem.

We distinguish between two cases: (a) The coefficient a(t) in (1.1) has no zeros. (b) The coefficient a(t) has zeros. Consider these cases separately.

{a). Suppose that a(t) ol 0 (though a(t) possibly tends to 0, as t-+ oo). We prove that under some additional conditions on a(t), every solution y(t) of (1.1), decreasing fast enough as t -+ oo, vanishes identically. Moreover, the faster a( t) tends to 0 (as t -+ oo) the more restrictive estimates should be

ON VERDUYN LUNEL CONJECTURE 151

imposed on y(t) to guarantee that y(t) = 0. (A precise formulation is given in Theorem 1). For instance, Corollary 1 of Theorem 1 (Section 2 below) states: If b(t) = 1/a(t + 1) satisfies for some k 2 0 the estimate

( 1.3) Jb(nl(t)J :S CBnnn>.tk, 1 :S t < oo; n = 0, 1, ... ,

then any solution y(t) of (1.1) satisfying the estimate

(1.4) Jy(t)l :S Cexp{·-rtlnt} t 2 T > 0

for some C > 0 and 1 > /l = k + 1 + max{1, >.}vanishes identically. Let us compare the latter result with Verduyn Lunel's conjecture. The

conclusion of Corollary 1 is weaker than the conclusion of the conjecture: under the assumptions of Corollary 2 we can not claim that every small solution is identically zero. The solution y(t) of (1.1) has to satisfy estimate (1.4) to be trivial, i.e. y(t) = 0.

On the other hand, the assumptions of Corollary 1 are in some sense less restrictive than the assumptions of the Conjecture. In fact, roughly speaking, condition (1.3) means that Ja(t)l = 1/lb(t -- 1)1 is bounded below by some power function:

Ja(t)l > C1t-k 0 < t < oo

and that its derivatives do not grow too fast. On the contrary, in the Conjecture it is supposed that a(t) is bounded away from 0: Ja(t)l > m > 0 and from oo also: Ja(t)l < M, i.e. 0 < m < Ja(t)l < M.

It is also instructive to compare Corollary 1 with Cooke-Verduyn Lunel's Theorem.

Again, the conclusion of Corollary 1 is weaker than conclusion of the Theorem: superexponential decrease of the solution y(t) is not sufficient to guarantee its triviality- estimate (1.4) is needed for that.

On the other hand, the assumptions of Corollary 1 are (in some sense) also weaker than Cooke--Verduyn Lunel's Theorem. Namely, no boundedness of a(t) on R is required in the corollary, moreover all conditions on a(t) are imposed on R+ only. In addition a(t) is not assumed to be analytic- it can be nonanalytic, when >. > 1 in (1.3).

{b). Suppose now that a(t) has zeros. Almost all existing results in this case, were derived under additional assumption that a(t) is 1-periodic function [12, 8, 11].

The following important result in this direction is due to Verduyn Lunel [11].

Theorem. ([11]) Suppose that a(t) is a continuous 1-periodic function with isolated zeros. then equation (1.1) has small solution if and only if a(t) has a sign change.

For example, the equation

y'(t) = cos(27rt)y(t- 1)

has nontrivial small solutions [12, 6]. The following question arises here: how small can such a small solution

be? An answer to this question is given in Theorem 2 below under some natural conditions on a(t).

If, for instance, a(t) is an analytic 1-periodic function with simple isolated zeros, then every solution of (1.1) satisfying the estimate

[y(t)[ ::; C exp{ -')'tInt}, t > T > 0,

for some C > 0 and I' > 2 vanishes identically. The paper is organized as follows . In Section 2 we formulate the main

results (Theorems 1 and 2). The proofs are based on the technique of the Shilov S spaces. Section 3 is auxiliary, and devoted to presenting all required results on S spaces (Sg, S~,A and S~,[c,dJ). In Senction 4 we provide proofs of Theorem 1 and 2.

2. Main results. We consider the equation

(2.1) y'(t) = a(t)y(t- 1), 0::; t < oo.

As in the introduction, we split the discussion into two cases:

2.a). Suppose that a E coo is a real-valued or complex-valued function and a(t) # 0 for all t E R+. (Note that we do not exclude the possibility that limt->oo a(t) = 0.) Denote

b(t) = 1/a(t + 1)

and rewrite (2.1) in the form

(2.2) y(t) = b(t)y'(t + 1).

The following theorem links together the rate of the decrease of a(t) and the one of the solution y(t) of (2.1), which guarantees that y(t) = 0.

THEOREM 1. Assume that b(t) satisfies the following estimates:

(2.3) [bCnl(t)[ S:: CBnnn"e9Ctl, 0 ::; t < oo, n = 0, 1, ... ,

for some C, B, .\ > 0 and continuous nondecreasing function g(t) 2: 0. Then any solution y(t) of (2.1), which for some C1 > 0 and

{20< .\<1

(2.4) 'Y > 'Yo= 1 + max{1, .\} = 1 '+ .\, .\; 1

satisfies the estimate

(2.5) {t+!

ly(t)l::; C1exp{-'Ytlnt- )1

g(x) dx}, t > T > 0,

vanishes identically. (Throughout the paper we assume that 0° = 1.)

Remark. Theorem 1 remains true, if (2.3) holds for sufficiently large t only.

The following two corollaries follow from Theorem 1 by direct calculation. COROLLARY 1. Let g(t) = k ln(1 + t), where k 2: 0, i.e.

(2.6) lb(n)(t)l::; CBnnn>.(l +t)k, 0 ::; t < oo, n = 0, 1, ...

Then any solution y(t) of (2.1), which for some C1 > 0 and

(2.7) { k +2,

'Y > /'l = k + 1 + max{1, .\} = k + 1 + .\,


0<.\::;1 .\>1

(2.8) iy(t)i::; C1 exp{ -')'t ln t}, t > T > 0,

vanishes identically. COROLLARY 2. Let g(t) = kt, where k 2: 0, i.e.

(2.9) lb(nl(t)l::; CBnnn>-ekt, 0 ::; t < oo, n = 0, 1 , ...

Then any solution y(t) of (2.1), which for some C1 > 0 and

(2.10) 'Y > 1'2 = k/2


(2.11) ly(t)l ::; cl exp{ -')'t2}

vanishes identically.

Proof of Corollary 1. We have:

{t+l {t+1 }

1 g(x) dx = k }

1 In(l + x) dx =::::: kt In(t + 2).

Therefore, for any c: > 0 and sufficiently large t ( t > T > 0 )

l+l g(x) dx:::; kt In [t ( 1 + ·D] = kt [rn t +In ( 1 +f)] :::; (k + c)t Int.

Hence the right-hand side of (2.5) satisfies

{t+l C exp{ -1t In t- }

1 g(x) dx} 2': C exp{ -(r + k + c)t In t}, t > T > 0,

Hence, it follows that, any solution y(t) of (2.1), which satisfies (2.8) (where 1 fulfills (2.7)) satisfies (2.5) as well (with 1 fulfilling (2.4)), and therefore y(t) = 0.

Proof of Corollary 2. For any c > 0 and and sufficiently large t

l, t+l /,t+1 k g(x)dx=k xdx=-[(t+1) 2 -1]

I 1 2

=Ht2(1+~r -1]::::: k(1~c)t2

Thus:

{t+l k(l + 2c) Cexp{-}'tlnt+ }

1 g(x) dx} 2': C1 exp{

2 t 2

}.

Hence, it follows that, every solution y(t) of (2.1), which satisfies (2.11) (with 1 fulfilling (2.10)) satisfies (2.5) where 1 fulfills (2.4) and therefore y(t) = 0.

The following Examples 1 and 2 demonstrate the sharpness of the Corollaries 1 and 2 (up to the constants 11 and 72). ( For the sake of convenience we choose in the following examples initial point t = 1 rather than t = 0.)

Example 1. Consider the equation

(2.12) y'(t) = { -krk ( 1 + D -kt [rn(1 + t) -1 ~ t + 1]} y(t- 1),

1::::: t < oo,

of the form (2.1). Here

1 1 ( 1)kt 1 b(t- 1) =- = --tk 1 +- .... a(t) k t · '- '

The assumptions of Corollary 1 are fulfilled, and b(t) satisfies (2.6) with A= 1. Indeed, b( t) can be analytically continued to the right half-plane Re z > 0, z = t + iy. Moreover in the right half-strip

1 ::; t < oo; \y\ :S C

the following inequality holds

\b(t+iy)\::; C1 (1 +tk).

A slight modification of Theorem 4' from ([5], V.7.6) (with J.L = 0) allows one to conclude that

\b(nl(t)\ :S CBnnntk, 1 ::; t < oo; n = 0, 1, ...

Therefore any solution y(t) of (2.1), which satisfies (2.8) with "! > k + 2, vanishes identically.

On the other hand the function

y(t) = e-ktln(t+l)

is a solution of (2.12) (as is easy to see by a direct calculation) and satisfies (2.8) with any 0 < "! ::; k. This shows the sharpness of Corollary 1 (up to the constant "!).

Example 2. Consider the equation

(2.13) y'(t) = -kte-k(t-f)y(t -1), 1::; t < oo

Here

b(t -1) = _1_ = -~rlek(<-t) a(t) k ·

The assumptions of Corollary 2 are fulfilled for (2.13) and b(t) satisfies (2.9) with A= 1. Indeed, b(t) can be analytically continued to the right half-plane Re z > 0, z = t + iy. In the right half-strip

1 ::; t < oo; \y\ ::; C

we have

jb(t + iy)j :::: cl ekt.

A slight modification of Theorem 4 from ((5], V.7.6) (with J.l = 0) gives

jb(nl(t)j :5: CBnnnekt, 1 :5: t < oo; n = 0,1, ....

Therefore any solution y(t) of (2.13), satisfying (2.11) with 1 > k/2, vanishes identically.

On the other hand, the function

k t' y(t) = e-2

forms a solution of (2.13) and satisfies (2.11) with 1 = ~· This proves the sharpness of the Corollary 2.

2. b). Suppose now that a(t) E coo is a 1-periodic function with simple zeros only (located within the interval (0, 1)) at the points 0 :5: t0 < t1 < · · · tr < 1. The set of all zeros of a( t) denote by S:

s = {ti +kli = o, ... r;k = 0,±1,±2, ... }.

Denote b(t) = 1/a(t + 1) = a(t) fortE R+ \ S. Then

(2.14) y(t) = b(t)y'(t + 1), t E R+ \ S.

In addition, suppose that

(2.15) jb(nl(t)l :5: CBnnn>-j(t- to)(t- tt) · · · (t- tr)l-(n+ll,

for

t E (0,1]; t-/= ti i = 0, ... , r; n = 0, 1, ....

THEOREM 2. Suppose the coefficienta(t) in (2.1) is a 1-periodicfunction with simple isolated zeros such that conditions (2.15) hold for b(t) = 1/a(t).

Then any solution y(t) of (2.1}, which for some C1 > 0 and

{ 2, 0 <.A :5: 1

(2.16) l>lo=1+max{1,.A}= 1 +-A, A> 1


(2.17) jy(t)l :5: C1 exp{-1tlnt}, t > T > 0,

vanishes identically. The following Example 3 demonstrates the sharpness of Theorem 2.

Example 3. Consider (following Zverkin [12]) the equation

(2.18) y'(t) = cos(211t) y(t- 1), 1:::; t < oo.

Here a(t) = cos(211t) is a 1-periodic function with zeros t0 = 1/4, t1 = 3/4 on [0,1]. Define b(t) = 1/ cos(211t) fortE [0, 1] \ u, 1}. Then

(2.19) \b(n)(t)\:::; CBnnn l(t- D (t- Dl-(n+1),

t E [0, 1] \ {1, n; n = 0, 1, ... ,

i.e. b(t) satisfies (2.15) with .A = 1. Hence, according to Theorem 2, any solution y(t) of (2.18), which satisfies (2.17) with"!> 2, vanishes identically.

On the other hand, according to Zverkin [12] the equation (2.18) has nontrivial small solution y(t) satisfying (2.17) with"!= 1- c: i.e.

(2.20) jy(t)\:::; Cexp{-(1-c:)tlnt}, t > T > 0,

which proves the sharpness of Theorem 2. For the sake of convenience we give here a sketch of Zverkin's construc

tion. Note first that (2.18) has the following solution of Floquet type:

Y1(t) = exp { 2~ sin(211t)}.

Secondly, note that (2.18), supplemented by the initial condition

y(t) = 1, 0 :::; t < 1,

has an additional solution

[t] Yz(t) = L ~ (sin(211t))k

k=O k. 211 ' 0:::; t < oo,

which can be constructed using the step-by-step method. The difference Yz(t) -y1 (t) is a solution of (2.18), as well. To estimate it, expand y1(t) into a Taylor series with the remainder term in Lagrange form. We obtain

Y = Y1- Yz = _1 _ (sin(211t)) [t+l] x II sin(211t) _ ( 1 )

[t + 1]! 211 e P 211 -0

(211)'f(t) '

where 0 < II < 1. By Stirling's formula, (2.20) follows.

3. The functional spaces of S type (Sg, S~,A> S~,[c,dJ) and their properties. In this section we present some basic definitions and properties of the Shilov spaces (henceforward S type spaces) to be used in Section 4 for the proofs of Theorem 1 and 2. Our presentation is rather formal. For a thorough treatment of the subject we refer the reader to Gel'fand and Shilov [5].

3.1. The space sg (a :::>: 0, (J :::>: 0). sg is defined as the set of all infinitely differentiable functions 0 are constants (and may depend on 'P).

3.2. The space s~.A (a :::>: 0, fJ :::>: 0). s~,A is the set of all functions 0 the inequality

(3.2) lxk<p(q)(x)l ~ C,(A + o)k Bqkk"qqfJ (k,q = 0, 1,2, ... ),

where C6, A, B > 0 are constants (and C6, B may depend on <p). In particular, sg A is the set of all compactly supported functions :A.

It is known that S~ A is nontrivial for any a, (J, A such that a+ (J > 1 ([5], IV.8.3). '

3.3. A Lagrange type lemma for sg,A. Let .

If, it follows from (3.4), that j(x) = 0 almost everywhere, then 1) is not rich in functions ([5], IV.8.4). Indeed, all functions <p E sg,A are supported on [-A, A]. Therefore any function f(x) ;j3. 0 on R but such, that f(x) = 0 on [-A, A] satisfies (3.4).

Nevertheless, if j(x) is assumed to be supported on [-A, A] , then (3.4) implies that f ( x) = 0 for all x E R.

In other words, the following Lagrange type lemma holds.

LEMMA 1. Let f(x) be locally integrable function such that supp f C [-A, A]. Assume also that

(3.5) Loo f(x)<p(x) dx = ;_: f(x)<p(x) dx = 0

for all 1). Then f(x) = 0 almost everywhere. The proof of the lemma is analogous to the proof of the corresponding

Lemma in ( [5], IV.8.4). It is based on the following two observations: (i) For any x0 E (-A, A) there exists a function 1), then for any a E R we have <p(x)eixu E St,A

([5], IV.4.2).

3.4. The space sg,[c,d]' Define

sg,[c,d] = { 1)1: R --t R, 1/!(x) = 'P [d2~c (x- c; d)] I 'P E sg,A}

In other words, functions 1)1 E sg,[c,d] are derived from the corresponding

functions <p E sg,A by means of an affine transformation of their arguments.

If 1/J E sg,[c,d] then supp 1/J <;;; [c, dj and

(3.6) \1/!(q)(x)\ :S CBqqq13, X E [c, dj, q = 0, 1, ...

Furthermore, sg,[c,dJ is nontrivial and Lemma 1 is valid in sg,[c,d] for {3 > 1.

3.5. The behaviour of functions from sg,A at the end points of [-A, A]. As noted above, elements of sg A are infinitely differentiable functions supported on [-A, A]. Therefore if, <p E sg,[c,d] then it tends (together with all its derivatives) to 0 as t --t ±A.

A quantitative version is given by

LEMMA 2. Let 0 (and may depend on <p ).

'•""''' ·~"-


Proof Apply Taylor's formula (with the reminder term in Lagrange's form) to the function <p(m)(x) on the interval [-A, A] at the point x = A. We obtain

(3.8) (m+l)(A)

<p(m)(x) = <p(m)(A) + 'P (x- A) 1!

'P(m+2)(A) 'P(m+q)(~) + ~, (x- A) 2 + · · · +

1 (x ,-- A)q,

. q.

where -A < ~ < A. Taking into account the fact, that <p(k)(A) = 0 for all k = 0, 1, ... , we get from (3.8)

(3.9) /'P(m)(x)/ ~ inf .!..,/'P(m+q)(~)/(A- x)q, m = 0, 1, ... q2:0 q.

Substituting (3.3) into (3.9) we find that

(3.10) /'P(m)(x)/ < inf .!..csCm+q)(m + q)I'Cm+q)(A- x)q - q:?:O q! '

m=0,1, ... , xE[-A,A].

Put q = 2n in (3.10). Then we get

(3.11) /'P(m)(x)/ ~ 2~!CB(m+2n)(m + 2n)i1(m+2n)(A- x)2n,

m, n = 0, 1, ... , x E [-A, A].

Similarly, applying to <p(m)(x) Taylor's formula at the point x = -A (rather than A) we get

(3.12) /'P(m)(x)/ ~ 2~1 csm+2n(m + 2n)i1(m+2n)(A + x)2n, x E [-A, A].

Combining (3.11) and (3.12) we obtain (3.7).

Remark. Let 'P E sg,[c,d]· Then for all m, n = 0, 1, ... :

(3 13) /'P(m)(x)/ < - 1-csm+2n(m + 2n)i1(m+2n)[(x- c)(d- x)Fn · - (2n)! ·


4. Proofs of Theorems 1 and 2.

Proof of Theorem 1. Let y(t) be a solution of (2.1) and tp(t) an arbitrary function in sg,[O,l] where {3 > 1. sg,[O,l] is nontrivial for {3 > 1 and Lemma 1 is valid for it. Consider the integral

(4.1) J = /_: y(t)tp(t) dt = l y(t)tp(t) dt.

Applying equation (1.2) and using integration by parts we can rewrite J in the form

J = f y(t)tp(t) dt = lal y'(t + 1) ( b(t)'P(t)) dt

- y(t + 1)b(t)tp(t) 1:- f y(t + 1) (b(t)tp(t))' dt

(4.2) = - { y(s) (b(s- 1)tp(s- 1) )' ds = h

Continuing this process over and over we obtain

(4.3) J = ( -1t f+l y(t) (b(t- 1) (b(t- 2) ...

')' · · · ( b(t- n)tp(t- n) )'- .. ) dt = ln.

'---v---' n times

If we perform all n differentiations in ( 4.3), then (before collecting similar terms) we obtain in the integrand a sum of (n + 1)! terms of the form:

b(kl)(t -1)b(k,)(t- 2) .. ·b(kn)(t- n)tp(kn+ll(t- n)

where 0 :S: k1 :S: 1, 0 :S: k2 :S: 2, ... , 0 :S: kn, kn+l :S: n, k1 + k2 + · · · + kn + kn+l = n. For brevity sake, denote the set of indices (k~o ... , kn, kn+l) E zn+l

described above by K and the above mentioned sum by E(k,, ... kn,kn+llEK· The sign E' means that some terms may appear in the latter sum more than once (with the total number of terms, before collecting similar terms, equals (n + 1)!). Thus

n+l

(4.4) J = Jn = (-l)n I y(t) L:' b(ki)(t -1)b(k,)(t- 2) • • • n (kl, ... kn,kn+l)EK

... b(kn)(t- n)<p(kn+ll(t- n) dt.

To prove Theorem 1 it suffices to show that

(4.5) lim Jn = 0. n-too

In fact, ( 4.5) implies that

(4.6) J = l y(t)<p(t) dt = 0

for all 'P E sg,[o,l]' By means of Lemma 1 (Section 3), it follows that y(t) = 0 for all t E [0, 1]. Applying (2.1) for stepcby-step continuation of the solution we obtain finally that y(t) = 0 for all 0::; t < oo.

