Nonlinear Differential Equations in Abstract Spaces

✍ Scribed by V. Lakshmikantham, S. Leela

• Publisher: Pergamon Pr
• Year: 1981
• Tongue: English
• Leaves: 269
• Series: International Series in Nonlinear Mathematics, V. 2
• Category: Library

✦ Synopsis

Many problems in partial differential equations which arise from physical models can be considered as ordinary differential equations in appropriate infinite dimensional spaces, for which elegant theories and powerful techniques have recently been developed. This book gives a detailed account of the current state of the theory of nonlinear differential equations in a Banach space, and discusses existence theory for differential equations with continuous and discontinuous right-hand sides. Of special importance is the first systematic presentation of the very important and complex theory of multivalued discontinuous differential equations

✦ Table of Contents

Contents

CHAPTER 1
1.0 Introduction 1
1.1 Preliminaries 1
1.2 Directional Derivatives 7
1.3 Mean Value Theorems 13
1.4 Measures of Noncompactness 17
1.5 Measure of Weak Noncompactness 24
1.6 Comparison Results 26
1.7 Notes and Comments 31

CHAPTER 2
2.0 Introduction 32
2.1 The Cauchy Problem 33
2.2 Successive Approximations 34
2.3 Types of Approximate Solutions 37
2.4 Dissipative Type Conditions 40
2.5 Compactness Type Condtions 44
2.6 Existence in Closed Sets 50
2.7 Existence in Closed Sets (Continued) 58
2.8 Nonlinear Semigroups 66
2.9 Invariance Criteria 70
2.10 Notes and Comments 75

CHAPTER 3
3.0 Introduction 77
3.1 Dissipative Operators 79
3.2 The Exponential Formula 83
3.3 Difference Approximations 91
3.4 Convergence of Difference Approximations 96
3.5 Integral Solutions 104
3.6 Existence of Limit Solutions 108
3.7 $m$-Dissipative Operators 118
3.8 Perturbation of Dissipative Operators 122
3.9 Invariance of Nonlinear Semigroups 125
3.10 Examples 132
3.11 Compactness-Type Condition 137
3.12 Notes and Comments 142

CHAPTER 4
4.0 Introduction 144
4.1 Global Existence 144
4.2 Fundamental Properties 149
4.3 Differential Inequalities in Cones 156
4.4 Flow-Invariant Sets 161
4.5 Method of Quasi1inearization 165
4.6 Asymptotic Equilibrium 169
4.7 Stability and Asymptotic Behavior 175
4.8 Perturbing Lyapunov Functions 179
4.9 Vector Lyapunov Functions 183
4.10 Notes and Comments 185

CHAPTER 5
5.0 Introduction 187
5.1 Existence of Solutions in Weak Topology 187
5.2 Equations with Delay 194
5.3 Boundary Value Problems 210
5.4 Monotone Iterative Methods 217
5.5 Integral Equations 229
5.6 Notes and Comments 243

REFERENCES 244

SUBJECT INDEX 257