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Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional PDF

335 Pages·1969·4.22 MB·English
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Preview Differential and Integral Inequalities: Functional Partial, Abstract and Complex Differential Equations v. 2: Theory and Applications: Functional

DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I1 FUNCTIONAL, PARTIAL, ABSTRACT, AND COMPLEX DIFFERENTIAL EQUATIONS This is Volume 55 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California A complete list of the books in this series appears at the end of this volume. DIFFERENTIAL AND INTEGRAL INEQUALITIES Theory and Applications Volume I1 FUNCTIONAL, PARTIAL, ABSTRACT, AND COMPLEX DIFFERENTIAL EQUATIONS V. LAKSHMIKANTHAM and S. LEELA University of Rhode Island Kingston, Rhode Island A C A D E MI C P RES S New York and London 1969 0 COPYRIGHT 1969, BY ACADEMIPCR ESSI, NC. ALL RIGHTS RESERVED. NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS. ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. Berkeley Square House, London WlX 6BA LIBRAROYF CONGRESCSA TALOCGA RDN UMBER68: -8425 AMS 1968 SUBJECT CLASSIFICATIO3N4S01 , 3501 PRINTED IN THE UNITED STATES OF AMERICA Preface The first volume of Differential and Integral Inequalities: Theory and Applications published in 1969 deals with ordinary differential equations and Volterra integral equations. It consists of five chapters and includes a systematic and fairly elaborate development of the theory and application of differential and integral inequalities. This second volume is a continuation of the trend and is devoted to differential equations with delay or functional differential equations, partial differential equations of first order, parabolic and hyperbolic types respectively, differential equations in abstract spaces including nonlinear evolution equations and complex differential equations. To cut down the length of the volume many parallel results are omitted as exercises. We extend our appreciation to Mrs. Rosalind Shumate, Mrs. June Chandronet, and Miss Sally Taylor for their excellent typing of the entire manuscript. V. LAKSHMIKANTHAM S. LEELA Kingston, Rhode Island August, 1969 V This page intentionally left blank Contents PREFACE V FUNCTIONAL DIFFERENTIAL EQUATIONS Chapter 6. 6.0. Introduction 3 6.1. Existence 4 6.2. Approximate Solutions and Uniqueness 9 6.3. Upper Bounds 13 6.4. Dependence on Initial Values and Parameters 18 6.5. Stability Criteria 21 6.6. Asymptotic Behavior 24 6.7. A Topological Principle 29 6.8. Systems with Repulsive Forces 32 6.9. Functional Differential Inequalities 34 6.10. Notes 42 Chapter 7. 7.0. Introduction 43 7.1. Stability Criteria 43 7.2. Converse Theoreins 49 7.3. Autonomous Systems 58 7.4. Perturbed Systems 62 7.5. Extreme Stability 66 7.6. Almost Periodic Systems 72 7.7. Notes 80 Chapter 8. 8.0. Introduction 81 8.1. Basic Comparison Theorems 81 8.2. Stability Criteria 87 8.3. Perturbed Systems 97 8.4. An Estimate of Time Lag 100 8.5. Eventual Stability 101 8.6. Asymptotic Behavior 105 8.7. Notes 110 vii ... Vlll CONTENTS PARTIAL DIFFERENTIAL EQUATIONS Chapter 9. 9.0. Introduction 113 9. I. Partial Differential Inequalities of First Order 113 9.2. Comparison Theorems 118 9.3. Upper Bounds 127 9.4. Approximate Solutions and Uniqueness 134 9.5. Systems of Partial Differential Inequalities of First Order 136 9.6. Lyapunov-Like Function 144 9.7. Notes 148 Chapter 10. 10.0. lntroduction 149 10. I. Parabolic Differential Inequaliies in Bounded Domains 149 10.2. Comparison Theorems 155 10.3. Bounds, Under and Over Functions 163 10.4. Approximate Solutions and Uniqueness 170 10.5. Stability of Steady-State Solutions 174 10.6. Systems of Parabolic Diffcrential inequalities in Bounded Domains 181 10.7. Lyapunov-Like Functions 186 10.8. Stahility and Boundedness 190 10.9. Conditional Stahility and Boundedness 200 10.10. Parabolic Differential Inequalities in Unbounded Domains 205 10.11. Uniqueness 210 10.12. Exterior Boundary-Value Problem and Uniqueness 21 3 10.13. Notes 219 Chapter 11. I 1.0. Introduction 22 1 1 1.1, Hyperbolic Diflerential Inequalities 22 1 1 1.2. Uniqueness Criteria 223 11.3. Upper Bounds and Error Estimates 229 11.4. Notes 233 DIFFERENTIAL EQUATIONS IN ABSTRACT SPACES Chapter 12. 12.0. Introduction 237 12. I. Existence 231 12.2. Norilocal Existence 24 1 12.3. Uniqueness 243 12.4. Continuous Dependence and the Method of Averaging 247 12.5. Existence (continued) 249 12.6. Approximate Solutions and Uniqueness 254 12.7. Chaplygin’s Method 258 12.8. Asymptotic Behavior 264 12.9. Lyapunov Function and Comparison Theorems 267 12.10. Stability and Boundedness 269 12.11. Notes 272 CONTENTS ix COMPLEX DIFFERENTIAL EQUATIONS Chapter 13. 13.0. Introduction 215 13.1. Existence, Approximate Solutions, and Uniqueness 215 13.2. Singularity-Free Regions and Growth Estimates 219 13.3. Componentwise Bounds 284 13.4. Lyapunov-like Functions and Comparison Theorems 286 13.5. Notes 288 Bibliography 289 Author Index 315 Subject Index 318

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