Theory of Integro-Differential Equations

375 Pages·1995·16.06 MB·English

by  V. Lakshmikantham

Theory of lntegro-Differential Equations
Stability and Control: Theory, Methods and Applications
A series of books and monographs on the theory of stability and control
Edited by A. Martynyuk, Institute of Mechanics, Kiev, Ukraine and V. Lakshmikantham, Florida
Institute of Technology, USA
Volume 1
Theory of Integro-Differential Equations
V. Lakshmikantham and M. Rama Mahana Rao
Additional Volumes in Preparation
Stability Analysis: Nonlinear Mechanics Equations
AA. Martynyuk
Stability and Motion of Nonautonomous Systems (Method of Limiting Equations)
J. Kato, AA. Martynyuk and A.A. Shestakov
This book is part of a series. The publisher will accept continuation orders which may be cancelled
at any time and which provide for automatic billing and shipping of each title in the series upon
publication. Please write for details.
Theory of Integro-Differential Equations
V. Lakshmikantham
Florida Institute of Technology, USA
and
M. Rama Mohana Rao
Indian Institute of Technology,
Kanpur, India
GORDON AND BREACH SCIENCE PUBLISHERS
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Copyright ©1995 by OPA (Amsterdam) B. V. All rights reserved. Published under license by Gordon
and Breach Science Publishers S.A.
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British Library Cataloguing in Publication Data
Lakshmikantham, V.
Theory of Integro-differential
Equations. - (Stability & Control:
Theory, Methods & Applications Series;
Vol. 1)
I. Title II. Rao, M. Rama Mohana
III. Series
515.353
ISHN 2-1111--1--i'l-()(lO-O
No part of this book may be reproduced or utilized in any form or by any means, electronic or
mechanical, including photocopying and recording, or by any information storage or retrieval
system, without permission in writing from the publisher. Printed in Singapore.
CONTENTS
Preface ......................................................................................................................... ix
Chapter 1. Basic Theory
1.0 Introduction ....................................................................... 1
1.1 Local and Global Existence ................................................. 2
1.2 Integro-differential Inequalities ........................................... 7
1.3 Existence of Extremal Solutions ......................................... 9
1.4 Comparison Results .......................................................... 13
1.5 Convergence of Successive Approximations ....................... 19
1.6 Continuous Dependence .................................................... 22
1.7 Linear Variation of Parameters ......................................... 24
1.8 Nonlinear Variation of Parameters ................................... 31
1.9 Monotone Iterative Technique .......................................... 36
1.10 Interval Analytic Method ................................................. 43
1.11 Notes and Comments ....................................................... 48
Chapter 2. Linear Analysis
2.0 Introduction ...................................................................... 49
2.1 Basic Properties of Linear Systems ................................... 50
2.2 Stability of Linear Convolution Systems ........................... 51
2.3 Stability Criteria for General Linear Systems ................... 58
2.4 Stability by Method of Reduction ..................................... 64
2.5 Stability in Variation ........................................................ 68
u
2.6 Lipschitz Stability ............................................................ 75
2. 7 Asymptotic Equivalence ................................................... 80
2.8 Ultimate Behavior of Solutions ......................................... 93
2.9 Difference Equations ......................................................... 99
2.10 Impulsive Integro-differential Systems ............................ 106
2.11 Periodic Solutions ........................................................... 115
2.12 Notes and Comments ..................................................... 122
Chapter 3. Lyapunov Stability
3.0 Introduction ................................................................... 125
3.1 Method of Lyapunov Functionals .................................... 126
3.2 Equations with Unbounded Delay ................................... 137
3.3 Perturbed systems .......................................................... 145
3.4 Method of Lyapunov Functions ...................................... 158
3.5 Lyapunov Functions on Product Spaces .......................... 171
3.6 Impulsive Integro-differential Equations ......................... 181
3.7 Impulsive Integro-differential Equations (continued) ...... 191
3.8 Notes and Comments ..................................................... 199
Chapter 4. Equations in Abstract Spaces
4.0 Introduction ................................................................... 201
4.1 Existence and Uniqueness ............................................... 202
4.2 Existence of Maximal and Minimal Solutions ................. 210
4.3 Well-Posedness of Linear Equations ................................ 218
4.4 Semigroups and Resolvent Operators .............................. 229
4.5 Evolution Operators and Resolvents ............................... 238
4.6 Asymptotic Behavior and Perturbations ......................... 250
4. 7 Stability of Solutions ...................................................... 254
4.8 Notes and Comments ..................................................... 268
Chapter 5. Applications
5.0 Introduction ................................................................... 271
5.1 Biological Population ...................................................... 272
5.2 Grazing Systems ............................................................. 281
vi
5.3 Wave Propagation .......................................................... 293
5.4 Nuclear Reactors ............................................................ 301
5.5 Viscoelasticity ................................................................ 310
5.6 Large-Scale Systems ....................................................... 324
5.7 Notes and Comments ..................................................... 337
References ................................................................................................. 339
Index ......................................................................................................... 357