Table Of ContentTheory of Differential Equations with Unbounded Delay
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume298
Theory of
Differential Equations
with Unbounded Delay
by
V. Lakshmikantham
F/orida Institute of Technology,
Department ofA pplied Mathematics,
Melboume, Florida, U.S.A.
Lizhi Wen
South China Normal University,
Department of Mathematics,
Guangwou, People's Republic of China
and
Binggen Zhang
Ocean University of Qingdao.
Department ofA pplied Mathematics,
Qingdao. People's Republic of China
Springer-Science+Business Media, B.V.
Library of Congress Cataloging-in-Publication Data
Lakshmikantham, V.
Theory of differential equations with unbounded delay 1 by
V. LakshMikantham, Lizhi Hen, Binggen Zhang.
p. CN. -- <Mathellat ies and its appl ications ; v. 298)
Includes bibliographical references and index.
ISBN 0-7923-3003-X <acid-freel
1. Delay differential equations. I. Hen. Lizhi. II. Zhang, B.
G. • 1934- III. Tit le. IV. Ser ies: Mathematics and its
applications <Kluwer Acade111c Publ1shersl ; v. 298.
OA371.L245 1994
515' .35--dc20 94-27710
ISBN 978-1-4613-6116-9 ISBN 978-1-4615-2606-3 (eBook)
DOI 10.1007/978-1-4615-2606-3
Printed on acid-free paper
AII Rights Reserved
© 1994 Springer Science+Business Media Dordrecht
Originally published by K1uwer Academic Publishers in 1994
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
CONTENTS
PREFACE ........................................................................................................ ix
CHAPTER 1. PRELIMINARIES ................................................................. 1
1.0 Introduction .......................................................................... 1
1.1 Classification and Description ................................................ 2
1.2 Notes and Comments ........................................................... 13
CHAPTER 2. EXISTENCE THEORY FOR p-TYPE NFDE ..................... 15
2.0 Introduction ........................................................................ 15
2.1 Existence and Uniqueness .................................................... 15
2.2 Continuous Dependence ....................................................... 33
2.3 Differentiability Relative to Initial Data .............................. 38
2.4 Continuation of Solutions .................................................... 41
2.5 Notes and Comments ........................................................... 45
CHAPTER 3. EXISTENCE THEORY OF NFDE WITH
INFINITE DELAY .............................................................. 47
3.0 Introduction ........................................................................ 47
3.1 Description of Phase Spaces ................................................. 48
3.2 Existence and Uniqueness .................................................... 50
3.3 Continuation of Solutions .................................................... 56
3.4 Implication to Admissible Phase Spaces ............................... 61
3.5 Notes and Comments ........................................................... 72
Vl Theory of Differential Equations with Unbounded Delay
CHAPTER 4. STABILITY AND BOUNDEDNESS FOR RFDE WITH
BOUNDED DELAY ............................................................ 75
4.0 Introduction ........................................................................ 75
4.1 Definitions ........................................................................... 76
4.2 Classical Results for Stability .............................................. 79
4.3 Razumikhin-type Theorems with Lyapunov Functionals ...... 82
4.4 Stability in Terms of Two Measures .................................. 101
4.5 Weak Exponential Asymptotical Stability .......................... 111
4.6 Boundedness Criteria ......................................................... 123
4.7 Notes and Comments ......................................................... 125
CHAPTER 5. STABILITY CRITERIA FOR p-TYPE NFDE .................. 127
5.0 Introduction ...................................................................... 127
5.1 Stability Criteria ............................................................... 127
5.2 Boundedness Results .......................................................... 149
5.3 Notes and Comments ......................................................... 174
CHAPTER 6. STABILITY AND BOUNDEDNESS FOR EQUATIONS
WITH INFINITE DELAY ................................................. 177
6.0 Introduction ...................................................................... 177
6.1 Notation and Definitions .................................................... 177
6.2 Stability in Terms of Two Measures .................................. 180
6.3 Boundedness in Terms of Two Measures ............................ 194
6.4 Razumikhin's Method for NFDE by Comparison Method .. 211
6.5 Notes and Comments ......................................................... 243
CHAPTER 7. ASYMPTOTIC BEHAVIOR ............................................. 245
7.0 Introduction ...................................................................... 245
7.1 In variance Principle ........................................................... 245
7.2 Convergence of Solutions ................................................... 257
7.3 Asymptotic Behavior ......................................................... 268
7.4 Notes and Comments ......................................................... 273
CHAPTER 8. OSCILLATION THEORY ................................................. 275