We return now to prove (4.5). From (4.4) we have •

(4.7)

n+l

IJnl :S:: I ly(t)l·l L 1 b(k,)(t- 1)b(k,)(t- 2) · · ·

n (kt, ... ,kn,kn+dEK

.. ·b(kn)(t- n)<p(kn+d(t- n)l dt :S:: max ly(t)l(n + 1)! n$t$n+1

X max lb(kl)(t -1) · · ·b(kn)(t- n)<p(kn+d(t- n)j. n:St:Sn+l

(kl,··· 1kn,kn+l)EK

Next, plug in (4.7) the estimates (2.3) and (3.3) for b(t) and <p(t), correspondingly. Taking into account Stirling's formula for (n+ 1)! we obtain

lJnl ~ max ly(t)IC1D?nn max e-k,eg(n)e·k,eg(n-1) ... e·k"eg(J)k!3k.+J n~t:Sn+l (kl, .. ,kn,kn+l)EK 1 2 n n+l

~ max ly(t)IC1Dnnn max n·*'+ +kn)eg(1)+g(2)+ .. +g(n)k!3kn+J

n<t<n+l 1 (k k k )EK n+J -- l, .. ,n,n+l

(4.8) ~ max ly(t)IC1Dnn(l+v)nef;+J g(t)dt

n:St.Sn+l 1 ,

where C1 , D1 > 0 and v = max{-\, ,8}. Substituting estimate (2.5) for y(t) in ( 4.6) we get

(4.9) IJnl ~ C2D~e(-Hl+v)nlnn,

where C2 , D2 > 0. It remains us to show that

( 4.10) _, + 1 + IJ < 0.

To this end we apply (2.4). Denote c: = (!- lo)/2 and put ,B = 1 + c:. Then, by (2.4)

1 > (o + c:?: 1 + max{l + c:, -\} = 1 + max{,B, -\} = 1 + v

i.e. (4.10) holds. This implies (4.5) and proves the theorem.

Proof of Theorem 2. The proof proceeds similarly to the proof of Theorem 1. The most important feature of the present case is the discontinuity of b(t) at the zeros of a(t). However, assuming the periodicity of b(t) and controlled rate of growth of b(t) at its singularity points (see (2.15)) we can treat this case similarly to the previous one. To do this, we shall choose the space sg,[c,dJ of test functions <p in such a way that the functions <p vanish together with all their derivatives at the singularity points of b(t). This will compensate the singularities of b(t) and assure the existence and convergence of the related integrals Jn.

We now turn to the proof itself. It is sufficient to demonstrate that every solution y(t) of (2.1), which satisfies (2.17), vanishes on each of the intervals (0, t0), (t0 , t1), ... , (t"' 1). By the continuity ofy(t), this implies that y(t) = 0 on [0, 1] and therefore y(t) = 0 for all t?: 0.

Without loss of generality we demonstrate that y(t) = 0 on [t0 , tl]. Let y(t) be a solution of (2.1) and <p(t) an arbitrary function in s{[to,t,]'

where ,B > 1. As in the proof of Theorem 1 denote

(4.11) !00 rt1 J = -oo y(t)<p(t) dt = lto y(t)<p(t) dt.

Applying (2.14) and then integration by parts we can rewrite J in the form

11!+1 ( )' (4.12) J =- y(s) b(s- 1)<p(s- 1) ds = h to +I

(Similarly to (4.2) in the proof of Theorem 1). Note that, despite the unboundedness of b(t) and its derivatives in the

vicinity of the set S (zeros of a(t)) integral J 1 is proper. The reason is that b(t), its derivatives and shifts appear in (4.12) in combination with <p(t), its derivatives and shifts only. Since the latter ones rapidly tend to zero as t approaches S, the integral J1 is proper.

Repeating the above procedure (application of (2.14) and subsequent integration by parts) n times we get

1n+tl ( ( (4.13) J = ( -l)n X y(t) b(t- 1) b(t- 2) · · ·

or

n+to

')' .. · (b(t- n)<p(t- n) )'- ·) dt = ln

n+h

~ n times

(4.14) J = Jn = (-l)n X J y(t) ,E' b(k,)(t -1)b(k2l(t- 2) · · · n+to (kt. ... ,kn,kn+l)EK

.. ·b(knl(t- n)<p(kn+ll(t- n) dt

similarly to (4.3) and (4.4) in the proof of Theorem 1. Note that the integrals in (4.13) and (4.14) are proper, as have been

explained already. Also the total number of terms (before collecting similar ones) in the right-hand side of (4.14) is (n + 1)!.

Herefrom we have

( 4.15)

X

IJnl :S:: max Jy(t)J(n + 1)! n+to$t$n+tt

max jb(kd(t -1)b(k2l(t- 2) · · ·b(knl(t- n)<p(kn+ll(t- n)l n+to$t$n+ 1

(kl, ... ,kn,kn+l)EK

similarly to (4.7). Taking into account the 1-periodicity of b(t), from (4.15) we obtain

( 4.16)

X

IJnl:; max ly(t)l(n + 1)! n+to:St::Sn+t1

max to::Ss:Stt

lb(k1l(s)b(k,)(s) ... b(kn)(s)<p(kn+tl(s)l·

(k1 ) ... ,kn,kn+l )EK

Plug the estimates (2.15) and (3.13) for b(t) and <p(t), correspondingly in (4.16). We get

IJnl:; max ly(t)l(n+1)! n+to:St:Sn+tt

max to:St::;t1

C nBnki-kt ki-k2 k!-kn 0 0 1 2 "' n

(k1 , ... ,kn,kn+l )EK

(4.17) x[(t2- t!) ... (tr-1- t1)(tr- t1)]-[(kt+1)+·+(kn+1)]

x[(t- t0)(t1 - t)tf(kt+l)+·+(kn+t+l)]

X (2~)!CB(kn+t+2n)(kn+l + 2n)fl(kn+t+2n)[(t _ to)(tl _ t)J2n.

Proceeding further we have

(4.18) X

II l()l(n+1)!cnn

Jn :; max Y t (2 )I 1 B1 n:St:Sn+l n .

max n>-(kt + .. +kn) (3n )fl(kn+l +2n)

(k1 , ... ,kn ,kn+l )EK

:; max ly(t)IC2B~n-n max n"'(kt+ .. +kn)+flkn+tn2fln n:S_t:Sn+l (k 1, ... ,kn,kn+i)EK

:; max ly(t)IC2Bnn(2{J+v-1)n n:St:Sn+l 2 ,

where C2, B2 > 0 and v =max{ A, ;J}. Next, substitute estimate (2.17) for y(t) in (4.18) to obtain:

(4.19) IJnl :; C3 B~n-(7-2fl-v+l)n.

It remains to show that under (2.16) we can choose fJ > 1 such that

-('y- 2(J- v + 1) < 0.

Indeed, denote c = ('y- 'Yo)/2 and put (J = 1 + c/4. Then, by (2.16)

'Y > 'Yo+ c = 1 + c + max{1, ,\} = (1 + c/2) + (c/2 + max{1, ,\})

( 4.20) :;::: (2(J- 1) + max{1 + c/4, ,\} = (2(J -1) + max{(J, ,\}

= 2(J + v- 1.

It follows from (4.20) and (4.19) that J = lim Jn = 0, which completes the n-+oo

proof.

Acknoledgements. The authors are thankful to D. Berend, K. Cooke, A. Gordon, S. Molchanov, S. Verduyn Lunel, F. Vogl for stimulating discussions and useful remarks.

REFERENCES

[1] Y. Cao, The discrete Lyapunov function for scalar differential delay equations, J. Diff. Equations 87 (1990), 365-390.

[2] K. Cooke and G. Derfel, On the sharpness of the theorem by Cooke and Verduyn Lunel, J. Math. Anal. Appl., 197 (1996), 379-390.

[3] G. Derfel, How small can a "small solution" be? Proceedings of the Equadiff 2003 (Editors: F. Dumortier, H. Broer, J. Mawhin, A. Vanderbauwhede and S. Verduyn Lunel), World Scientific, 2004, 750-756.

[4] K. Cooke and S. Verduyn Lunel, Distributional and small solutions for linear differential delay equations, Differential and Integral Equations, 6 (1993), 1101-1117.

[5] I. M. Gel'fand and G. E. Shilov, Generalized Functions II, Academic Press, New York and London, 1968.

[6] J. K. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, New York, 1993.

[7] D. Henry, Small solutions of linear autonomous functional differential equations, J. Diff. Equations 8 (1970), 494-501.

[8] Yu. Lyubich and V. Tkachenko, Floquet's theory for equations with retarded arguments argument, Diff. Equations, 5 (1969), 648--656.

[9] S. Verduyn Lunel, A sharp version of Henry's theorem on small solutions, J. Diff. Equations 62 (1986), 266-274.

[10] S. Verduyn Lunel, Small solutions and completeness for linear functional differential equations, 127-152 in "Oscillations and Dynamics in Delay Equations" (ed. J. Graef and J. Hale) Contemp. Math. 129 Amer. Math. Soc., Providence, RI 1992.

[11] S. Verduyn Lunel, About completeness for a class of unbounded operators, J. Diff. Equations 120 (1995), 108-132.

[12] A. M. Zverkin, The completeness of a system of Floquet type solutions for equations with retardations, Diff. Equations, 4 (1968), 249-251.


VOLUME 12 2005, NO 1-2 PP. 167-173

FLOATING AND PARTLY IMMERSED BALLS IN A WEIGHTLESS

ENVIRONMENT '

R. FINN t

Dedicated to Anatolii Dmitrievich Myshkis, on the occasion of his eighty fifth anniversary

Abstract. It is shown that a ball floating in symmetric capillary equilibrium in an infinite liquid bath in the absence of gravity cannot symmetrically be pushed deeper into the liquid. This phenomenon is discussed from the point of view of approximation of the infinite bath by cylindrical tanks of increasing radius. A singular perturbation appears in the limit of infinite radius, of a sort that seems not easy to reconcile with physical experience.

Key Words. Capillarity.

AMS(MOS) subject classification. 76D45, 53Al0

1. Initial remarks and configuration. Capillarity phenomena can manifest themselves in idiosyncratic ways, which would not ordinarily be predicted. Some of these occurrences are described in the survey article [1]. In the present article I point out yet another phenomenon that seems at variance with everyday experience, but which is clearly predicted by the

' I wish to thank Gerald Fuller for very helpful comments on the physical background and for informing me of relevant literature. I am indebted to the Max-Planck-Institut fiir Mathematik in den Naturwissenschaften, in Leipzig, for its hospitality during preparation of the paper.

t Mathematics Department, Stanford University, Stanford, CA 94305-2125, [email protected]

167

168 R. FINN

FIG. 1. Floating ball in absence of gravity

formal equations that describe the free surface interfaces separating adjacent fluids that do not mix. Experiments have verified such predictions in the past, and may be expected to do so also in the present instance. Background information as to the general equations and boundary conditions can be found in [2]. We take here as starting point that a capillary free surface interface S in the absence of an external force field (such as gravity) is a surface of constant mean curvature H. When described as a graph in local coordinates, the height u(x, y) satisfies the equation

(1) divTu = 2H, Tu = \Tu - )1 + 1Vul2 ·

The constant H is determined by the physical constraints and by the boundary conditions. If S abuts on a rigid (support) surface Z of homogeneous material, then it meets Z in a contact angle 'Y depending only on the materials, and in no other way on the conditions of the problem. For surfaces S of common experience separating a liquid with a gas or vacuum, it is customary to measure 'Y within the liquid. We consider first the particular configuration that appears when a solid spherical ball of homogeneous material floats freely in equilibrium, in a fluid (such as water) of infinite extent, in the absence of gravity. We assume that the fluid surface is asymptotically flat at infinity, at a known finite height, which we may take to be u = 0. Figure 1 illustrates a particular solution to the problem, for a value 'Y in the range 1r /2 < "( < 1r.

We assert first: The solution u = 0 of Figure 1 is unique among all symmet-

ric configurations for which the ball meets the liquid surface in the constant angle 'Y· To see that, observe that under our hypotheses there will be a

neighborhood fl of infinity in the (x, y) plane, in which the surface interface

PARTLY IMMERSED BALLS 169

z

a c r

FIG. 2. Nodoid

becomes a graph u(x, y) satisfying (1). Consider a disk BR C n of radius R. Integrating (1) over BRand applying the divergence theorem, we find

(2) 271" R2 H = f v · Tu ds 8Bn

where v denotes the unit exterior normal on 8BR. Since \Tu\ < 1, we find from (2) that \H\ < 1/ R, and since R can be chosen arbitrarily large, there follows H = 0. That is, u(x, y) describes a minimal surface symmetric about the z axis. The only such surfaces other than horizontal planes are catenoids, which are unbounded at infinity. Thus, u(x, y) is identically constant, from which the statement follows.

2. The anomaly; attempt at clarification. Starting with a ball of radius R in equilibrium as in Figure 1, we push it rigidly downward into the fluid, a distance h Jess than the height of the ball above the fluid. We obtain a "partial immersion" problem, for which the ball is held rigidly at a prescribed level. The same reasoning as above shows that the fluid must be a horizontal plane. But at this new level, the ball does not meet the plane in the correct angle. We conclude that there is no symmetric solution for the partial immersion problem, for which the fluid maintains its original level at

170 R. FINN

infinity. This result seems at first to conflict with what one would anticipate on the basis of everyday physical experience, and suggests that the problem may be incorrectly posed. In order to determine what actually happens, we consider an intermediate configuration, in which a prescribed volume of liquid partially fills a cylindrical container of (large) radius c that is closed at the bottom, and we allow the ball to float in symmetric equilibrium. We suppose the material of the container to make a contact angle 1r /2 with the liquid. We again find a horizontal surface as an exact solution, as indicated by the dashed lines of Figure 3, which refer to the floating configuration. Again we push the ball rigidly downward the same distance h. In this finite configuration, the fluid is no longer constrained to be horizontal, and in fact we can find an explicit solution in terms of elliptic integrals, as a portion of a rotation surface of constant mean curvature known as a nodoid, see, e.g. [3],[4]. Figure 3 indicates the ball in that constrained condition. Figure 2 illustrates a typical profile curve C of a nodoid, which we denote as a nodary. The curve is periodic in the vertical coordinate, and is completely determined up to rigid vertical motion by specifying the distances r = a of the inner vertical, and r = c of the horizontal point of the inner loops. In the configurations we will encounter, a will be small and c large, yielding inner loops that are much flatter and longer than those of the figure. We seek to choose an upper segment of an inner loop, in an interval a* < r < c, a* > a, as profile for the surface interface, as indicated in Figure 3. From (1) we find the polar representation

(3) (rsin !f)r = 2Hr

where 7{1 is the inclination angle with the horizontal direction. At the inner vertical, we may assume 7{1 = 1r /2. Integrating from that point, we obtain

(4) rsin!f- a= H(r2- a2

)

from which

(5) H= a

c2- a2 .

Solving the quadratic (4) for r, we obtain

(6)

The substitutions

(7)

2Hr =sin 7{1- /sin2 7{1 + 4fl2c2.

7{1 = 7[ /2 - <p, k = c2- a2

c2 + a2

PARTLY IMMERSED BALLS

FIG. 3. Depressed ball in a finite tank; reference floating configuration

lead to

(8) r = 1 ~ k { -k cos <p + )1- k2 sin2 <p}

in an interval of the form 0 < 'Po < <p < 1r /2. We have

(9) du = tan'lj; = cot<p, dr

du dr - = cot<pd<p d<p

from which we obtain

(10) du { k 1 ; . 1 + k } - =a --cos <p- --y 1- k2 sm2 0, dr / d 0; thus the height u(r) is strictly increasing on the entire interval a < r < c. One shows formally that d2u/ dr2 < 0 on such a loop. The total change in height from the vertical point r = a to the horizontal point r = c is exactly

(11) 5 =a {(1 + k)K(k) _ E(k)- k} 1-k

where K and E are complete elliptic integrals of the first and second kind. This quantity bounds from above the total change of height on the free surface S. We must adjust Cto respect the volume constraint and also the boundary angles on the container and on the ball. By choosing c to be the radius of the container, the prescribed angle 1r /2 at that position is automatically achieved. Since C must intersect the meridian profile on the ball, a is restricted to the interval 0 < a < R. We are interested in configurations for which c is large. By (7) and by the bound on a, we are restricted to configurations for which

172 R. FINN

k is near unity. Using known representations for elliptic integrals, we find asymptotically as k -+ 1 that

(12) 4

0"' a(1 + k)ln v1- k2

Considering the volume determined under the entire curve, the amount lying interior to the ball cannot exceed 27r R3 /3, and becomes small relative to the fluid volume as c -+ oo. But as c -+ oo we find k -+ 1 since a is bounded. Thus if a is bounded from zero we would have o -+ oo by (12), and the total fluid volume would become unbounded relative to the volume for the reference solution. Thus, the volume constraint imposes the requirement a -+ 0 as c -+ oo. From the contact angle requirement we conclude in particular that a fixed segment a < r < r0 of C lies interior to the ball, for all k close enough to unity. Now for any fixed a and rp < 1r /2 we find from (8) that

(13) lim r(a; k; rp) = (cosrp + sinrp)a < av'2 k-t!

Thus, for any such rp, the corresponding point on C will lie interior to the ball, whenever 1 - k is small enough. That is, for k close enough to unity, any given interval1r /2- rp < c: will cover the entire curve exterior to the ball. By (9), the slope of the curve must tend uniformly to zero exterior to the ball, as the size of the container becomes infinite. We conclude that if the ball is pushed downward a distance h > 0, then interior to any fixed sphere E the free surface moves ultimately downward a distance arbitrarily close to that same value. It remains to complete the proof that by choice of a and by rigid vertical motion of C, the prescribed boundary angle 1 and the prescribed volume of fluid can be achieved. To prove that, we start by choosing o, c > R and choose a to be the unique solution of

(14) c = ae/ila

By (12), if c and o are large enough, then the fluid volume determined by the nodoid will exceed the initial volume Vo "' 1rc2u0 . That is so, since outside B the nodary height lies above the conical surface determined by the line joining (xo, uo-h) with (c, u 0 -h+o). Here we have neglected the volumes contained in and under B, which are bounded independent of c and hence do not affect the substance of the statement when c is large. We assume such a choice and proceed iteratively, setting h1 = h and shifting the entire horizontal fluid surface rigidly downward that distance. This maintains correct contact

PARTLY IMMERSED BALLS 173

angles, but makes the volume too small. Let I denote a horizontal segment of non-zero length less than r0 along the r-axis, centered at (r0 , u0 - h). By the above comments, the rate of change of slope with height for the local family of nodaries with vertical points on the limiting straight line through (ro, uo -h) tends to zero with increasing c. But the rate of change of slope with height for the profile curve of B is bounded from zero at that level. Thus, if we take as next step in our procedure a choice of a (with c fixed) that makes the volume correct (which we can do by decreasing a, making the nodary slope still smaller) then the contact angle will be too small. We now move the nodoid rigidly downward, to correct the contact angle, and continue in that fashion. We obtain a decreasing sequence a1, bounded from zero, and an increasing sequence hl, bounded above by R, both of which converge to yield the required configuration.

3. Comments. On considering the global effects of depressing the ball a fixed distance h with increasing c, it is clear that as c -+ oo fluid displacements of unboundedly large volume occur. That behavior appears to suggest that the force needed to achieve the motion becomes unboundedly large, unless the motion becomes arbitrarily slow. Both these eventualities seem at variance with experience, and it therefore appears that dynamically complex time dependent motion could occur, as c -+ oo. An alternative is that the symmetric solutions constructed here could branch into asymmetric surfaces, in which the disturbances are better localized. Such a circumstance could open up new and general questions, as very little is known about minimal surfaces with the required asymptotic behavior in exterior domains, see the discussion in [5], p. 113-116. In the absence of a precise analytical investigation on this point, it would seem desirable to conduct experiments.

REFERENCES

[1] R. Finn, Eight remarkable properties of capillary surfaces, Math. Intelligencer, 24(2002), 21-33.

[2] R. Finn, Equilibrium capillary surfaces, Grundlehren der Mathematischen Wissenschaften, 284, Springer-Verlag, New York, 1986.

[3] C. Delaunay, Sur la surface de revolution dont la courbure moyenne est constante, J. Math. pures et appl, 1(6) (1841), 309-320.

[4] R. Finn, and T. I. Vogel, On the volume infimum for liquid bridges, z. Anal. Anwendungen, 11 (1992), 3-23.

[5] R. Ossermann, A survey of minimal surfaces, Dover Publications, (1986).


VOLUME 12 2005, NO 1-2 PP. 175-185

THE AIZERMAN-MYSHKIS PROBLEM FOR FUNCTIONAL-DIFFERENTIAL EQUATIONS WITH CAUSAL

NONLINEARITIES

M.GIL' •

Abstract. We consider a class of nonlinear scalar functional differential equations of an arbitrary order. The equations have separated linear parts and sublinear causal nonlinearities. Explicit absolute stability conditions are established, provided Green's functions of the considered equations are positive. It is also shown that these equations satisfy the Aizerman - Myshkis hypothesis. In addition, new conditions providing existence of positive solutions are established. Applications of the above mentioned results to functional differential equations with nonlinear Volterra operator are also discussed.

