Second Edition
T.A. Burton
DEPARTMENT OF MATHEMATICS
SOUTHERN ILLINOIS UNIVERSITY
CARBONDALE, ILLINOIS
USA
Contents
Preface v
Preface to the second edition vii
0 Introduction and Overview 1
0.1 Statement of Purpose 1
0.2 An Overview 2
1 The General Problems 5
1.1 Introduction 5
1.2 Relations between Differential and Integral Equations . . . 7
1.3 A Glance at Initial Conditions and Existence 13
1.4 Building the Intuition 15
1.5 Reducible Equations 20
2 Linear Equations 23
2.1 Existence Theory 23
2.2 Linear Properties 27
2.3 Convolution and the Laplace Transform 30
2.4 Stability 36
2.5 Liapunov Functionals and Small Kernels 40
2.6 Uniform Asymptotic Stability 51
2.7 Reducible Equations Revisited 65
3 Existence Properties 69
3.1 Definitions, Background, and Review 69
3.2 Existence and Uniqueness 76
3.3 Continuation of Solutions 82
3.4 Continuity of Solutions 95
4 History, Examples, and Motivation 103
4.0 Introduction 103
4.1 Volterra and Mathematical Biology 104
4.2 Renewal Theory 120
4.3 Examples 123
5 Instability, Stability, and Perturbations 133
5.1 The Matrix A
T
B + BA 133
5.2 The Scalar Equation 142
5.3 The Vector Equation 154
5.4 Complete Instability 163
5.5 Non-exponential Decay 167
6 Stability and Boundedness 171
6.1 Stability Theory for Ordinary Differential Equations . . . . 171
6.2 Construction of Liapunov Functions 183
6.3 A First Integral Liapunov Functional 191
6.4 Nonlinearities and an Annulus Argument 198
6.5 A Functional in the Unstable Case 211
7 The Resolvent 217
7.1 General Theory 217
7.2 A Floquet Theory 223
7.3 UAS and Integrability of the Resolvent 233
8 Functional Differential Equations 243
8.0 Introduction 243
8.1 Existence and Uniqueness 244
8.2 Asymptotic Stability 254
8.3 Equations with Bounded Delay 264
8.4 Boundedness with Unbounded Delay 293
8.5 Limit Sets 308
8.6 Periodic Solutions 316
8.7 Limit Sets and Unbounded Delays 330
8.8 Liapunov Theory for Integral Equations 339
References 340
Author Index 349
Subject Index 351
Integral and Differential Equations - School of Mathematics
Differential and Integral Equations - Aftabi
