From the authors' preface: "This volume constitutes the first part of a monograph on theory and applications of differential and integral inequalities. The main tools for applications are the Ljapunov functions. The material of the present volume is divided into two sections. The first section, consisting of four chapters, deals with ordinary differential equations while the second section is devoted to Volterra integral equations.'' AMS Review by A. Halanay
Author(s): V. Lakshmikantham and S. Leela
Publisher: Academic Press
Year: 1969
Language: English
Pages: 404
Tags: Математика;Дифференциальные уравнения;
Contents
PREFACE v
ORDINARY DIFFERENTIAL EQUATIONS
Chapter 1. 1.0. Introduction 3
1.1. Existence and Continuation of Solutions 3
1.2. Scalar Differential Inequalities 7
1.3. Maximal and Minimal Solutions 11
1.4. Comparison Theorems 15
1.5. Finite Systems of Differential Inequalities 21
1.6. Minimax Solutions 25
1.7. Further Comparison Theorems 27
1.8. Infinite Systems of Differential Inequalities 31
1.9. Integral Inequalities Reducible to Differential Inequalities 37
1.10. Differential Inequalities in the Sense of Caratheodory 41
1.11. Notes 44
Chapter 2. 2.0. Introduction 45
2.1. Global Existence 45
2.2. Uniqueness 48
2.3. Convergence of Successive Approximations 60
2.4. Chaplygin's Method 64
2.5. Dependence on Initial Conditions and Parameters 69
2.6. Variation of Constants 76
2.7. Upper and Lower Bounds 79
2.8. Componentwise Bounds 84
2.9. Asymptotic Equilibrium 88
2.10. Asymptotic Equivalence 91
2.11. A Topological Principle 96
2.12. Applications of Topological Principle 100
2.13. Stability Criteria 102
2.14. Asymptotic Behavior 108
2.15. Periodic and Almost Periodic Systems 120
2.16. Notes 129
Chapter 3. 3.0. Introduction 131
3.1. Basic Comparison Theorems 131
3.2. Definitions 135
3.3. Stability 138
3.4. Asymptotic Stability 145
3.5. Stability of Perturbed Systems 155
3.6. Converse Theorems 158
3.7. Stability by the First Approximation 177
3.8. Total Stability 186
3.9. Integral Stability 191
3.10. $L^p$-Stability 199
3.11. Partial Stability 205
3.12. Stability of Differential Inequalities 209
3.13. Boundedness and Lagrange Stability 212
3.14. Eventual Stability 222
3.15. Asymptotic Behavior 229
3.16. Relative Stability 241
3.17. Stability with Respect to a Manifold 244
3.18. Almost Periodic Systems 245
3.19. Uniqueness and Estimates 254
3.20. Continuous Dependence and the Method of Averaging 257
3.21. Notes 264
Chapter 4. 4.0. Introduction 267
4.1. Main Comparison Theorem 267
4.2. Asymptotic Stability 269
4.3. Instability 273
4.4. Conditional Stability and Boundedness 277
4.5. Converse Theorems 284
4.6. Stability in Tube-like Domain 293
4.7. Stability of Asymptotically Self-Invariant Sets 297
4.8. Stability of Conditionally Invariant Sets 305
4.9. Existence and Stability of Stationary Points 308
4.10. Notes 311
VOLTERRA INTEGRAL EQUATIONS
Chapter 5. 5.0. Introduction 315
5.1. Integral Inequalities 315
5.2. Local and Global Existence 319
5.3. Comparison Theorems 322
5.4. Approximate Solutions, Bounds, and Uniqueness 324
5.5. Asymptotic Behavior 327
5.6. Perturbed Integral Equations 333
5.7. Admissibility and Asymptotic Behavior 340
5.8. Integrodifferential Inequalities 350
5.9. Notes 354
Bibliography 355
Author Index 385
Subject Index 388