Key words: nonlinear equations with delay, causal mappings, absolute stability, the Aizerman - Myshkis problem, Volterra integra-differential equations.

AMS (MOS) subject classification: 34 K 20

1. Introduction and notation. In 1949 M. A. Aizerman conjectured the following hypothesis: let A, b, c be an n x n-matrix, a column-matrix and a row-matrix, respectively. Then for the absolute stability of the zero solution of the equation x == Ax+ bf(cx) in the class of nonlinearities f : R 1 -t R 1

,

satisfying the condition

0::; j(s)js::; q (q ==canst> O,s E R 1, s # 0),

it is necessary and sufficient that the linear equation x == Ax+ q1bcx be asymptotically stable for any q1 E [0, q] [1].

• Department of Mathematics Ben Gurian University of the Negev P.O. Box 653, Beer-Sheva 84105, Israel. This research was supported by the KAMEA program.

175

176 M. GIL'

These hypothesis caused the great interest among the specialists. Counterexamples were set up that demonstrated it was not, in general, true, cf. [16]. Therefore, the following problem arose: to find the class of systems that satisfy Aizerman's hypothesis. The author has showed that any system satisfies the Aizerman hypothesis if its impulse function is non-negative [6]. The similar result was proved for multivariable systems, distributed ones and in the input-output version. For the the details see [7]. On the other hand, A.D. Myshkis [14, Section 10] pointed out at the importance of consideration of the generalized Aizerman problem for retarded systems. In [8] that problem was investigated for the retarded systems, whose nonlinearities have discrete constant delays; in [9], more general systems with nonlinearities acting in space C were considered. In this paper we investigate the pointed problem for equations with causal nonlinearities acting in V. In addition, new conditions providing existence of positive solutions are established. They supplement the well-known results, cf. [5, 11]. Denote R+ = [O,oo), C_ = {z E C: Re z < 0} and~= [-17,00) for a positive 1). C ( w) is the space of continuous scalar-valued functions defined on a set w with the sup-norm. V(w) (1 ::::; p::::; oo) is the space of scalar-valued functions defined on w with the usual norm. A mapping F: V(~) -+ £P(R+) is called a causal nonlinearity, iffor a given t:::: 0 and arbitrary u1, u2 E V(~), satisfying condition u1 ( s) = u2 ( s) for almost all -1) ::::; s ::::; t, we have the equality [Fu1](s) = [Fu2](s) for almost all 0::::; s ::::; t, cf. [3]. In this paper we investigate the scalar equation

x(n)(t) + ~ [ x(k)(t- r)drk(r) = Q(D)[Fx](t) (t > 0), (1.1)

where rk (k = 0, ... , n -1) are bounded nondecreasing functions defined on a finite segment [0, 1)], D = d/dt,

Q(>.) = 1 + ... + bmAm (bk =canst; k = 1, ... , m < n)

is a real polynomial; F: V(~) -+ V(R+) is a causal nonlinearity. Let K(.) be the characteristic function of the left part of ( 1.1):

n-1 1J

K(>.) =An+ L ).k 1 e->-r drk(r). k=O O

Throughout this paper it is assumed that all the zeros of K (.) are in C _. Introduce the function

1 100

. G(t) :=- e1'wK- 1 (iw)Q(iw)dw (t ::0: 0). 211' -00

THE AIZERMAN - MYSHKIS PROBLEM 177

Let us take the initial conditions

x(kl(t) = ¢k(t) ( -ry::; t::; 0; k = 0, ... , n- 1) (1.2)

with given continuous functions ¢k· A mild solution of problem (1.1}, (1.2) is a continuous function x defined on Rry and satisfying (1.2} and the equation

x(t) = w(t) + l G(t- s)[Fx](s)ds (t > 0), (1.3)

where w is a solution of the linear equation

n-1 "1

w(n)(t) + ~ 1 w(k)(t- T)drk(T) = 0 (1.4)

with the initial conditions (1.2). This definition is motivated by the fact that a solution of the equation

n-1 17 x(n)(t) + ~ 1 x(k)(t- T)drk(T) = Q(D)j(t) (1.5)

with a given smooth function f can be written as

x(t) = w(t) + l G(t s)f(s)ds.

The existence of mild solutions is assumed. About various existence results see [2, 3, 13, 10]. We need the following simple result.

LEMMA 1.1. Let G(t) be a non-negative function. In addition, let

[Fv](t) 2 0 for almost all t 2 0 and any nonnegative v E LP. (1.6)

Then (1.1} has nonnegative mild solutions. Proof. Take the initial conditions in such a way that w(t) = G(t). Then

there is a t0 > 0, such that from (1.3) it follows that x(t) 2 0 (t ::; to) and x(t) 2 G(t) (t::; t0 ). Extending this inequality to R+, we get the result. D

2. The main result. It is assumed that there is a constant q 2 0, such that, for a p 2 1,

IIFv!ILP(R+) = [[" I[Fv](t)IPdt] 1iP :S qllviiLP(R") (v E LP(R.ry). (2.1)

178 M. GIL'

DEFINITION 2.1. The zero solution to equation {1.1) is said to be absolutely LP-stable in the class of nonlinearities (2.1}, if there is a positive constant M0 independent of the specific form of functions F (but dependent on q), such that

n-l

llxlb(R+) :S MoL II<Pklbr-~.o] (2.2) k=O

for any mild solution x(t) of problem (1.1}, (1.2). Let us consider the following (generalized Aizerman- Myshkis) problem: Problem 1: To separate a class of systems (1.1), such that the asymp

totic stability of the linear equation

n-!1~ x(n) + L x(k)(t- r)drj(r) = ijN(D)x(t)

k=O O

(2.3)

with some ij E [0, q] provides the absolute LP -stability of the zero solution to (1.1) in the class of nonlinearities (2.1).

THEOREM 2.2. Let

G(t) ;:: 0 (t;:: 0) (2.4)

and

q < K(O). (2.5)

Then the zero solution to equation (1.1) is absolutely LP-stable in the class of nonlinearities (2.1).

Proof Due to the Young's inequality [15, p. 192],

11[ G(t- s)f(s)dsiiL" :S 1'"" IG(s)lds IIJIIL" (f E LP(R+)).

The Laplace transform implies

Therefore

100

e-zsG(s)ds = Q(z). o K(z)

100

1G(s)lds= 1'>0 G(s)ds= ~~~~·

THE AIZERMAN - MYSHKIS PROBLEM

But Q(O) = 1. Thus

I [' ( ( ) II IIJI!Lr I Jo G t- s)f s ds LP:::; K(O).

Now from (1.3) it follows that

IIFxiiLr(R+) llxiiLr(R+) :::; llwiiLP(R+) + -· · · ·

Relation (2.1) yields

where

But

Therefore,

llxiiLr(R+) :S: llwiiLr(R+) + BlllxiiLr(R")'

q B1 = K(O).

llxi!Lr(R") :S: llxllu[-~,0] + llxiiLr(R+) =

ll¢oiiLr[-~,o] + llxi!Lr(R+)·

llxllu(R+) :S: llwlb(R+) + Blll¢oiiL•[-ry,o]+

Blllxi!Lr(R+)· Condition (2.5) implies that B1 < 1. So

II II < (llwiiLr(R+) + BJ!I<Poi!Lr[-ry,o]) X LP(R+) - 1 - Bl .

Since the linear homogeneous equation (1.4) is asymptotically stable,

n-1

llwlb(R+) :S: canst 2::: ll¢kliu[-ry,O]· k=O

Hence, estimate (2.2) follows. As claimed. D

179

As it was proved in [9, Theorem 6.2], under (2.4), condition (2.5) holds if and only if the roots of qQ(z)- K(z) are inC_. In other words, under (2.4) the linear equation (2.1) is asymptotically stable with q = q if and only if (2.5) holds. That is, Theorem 2. 2 separates a class of equations solving Problem 1.

180 M. GIL'

3. The first order equations. Consider the equation

x(t) + [ x(t- s)dr(s) = [Fx](t) (t > 0), (3.1)

where r is a nondecreasing function having a bounded variation Var (r). THEOREM 3.3. Let the conditions

e'r) Var (r) < 1 (3.2)

and V ar (r) > q hold. Then the zero solution to equation (3.1) is absolutely Il'-stable in the class of nonlinearities (2.1). Moreover, under conditions (1.6} and {3.2}, equation (3.1) has non-negative solutions.

Proof. With a constant b > 0, let us consider the equation

u(t) + bu(t- 77) = 0 (t > 0). (3.3)

The Green function Gb(t) of this equation is a solution of (3.3) with the initial conditions

u(O) = 1; u(t) = 0 (t < 0).

We need the following result: let the condition

eryb < 1 (3.4)

hold. Then the Green function to equation (3.3) is non-negative. For the proof see for instance [8]. Let us consider the equation

u(t) + Ia~ u(t- s)dr(s) = 0 (t > 0). (3.5)

The Green function Gr of (3.5) is defined as the one for (3.3). Then the Green function Gr(t) of equation (3.5) is nonnegative. Moreover,

Gr(t) ~ G+(t) ~ 0 (t ~ 0) (3.6)

where G+(t) is the Green functions of equation (3.3) with b = Var(r). Indeed, according to the initial conditions, for a sufficiently small t 0 > 7],

Gr(t) ~ 0, Gr(t) :S 0 (0 :S t :S to).

Thus,

Gr(t- 77) ~ Gr(t- s) (s :S 'r) :S t :S to).

Hence,

r Var (r)Gr(t- r1) 2: Jo Gr(t- s)dr(s) (t::; to).

According to (3.5) we get

Gr(t) + Var (r)Gr(t- ry) = j(t)

with

f(t) = Var (r)Gr(t- ry)- [ Gr(t s)dr(s) 2:0 (0::; t::; to).

Hence, by virtue of the variation of constants formula, we arrive at the relation

Gr(t) = G+(t) + l G+(t- s)f(s)ds 2: G+(t) (0::; t::; to).

Extending this inequality to the whole axis, we get inequality (3.6). Now the absolute stability is due to Theorem 2.2. Existence of non-negative solutions is due to Lemma 1.1. D

ExAMPLE 3.4. Consider the equation

x(t) + bx(t-:- ry) = [Fx](t), (3.7)

where b is a positive constant,

[Fx](t) = l a(s)T(t s)1J;(x(s- ry))ds (t 2: 0),

the functions a and T belong to L2 (Rc), and the function 1/J : R 1 -+ R1 has the property

11/J(y)i::; qtiYI (y E R1). (3.8)

Due to the well-known Theorem 4.1.6 [15],

IIFxiiL2(R+)::; IITIIL'(R+)Iia(.)1j;(x)llv(R+) (x E L 2 [-ry, oo)).

By the Schwarz inequality and (3.8)

iia(.)1/;(x)llv(R+) ::; iia(.)iiL'(R+lli1/!(x)iiL'U<+l ::;

182 M. GIL'

II a(.) II L2(R+)qlllxii£2 (R,) ·

Thus (2.1) holds with p = 2 and

q = IITII£ZcR+)IIa(.)II£2(R+Wl·

Now Theorem 3.1 implies the following result: let condition (3.4) hold. Then the zero solution to equation (3. 7) is absolutely £ 2-stable in the class of nonlinearities (3.8), provided the condition

IITII£'11all£'ql < b

holds. Moreover, equation (3.7) has non-negative solutions, provided the conditions (3.4),

a(s) 2: 0, T(s) 2: 0 and '1/J(s) 2: 0 (s 2: 0)

hold.

4. Higher order equations.

4.1. Second order equations. Let us consider the second order equa-tion

u(t) + Au(t) + Bu(t- 1) + Cu(t) + Du(t- 1)+

Eu(t- 2) = [Fu](t) (4.1)

with non-negative constants A, B, C, D, E. Assume that

B 2/4>E, A2/4>C, (4.2)

and denote

R±(A, C) = A/2 ± (A2 /4- C) 112

and

R±(B, E) = B/2 ± (B 2 /4- E) 112•

THEOREM 4.5. Let the conditions (4.2),

D::; min{R+(B, E)R_(A, C), R_(B, E)R+(A, C)} and

-

•

R+ (B, E)eR+(A,C)+l < 1 (4.3)

hold. If, in addition,

C+D+E > q,

then the zero solution to equation (4.1) is absolutely LP-stable in the class of nonlinearities (2.1). Moreover, under conditions (1.6), (4.2) and (4.3), equation (4.1) has non-negative solutions.

Proof Indeed as it was proved in [8, Lemma 3.4], under conditions (4.2), ( 4. 3) the linear part of ( 4.1) is stable and the Green function

1 loo etiw dw 21T _

00 -w2 + iw(A + Be-•w) + C + De-•w + Ee-2iw

of the linear part of (4.1) is non-negative. Now the absolute stability is due to Theorem 2.2. Existence of non-negative solutions is due to Lemma 1.1. D

4.2. Equations of an arbitrary order. For an arbitrary continuous function v: [-1), oo)-+ R, let us define an operator Sk by

rk (Skv)(t) == Jo v(t- s)drk(s) (k == 1, ... ,n; hk = const > 0; t:::: 0)

where rk : [0, hk] -+ R+ are nondecreasing functions with bounded variations Var (rk). Besides

h1 + ... + hn == rJ.

Let us consider the equation

n

IJ(d/dt + Sk)x(t) == [Fx](t). ( 4.4) k=l

THEOREM 4.6. Let the conditions

ehkVar (fk) < 1 (k == 1, ... ,n) (4.5)

and

n

IJVar (i\) > q k=l

184 M. GIL'

hold. Then the zero solution to equation (4.4) is absolutely Il'-stable in the class of nonlinearities (2.1). Moreover, under conditions (1.6) and (4.5) equation (4.4) has non-negative solutions.

Proof Clearly, under consideration Q(.\) = 1 and

co eitwdw G(t) = (27r)-

11 nn (' + t• e siwdfk(s)) -co k=1 ZW 0

(t ;::::: 0).

According to the properties of the convolution, we have

t r t·-' G(t) = Jo G1(t- ti) Jo G2(t1- t2) ... Jo Gn(tn-ddtn-1 ... dt2 dtt,

where Gk(t) (k = 1, ... , n) are the Green function of equation (3.5) with r = i'k· As it was shown in the proof of Theorem 3.1, under (4.5), Gk are nonnegative. Therefore G is nonnegative. Moreover, under consideration

n rh· n K(O) = ITJo di'k(s) = I1Var (rk).

k=1 0 k=1

Now the absolute stability is due to Theorem 2.2. Existence of non-negative solutions is due to Lemma 1.1. D

REFERENCES

[1] M. Aizerman, On a conjecture from absolute stability theory, Ushechi Matematicheskich Nauk, 4(1949), 25-49 (In Russian).

[2] T. Burton, Volterra Integral and Differential Equations, Ac. Press, New York, 1983. [3] C. Corduneanu, Functional Equations with Causal Operators, Taylor and Francis,

London, 2002. [4] Yu. Daleckii, and M. Krein, Stability of Solutions of Differential Equations in Banach

Space, Amer. Math. Soc., Providence, R.I., 1974. [5] L. Erbe , Q. Kong, and B.Zhang, Oscillation Theory for Functional Differential

Equations, Marcel Dekker, New York, 1995. [6] M. Gil', On a class of one-contour systems which are absolutely stable in the Hurwitz

angle, Automation and Remote Control,l0(1983), 70-76. [7] M. Gil, Stability of Finite and Infinite Dimensional Systems, Kluwer Academic Pub

lishers, Boston-Dordrecht-London, 1998. [8] M. Gil', On Aizerman-Myshkis problem for systems with delay, Automatica,

36(2000), 1669-1673. [9] M. Gil', Boundedness of solutions of nonlinear differential delay equations with pos

itive Green functions and the Aizerman - Myshkis problem, Nonlinear Analysis, TMA , 49(2002), 1065-168.


[10] G. Gripenberg, S.-0 Londen, and 0. Staffans, Volterra Integral and Functional Equations, Cambridge University Press, Cambridge, 1990.

[11] I. Gyiiri, and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon, Oxford, 1991.

[12] J. Hale, and S. Verduyn, Introduction to Functional Differential Equations, SpringerVerlag, New York, 1993.

[13] V. Kolmanovskii, and A. Myshkis, Introduction to the Theory and Applications of Functional Differential Equations, Kluwer Academic Publishers, Boston, 1998.

[14] A. Myshkis, On some problems of theory of differential equations with deviation argument. Uspechi Matemat. Nauk, 32(194)(1977), 173-202 (In Russian).

[15] G. Okikiolu, Aspects of the Theory of Bounded Integral Operators, Ac. Press, London, 1971.

[16] M. Vidyasagar, Nonlinear Systems Analysis, second edition. Prentice-Hall, Englewood Cliffs, New Jersey, 1993.


VOLUME 12 2005, NO 1-2 PP. 187-208

ASYMPTOTICALLY ORDINARY DELAY DIFFERENTIAL EQUATIONS

ISTVAN GYORI' AND MIHALY PITUKl

Abstract. According to a result due to Ryabov, a system of linear differential equations with small delays is asymptotically equivalent to a linear ordinary differential equation of the same dimension. In this paper, we present some improvements, simplify the proofs and summarize some earlier results on this subject.

Key Words. delay differential equations, special solution, asymptotic behavior

AMS(MOS) subject classification. 34K25

Dedicated to Professor' A. D. Myshkis on the occasion of his 85th biTthday

1. Nonlinear Systems.

1.1. Introduction. In [21], Ryabov gave an asymptotic characterization of the solutions of differential systems with small delays with certain special solutions. Ryabov's results have been extended and improved by several authors, notably by Uvarov [24], Driver [4, 5], Jarnik and Kurzweil [12], Myshkis [16], Arino and the authors [1, 2, 3, 7, 8, 9, 10, 17, 18, 19, 20].

In the present paper, we offer some further improvements, simplify the proofs and summarize our earlier results on this subject. We shall demonstrate the main ideas on the example of a differential equation with one single

• Department of Mathematics and Computing, University of Veszpn\m, Veszpn\m, Hungary. Supported by the Hungarian National Foundation for Scientific Research Grant No. T 046929.

t Department of Mathematics and Computing, University ofVeszprem, Veszprem, Hungary. Supported by the Hungarian National Foundation for Scientific Research Grant No. T 046929.

187

188 I. GYORI AND M. PITUK

delay

(1.1) ±(t) = f(t,x(t),x(t- 7(t))),

where 7 is a nonnegative function on lR and j is a function mapping lR x JRn x JRn into JRn. Here lR denotes the real line and JRn is the n-dimensional space of real column vectors with any convenient norm 1·1.

The initial condition associated with Eq. (1.1) has the form

(1.2) x(t) = tj;(t), L1 :":: t :"::to,

where t0 E JR, L 1 = inft>to(t- 7(t)) and ¢;: [L1, t0] -+ lR" is a given continuous initial function.

Throughout the paper, we shall assume that 7 is measurable, limt-t00 (t-7(t)) = oo, f satisfies locally the Caratheodory conditions and, in addition, the following hypotheses are satisfied:

(1.3) lf(t,x~>YI)- f(t,xz,Yz)l :":: L1(t)lx1- xzl + Lz(t)IYI- Yzl

(1.4)

(1.5)

(1.6)

(1. 7)

(1.8)

LI(t):::: q!,\(t),

Lz(t) :":: qz,\(t),

IJ(t,O,O)I:::: K,\(t),

l t ,\(s)ds::;r, t-T(t)

for t E lR, X;, Yi E lRn , i = 1, 2,

t E lR,

t E lR,

t:::: 0,

t E lR,

ql + qze"" < J-t,

where L 1, L 2 and A are nonnegative locally integrable functions on lR and q1 ,

qz, K, r and J-t are positive constants. It follows by known existence theorems (see, e.g., [11]) that under the

above hypotheses the initial value problem (1.1)-(1.2) has a unique solution x on [L 1,oo). That is, a unique continuous function x: [L1,oo)-+ JRn exists, which is locally absolutely continuous on [t0 , oo), Eq. (1.1) is satisfied for almost every (a.e.) t :::: t0 and (1.2) holds.

It is easily shown that hypothesis ( 1.8) is satisfied for some 1-' > 0 if and only if

(1.9) qzrel+q, r < 1.

The latter condition certainly holds if r is sufficiently small. Thus, the above hypotheses can be regarded as a kind of smallness condition on A (and hence on f) and/or on the delay 7.

DELAY DIFFERENTIAL EQUATIONS 189

1.2. Special solutions. In the next theorem, we prove the existence and uniqueness of certain global solutions of ( 1.1).