8.0 Introduction ...................................................................... 275
8.1 Oscillation of Systems ........................................................ 275
8.2 Oscillation of Scalar Equations .......................................... 283
8.3 Comparison Theorems ....................................................... 289
8.4 Nonoscillatory Solutions .................................................... 296
8.5 Notes and Comments ......................................................... 304
Contents vii
CHAPTER 9. PERIODIC SOLUTIONS .................................................. 307
9.0 Introduction ...................................................................... 307
9.1 Extension of ODE Methods ................................................ 308
9.2 Periodic Solutions Generated by ODEs .............................. 317
9.3 Nussbaum's Fixed Point Theorem ..................................... 328
9.4 Periodic Delay Logistic Equations ...................................... 332
9.5 Volterra Equations with Infinite Delay .............................. 341
9.6 Method of Lyapunov Functionals ....................................... 349
9.7 Notes and Comments ......................................................... 354
REFERENCES ............................................................................................... 355
INDEX ........................................................................................................... 383
Preface
It is Volterra to whom we must pay tribute as the founder of equations in which
unbounded delay occurs. In fact, he is to be credited for applications of such
equations in population dynamics and materials with memory. Since the
necessary tools of functional analysis were not available at that time, Volterra
himself made the investigation more accessible by studying the equations with
finite delay. Naturally, the development of the theory of differential equations
with finite delay progressed dramatically and the first book dedicated to this
subject was published by Myshkis [3] in 1951. Several monographs concerned
with the general theory of equations with delay, including qualitative aspects,
appeared in succession, namely, El'sgol'ts [1], Krasovski [1), Pinney [I), Bellman
and Cooke [I], Halanay [IJ, El'sgol'ts [2), Norkin [IJ, Oguztoreli [1],
Lakshmikantham and Leela [3], Mitropoliskii and Martynyuk [I), Hale [4),
El'sgol'ts and Norkin [1), and Driver [4].
The theory of differential equations with unbounded delay lagged behind
until recently. When the old problem of Volterra, namely, to create an adequate
basic theory for materials with memory surfaced again, it is the Carnegie-Mellon
group which provided needed impetus in setting up necessary framework through
axiomatic approach. See Coleman [1]. Since a large variety of phase spaces
could be utilized to build an appropriate theory for equations with unbounded
delay, it became desirable to approach the problem purely axiomatically. Hale
[2] undertook such a task which gave rise to further contributions by Hino [1-4,
7], Naito [2, 3], Hale and Kato [1], Schumacher [1], Hino and Murakami [I],
zx
Theory of Differential Equations with Unbounded Delay
X
Kaminogo [2], Sawano [1] resulting in a survey of equations with unbounded
delay by Corduneanu and Lakshmikantham [1]. Recently, the lecture notes by
Hino, Murakami and Naito [1] reported the developments of this important
branch of functional differential equations.
A careful analysis shows that even though the delay terms occurring in
the equations are unbounded, the domain of the initial data (past history or
memory) may be finite or infinite. Consequently, these two cases need to be
investigated independently. Moreover, one can consider differential equations
with delay so that the delay terms also occur in the derivative of the unknown
solution. Since thegeneral formulation of such a problem is difficult to state, a
special kind of equations called neutral functional differential equations (NFDE)
have been introduced. The study of the theory of NFDE with bounded or
unbounded delay terms, has also progressed well.
In view of the existence of the theory of equations with delay terms
occurring in a variety of ways, it becomes important to provide a framework,
whenever possible, to handle as many cases as possible simultaneously so as to
bring out a better insight and understanding of the subtle differences of the
various equations with delays. Furthermore, such a unified theory would avoid
duplication and expose open questions that are significant for future research.
It is with this spirit that we see the importance of our monograph. Its
aim is to present a systematic and unified theory of recent developments of
equations with unbounded delay, describe the current state of the useful theory
by showing the essential unity achieved and provide a general structure
applicable to a variety of problems.
Some important features of the monograph are as follows:
It is the first book that
(i) presents a unified framework to investigate the basic existence theory
for a variety of equations with delay;
(ii) treats the classification of equations with memory precisely so as to
bring out the subtle differences between them;
(iii) develops a systematic study of stability theory in terms of two
different measures which includes several known concepts; and
( iv) exhibits the advantages of employing Lyapunov functions on product
spaces as well as the method of perturbing Lyapunov functions.