THEOREM 1. Suppose conditions { 1. 3 )-( 1. 8) hold. Then for every t 0 E

1ft and x0 E lftn Eq. { 1.1) has a unique solution x on 1ft such that

(1.10) x(to) = Xo

and

(1.11) sup lx(t)l exp(-t-Lio >.(u) du) < oo. t:SO t

Proof Let B denote the space of continuous functions x: ( -oo, t 0] -+ lftn

such that

llxiiB =sup lx(t)l exp(-t-Lito >.(u) du) < oo. t~to t

Clearly, (B, II ·11 8 ) is a Banach space. Define

ito

:Fx(t) = xo- t f(s, x(s), x(s- T(s)) ds

for x E Bandt ::; t0 . Obviously, :Fx is continuous on ( -oo, t0] and fort ::; t0 ,

ito

I:Fx(t)l::; lxol + t (lf(s,O,O)I + lf(s,x(s),x(s- T(s)))- f(s,O,O)I)ds

i to ito ::::lxol+ t K,\(s)ds+ t (Ll(s)lx(s)I+L2(s)lx(s-T(s)l)ds

::; lxol + K [o ,\(s) exp(f-L [o ,\(u) du) ds

ito

+ t >.(s)(qllx(s)l + q2lx(s- T(s))l) ds.

Since, for s ::; t0 ,

lx(s)l::; llxll 8 exp(f-L [o >.(11) du)

190

and

I. GYORI AND M. PITUK

lx(s- r(s)l :S: llxll 8 exp(J-t[to A(u) du) s-r(s)

= llxll 8 exp(J-t[' A(u)du) exp(J-t[to A(u)du) s-r(s) s

:S: llxll 8 e~'" exp(J-t [o A(u) du).

it follows for t ::; t 0 ,

IFx(t)l :S: lxol + [K + (ql + q2e~'")llxll 8 ] [

0

A(s) exp(J-t [0

A(u) du) ds

:S: lxol + [KJ.!- 1 + kllxll 8 ]exp(J.l [0

A(u)du),

where

(1.12) k = ( q1 + q2e~'") / J.l.

This shows that Fx belongs to B.

--~·- ""''-

If x, y E B and t ::; t0 , then by similar estimates as before one can show that

IFx(t)- Fy(t)l :S: kllx- Ylla exp (J.l [o A(u) du).

Hence IIFx- Fyll 8 ::; kllx- yll 8 , and, by virtue of (1.8), k < 1. By Banach's fixed point theorem, there exists a unique x E B such that Fx = x. This fixed point is obviously a solution of (1.1) on ( -oo, to] with the required properties. The solution x can be uniquely extended to lR by known existence theorems [11]. 0

DEFINITION 1. We shall call the solution x from Theorem 1 the special solution passing through (t 0 , x 0 ) and denote it by x = x(to, xo).

COROLLARY 1. Under the hypotheses of Theorem 1, the totality of all special solutions is only an n-parameter family.

Proof. The mapping x H x(O) is a one-to-one correspondence between the set of special solutions and !Rn. 0

REMARK 1. The roots of the equation

(1.13) A = ql + q2e>.r

will play an important role in the sequel. Since the function

x(>-) = ql + qze>.r--\

is strictly convex on [0, oo) and x(p,) < 0 (see (1.8)), it follows that x and thus Eq. (1.13) has exactly two positive roots JL1, JL2 , 0 < /h < p, < ftz, and :\:(p,l) < 0. Equivalently,

(1.14) qzre"'r < 1.

Now we prove a useful lemma about the distance of two special solutions.

LEMMA 1. Let the hypotheses of Theorem 1 be satisfied. Let xi, i = 1, 2

be special solutions of {1.1). Then for all t::; t 0 ,

lx1(t)- xz(t)l::; lxl(to)- xz(to)lexp(ftl [' >-(s)ds),

wher·e JL1 is the smaller positive root of Eq. {1.13). Proof. Put

uo(t) = x1 (to), rto

Ui+!(t) = X!(to)- lt f(s,ui(s),ui(S- r(s))ds

and

Vo(t) = Xz(to),

ito

Vi+!(t) = xz(to)- t f(s,vi(s),vi(s- r(s))) ds

for t ::; to and i = 0, 1, .... We shall show by induction that

(1.15) lui(t)- vi(t)l::; lxl(to)- xz(to)lexp(Jll[' >-(s)ds)

fort::; to and i = 0, 1, .... Clearly, (1.15) holds fori= 0 and its validity for


some i implies

ito

Jui+l(t)- Vi+l(t)J :S: Jxl(to)- X2(to)l + t (ql.\(s)Jui(s)- Vi(s)J

+ q2.\(s)iui(s- r(s))- vi(s- r(s))l) ds

:S: Jxl(to)- x2(to)J[1 +to ql.\(s)exp(/11 [o .\(u)du) ds

+ [0

q2.\(s)exp(J11 {r(s) .\(u)du) exp(/1! [0

.\(u)du) ds]

:s: lx!(to)- x2(to)l [1 + (ql + q2e'"'r) [0

.\(s) exp(/11 ito .\(u) du) ds]

= Jx1(to) -x2(t0 )Jexp(J11 [o .\(u)du).

Thus, (1.15) is confirmed for all i = 0, 1, .... Referring to the proof of Theorem 1, we have ui -t x 1 and vi -t x2 as

i -t oo. The proof is now completed by letting i -t oo in (1.15). 0

COROLLARY 2. Under the hypotheses of Theorem 1, the special solutions of {1.1} satisfy an ordinary differential equation.

Proof Define g: lR x lRn -t lRn by

g(t,x) = f(t,x,x(t,x)(t- r(t)) for t E lR and x E JRn.

Since x(t, x(t)) = x for every special solution of (1.1), the special solutions satisfy the ordinary differential equation

(1.16) i; = g(t,x) for a.e. t E JR.

It follows easily from (1.3) and Lemma 1 that g is locally Lipschitz continuous with respect to the second variable. Therefore, the solutions of (1.16) are uniquely determined by their initial values. This means that the set of solutions of (1.16) coincides with the family of special solutions of (1.1). 0

1.3. Asymptotic behavior of arbitrary solutions. The reason for the interest in the special solutions will be clear from the next theorem. It shows that under a mild additional assumption every solution of (1.1) approaches some special solution as t -t oo.

THEOREM 2. In addition to the hypotheses of Theorem 1, suppose that

(1.17) l"' .\(s)ds = oo.

Let x be the solution of (1.1} with initial condition (1.2}. Then there exists exactly one special solution x of (1.1) such that

(1.18) sup lx(t) - x(t) I exp (f-L 1\(s) ds) < oo. t?_to to

In particulaT, the solutions x and x are asymptotically equivalent, i.e.,

lim lx(t)- x(t)l = 0. t-;oo

Furthermore,

(1.19) x(to) =lim x(t,x(t))(to). t->oo

Proof Uniqueness of x. Let Xi, i = 1, 2 be special solutions of (1.1) satisfying (1.18). Then

C =sup lx1(t)- x2(t)l exp(f-Llt .\(s) ds) < oo. t?.to to

By Lemma 1, we have fort;:: t 0 ,

lxl(to)- x2(to)l:::: lxl(t)- x2(t)1 exp(f-Ll lot .\(s) ds)

::0: C exp ( (J-L1 - J-L) lot .\(s) ds).

Since J-L 1 < J-L (see Remark 1), letting t -> oo, we get x1(t0 ) = x2 (t0 ), and, by virtue of the unicity of the special solutions, x1 = x2 .

Existence of x. Let B denote the set of continuous functions y: [L1, oo) -+ Rn such that

IIYIIs = sup ly(t)l exp(f-Llt .\(u) du) < oo. t?.L1 t0

Clearly, (B, II· 11 8 ) is a Banach space. For y E B, we define

:Fy( t) =

100

(f(s, x(s)- y(s), x(s- r(s))- y(s- r(s)))

- f(s, x(s), x(s- r(s))) ds

x(t)- x(to,x(to)- :Fy(to))(t)

for t ;:: to,

for L 1 ::0: t <to.

If y E B and t ~ t0 , then

j[(J(s, x(s)- y(s), x(s- r(s)) - y(s- r(s)))- f(s, x(s), x(s- r(s)))) dsl

:S:: IIYII 8 (q1 +q2e~'r) l:rv -X(s)exp( -111: -X(u)du) ds

= kiiYII 8 exp ( -111: -X(u) du),

where k is defined by (1.12) Thus, :F(B) C B. (The continuity of :Fy at t =to follows from the fact that x(to)- x(to, x(to)- :Fy(to))(to) = :Fy(to).)

We shall show that :F is a contraction. If y, z E B, then for t ~ t0 ,

/:Fy(t)- :Fz(t)l::; (q! + q2e~'")lly- zll 8 [o -X(s) exp( -11 l: -X(u) du) ds

= kl/y- zl/ 8 exp( -11 l -\(u) du),

while for L 1 ::; t < t0 we have , according to Lemma 1,

/:Fy(t)- :Fz(t)l

::; exp (111 lo -X(u) du) lx(to, x(to)- :Fy(to))(to) - x(to, x(to)- :Fz(to))(to)l

= exp(/11 lo -X(u) du) j:Fy(to)- :Fz(to)l

::; exp(fJ. lo -X(u) du )kl/y- zl/ 8 ,

the last inequality being a consequence of the fact that 111 < /1· Consequently,

1/:Fy- :Fzl/8 :S:: kl/y- ziiB,

and, of course, k < 1 (see (1.8)). From this, we conclude that there is a unique y E B such that :Fy = y. This fixed point y is a solution of the equation

(1.20) y(t) = j(t,x(t),x(t- r(t)))- f(t,x(t)- y(t),x(t- r(t))- y(t- r(t)))

on [t_~> oo) satisfying the initial condition

(1.21) y(t) = x(t) - x(t) for L1 ::; t ::; to,

where x = x(to, x0 ) with x0 = x(t0 ) - y(t0). But, x - x is also a solution of (1.20) with the same initial condition (1.21). By virtue of the unicity of the solutions of (1.20) (the function on the right-hand side of (1.20) satisfies (1.3)), we have y(t) = x(t)- x(t) for all t 2: t-1- Assertion (1.18) now follows from the fact that y E B.

It remains to show (1.19). Denote

C =sup \x(t)- x(t)\ exp(J.L t .\(s) ds); t?:to J to

c < 00.

By Lemma 1, we obtain

jx(to) - x(t, x(t))(to)\ ::; jx(t) - x(t)\ exp (J.L1 1: .\(s) ds)

::; Cexp((J.L1 - J.L) 1: .\(s)ds)--+ 0 as t--+ oo,

which completes the proof. 0

Combining Theorem 2 with Corollary 2, we obtain

COROLLARY 3. Under the hypotheses of Theorem 2, the solutions of Eq. (1.1) are asymptotically equivalent to those of an ordinary differential equation.

2. Linear Systems.

2.1. The special matrix solution. In this section, we shall deal with the linear system

(2.1) x(t) = A(t)x(t) + B(t)x(t- r(t)),

where A, B : lR --+ !Rnxn are locally integrable matrix functions and the delay function r satisfies the same hypotheses as in Section 1. Eq. (2.1) is a special case of (1.1) when j(t,x,y) = A(t)x+B(t)y fortE !R, x, y E !Rn Hypotheses (1.3)-(1.8) reduce to

(2.2) \A(t)\ :S q1.\(t), t E !R,

(2.3) \B(t)\ :S q2.\(t), t E !R,

(2.4) [ .\(s)ds::; r, t E lR t-T(t)

(2.5) ql + qzeM < J.L,

196 I. GYORl AND M. PITUK

where the symbols >., q1 , q2 , r and J.t have the same meaning as in Section 1. Here and in the sequel the matrix norm is the one induced by the norm used in lR".

As an immediate consequence of Theorem 1, we have

THEOREM 3. Suppose conditions (2.2)-(2.5) hold. Then there exists a unique locally absolutely continuous matrix function X: lR ~ JRnxn such that

(i) each column of X is a solution of (2.1) on JR, (ii) X(O) =I, I being then x n identity matrix, and (iii) sup1:So /X(t)/exp(-!1ft

0 >.(s)ds) < oo.

This matrix function X, called the special matrix solution of (2.1), has the following properties:

(iv) for each t E JR, the matrix X ( t) is nonsingular, (v) /X(s)X- 1(t)/ ::; exp(J.!I J: >.(u) du) for all s ::; t, where J.t1 is the

smaller positive root of Eq. (1.13}, (vi) the special solution of (2.1) passing through (t0 ,x0 ) E JRxJRn is given

by

x(to, Xo)(t) = X(t)X- 1(to)Xo, t E JR.

Proof Let e; be the ith column of the n x n identity matrix I. Then in order to satisfy conditions (i)-(iii), the ith column of X must be x(O, e;). The asserted properties now follow from Theorem 1 and Lemma 1. 0

2.2. Asymptotic representation of all solutions. The next theorem is a reformulation of Theorem 2 for the linear case. It states that every solution of (2.1) is asymptotic to a linear combination of n special solutions, the columns of the special matrix solution X described in Theorem 3.

THEOREM 4. In addition to the hypotheses of Theorem 3, suppose condition (1.17} holds. Let x be the solution of (2.1} with initial condition (1.2}. Then there exists a unique vector l = l ( ¢>) E lR" such that

(2.6) sup /x(t)- X(t)l/ exp(J.t11

>.(s) ds) < oo. t~to to

In particular,

lim /x(t)- X(t)l/ = 0. l-+00

This vector l can be written in the form

(2.7) l = lim X(t0 )X-1(t)x(t). t->oo

Proof. Theorems 2 and 3. 0

2.3. The asymptotic ordinary differential equation. In view of Theorem 3 (iv), the special matrix solution X of (2.1) is a fundamental matrix of a linear homogeneous ordinary differential equation

(2.8) x = M(t)x,

where M: lR:.-+ !R:."xn is a locally integrable matrix function. Moreover, the delay differential equation (2.1) is asymptotically equivalent to this ordinary differential equation. In the next theorem, we describe the "asymptotic" ordinary differential equation (2.8) in terms of the coefficients and the delay of the original equation (2.1).

THEOREM 5. Let the hypotheses of Theorem 3 be satisfied. Define a sequence of matrix functions {Mi(t, s)};:;0 , -oo < s:::; t < oo, by

Mo(t, s) = A(s) + B(s),

and

Mi+1 (t, s) = -A(s) lt Mi(t, u) du- B(s) lt Mi(t, u) du S S-T(S)

for s:::; t and i = 0, 1, .... Then the special matrix solution X of (2.1) is a

fundamental matrix of the ordinary differential equation (2. 8) with a locally integrable coefficient M given by

00

(2.9) M(t) = L Mi(t, t), i=O

the last series being absolutely convergent for each t E lR:.. Proof. First we show that the series (2.9) is absolutely convergent. Let

t E lR:. be fixed. Clearly,

[Mo(t,s)[:::; C>.(s):::: C>.(s)exp(/L [ >.(v)dv),

where C = q1 + q2. Assume for induction that for some i,

(2.10) IM;(t,s)i::; CkiA(s)exp(Jl[ A(v)dv ),

where k is given by (1.12). Then for s ::; t,

s ::; t,

IMi+l(t,s)i::; Ckiq1A(s) [ A(u)exp(Jl1t A(v)dv) du

+Ckiq2A(s) t A(u)exp(Jl1tA(v)dv)du ls-r(s) u

::; Ckiq2A(s)Jl-l exp(Jl [t A(v) dv)

+ Ckiq2A(s)Jl-l exp(Jl t A(v) dv) ls-r(s)

::; CkiA(s)[ql +q2e~'r]JI- 1 exp(Jl[ A(v)dv)

= Cki+l A(s) exp(Jl J.t A(v) dv).

the last but one inequality being a consequence of (2.4). Thus, (2.10) is confirmed for all i = 0, 1, ....

From (2.10), we obtain

IM;(t, t)i ::; Ck; A(t).

By virtue of (2.5), k < 1. Thus,

00

IM(t)l ::; .E IM;(t, t)i ::; C(1- k)-1 A(t). i=O

Since A is locally integrable, so is M. Define a sequence of matrix functions {N;(t, s)}~0 , -oo < s ::; t < oo,

by

N0(t, s) = A(s)X(s)X-1(t) + B(s)X(s- r(s))X-1(t)

and

Ni+l(t,s)=-A(s) tN;(t,u)du-B(s) t N;(t,u)du }, ls-r(s)

•

for s :S t and i = 0, 1, . . . . One can prove by easy induction on i using Eq. (2.1) that

(2. 11) Mi(t,s) + Ni+l(t,s) = Ni(t,s), s :S t

for i = 0, 1, ....

Now we prove that

(2.12) X(t) = (~Mi(t,t)+Nm+J(t,t))X(t) for a.e. t E JR.

form= 0, 1, .... By virtue of (2.11), it suffices to verify (2.12) form= 0. Using (2.1) and (2.11), we find

(M0 (t,t)+N1 (t, t))X(t) = N0 (t, t)X(t) = A(t)X(t)+B(t)X(t-T(t)) = X(t)

for a. e. t E JR. Consequently, (2. 12) holds for all m.

Our next aim is to show that, for each t E JR., Nm(t, t) -+ 0 as m-+ oo. To this aim, we shall show by induction on m that

(2.13) INm(t,s)l :S f.LkmA(s)exp(f.L l A(v)dv), s :S t

form= 0, 1, .... By Theorem 3 (v), we have for s :S t,

INo(t, s)l :S IA(s)IIX(s)X- 1(t)1 + IB(s)IIX(s- T(s))X-1(t)l

:S q1A(s) exp (f.L1 t A( v) dv) + q2 A(s) exp (/JJ t A( v) dv) }, J.,_~(s))

:S A(s)[ql + q2eJ.t'r]exp(f.Ll/,t A(v) dv)

:S A(s)f.Lexp(f.L l A(v) dv),

the last inequality being a consequence of (2.5) and the fact that /JJ < f.L· Thus, (2.13) holds form= 0.

Assume that (2.13) holds for some m. Then for s::::; t,

/Nm+ 1(t, s)/::::; /A(s)/ {t /Nm(t, u)/ du + /B(s)/ ft /Nm(t, u)/ du }, ls-T(s)

::::; qr.A(s) [ ttkm.A(u) exp(f.t [ .A(v) dv) du

+q2.A(s) t ttkm.A(u)exp(f.t r .A(v)dv) du ls-T(s) Ju

::::; qr.A( s )km exp (tt [A( v) dv)

+ q2.A(s)km exp (tt r .A(v) dv) ls-T(s)

::::; .A(s)km[q1 + q2e~""]exp(f.t [ .A(v) dv)

= ttkm+l .A(s) exp(f.t J.t .A(v) dv).

Thus, (2.13) is confirmed for all m = 0, 1, .... From (2.13), we find fortE lR and m = 0, 1, ... ,

/Nm(t, t)/ ::::; ttkm A(t).

Since k < 1, Nm(t, t)-+ 0 as m-+ oo. The proof is now completed by letting m-+ oo in (2.12). 0

REMARK 2. The description of the limiting ordinary differential equation given in Theorem 5, combined with Theorem 4, can be used to obtain sharp explicit stability criteria. For example, in [9], the authors have shown that if in Eq. (2.1) A is identically zero, r is bounded and J0

00 /B(s)/mds < oo for some positive integer m, then Eq. (2.1) has essentially the same stability properties as the ordinary differential equation

m-1

x = 2::: M;(t, t)x i=O

(see [9] for details).

2.4. A more precise asymptotic characterization. Since formulae (2.6) and (2.7) give a genuine asymptotic characterization of the solution x of (2.1) with initial condition (1.2) provided l = l(¢) # 0, it is important to

be able to express 1(¢) explicitly in terms of the initial function¢. In this paragraph, we shall deal with this problem. In order to simplify the proofs, we shall assume that the initial point to is zero and the delay in Eq. (2.1) is constant,

(2.14) i(t) = A(t)x(t) + B(t)x(t- r).

(The general case can be treated in a similar way; see [3].) We shall show that 1(¢) in Theorem 4 can be computed with a special matrix solution of the formal adjoint equation associated with (2.14),

(2.15) y(t) = -y(t)A(t) - y(t + r)B(t + T),

where y(t) is ann-dimensional row vector.

THEOREM 6. For Eq. (2.14), suppose the hypotheses of Theorem 3 are satisfied. Then there exists a unique locally absolutely continuous matrix function Y: [0, oo) --+ JR.nxn such that

{i) each row of Y is a solution of the formal adjoint equation (2.15) on [0, oo),

{ii) for all t 2: 0,

ft+r

(2.16) Y(t)X(t) + t Y(s)B(s)X(s- T) ds =I,

where X is the special matrix solution of (2.14), and

(iii) sup1>o IY(t)i exp( -M1 J; ,\(v) dv) < oo, where f.L 1 is the smaller positive root of Eq. (1.13).

Proof Let B be the vector space of those continuous matrix functions Y: [0, oo) --+ JR.nxn for which

1\YIIe =sup \Y(t)i exp(-f.L1 t ,\(v) dv) < oo. t?:O lo

Clearly, (B, ll·lle) is a Banach space. For Y E B and t 2: 0, define

ft+T

FY(t) = x-'(t)- t Y(s)B(s)X(s- r)X-1(t) ds.

By Theorem 3 (v), we have fort<': 0,

it+T

IFY(t)i::; IX-1(t)i + t IY(s)IIB(s)IIX(s- r)X- 1(t)i ds

where

(2.17)

::; exp (t.ti l ,\( v) dv)

+ it+r IIYIIBexp(t.ti [ ,\(v)dv)q2,\(s)exp(t.ti {r ,\(v)dv)ds

= exp(t.t1 l ,\(v) dv)

+exp(t.tll ,\(v)dv)i1YiiBQ21t+r ,\(s)exp(t.tl rr ,\(v)dv) ds

::; exp (t.t1 l ,\( v) dv)

+exp(t.tll ,\(v)dv)i1YIIBq2e~'17 it+r ,\(s)ds

::; exp(t.tll ,\(v)dv) +exp(t.tll ,\(v)dv)IIYIIBx,

x = q2re~'17 < 1

(see (1.14)). The last estimate shows that FY E B. If Y, Z E Bandt<': 0, then by similar estimates as before one can show

that

IFY(t)- FZ(t)i ::; xliY- ZIIB exp (t.ti l ,\(v) dv).

Hence IIFY- FZIIB::; xiiY- ZIIB· Thus, we have shown that F: B-+ B is a contraction. Evidently, the unique solution Y E B of FY = Y satisfies assertions (ii) and (iii) of the theorem.

Differentiating (2.16) with respect tot and using Eq. (2.14), we find for a.e. t <': 0,

[Y(t) + Y(t)A(t) + Y(t + r)B(t + r)]X(t) = 0.

Since X(t) is nonsingular (see Theorem 3 (iv)), this proves assertion (i). 0 We are in a position to state a refinement of Theorem 4.

THEOREM 7. For Eq. (2.14), suppose the hypotheses of Theorem 4 are satisfied. Let x be the solution of (2.14) with initial condition

x(t) = ¢(t), -T :S t :S 0,

where ¢: [ -T, 0] -+ IR.n is continuous. If X and Y are the special matrix solutions of Eqs. (2.14) and (2.15 ), respectively, then the limit

(2.18) 1(¢) = lim x- 1(t)x(t) t-HX>

exists in IR.n and its value is given by

(2.19) 1(¢) = Y(0)¢(0) + [ Y(s)B(s)</>(s- T) ds.

Before we present the proof of Theorem 7, we need the following lemma.

LEMMA 2. Let x andy be solutions of Eqs. (2.14) and (2.15) on [-T, oo) and on [0, oo), respectively. Then

(2.20) 1t+7

y(t)x(t) + t y(s)B(s)x(s- T) ds =constant fort 2: 0.

Proof Differentiating the last expression and using Eqs. (2.14) and (2.25), it follows that the derivative of the locally absolutely continuous function on the left-hand side of (2.20) is zero for a.e. t 2: 0. 0

Proof of Theorem 7. By Lemma 2,

Jt+T

Y(t)x(t) + t Y(s)B(s)x(s- T) ds = 1(¢) for t 2: 0,

where 1(¢) is given by (2.19). From this, taking into account that

Jt+T

Y(t) = x- 1(t)- t Y(s)B(s)X(s- T)X- 1 (t) ds, t 2: 0

(see (2.16)), we find fort 2:0,

Jt+T

z(t)- 1(¢) = t Y(s)B(s)X(s- T)[z(t)- z(s- T)] ds,

...... .--..


where

(2.21) z(t) = x-1(t)x(t), t:::: -T.

Consequently, for t ;::: r,

i t+r jt z(t) -I(¢)=

1 Y(s)B(s) s-r X(s- r)X- 1(u)X(u)z(u) duds,

and hence

(2.22)

i t+T jt iz(t) -1(¢)1:::;

1 jY(s)iiB(s)i s-r IX(s- r)X-1(u)iiw(u)i duds,

where

w(t) = X(t)z(t).

Since x(t) = X(t)z(t), we have for a.e. t;::: 0,

x(t) = X(t)z(t) + X(t)i(t),

and hence

w(t) = X(t)i(t) = x(t)- X(t)z(t) = B(t)[x(t- r)- X(t- r)z(t)) = B(t)[X(t- r)z(t- r)- X(t- r)z(t)J.

Consequently, for t ;::: r,

w(t) = -B(t)X(t- r) [r i(s) ds

= -B(t) [r X(t- r)X-1 (s)w(s) ds.

From this, for t ;::: r,

iw(t)i:::; IB(t)i [r iX(t- r)X-1(s)iiw(s)i ds

:::; q2.\(t) [r exp(J.tl [r -\(u)du)iw(s)ids.

Multiplying the last inequality by exp(fl1 J; :\(u) du) and using (2.4), we

obtain for t 2:: T,

(2.23) a(t) :<; q2 ei'lr :\(t) 1~7 a(s) ds,

where a is defined by

a(t) = exp (Ill l :\(u) du )iw(t)l for t 2:: 0.

Now we show that J~oo a(t) dt < oo. Let T > T be fixed. Integrating (2.23)

from T to T and interchanging the order of integration on the right-hand side, we obtain

T T t 1 a(t) dt :<; q2 eM17 1 :\(t) L7

a(s) ds

1T 1s+7 :<;q2 eM17

0 a(s) s :\(t)dtds

T

:<; x 1 a(s) ds,

where x is given by (2.17). Since x < 1, this implies

1T X 17 a(t)dt :<; -- a(s)ds. 7 1- X 0

Letting T -too, we see that f000 a(t) dt < oo. In particular,

(2.24) 1~7 a(s) ds -t 0 as t -> oo.

From (2.22), using (2.3), (2.4) and the estimates for the growth of the special matrix solutions X and Y, we find for t 2:: T,

lz(t)- l(¢)1

:S [+7

IIYIIBexpG1 [:\(v)dv)q2:\(s) {;xpG1 {7

A(v)dv)iw(u)lduds

:S IIYIIBq2eM'7 1* :\(s) 1~7 a(u) duds

1t 1t+7 :S IIYIIBq2e~ 17

t-7

a(u) du t :\(s) ds

:S IIYIIsq2eM17T Lr a(u) du,


where JJYJJ 8 has the meaning from the proof of Theorem 6. In view of (2.24), the right-hand side of the last inequality tends to zero as t -+ oo. Consequently,

lim z(t) =lim x-1(t)x(t) = 1(¢) t-too t~oo

with 1(¢) given by (2.19). 0

3. Discussion. The special solutions used in this paper were introduced by Ryabov [21] in the special case .X(t) = 1/T, where f denotes the maximum delay. For linear delay differential equations the asymptotic relation (2.7) was proved by Uvarov [24] and Driver [5]. The asymptotic characterization of all solutions by Ryabov's special solutions (.X(t) = 1/T) in the nonlinear case was proved by Jarnik and Kurzweil [12]. Their proof is based on a general lemma concerning invariant sets and on a construction of a Lyapunov functional. The simple proof of Theorem 2 presented here is due to the second author [19]. In [10], using a similar approach, the results of Section 1 have been extended to neutral delay differential equations. For linear delay differential equations the representation of the limiting ordinary differential equation was obtained by Arino and the authors [2]. As we have already mentioned, in [9], using the above series expansion, the authors obtained sharp stability criteria. The computation of the limit (2. 7) in terms of the initial function is due to Arino and the second author [3]. Note that the term asymptotically ordinary delay differential equation was introduced in [7].

Finally, let us mention some works, which are relevant to our study. In the autonomous case (J and r are independent of t), the smallness condition (1.8) appears in the work of Smith and Thieme [23]. They have shown that the theory of monotone dynamical systems applies to these kind of equations, and this fact has interesting dynamical consequences. For example, the local stability properties of an equilibrium are essentially the same as for the ordinary differential equation obtained from (1.1) by ignoring the delay. Furthermore, under some additional assumptions, most of the solutions tend to equilibrium as t -+ oo. For some improvements and further related results, see [13, 14, 15, 16, 20, 25, 26].

REFERENCES

[1) 0. Arino and I. Gy6ri, Delay differential systems asymptotically equivalent to ordinary differential equations, Qualitative Properties of Differential Equations, Proceedings of the Edmonton Conference 1984 (Eds.: Allegretto W., Butler G.J.) 1987,27-37.


[2] 0. Arino, I. Gyori, and M. Pituk, Asymptotically diagonal delay differential systems, J. Math. Anal. Appl., 204, (1996), 701-728.

[3] 0. Arino and M. Pituk, More on linear differential syste~s with small delays, J. Differential Equations 170, (2001), 381-407.

[4] R. D. Driver, On Ryabov's asymptotic characterization of the solutions of quasilinear differential equations with small delays , SIAM Review 10, (1968), 329-341.

[5] R. D. Driver, Linear differential systems with small delays, J. Differential Equations 21, (1976), 149-167.

[6] R. D. Driver, D. W. Sasser, and M. L. Slater, The equation x'(t) = ax(t) + bx(t- r) with "small" delay, American Math. Monthly 80, (1973), 990-995.

[7] I. Gyori, On existence of the limits of solutions of functional differential equations, Colloq. Math. Soc. Janos Bolyai, 30. Qualitative theory of differential equations, Szeged (Hungary) {1979), North Holland Publ. Company, (1980), 325-362.

[8] I. Gyori, Necessary and sufficient stability conditions in an asymptotically ordinary delay differential equation, Differential Integral Equations 6, (1993), 225-239.

[9] I. Gyori and M. Pituk, Stability criteria for linear delay differential equations, Differential Integral Eq1tations 10, (1997), 841-852.

[10] I. Gyori and M. Pituk, Special solutions of neutral functional differential equations, J. Inequal. Appl. 6, (2001), 99-117.

[11] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New-York, 1977.

[12] J. Jarnik and J. Kurzweil, Ryabov's special solutions of functional differential equations, Boll. Un. Mat. !tal. 11, (1975), 198-218.

[13] T. Krisztin and ,). Wu, Monotone semiflows generated by neutral equations with different delays in neutral and retarded parts, Acta Math. Univ. Comenianae 63, (1994), 207-220.

[14] T. Krisztin and J. Wu, Asymptotic periodicity, monotonicity and oscillation of solutions of scalar neutral functional differential equations, J. Math. Anal. Appl. 199, (1996), 502-525.

[15] T. Krisztin and J. Wu, Asymptotic behaviors of solutions of scalar neutral functional differential equations, Differential Equations Dynarn. Systems 4, (1996), 351-366.

[16] A. D. Myshkis, The phase portrait of the set of special solutions to autonomous functional differential equations, Differential Equations 30, (1994), 526-535.

[17] M. Pituk, Asymptotic characterization of solutions of functional differential equations, Boll. Un. Mat. !tal. 7-B, (1993), 653·-689.

[18] M. Pituk, Asymptotic behavior of solutions of a differential equation with asymptotically constant delay, Nonlinear Anal. 30, (1997), 1111-1118.

[19] M. Pituk, Special solutions of functional differential equations , Studies of the University of Zilina, Mathematical Series 17, (2003), 115-122.

[20] M. Pituk, Convergence to equilibria in scalar nonquasimonotone functional differential equations, J. Differential Equations 193, (2003), 95-130.

[21] Yu. A. Ryabov, Certain asymptotic properties of linear systems with small time lag, Trudy Sem. Teor. Differencial. Uravnenii s Otklon. Argurnentorn Univ. Druzby Nar·odov Patrica Lumurnby 3, (1965), 153-164, in Russian.

[22] H. L. Smith, Monotone Dynamical Systems. An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Volume 41, American Mathematical Society, Providence, Rhode Island, 1995.

[23] H. L. Smith and H. R. Thieme, Monotone scmiflows in scalar non-quasi-monotone functional differential equations, J. Math. Anal. Appl. 150, (1990), 289-306.


[24] V. B. Uvarov, Asymptotic properties of the solutions of linear differential equations with retarded argument, Differencialnye Uravneniya 4, (1968), 659-663, in Russian.

[25] J. Wu, Global dynamics of strongly monotone retarded equations with infinite delay, J. Integral Eqns and Appl. 4, (1992), 273-307.

[26] J. Wu and H. I. Freedman, Monotone semiflows generated by neutral equations and application to compartmental systems, Canadian J. Math. 43, (1991), 1098-1120.


VOLUME 12

2005, NO 1-2

PP. 209-244

RELATIONS BETWEEN STABILITY AND ADMISSIBILITY FOR STOCHASTIC LINEAR FUNCTIONAL DIFFERENTIAL

EQUATIONS*

R. KADIEV t AND ARCADY PONOSOV t

Key Words. Stability, admissibility, stochastic functional differential equations, integral transforms

Abstract. A stochastic version of Azbelev's W-method is described and justified in this paper. The method is based on relations between asymptotic properties of the given functional differential equation and the property of admissibility of certain pairs of functional spaces. The W-transform is known to be one of the main tools to study admissibility and stability. In contrast to our previous papers, where we applied this transform to the equation from the left., we suggest here a different approach, where the transform is applied from the right. Using this modification we prove some general results explaining how and why this method works and give some applications of the method to stability theory.

1. Introduction. The idea to study Lyapunov stability of linear differential equations with the help of stability under constantly acting perturbations was introduced by P. Bohl and 0. Perron in 192? and later developed by S. Barbashin [3], J. Massera & J. Schafer [13], S. Dalecky & M. Krein [4] and many others in the case of ordinary differential equations, and by N. Azbelev and his students for the case of functional differential equations (see e. g. [1] and [2] and references therein).

In the present paper we explain how to use theW-method to study linear stochastic functional differential equations. The idea is as follows. Given an

' The work is partially supported by the Norwegian Council of Universities' Committee for Development Research and Education (NUFU)

t Department of Mathematics, Dagestan State University, Republic ofDagestan ,Russia

Institutt for matematiske realfag og teknologi, NLH, postboks 5003, N-1432 As, Norway

209

210 R. KADIEV AND A. PONOSOV

asymptotic property (for instance, asymptotic stability) we first of all have to identify the functional space which correspond to the asymptotic behavior of solutions we are interested in and which the solutions of the equation should therefore belong to. In order to prove the last property we choose another, simpler equation (called in the sequel "the reference equation") which already has the needed asymptotic property so that its solutions belong to the identified functional space. The reference equation gives rise to the W-transform, associated with this equation, and it is this transform which we apply to the original equation. By this, we arrive at a certain integral equation of the form X- ex = f. The operator e is uniquely determined by the original equation (and the W-transform generated by the chosen reference equation. If we now manage to solve this equation in the identified functional space, then the desired asymptotic property will be proved. To check invertibility of I- e we can, for example, estimate the norm of the operator e.

In some sense, the W-method is similar to Lyapunov's method. But instead of seeking a Lyapunov function(al) one aims to find a suitable reference equation which possesses the prescribed asymptotic properties. It is important to stress that this approach, like Lyapunov's method, also provide necessary and sufficient stability conditions (also in the case of nonlinear equations which are not studied in this paper).

As we demonstrate here, the W-method can be used when the classical methods fail. We show, for instance, how the method can produce efficient stability conditions in terms of parameters of the studied equations as well as to investigate some equations that can hardly be covered by the Lyapunov-Krasovskii-Razumikhin framework (like equations driven by arbitrary semimartingales, equations with unbounded delay, or equations with random coefficients).

Essentially, this paper mimic our previous paper [9], where we exploited the W-transform in a different manner (applying it from the left, and not from the right as here). Also the technique was similar. However, in many cases we have to use different spaces of stochastic processes, different norms and make different assumptions and estimates. Therefore only a few results from the paper [9] are directly applicable to the results presented here.

2. Notation and main assumptions. Let (D, :F, (:Ft)t>o, P) be a complete filtered probability space (see e. g. [10, p. 9]), Z := (z1, •.. , zmf be a m-dimensional semimartingale [10, p. 73] on it (we distinguish column vectors (a1 , ... an)T and row vectors (a1 , ... an)).

In the sequel, we let I · I denote the norm in Rn; Rkxn will be a linear space consisting of all real k x n-matrices with the norm 11.11 that agrees with

STABILITY AND ADMISSIBILITY FOR LINEAR STOCHASTIC FDE 211

the chosen vector norm in Rn. We write 0 for the zero column vector in Rn, the symbol E denotes the unit matrix, while E stands for the expectation.

For convenience, by A* we denote the complete measure on an interval I, generated by a nondecreasing function >-(t), (t E I).

The following linear spaces of stochastic processes will be used in the sequel:

- Ln(z) consists of predictable n x m-matrix functions defined on [0, oo) with the rows that are locally integrable w. r. t. the semimartingale Z, see e. g. [5];

- kn consists of n-dimensional .:F0-measurable random variables (we set also k := k1 );

- Dn consists of n-dimensional stochastic processes on [0, oo), which can be represented in the following form:

t

x(t) = x(O) + j H(s)dZ(s), 0

where :c(O) E kn, HE Ln(z); -L; consists of scalar functions defined on [O,oo), which are q-integrable

(1 S q < oo) w. r. t. the measure,),*, generated by a nondecreasing function .>,(t), (t E [O,oo));

- L~ consists of scalar functions defined on [0, oo ), which are measurable and a. s. bounded w. r. t. the measure A', generated by a nondecreasing function >-(t), (t E [O,oo));

- Lq stands for L; in the case when >-(t) = t (1 S q S oo). In addition, we will always silently assume that the real numbers p, q

satisfy the inequalities

1 ::; p < oo, 1 ::; q ::; 00.

The following notational agreement will be used in the sequel:

b

J = r otherwise we will write 1 ' 1 etc. J[a,b] (a,b] (a,b)

a

The variation of a function over a closed interval [a, b] will be denoted by V , and we will also write V for lim V and V for lim V , re-

[a,bJ (a,b] 0->0+ [a+O,b] [a,b) 0->0+ [a,b-o] specti vely.

We are now able to formulate the main assumption on the semimartingale Z(t). In what follows we always assume that the semimartingale Z(t) (t E

[0, oo)) can be decomposed as a sum

(1) Z(t) = b(t) + c(t),

where b(t) is a predictable stochastic process oflocallyfinite variation and c(t) is a local squire-integrable martingale [10, p. 28] such that all the components of the process b(t) as well as the predictable characteristics < ci, d > (t), 1 :::; i, j :::; m of the process c(t) [10, p. 48] are absolutely continuous w. r. t. to a nondecreasing function A: [O,oo) ~ R+· In this case, we can write

(2) bi = j aidA, < ci, d >= j AiidA, i, j = 1, ... , m. 0 0

For example, A(t) = t for ItO equations. Without loss of generality, it will be convenient in the sequel to assume that the first component of the semimartingale Z(t) coincides with A(t), i.e. z1 (t) = A(t). Clearly, we can always do it adding, if necessary, a new, (m+ 1)-th component to them-dimensional semimartingale Z ( t) .

It is known [5] that under the assumption (1) the space Ln(z) can be described as a set of all predictable n x m-matrices H( t) = [Hii ( t)], for which

t

(3) j(IHal + IIHAHTII)dA < oo a. s. 0

for any t 2: 0. Here

(4) a ·- (a1 am)T A ·- [Aii] .- , ... , ' .- .

Note that a is an m-dimensional column vector and A is a m x m-matrix. t t

Under the above assumptions we can also write f HdZ = f Hdb + 0 0

t f H de. Moreover, we can describe the space Dn as a set consisting of all no dimensional adapted stochastic processes on [0, oo), the trajectories of which are right continuous and have left hand limits for all t E [0, oo) and almost all w (the so-called 'cadlag processes'). Moreover, the following estimate holds:

( E I HdZ

2

p) tp:::; ( E (/ IHaldA rp) tp + Cp ( E (/ IIHAHTIIdA r) tp'

(5)

where Cp is a certain positive constant depending on p and, possibly, on Z (see e. g. [10, p. 65], or [12, p. 7]).

Given H = [HiJ] E Ln(z) and a, A defined in (4), we will write

(6) a+:= (la1l, ... ,lam if, A+ := [IAiJIJ, H+ := [IHiJIJ.

Studying different kinds of stochastic stability requires different spaces of stochastic processes which are listed below.

Main spaces.

We are assume the following to be given - a scalar nonnegative function~, defined on [0, oo) and integrable w.

r. t. the measure A* generated by the function .\ (.\is the same as in 2); - a positive scalar function 'Y(t) (t E [O,oo)). REMARK 1. In what follows we silently adopt the following convention:

if in a definition, a theorem etc. 'Y(t) is mentioned without any comments, then it is only assumed to be a positive scalar function. Otherwise, additional properties of 'Y will be explicitly described.

These functions are involved in the definitions of almost all spaces we are going to use in the sequel. Both are crucial for our considerations as they are responsible for the asymptotic behavior of the solutions.

k; ={a: a E kn, llallk;; := (EiaiP)l/p < oo};

M~ p = {x: x E Dn, llxiiM" := (supEI'Y(t)x(t)IP)1/P < oo}

p t:2:0

A.;,q(~) = {H: HE Ln(z), (EIHaiP) 11P~q-1

- 1

+(EIIHAHTIIp/2)1/p~q-1-05 E L;};

i\;~(~) = {H: HE Ln(z), (EIWa+iP) 1 1P~q- 1 -1

+ (EIIH+ A+(H+)TIIp/2)1/p~q-1-05 E L;}.

The following parameters are involved in the above definitions:

(M~ = Mp);

- the numbers p, q are assumed to satisfy the inequalities 1 :<:; p < oo, 1 :<:; q :<:; oo;

··a, A are defined by (4); -a+, A+, H+ are given by (6).

In the two last spaces the norm are given by

IIHIII\n (<) := II(EIK1H aiP) 1 /P~q-1

- 1 l !£> p,q q

+ II(EIIK2HAHTIIP12 ) 1/p~q-l-o 5 IIL~>

IIHIIJ\n+(<;) := II(EIK1H+a+IP) 1/p~q- 1 - 1 IIL> p,q q

+ II(EIIK2H+ A+(H+) TIIP12 ) 1/p~q-'-O.sll£~,

respectively. Here K 1 and K 2 are two given positive constants. Remark that we use the notation A in order to distinguish these spaces from similarly defined spaces A which we used in the left W-transform (see (13)). The spaces A and A consist of the same stochastic processes, but they are equipped with different norms.

Finally, we introduce the following general notation: -Given a linear normed space B with the norm II·IIB, we write

(7) B'' := {f : f E B, 'Yf E B}

with the norm llfiiB, = II'YfiiB· This notation is used for instance when B is equal to one of the above

spaces of stochastic processes.

Operators and equations.

DEFINITION 1. An operator V: Dn-+ Ln(z) is called Volterra (see [8}) if for any stopping time [1 0, p. gj r E [0, oo) a. s. and any x, y E Dn such that x(t) = y(t) (t E [0, r] a.s.) one has (Vx)(t) = (Vy)(t) (t E [0, r] a.s.).

DEFINITION 2. An operator V: Dn-+ Ln(z) is called k-linear if

V(a1x1 + a2x2) = a1 Vx1 + a2Vx2

for any a; E k, x; E Dn, i = 1, 2. This property exclude 'global' operations, like expectation, from the coefficients of the equation, and therefore determines the pathwise way of describing solutions.

REMARK 2. If V is continuous w.r.t. natural topologies in the spaces Dn and Ln(z), then one can show that k-linearity follows from the usual linearity (w.r.t. R).

The central object of this paper is a stochastic functional differential equation

(8) dx(t) = [(Vx)(t) + f(t)]dZ(t) (t;::: 0),

•

where f E Ln(z) and V : Dn -+ Ln(z) is a k-linear and Volterra (in the sense of Definition 1) operator.

In [15] it is shown that the equation (8) covers linear stochastic delay equations, linear stochastic integra-differential equations, linear stochastic neutral equations - all with driven semimartingales etc. It can look a little bit confusing as the equation (8) does not depend on the values x(t) fort < 0. In fact, this dependence can be incorporated into the right-hand side as it is demonstrated in the following example.

EXAMPLE 1. Consider a linear scalar stochastic differential equation of the form

(9) dx(t) = [a(t)(Tx)(t) + g(t)]dZ(t) (t?:: 0)

with the prehistory condition

(10) x(s) = <p(s), s < 0,

where (Tx)(t) = fc-oo,t) d,R(t, s)x(s) is the distributed delay. Under natural assumptions on the right-hand side (see, {15}) this equa

tion can be reduced to the for-m (8) if one sets

(Vx)(t) := a(t) ( d,R(t, s)x(s) and f(t) := a(t) ( d,R(O, s)<p(s)+g(t). l[o,t) lc-oo,o}

(11)

In addition to the equation (8) we consider the associated homogeneous equation (f = 0).

(12) dx(t) = (Vx)(t)dZ(t) (t?:: 0).

Using k-linearity of the operator V, we immediately obtain the following LEMMA 1. Let for any x(O) E kn there exists the only solution (up to a

P-null set) x(t) of the equation (8). Then one has the following representation ('the Cauchy representation') of the solutions

(13) x(t) = X(t)x(O) + (Kf)(t) (t?:: 0),

where X(t) (X(O) = E) is ann x n-matrix, the columns of which are the solutions of the linear homogeneous equation (12) ('the fundamental matrix'), while K: Ln(z) -+ Dn is a k-linear operator ('the Cauchy opemtor') such that (K f)(O) = 0 and K f satisfies the equation (8). D

In what follows we will always consider the equation (8) under the uniqueness assumption, i.e. existence, for any :r(O) E kn, of the unique (up to a ?-null set) solution x(t) of this equation. In other words, according to Lemma 1, the representation (13) will silently be assumed in all further considerations.

3. Ml - stability. We start with the classical definitions of stochastic stability.

DEFINITION 3. The zero solution of the linear homogeneous equation {12) is called

a) p-stable if for an arbitrary c > 0 there exist 'f/ = 'fl(c) > 0 such that

E/X(t)x(O)IP :5: c

for all x(O) ERn, /x(O)/ < ry, t <': 0. b) asymptotically p-stable if it is p-stable and lim E/X(t)x(O)/P = 0

t....:;+oo for all x(O) E Rn.

c) exponentially p-stable if there exist c > 0, f3 > 0 such that

E/X(t)x(O)/P:::; cJx(O)/ exp{ -f3t} (t;::: 0)

for all x(O) ERn. Similarly, we can define stability of solutions to the inhomogeneous equa

tion (8). Clearly, the representation (13) implies that all solutions to (8) are p-stable (asymptotically p-stable, exponentially p-stable) if and only if the zero solution to the homogeneous equation (12) is p-stable (asymptotically p-stable, exponentially p-stable). In the sequel we shall therefore say the inhomogeneous equation (8) is stable (in a proper sense) if the zero solution to the homogeneous equation (12) is stable in the same sense.

The following result was proved in [9]. THEOREM 1. A}. The equation (8} is p-stable if and only if X()x(O) E Mp for all

.. x(O) ERn.

B). The equation {8} is asymptotically p-stable if and only if there exists a function /'(t), for which /'(t) ;::: o > 0 and lim J'(t) = +oo, so that

t-t+oo X(·)x(O) E M7 for all x(O) ERn.

C). The equation ( 8) is exponentially p-stable if and only if there exists a number f3 > 0 such that X(-)x(O) E MJ for all x(O) E Rn, where /'(t) = exp{/3t}. D

This theorem says that to check p-stability (asymptotic, exponential pstability) of the equation (8) we can prove that the solutions of the homogeneous equation (12) belong to a certain space of stochastic processes (Mp or MJ).

Due to Theorem 1, we can now introduce a new, unified definition of stability which is more convenient for our purposes.


DEFINITION 4. The equation {8) is called MJ -stable, if for any x(O) E k; we have X(·)x(O) E MJ.

Due to Theorem 1 we can now say that - Mp-stability (i. e. MJ-stability with 'Y = 1) of the equation (8) implies

the Lyapunov p-stability of the equation (8); - MJ-stability of the equation (8) with 'Y satisfying 'Y(t) ?:: o > 0 and

lim 'Y(t) = +oo) implies the asymptotic p-stability of the equation (8); t~+oo

- MJ-stability of the equation (8) with 'Y(t) = exp{,6t} (for some ,6 > 0) implies the exponential p-stability of the equation (8).

Thus, we have replaced stability analysis of the equation (8) to the problem of how resolve this equation in a certain space of stochastic processes. This observation is crucial for applying the method of the W-transform.

To prove the MJ-stability of the equation (8), it is sufficient to check that all solutions x(t) of the homogeneous equation corresponding to (8) are contained in the space MJprovided x(O) E k;. We shall verify this property by applying the right W-transform to the equation (8).

As we have already mentioned any W -transform comes from an auxiliary equation, which we call a reference equation. That is why we assume given another equation, similar to (8), but 'simpler'. In addition, we assume the asymptotic properties of the reference equation to be known.

Let the reference equation have the form

(14) dx(t) = [(Qx)(t) + g(t)]dZ(t) (t?:: 0),

where Q : D" -+ L"(Z) is a k-linear Volterra operator, and g E L"(Z). Also for the equation (14) it is always assumed the existence and uniqueness assumption, i. e. for any x(O) E k" there is the only (up to a P-null set) solution x(t) of the equation (14). Then, according to Lemma 1, for this solution we have 'the Cauchy representation' x(t) = U(t)x(O) + (W g)(t), t?:: 0, where U ( t) is the fundamental matrix of the related homogeneous equation, and W is the corresponding Cauchy operator.

Substituting this expression for x(t) into the equation (8) gives

[(QUx(O))(t) + (QWg)(t) + g(t)]dZ(t) = [(V(Ux(O) + Wg)) (t) + f(t)]dZ(t)

(t?:: 0).

Denoting (V- Q)W = 8r, we obtain the operator equation

(15) (I- 8r)g = (V- Q)Ux(O) +f.


Let B be a .linear subspace of the space Ln(z). In the sequel the property of invertibility of the operator (I - 6r) : B --t B means that this operator is a bijection on the space B (i. e. the inverse operator is not necessarily continuous). The following simple result is, however, an important step in our further considerations.

THEOREM 2. Let the reference equation (14) be M;-stable and there exists a linear subspace B c Ln(z) such that W : B --t Ml, Q, V : Ml--t B. If now the operator

(I - er) : B --t B

is invertible, then the equation (8) is Ml-stable. D

Proof As the operator (I - 6r) : B --t B is invertible, the equation (I -6r)9 = f has the unique solution g E B for any fEB, i.e. (I -er)-1 f E B. From this and the representation X(t) = U(t)+(W(I -6r)-1(V -Q)U)(t) we obtain X(·)x(O) E Ml if x(O) E k~ as (V- Q)Ux(O) E B for these x(O). 0

Let us remark that, in general, there is no direct dependence between stability of the equation in question and invertibility of the operator equation (15). For instance, in certain situations the operator er even does not act in the corresponding space of stochastic processes, while both equations (8) and (14) are stable in this space.

Nevertheless, if the equation (8) is stable, then there always is at least one (in fact, infinitely many) stable reference equations, for which the operator er will act in the related space of stochastic processes and I - 6r will be invertible there. For instance, one can choose the equation (14) be identical to (or be in the vicinity of) the initial equation (8).

This observation and Theorem 2 imply the following stability criterion based on the W-transform.

COROLLARY 1. The equation (8) is M;-stable if and only if there exist a linear subspace B C Ln(z), where V: Ml--t B, and an Ml-stable reference equation (14), where W : B --t Ml, Q : Ml --t B, which give rise to the invertible operator (I - 6r) : B --t B. D

Among all the assumptions imposed on the initial equation (8) and the reference equation (14) one is more involved than the others when applying Theorem 2. It is invertibility of the operator (I- er) : B --t B. A reasonable method to check this requirement is to estimate the norm of the operator 6r in the space B, provided B is a normed space (which is always the case in practice).

Thus, from Theorem 2 we obtain the following simple proposition.

•

COROLLARY 2. Assume that there exists a linear Banach subspace B c Ln(z) and an M;-stable reference equation (14), such that W: B---+ M;, Q : M7---+ B. Assume also that V: M;---+ B and ll8riiMJ < 1. Then the equation ( 8) is M;-stable. D

In what follows we will need a more explicit description of the Wtransform (and the corresponding reference equation (14)), which is summarized in the assumptions Rl and R2 below.

Rl. The fundamental matrix U(t) to (14) satisfies IIU(t)il ::; c, where cE R+·

R2. TheW-transform coming from (14) has the form

t

(16) (W g)(t) =I C(t, s)g(s)dZ(s) (t;::: 0), 0

where C(t,s) is ann x n-matrix defined on G := {(t,s): t E [O,oo), 0::; s ::; t}, and satisfies

(17) IIC(t, s)ll ::; cexp{ -aD.v },

where

t

v(t) =I ~(()d.X((), bJ.v = v(t)- v(s) 0

for some a > 0, c > 0. For some specific reference equations (14), which give rise to the W

transform of the form (16) and which satisfy all the additional assumptions listed above, see [7].

REMARK 3. Let the reference equation be given by

m

(18) dx(t) = (A(t)x(t-) + g0 (t)) d.X(t) +I; gi(t)dzi(t), i=2

where A (t) is an n x n-matrix with locally A-integrable entries (in this case (Qx)(t) = (A(t)x(t),O, ... ,0)). In this case it is straightforward that the kernel C(t, s) in (16) is of the form C(t, s) = U(t)U- 1(s). Notice also that if we set A(t) = -a~E, then the conditions Rl-R2 will be fulfilled.

A more involved example of the reference equation is given by

(19) dx(t) = Uo,t) d,R(t, s)x(s) + g0 (t)) d.X(t) + ~9i(t)dzi(t),

where the entriesrjk(t, s) of a (non-random) ri x ncmatriX 1?-(t, s), defined on the set G from R2, are of bounded variationin s and, in addition, · V rjk(t,s) are locally >.-integrable for each t E [O,oo). In this case, the sE~,ij .

representati()n (16) .is again valid, .but there .is no direct relations between C(t,s) and thefundamenta[matr:ix U(t). The estimate (17) can be obtained in special cases {see [2} for details) .

. . In the rest of th)s section we will be concerned with the M7 -stability of the equation (8), where

t

(20) 1(t) = exp{,B j ~(s)d>.(s)}, . 0

where ,B is some positive number satisfying ,B <a (see the assumption R2). Choosing this specific weight 1 is therefore determined by the specific Wtransform described in Rl-R2. We wish to use this W-transform and the weight 1 in order to prove two main results .of this section (Theorems 3 and 4). The first theorem justifies the W-method in connection with Mlstability (and by this to the Lyapunov stability of the equation (8) .w .. r. t. the initial value x(O)). The second theorem deals with the following fundamental problem which is also well-known for deterministic functional differential equations (see e. g. [2]): find conditions, under which the p

stability implies the exponential p-stability. We shall prove that it is the case if the delay function satisfies the so-called 'Ll-condition' (see Definition 6 below). The Ll-condition is fulfilled if for instance the delays are bounded (see Lemma 3). Apart from the importance of these two genera!factsfor the theory of stochastic functional differential equations, the technique we use to prove them is itself a good illustration of how the W•transform·Works in practice.

For further purposes we will .need the following technical lemma. LEMMA 2. Assume that the function 1(t) is given by {20) for some ,B

(0 < ,B < a, wherectistaken from (l'l)) .. Jfthe refer:e'f}Ce .eq~ation{J4) satisfies the assumptior! Jt2, then H; givenby (16) is a. continuous operator from (A2p,q(~))' to M:j1, where 2p::::; q::::; oo.

Proof To prove the lemma it suffices to. check that

(21) JJWgJJM;rp::::; cJJgJJ(A.~p.,«JJo (c E R+)

if g E (A2v,q(~))'.


We have

IIWgiiMir = lbWgiiM,r =Ill J C(·,s)g(s)dZ(s)IIM,r· 0

We need to prove that

lg :=II J C(-, s)g(s)dZ(s)IIM,r::; clb9llx~r.,«l' 0

where cis a positive number.

Now

t l

lg::; sup(E(fexp{-(a- J])~v}b(s)g(s)a(s)ld,\(s)) 2P)2P t2:0 0

t l

+cp sup(E(f exp{ -2(a- {J)~v} b(s)g(s )A(s)(r(s )g(s)) Tld,\(s))P) 2P t2:0 0

t 2p-l

::;sup(fexp{-(a-{J)~v}dv(s)) 'r t2:0 0

t l

x(J exp{ -(a- {3)~v}(.;(s)) 1 -2PEil(s)g(s)a(s)I 2Pd,\(s))2P 0

t +cp sup(J exp{ -2(a - {J)~v }dv(s) )(P- 1)2P

t2:0 0

t l

x (f exp{ -2(a - {J)~v W(s)) 1-PEI!r(s )g(s )A(s )('y(s )g(s)) TIIPd,\(s )) 2P 0

t l

:S c{ sup(f exp{- (a - {J)~v }( s( s)) 1- 2PEir(s) g(s )a(s) I2Pd,\(s)) 2P + t2:0 0

t 1

+cp sup(J exp{ -2(a- {J)~v }(S"(s) )1-PEI!r(s )g(s )A(s)(r(s )g(s)) TWd,\(s)) 2P}, 12:0 0

where c is some positive number.

Here we used the inequality

t (EI'Y J C(t, 8)g(8)dZ(8)12P)1/2p

0

t ::; (E(f exp{- (a - ,6)~v }I'Y( 8) g( 8 )a( 8) ld>-( 8) )2P)lf2p

0

t +cp(E(f exp{ -2(a- ,6)~v }II'Y(8)g(8)A(8)1('Y(8)g(8WIId>-(8))P)112P,

0

which follows from (5) and (17). To be able to obtain further estimates we have to consider three cases

separately: 1) qj2p > 1, q i- oo; 2) qj2p = 1; 3) q = oo. Let first qj2p > 1, q i- oo. Then

t

t

lg :::: c{ sup[(/ exp{ (-(a - ,6)qj(q- 2p))~v }dv(8))(q-2p)(2pq t2'0 0

t

x (j ( (EIK! 1 K 1 "!( 8 )g( 8 )a( 8) I2P) 112P( ~( 8) )1/q-l )qd>-( 8) )1fq] 0

t

+cp sup[(/ exp{( -2(a- ,6)qj(q- 2p))~v }dv(8))(q-2P)(2pq t>O - 0

x (j ( (EIIK21 Kn( 8 )g( 8 )A( 8) ( 'Y( 8 )g( 8)) TIIP) 1 12P(~( 8)) lfq-lf2)qd>-( 8 ))lfq]} 0

:::: cii'Y9IIi\~".q<<J·

Assume now that qj2p = 1. In this case we derive the following estimates:

t

lg :S c{ sup(/( (EIKj1 K 1 'Y( 8 )g( 8 )a( 8) 12P) 1 12P(~( 8) )1/q-l )qd.>,( 8) )1

/q t2'0 0

t

+cv sup(/( (EIIK2 1 K21( s) g (s) A( s) ( 1( s )g( s )fliP) 1/2r ((( s)) IJHI2)qd>.( s)) l/q}

co 0

s: clhgllii~P.,<<l· Finally, if q = oo, then we have

lg :; c

t x { sup(J exp{- (a - ,6) L'.v }(( s) [(Ell( s) g( s )a( s )12P) 112P (( ( s)) -I j2Pd).( s)) I/2P

t?:O 0

t +cp sup J exp{ -2(a - ,6)6.v }((s)[(EIII(s )g(s)A(s) ('Y(s )g(s WW

t?:O 0

X ( ( ( S)) -1/2) lj2pj2Pd).( S)) I/2P}

:; c{ vrai sup[ (EI K)1 K 1 1( t) g( t)a( t)IZP) 112P (((t) )- 1] (1 j a) 112P

ost:Soo

+cr vrai sup[ (EIIK2 1 K 21( s) g( s )A( s) ( 1( s )g( s)) TIIP) 112P ( (( t)) -l/2] (1 /2a) 112P} O:St:Soo

:; clbgiiA~p,,(O· The lemma is proved. 0

CoROLLARY 3. Assume that the reference equation (14) satisfies R2. Then W given by (16) is a continuous operator from A2r,q(() to M2p, where 2p :; q :; 00. 0

From Lemma 2 and Corollary 2 we obtain THEOREM 3. Let the function 1(t) be given by (20) for some ,6 (0 <

,6 <a, where a is taken from (11)}. Assume that the reference equation (14) is Miv-stable. Assume also that the operators V and Q from (8) and (14), respectively, act from Mfr, to (li.2r,q(())'. Then the estimate ll8rll(ii~p,,(l;))• < 1 implies the Mir-stability of the equation (8), where 2p S: q:; oo. 0

This theorem offers a formal justification of stability analysis based on applying the right W-transform given by (16).

To be able to formulate and prove the second main result of this section we need some preparations. Below mp stands for the space Mp in the scalar case. We assume that the k-linear operator V in (8) satisfies

V : Mv-+ A;,q(~).

We will also use the following notation related to the operator V: - Vx = (Vix, ... , Vmx); - (Vfix)(t) := 'Y(t)(V(xh))(t) , where 'Y(t) is defined in (20). DEFINITION 5. We say that a k-linear Volterra operator V : mp -t

li.~~ ( ~) dominates a Votter::a operator V : Mp -t li.~,q ( ~) , if 1) V is_ positive, i. e. x;::: 0 a. s. implies Vx;::: 0 a. s., and 2) (iVixl, ... , JVmxiJ:::; VJxJ a. s. for any x E Mp.

DEFINITION 6. We say that a k-linear Volterra operator V : Mp -t li.~,q(~) satisfies the !:!.-condition, ifV is dominated by some k-linear Volterra operator V : mp -t li.~~ ( ~) with the following additional assumption: there exists a number (3 > 0, for which the operator

(V11x)(t) := 'Y(t)(V(xh))(t)

acts continuously from the space mp to the space li.~~(~). DEFINITION 7. Let X, Y be two linear spaces consisting of predictable

stochastic processes on [0, oo), and T : X -t Y be a k-linear Volterra operator. We say that the operator T satisfies a-condition if there exist two positive numbers 01

' 011' 01 > 011

' providing the following implication for all t E [0, oo): any x E X, satisfying x( (') = 0 for all (' E [0, t] such that t f e(s)dA(s) < 8', also satisfies (Tx)(('') = 0 for all ( 11 E [O,t] such that ('

t J e(s )dA(s) < 8". ("

LEMMA 3. Assume that a k-linear Volterra operator V: Mp -t li.~,q(O is dominated by a k-linear, bounded and positive operator V : mp -t li.~~(e) satisfying the a-condition. Then the operator V satisfies the !:!.-condition.

Proof According to our notation

(VIi•x)(t) = (v (exp{/30 J e(s)dA(s)}x)) (t).

The a-condition from Definition 7 implies that the value (Vy)(t) depends t

only on the values y(("), where J e(s)dA(s) < 8" (where a" is again taken ("

t from Definition 7, (" E [0, t]), and for these(" we have exp{/30 f e(s)dA( s)} :::;

("

exp{,8oc5"}. This leads to the following estimate

v11• X :::; V( exp{f3oc5"} )xi) = exp{f3c5"} VJxl

almost surely. 0 In examples we will use equations with a discrete delay as reference

equations. The next definition describes the operators to be involved. DEFINITION 8. We are given a measurable function g : [0, oo) -+ R

such that g(t) :S t (t E [O,oo)) and a. row vector G = (G1,G2 , ... ,Gm), where Gi = Gi(t) are all predictable and nonnegative stochastic processes. We define the weighted shift operator GS9 by (GS9x)(t) = G(t)(S9x)(t), where

(22) (S :r;)(t) = { x(g(t)), if g(t) :?: 0, g 0, ~f g ( t) < 0 ,

Clearly, GS9 : D 1 -+ U(Z). To be able to check the <~-condition from Definition 7 for weighted shifts

we will use special conditions on g. We will also need some new notation: for a given measurable function g : [0, oo) -+ R we will write

(23) (t) = { 1, if g(t) 2: 0, Xg 0, if g(t) < 0.

DEFINITION 9. We say that a measurable function g : [0, oo) -7 R t

satisfies the 6-condition if there exists /j > 0 such that J ~(s)d>.(s) < fJ x,(t)g(t)

for all t E [0, oo). EXAMPLE 2. If a measurable function g : [0, oo) -+ R satisfies the 8-

condition from Definition 9, then the weighted shift operator GS9 : D 1 -> L1 (Z) satisfies the 8-condition from Definition 7.

To see this, we notice that according to Definition 9 there exists 8 > 0 such that

t

j ~(s)d>.(s) < /j for allt E [0, oo) x9 (t)g(t)

Sett·ing /j' = 28, li" = /j and taking arbitrary t E [0, oo) and x E D 1, for

t which y((') = 0 for all(' E [0, t] satisfying J ~(s)d>.(s) < 6', we have to

(' t

check that (GS9x)((") = 0 a.s. for all(" E [0, t] such that f ~(s)d>.(s) < 8". ("

This follows from the equality (S9 x)((") = 0 a.s., or equivalently, from the t

estimate f ~(s)d>.(s) < 6'. But this is implied by Xg((")g((")

t ( 11 t

j ~(s)d>.(s) = j ((s)d>.(s) + j ~(s)d>.(s) < /j + li" = li'. x,((")g((") x,((")g((") <"


Notice also that if A = t (i.e. ,\* is the standard Lebesgue measure}, then ~(t) = 1, and the 6-condition for g takes the following form: t- g(t) :s; 6, that is, the delay will be bounded.

The concluding result of this section explains when the usual stability of solutions implies the exponential and asymptotic stability. We present here only a general principle, postponing all further discussions and examples until the last section.

THEOREM 4. Let the equation (8) and the reference equation (14) satisfy the following assumptions:

- the operators V, Q act as follows: V, Q : M2p --7 A~p,,,:J ~); - the reference equation ( 14) is M2p -stable and satisfies the condition

R2; - the operator V satisfies the !1-condition; If now the operator (I- 8r) : M2p --7 M 2p is continuously invertible, then

the equation {8) is Ml.P- stable, where 'Y is defined in (20} with some j3 > 0. Proof First of all, we notice that the equation (8) is Ml.P-stable if and

only if the equation

dy(t) = exp{/3[ ~(()d-X(()} [w exp{ -!3 [ ~(()d-X(()}y)(t) + f(t)] dZ(t)

(24)

+f3~(t)y(t)d-\(t) (t ~ 0)

M2p-stable. Hence, in order to prove the theorem it is sufficient to show the existence of a positive number /3, for which the equation (24) will be M2p

stable. From Theorem 2 it follows that if the operator 8~ acts in the space A.~p,oo(~), and the operator (I- e~) : A.~p,oo(~) --t A.~p,oo(~) is invertible for some /3, then the equation (24) will be M2p-stable for this j3. Here 8~ is a k-linear operator defined, according to our previous notational agreements, by

(25) (e?x)(t) := 'Y(t)(8r(xh))(t),

so that, in particular, e~ = 8r. Using the assumption of the theorem saying that the operator V satisfies

the /1-condition, we obtain a number j30 > 0, for which the operator 8~ acts continuously in the space Mp,oo(~) for all 0 :s; j3 :s; f3o. This fact follows from Corollary 3 and a simple observation that if the operator V satisfies the /1-condition, then the operator ifil acts from the space mp to the space A~t,00 (~) and it is boundedforallO :s; j3 :s; f3o. If now we check that ll8f-8diK• (~) --7

2p,oo

o where fl -+ o, then the operator (I- 8f) : A2p,oo(~) -+ A2p,oo(~) will also be invertible for some fl > 0.

Notice that the operator (8~- 8,) is given by

((8f- 8,)g)(t) = (V('y(t)h(.)- 1) J C(-, s)g(s)dZ(s))(t) 0

I

+(fl~ J C(t, s)g(s)dZ(s), 0, ... , 0). 0

Then

ll(8f- 8,)giiJ.n (<l::; IIV(rh(-)) -1) jC(·,s)g(s)dZ(s)IIJ.n (<l 2p,oo O 2p,oo

+ll(fl~ J G(-, s)g(s)dZ(s), o, ... , O)IIJ.~ (<)· 0 p,oo

Using the inequality')'( t) h( s) -1 ::; (fl I flo) exp{flot.v}, s E [0, t], 0 ::; fl S flo, which follows from the estimate

flv + fl2v2 /2! + fl3v3 13! + ... ::; fJ! flo+ flv + fl2v2 12! + ...

:C: (fll flo)(1 + flov + fl5v2 12! + ... )

(v > 0) and minding the fact that the operator V satisfies the C.-condition, we obtain

ll(8f- 8,)giiJ.~p.oo(0::; liV(I"YhU -lii[C(-,s)g(s)dZ(s)lliiJ.;~,oo(<l

+llfJ(EI J I<1C(-, s)g(s)dZ(s)I2P) 1 1 2PIIL~ 0

::; II (fl I flolCVflo (I J cc, s )g(s )dZ(s) I) IIJ.'+ m O 2p,oo ·

+fJII J I<1C(-, s)g(s)dZ(s)IIMzp 0

::; (fJI fJo)cll J C(·, s)g(s)dZ(s)IIM,p + flll J I<1C(·, s)g(s)dZ(s)IIM,p, 0 0

where c is some positive number.

Due to Corollary 3 we have

[[(ef- 6t)gffx~P.=(<) :S ,Bdffgffx~p.=(<)'

where dis some positive number. From this we obtain rref - et[ [xn ({) -+ 0 2p,oo

as ,B -+ 0. As we noticed before, this is enough to prove the theorem. 0

4. Admissible pairs of spaces and stability w. r. t. the initial function. Another name for admissibility of pairs of spaces is stability under constantly acting perturbations. Roughly speaking, given a pair (B~, B2) of spaces of stochastic processes, one calls it admissible for a linear stochastic differential equation if any solution of the equation lies in B 1 as soon as the right-hand side of the equation ("perturbation") lies in B2 . This terminology goes back to J. L. Massera and J. J. Schafer [13] who studied admissibility for ordinary deterministic differential equations in Banach spaces. The main idea of this theory is to connect admissibility and Lyapunov stability (or the dichotomy of solution spaces). This approach proved to be particularly useful for deterministic junctional differential equations [2]. For stochastic functional differential equations admissibility was studied in [8]-[7]. In this paper we develop these studies.

To outline this method in brief, let us again look at Example 1. Suppose we want to study Lyapunov stability of the solutions of the equation (9) w. r. t. the initial function (10). The usual Lyapunov-Krasovskii method suggests that we rewrite (9) as an equation in a Banach space of all initial functions <p (usually it is the space C[ -h, OJ), A detailed description of this approach in the case of stochastic differential equation can e.g. be found in the monographs [12], [16].

Another way is presented in [2] and developed in [8]-[7] for the case of stochastic delay differential equations. The idea is to rewrite the equation (9) in a different manner, namely in the form (8) with V and f defined in (11), as it is described in Example 1. By this, the initial function <p will be included in the right-hand side of the equation, and stability of the equation (9) w. r. t. <p will be reduced to a particular case of the general admissibility problem for the functional differential equation (8). This approach is flexible and efficient, especially in the case of linear equations. In its practical use, it is common to exploit the W-transform as an additional tool.

The main objective of this section is to demonstrate how this approach, in combination with the general results and techniques developed in the previous section, can be utilized to derive stability of stochastic delay differential equations with respect to the initial function <p.


We start with some more notation. Let B be a linear subspace of the space Ln(z) (defined in Section 1).

The space B is assumed to be equipped with a norm 11-l\ 8 . Given a weight r(t) (t E [0, oo)) we define B~ by the formula (7).

For the sake of convenience we will also write x J(t, x0 ) for the (unique) solution of the equation (8). Here f is the right-hand side of the equation (8) and x0 is the initial value of the solution, i. e. x1(0, x 0 ) == x0 •

DEFINITION 10. We say that the pair (MJ, B) is admissible for the equation {8) if there exists c > 0, for which x0 E k; and f E B imply xj(., x0 ) E MJ and the following estimate:

llxt(-,xo)IIM;::; c(ilxollk;; + llfiiBl·

This definition says that the solutions belong to MJ whenever f E B and x0 E k; and depend continuously on f and x0 in the appropriate topologies. The choice of spaces is closely related to the kind of stability we are interested Ill.

The first two results in this section explain what assumptions on the reference equation are to be checked if one wants to utilize the W -transform in the admissibility problem.

THEOREM 5. Assume the equation {8) and the reference equation {14) satisfy the following conditions:

1) the operator-s V, Q acts continuously from MJ to B~, 2) the reference equation {14) satisfies llr(t)U(t)ll::; c, where c ER-r, S) the operator W acts continuously from from B' to MJ. If now the operator I - 8r : B~ --+ B' has a bounded inverse in this

space, then the pair (MJ, B') is admissible for- the equation {8). Proof. Under the assumptions of the theorem we have

XJ(t, xo) == U(t)xo + (W(I- 8r)-1(V- Q)Uxo)(t) + (W(I- 8r)-1f)(t)

for an arbitrary x0 E k;, f E B~. Taking the norms and using again the assumptions of the theorem, we arrive at the inequality

llxtC Xo)IIM; ::; c(llxollk;; + llfiiBo ),

which holds for any x0 E k;, f E B~. Here cis some positive number. This means that the pair (MJ, B~) is admissible for the equation (8). D

If, in addition, we have the b.-condition from Definition 6, then we can prove more.

THEOREM 6. Assume that the weight 1(t) is defined by {20), the equa-tion {8) and the referenr:e ~quation {14} satisfy the following conditions:

1} the operators V, Q ~cfs eonti:wously from M2p to A~p,oo(~), 2} the reference equation (14) has the properties Rl-R2, 3} the operator V satisfies the 6.-condition (6). If now the operator I - 8r : A~,oo ( ~) ---+ A~p,oo ( ~) has a bounded inverse

in this space, then the pair (MJP, (A~p,oo(~))~) is admissible for the equation (8) for some (3 > 0.

Proof By construction, the pair (MJP, ~A~p,oo(~))~) is admissible for the equation (8) if and only if the pair (M2p, A~p,oo(~)) is admissible for the transformed equation (24), which we prove by checking that under the assumptions of Theorem 6 all the conditions of Theorem 5 are satisfied for the transformed equation (24), where we put 1 = 1 and use 2p instead of p (so that MP becomes M2p) and A2p,oo(~)) instead of a general B.

Indeed, the second condition of Theorem 5 and the fact that Q acts from M2p to A2p,oo(~)) trivially follow for the assumptions 1) and 2) of Theorem 6. The operator VIi, defined for the equation ( 24) by

Vlix = ,v(x/1),

corresponds to the operator V from the general equation (8). From the assumption 3) of Theorem 6 it follows that there exists a pos

itive (30 such that the operator VIi acts continuously from M2p to A~p,oo(~) for all 0 < (3 < f3o.

Now, applying Corollary 3 to the assumption 2) of Theorem 6 we conclude that the operator W acts continuously from A~p,oo(~) to M2p.

Finally, we check that there exists (31 > 0 such that the operator

I- 8~ : A~p,oo(~) ---+ A~p,oo(~)

is continuously invertible for all 0 < (3 < (31• Here 8?, defined in (25), corresponds to the general operator 8r from (15). According to the proof of Theorem 4 we have ll8f- 8zllxn (<) ---+ 0 as (3---+ 0, so that the operator

2p,oo

I - 8? for sufficiently small (3 > 0. D We are now ready to investigate Lyapunov stability w. r. t. the ini

tial function <p. In the previous section we studied stability w. r. t. the initial value x(O). The difference between these two stabilities can again be explained by virtue of Example 1. In the initial condition (10) there is no formal difference between all the 'prehistory' values of the solution x(s), s :5: 0. In fact, if we change the value of the initial function cp(s) for one (or even

countably many) s < 0, then the solution x(t), t > 0 will not be changed. If we, however, change the value <p(O), then the solution will be different, that is the instants s = 0 and s < 0 are different. This observation explains roughly why it is reasonable to treat the function <p(s), s < 0 and <p(O) = x(O) separately. That is why we rewrite the delay equation (9) with the initial condition (10) as the functional differential equation (8). This idea proved to be fruitful in many cases (see e. g. [2], [7] and references therein).

In our paper we exploit this approach to study Lyapunov stability w. r. t. the initial function with the help of the theory of admissible pairs of spaces and the W-method.

Generalizing Example 1 we consider a linear stochastic differential equation with distributed delay of the form

dx(t) = (Vx)(t)dZ(t), t 2 0, (26)

x(v) = <p(v), v < 0,

where

(V x) (t) = Uc-oo,t) d, R1 (t, s )x(s ), ... , fc-oo,t) d, Rm(t, s)x( s) ),

ffii R;(t, s) = ~ Q;J(t)r;i(t,s).

J~O

The equation (26) can be rewritten in the form (8) by putting

(Vx)(t) = (J[o,t) d,Rt(t, s)x(s), .... , f[o,t) d,Rm(t, s)x(s)), (27)

f(t) = (J(-oo,o) d, Rt (t, s)<p( s ), ... , fc-oo,O) d,Rrn(t, s)<p(s)),

where Q;i are n x n-matrices with the entries being predictable stochastic processes and r;i are scalar functions defined on {(t, s) : t E [0, oo), -oo < s :'S: t}.

Let

H0(t)= I: IIQ;i(t)il V r;i(t, s), J~O sE( -oo,D)

m Hi(t) = ~ IIQ;i(t)il V TiJ(t,s) (i = 1, ... ,m),

J~O sE[D,t)

Hi= (H}, ... , Hj") (j = 0, 1).

The equation (8) will be considered under the assumption that t J (!Hia+! + !!HiA+ HJ!I)d.A < oo a. s. for any t?: 0, j = 0, 1.

0

This implies, in particular, that Hi E Ln(z) (compare the last inequality with (3)). The initial function <p will be a stochastic process such that f E Ln(z). An example of such <pis given by a stochastic process on ( -oo, 0) which is independent of the semimartingale Z(t) and which has a. s. essentially bounded trajectories w. r. t. to the measure .A*, generated by the function .A(t). If these assumptions are satisfied, then the operator V in the equation (8), defined by the first formula in (27), will be k-linear and Volterra and act from the space nn to the space Ln(z). In addition, for any x(O) E kn there will be the unique (up to a P-null set) solution of the equation (8) (remember that the equation (8) is equivalent to the equation (26)). For the proofs of these results see [7].

As a particular case of the equation (26) we obtain stochastic differential equations with "ordinary", or concentrated delay. Another name is difference-differential stochastic equations. By this we mean the following object:

dx(t) = (Vx)(t)dZ(t) (t?: 0), (28)

x(v) = cp(v) (v < 0),

where

(

ml mm ) (29) (Vx)(t) = _f;Q!j(t)x(h!j(t)), ... , t;Qmj(t)x(hmj(t)) .

Here h;j are .A* -measurable functions, for which

hij(t)::; t (.A*- a. e.) for t E [O,oo), i = 1, ... , m, j = 0, ... , m;;

Q;j are n x n-matrices with the entries that are predictable stochastic processes for all i = 1, ... , m, j = 0, ... , m;; <p is a stochastic process which is independent of the semimartingale Z(t).

The assumptions imposed on the general delay equation (26) can easily be adjusted to its particular case (28). The details can be found in [7]. Here we just outline briefly how the equation (28) can be represented in the form (26) and then formulate the assumptions on the coefficients. We set

m;

R;(t,s) = I:Q;j(t)r;i(t,s), j;Q

where QiJ are the matrices from (29) and rij is the indicator (the characteristic function of the set

{(t,s): t E [O,oo), -oo < s :S t, h;J(t) :S t},

defined for i = 1, ... , m, j = 0, ... ,mi. By this, the equation (28) is rewritten in the form (26) and this leads automatically to the following assumptions on the coefficients of the equation (28):

where

t

j(\HA+\+IIHA+HT\\)d,\<oo a. s.forany t;:o:O, 0

m,

H = (H\ ... , Hm), Hi:= 2:: \\Qi1 \\ (i = 1, ... , m); j~o

the initial function <p is a stochastic process with trajectories which are a. s. essentially bounded on [0, oo) w. r. t. the measure,\*.

In what follows we treat the equation (28) as a special case of the equation (26).

REMARK 4. The assumptions on the initial function <p does not imply, in general, that <p should be cadlag. It is an important observation for what follows as we are going to use a weaker topology {the IJ' -topology) in the set of all <p. Moreover, we do not treat the solution x(t) on t E [0, oo) as a continuation of the stochastic process <p. This is an essential feature of the theory of functional differential equations presented in [2} as it offers more possibilities to choose a suitable topology in the space of initial functions. A similar idea was also used in [14} to define the Lyapunov exponents for stochastic flows associated with certain linear stochastic functional differential equations. In this case, in order to apply Ruelle's multiplicative ergodic theorem, one needs the topology of a Hilbert space which is strictly weaker as the usual unifor·m topologies in the space of initial functions.

If we, nevertheless, want the solutions x(t) of the equation {26) {or {28}) to be continuations of the initial functions <p(t), then we can easily treat this situation as a particular case of the more general setting described above. First of all we have to require that <p(t) should be cadlag {or continuous, if the semimartingale Z(t) is continuous). In addition, we set the continuity condition at t = 0, i. e. we demand that

x(O) = lim <p(8). 8-tO-

'"'" ---·- '- '"'M


By this, the solution will be cadlag (or continuous) for both positive and negative t.

Now we describe different kinds of stability of solutions of the equations (26) and (28) which we intend to study in this paper. The definitions below are classical, up to some small adjustments, and can be found in many monographs (see e. g. [11], [16], [14]).

In the next definition we use the following notation: x(t, x 0 , rp) stands for the solution of (26), with the initial function rp, such that x(O, x0 , rp) = x0 .

DEFINITION 11. The zero solution of the equation {26} (or its particular case {28}) is

called: - p-stable w. r. t. the initial function, if for any E > 0 there exists 77( E) > 0 such that the inequality

ElxoiP+ vrai sup Ejrp(v)IP< 7J v<O

(vrai sup is the essential sup w. r. t. the measure A*) implies the estimate

Ejx(t,x0 ,rp)jP:<:;E for t::::o

for any rp(v), v < 0 and Xo E k; ; - asymptotically p-stable w. r. t. the initial function, if it is p-stable w. r. t. the initial function and, in addition, for any rp( v), v < 0 and xo E k; such that

one has

ElxoiP+ vrai sup Ejrp(v)IP < oo v<O

lim Ejx(t, xo, rp)IP = 0; t-t+co

- exponentially p-stable w. r. t. the initial junction, if there exist positive constants c, (3 such that

Ejx(t,xo,rp)jP :<::: c(EixoiP+ vraisup Ejrp(v)IP) exp{ -(3t} (t:::: 0) v<O

for any rp(v), v < 0 and x0 E k;. It is easy to see that p-stability (resp. asymptotic p-stability, exponential

p-stability) of the zero solution of the equation (26) w. r. t. the initial function implies p-stability (resp. asymptotic p-stability, exponential p-stability) of the zero solution of the homogeneous equation (12) with respect to the

initial value :r;( 0). The converse is, in general, not true, even in the case of deterministic delay equations (see e. g. [2]).

The notions of admissibility and stability w. r. t. the initial function are close to each other. In the following lemma we assume, when treating admissibility, that the equation (26) is rewritten in the form (8).

LEMMA 4. Assume that for any <p such that vrai sup El<p(v)IP < oo the v<O

stochastic process f defined in (27} belongs to a normed subspace B of the space Ln(z), the noTm satisfying

II/IlB :'0 K vr·aisup (EI<p(v)IP) 11P, v<O

where K is a positive constant. If the pair (Mp, B) is admissible for the equation (8), corresponding to the equation (26}, then the zero solution of the equation ( 26) is p-stable w. r. t. the initial function.

Proof. Under the assumptions of the lemma we have

llxJ(-, Xo)IIMp::; c(ilxollk~ + IIIIIs)::; c(ilxollk;

+K vrai sup (EI<p(v)IP) 11P) :'0 c(llxollk~+ vrai sup (EI<p(v)IP) 1/P), v<O v<O

where c, c, K are some positive numbers. From this, using the identity x(t, 2:0 , 'P) = x1(t, x0), we obtain

sup(Eix(t, xo, <p)IP) 1/P :'0 c(llxollkn+ vrai sup (EI<p(v)IP)11P).

t2:0 P v<O

This implies p-stability of the zero solution of the equation (26) w. r. t. the initial function. 0

REMARK 5. Evidently, in Lemma 4 one can replace the space B by the space B' for any reasonable weight r. Then admissibility of the pair ( MJ, B') for the equation (8) with r(t) = exp{iJt}, !3 > 0 will imply the exponential pstability of the zero solution of the equation (26} w. r. t. the initial function. The asymptotic p-stability of the zero solution of the equation (26} w. T. t. the initial function can be derived from admissibility of the pair (MJ, B') for the equation (8), if lim r(t) = +oo and r(t) ;:: 8 > 0, t E [0, oo) for some

t-t+oo 8.

DEFINITION 12. We say that the semimartingale Z(t) satisfies the condition (Z) if< ci, c1 >= 0 fori# j, so that,\* x P- almost everywhere AiJ = 0 (i # j, i,j = l, ... ,rn).

We will subsequently use only semimartingales with the condition (Z).

We first treat the equation (26) including distributed delays. Wishing to use the W-transform and the related operator 8 7 we have to rewrite (26) in the form (8). It is easily done via the formulas (27).

We begin with listing some technical conditions the set of which in the sequel will be addressed as dl.

t dl. sup (v(t)- v(t- 1)) < oo, where v(t) = J ~(s)d,\(s);

tE[l,oo) 0

1\Q;j\\\ai\ :S: a], \IQ;JIIIAiil 0·5 :S: h} (,\* x P)-almost everywhere,

a; X V r;J{-, s)~-l E L~, h; X V rij(·, s)~-0·5 E L~ ( -oo,-] ( -oo,-]

(i = 1, ... , m, j = 0, ... , mi)· THEOREM 7. Let the semimartingale Z(t) satisfy the condition (Z), the

reference equation {14} satisfy Rl-R2, the operator Q: M2p---+ i\~p,oo(~) is bounded, and the equation {26} satisfy d1. If now the operator (I - 8 7 ) :

i\~p,oo(~) ---+ i\~p,oo(~) {constructed for the equation (8) corresponding to the equation {26}} has a bounded inverse, then the zero solution of the equation {26) is 2p-stable w.r.t. the initial function.

REMARK 6. Due to Corollary 3 the operator W, under the assumptions of Theorem 'l, acts continuously from the space i\~p,q(~)_ to the space M2p, while the operator e"' defined in {15}, acts in the space A~poo(~). Proof of Theorem 7. By Remark 6, W : i\~p,q(~) _---+ M2p is b~unded, and due to the assumptions of the theorem V : M2p ---+ A~p,oo ( ~) ---+ M2p is ~ounded, too. This enables us to apply Theorem 5 stating that the pair (M2p, A~p,oo(~)) is admissible for the equation (8), corresponding in our case to the equation (26), where the transition formulas are given by (27).

If we now manage to show that for any rp such that vrai sup Elrp(v) IP < v<O

oo the function f defined in (27) belongs to the space B := A~p,oo(e) and IIliis :S: K vrai sup (E\rp(v)I 2

P)112

P, where K is a positive number, then v<O

Lemma 4 would give us the 2p-stability of the zero solution of the equation (26) w.r. t. the initial function.

It suffices therefore to estimate the norm II f II B. We have

llflls = II(EKdfai 2P) 112PC1 1!£~ + !I(EIIKzf AfTW) 1f2pCD.511L~

( 2p) 1/2p

:S: I\~~ E (lr-oo,O) Klaj(·)lrp(r)ldr sEYoo,rJr;j(·,s)) C1 1!£~

( ( ) p)l~ +II E ~ E Loo,O) (K2hj(·)l<p(r)1)2il, sE(Yoo,/iJ{, s) c0511L;,

rn mi

-:::.vrai sup (EI<p(vWP)112P(L L K1llaj V Tij(·, s)C1IIL;, v<O i=l j=O sE(-oo,O)

m mi

+ L L K2llhj V r,k s)C0 5IIL;,)-:::. K vr-m sup (EI<p(v)I 2P)

112P, i=l i=l sE( -oo,O) IJ<O

where K is a positive constant. Thus, f E Band llfiiB -:::. K vr-aisup v<O

(EI<p(v)l2P) 112P. This completes the proof. D Let us now consider the case of discrete delays, that is the equation (28)

with the operators (29). The following assumption will be used.

t d2. sup (v(t)- v(t- 1)) < oo, where v(t) = J ((s)d>-(s);

tE[l,oo) 0

IIQiJIIIail-:::. ilj, 11QiJIIIAiil05 -:::. hj (>-' x P)-almost everywhere,

-ic-1 LA h-ic-0.5 LA ('- 1 ·- 0 ·) ajs E 00 , JS E ..~00 2 - , ... , rn, J - , ... , m,z .

From Theorem 7 we have COROLLARY 4. Let the semimar-tingale Z(t) satisfy the condition (Z),

the r-efer-ence equation (14) satisfy Rl-R2, the opemtor Q : M2p--+ li.~p,oo(() is bounded, and the equation (28) satisfy d2. If now the operator (I- 8r) : li.~p,oo(O --+ li.~p,oo(~) (constructed for the equation (8) corresponding to the equation (28)) has a bounded inverse, then the zero solution of the equation (28) is 2p-stable w.r.t. the initial function.

DEFINITION 13. The equation (26) (the equation (28)) is called MJstable w. r. t. the initial function, if for all x0 E k~ and <p such that vrai sup

v<O El<p(v)IP < oo one has x(·, x0 , <p) E MJ and

llx(·, Xo, 'P)IIMo-:::. c(llxollkn+ vrai sup (EI<p(v)IP) 11P), P P v<O

where c E R+. Let us stress that, as before, the notion of MJ -stability of the equation (26) w.r.t. the initial function covers the classical notions

'" ---·- '"'""'


of p-stability, exponential p-stability and asymptotical p-stability of the zero solution w.r. t. the initial function. It is also evident that MJ-stability of the equation (26) w.r.t. the initial function implies MJ-stability of the associated equation in the form (8).

THEOREM 8. Let the semimartingale Z(t) satisfy the condition (Z), the reference equation (14} satisfy Rl-R2, the operator Q : M2p-+ i\~p,oo(~) is bounded, and the equation ( 26} satisfy d1. If now the operator (I - 8r) : A~p,oo(~) -+ A~p,oo(~) {constructed for the equation (8} corresponding to the equation (26)} has a bounded inverse and there exist numbers O;j > 0 such that r;j(t, s) = 0, where -oo < s::; t- O;j < oo, i = 1, ... , m, j = 0, ... , m;, t E [0, oo), then the equation (26} is MiP-stable w.r.t. the initial function, where ")'(t) = exp{iJv(t)} for some !3 > 0.

Proof First of all, we have to rewrite the equation (26) in the form (8). Then we observe that under the assumptions of the theorem, the operator V in (8) will act continuou~ly from M 2p to A~p,oo(~). Due to Lemma 3, the operator V : M 2p -+ A~p,oo(~) satisfies the l!.-condition. Hence the assumptions of Theorem 6 are satisfied.

We proceed now as in the proof of the preceding theorem, i.e. we show that for any <p such that vrai sup El<p(v)I2P < oo the function f in the

v<O equation (26), given by the formulas (27), belongs to the normed space B'Y, where B := (A~p,oo(~)), and the following estimate holds

llfllso ::; K vrai sup (EI<p(v)I 2P)

1/2P,

v<O

K being a positive number. In this case the Miv-stability of the equation t

(26) w.r.t. the initial function, where 'l'(t) = exp{/3 H(v)dil(v)} (for some 0

!3 > 0) is implied Theorem 6 and Lemma 4. In order to check the above estimate on f we observe that

llfllso = II(E1Knfai2P)

112PC1Ib + II(EIIKz')'fA('I'f)TI!P) 11

2PC05 II£>. 00 00

Now we have

II(EIKn fai2P)fp~-%.' + II(EIIKzif A(Jf)T\IP)f;;~-11lz,:' 2p,oo 2p,oo

( ( ) ~)+,; II f. I: E J Kli(·)aj(·)lcp(T)Idr V riA,s) (~(·))-%;, t=l ]=0 ( -oo,O) sE( -oo,r]

1

+II f. I: (E ( f K2(i(·)hj(-)lcp(T)I) 2~ V ri1(·, s))P)-;;; (~(·))-1llz,;, t=l J=O (-oo,O) sE(-oo,r]

m mi O;J' . :S [L: L: exp{f:J f ~(v)d>.(v)}IIK1aj(·) V riA, s)(~(·))-%;,

t=l J=O 0 sE( -oo,O)

m mi Oij . 1

+ ~ L: exp{f:J f ~(v)d>.(v)}IIK2 hj(·) V ri1(·, s)(~(·))-zllz,;,] t=l ]=0 0 sE( -oo,O)

x vraisup (Eicp(v)I 2P)fp :S K vraisup (Eicp(v)I 2P)f;;, v<O v<O

where K is some positive number. This gives f E B7 and

llfllso :S K vraisup (Eicp(vWP) 112v. v<O

The theorem is proved. D From Theorem 8 for the equation (28) we obtain COROLLARY 5. Let the semimartingale Z(t) satisfy the condition (Z),

the reference equation (14) satisfy Rl-R2, the operator Q: M2p--+ li.~p,oo(O is bounded, and the equation (28} satisfy d2. If now the operator (I- 8 7 ) :

li.~p,oo(O --+ A2p,oo(~) (constructed for the equation (8) corresponding to the equation (26)) has a bounded inverse and there exist numbers 5iJ > 0 such

t -that J E(v)dv :S 8ij (t E [O,oo)), where i = l, ... ,m, j = o, ... ,mi,

Xhii (t)hij (t)

and x9 (t) was defined in (23). Then the equation (28) is M]p-stable w.r.t. t

the initial function /(t) = exp{f:J J ~(v)d>.(v)} for some {3 > 0. 0

Proof. To apply Theorem 8 we notice that under the assumptions, listed in Corollary 5, the equation (28) in the form (26) has the following properties:

t

rij(t, s) = 0 if -oo < s :S t-8iJ < oo, where 8iJ = inf{t E [0, oo): J E(v)dv > 0

<li1} (i = 1, ... , m, j = 0, ... , mi). D

For the sake of completeness we also observe that the estimates on hij(t) in the corollary's assumptions means nothing, but the a-condition on hij(t) (see Definition 9).

5. An example. To illustrate the developed theory we consider a linear functional differential equation with driven Wiener processes. We will use the following condition.

(B) Z(t) = (t, B 1(t), ... , sm-1(t)f, where Bi, i = 1, ... , m- 1 are independent standard Wiener processes.

REMARK 7. It is easy to see that the semimartingale described in (B) satisfies the condition ZO as in this case .\(t) = t, a= (1, 0, ... , o)r, and the m x m-matrix A is given by Aii = 1 if i = 2, ... , m, and Aii = 0 otherwise (see (4)). In addition, we have that Ln(z) is a linear space consisting of. n x m-matrices, where all the entries are stochastic processes on [ 0, oo) that are adapted w. r. t. the given filtration, and the first column in the matrix are a.s. locally (Lebesgue) integrable, while the other columns are a.s. locally squire integrable. The space Dn consists now of adapted stochastic processes on [0, oo) with a.s. continuous trajectories. The next assumption will be used in the sequel.

d3. Qij(t) := Qii(t)xh,; (t) (i = 1, ... , m,j = 0, ... , m;), where Xu

is given by (23),

Q10 (t) is non-random, with the entries from L00 , h10 (t) = t

(t E [0, oo)),

1\Qijll ::; qij (,\* x F)-almost everywhere, q;j E Leo

(i=1, ... ,m, j=O, ... ,mi).

For ann X n-matrix function M we will write IIJM!I!Loo := IJIIMJJI!Loo· THEOREM 9. Let the semimartingale Z(t) satisfy the condition (B), the

equation (28) satisfy the condition d3, and there exists a positive number a such that

p :=a-! (IIIQ!O + aEII!Loo + ~ llqlji!Loo) + c,(2a)-0'5 ~~ II%11Loo < 1,

(30)


where the constant cP is defined in (5). Then the zero solution of the equation {28) is 2p-stable w. r. t. the initial function. If, in addition, there exist positive numbers O;j such that t- h;j(t) :::; o;j fort E [0, oo), i= 1, ... , m, j = 0, ... , m;, then the zero solution of the equation {28} will be exponentially 2p-stable w.r.t. the initial function.

Proof. Put Wl = 1 (t E [O,oo)), B := Azp,oo(l) and take (Qx)(t) = ( -&x(t), 0, ... , 0) in the reference equation (14)). Then the kernel C(t, s) in the representation (16) will be a diagonal n x n-matrix where Cii(t, s) = exp{-&(t- s)} (i = 1, ... ,n). Evidently, the assumptions Rl-R2 will be fulfilled in this case. Notice also that the condition (Z) is implied by the condition (B), and this ensures that Q: M2p -+B.

Considering now the operator I- 8r : A2p,oo(1) -+ B we observe that if it is invertible, then we obtain the 2p-stability of the zero solution of the equation (28) w.r.t. the initial function due to Corollary 4 and the main assumptions of Theorem 9. If we also add the auxiliary assumptions of Theorem 9 and use Corollary 5, then we arrive at the exponential 2p-stability of the zero solution of the equation (28) w.r.t. the initial function.

To prove invertibility of the operator I- 8r we estimate the norm of the operator 8r in the space B. To do it we observe that 8r can be described explicitly, namely by

- _ m1 -

(8rg)(t) = [(QJO(t) + &E)(Wg)(t) + 2::: QlJ(t)(Sh,Wg)(t), j=l J

m2 - mm -2::: QzJ(t)(Sh,Wg)(t), ... 2::: QmJ(t)(ShmWg)(t)], j=O J j::::::O 3

where 59 is defined in Definition 8.

We choose K1 := o;-1, K 2 = cp(2&)-0·5 . These constants are used in the definition of the norm in the space B = A2p,oo(1) (see 'Main spaces' in Section 2).

Now we can estimate the norm of the operator 8r in the space B. We

do it as follows:

- - ffil -

JJ8rJJs ::; u-1JJ(EJ(Qw + &E)W g + L Q!jShlj W gJ2P) 112PJJL= j=l

m mi -

+cv(2u)-0·5 L L JJ(EJQ;jSh,;Wgj2P) 112PJJL= i=2j=O

::; &-1 (JJJQw + aEJJJL= + j~1 JJqljJJL=) JJWgJJM,v

m mi

+cp(2a)-05 L L JJ%JJLoo JJW gJJM2p = PJJW gJJM,p· i=2 j=O

Since

t

JJW gJJM,p ::; sup(EJ J C(t, s)g(s)dZ(s)J2P)

112P

t~O 0

t ::; sup(E(J exp{ -a(t- s)}Jg(s)a(s)Jd(s))2P) 112P

t~O 0

t +cvsup(E(f exp{ -2a(t- s)}JJg(s)A(s)g(s)TJJd(s))P) 112P

t~O 0

::; u-lJJ (EJgaJ2P)l/2vJJL= + cp/(2a)-o.5JJ(EJJgAg TJJP)l/2PJJL=

= JJgJJs,

we obtain JJ8rgJJs ::; pJJgJJs. As p < 1, the operator 8r has a bounded inverse in the space B. The theorem is proved. 0

Consider now the equation

m . dx(t) = Q1(t)x(t)dt +I: Q;(t)x(h;(t))dB'-1(t) (t 2': 0)

(31) i=2

x(v) = cp(v) (v < 0),

Bi (i = 1, ... , m- 1) are independent standard Wiener processes, h;(t) ::; t (t E [0, oo)) are ..\*-measurable, cp(t) is a stochastic process that is independent of the Wiener process Bi, i = 1, ... , m- 1.

COROLLARY 6. Assume that a n x n-matrix Q1 is deterministic and belongs to L00 , while the other matrices Q; (also n x n) can be random and

satisfy IIQi(t)ll S: q;(t), qi E Loo (i = 2, ... , m). If there exists a positive number a such that

- m £ := a-liiiQI + aEIIILoo + Cp(2a)-OS 2:: llqili£oo < 1,

i=l

then the zero solution of the equation (31) is 2p-stable w.r.t. the initial function <p. If, in addition, there exist positive numbers r51 such that t-hi(t) S: /ji fortE [0, oo), i= 2, ... , m, then the zero solution of the equation (31) will be exponentially 2p-stable w. r. t. the initial function.

Proof. The equation (31) is a particular case of the equation (28), where the semimartingale Z(t) satisfies the condition (B), and the condition d3 is satisfied due to the assumptions of Corollary 6. Clearly, also the constant £ in the corollary will be identical with pin (30). Thus, the result immediately follows from Theorem 9. 0

CoROLLARY 7. Assume that Q 1 (t) = diag[-b, ... , -b], where b > 0 and the matrices Qi (which can be random) satisfy IIQi(t)ll S: qi(t), qi E L 00

(i = 2, ... , m). If

m Cp(2b)-05 2:: llqiiiLoo < 1,

i=l

then the zero sol·ution of the equation (31) is 2p-stable w.r.t. the initial function <p. If, in addition, there exist positive numbers Si such that t-hi(t) S: 6; fortE [0, oo), i= 2, ... , m, then the zero solution of the eq1tation (31) will be exponentially 2p-stable w. r. t. the initial function.

Proof This follows trivially from Corollary 6 if we put a = b. 0

6. Acknowledgement. We would like to thank the anonymous referee for very constructive comments and suggestions.

REFERENCES

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