Because the theory of equations with delay terms occurs in a variety of contexts, it is important to provide a framework, whenever possible, to handle as many cases as possible simultaneously so as to bring out a better insight and understanding of the subtle differences of the various equations with delays. Furthermore, such a unified theory would avoid duplication and expose open questions that are significant for future research.
It is in this spirit that the authors view the importance of their monograph, which presents a systematic and unified theory of recent developments of equations with unbounded delay, describes the current state of the theory showing the essential unity achieved, and provides a general structure applicable to a variety of problems.
It is the first book that:
(i) presents a unified framework to investigate the basic existence theory for a variety of equations with delay;
(ii) treats the classification of equations with memory precisely so as to bring out the subtle differences between them;
(iii) develops a systematic study of stability theory in terms of two different measures which includes several known concepts; and
(iv) exhibits the advantages of employing Lyapunov functions on product spaces as well as the method of perturbing Lyapunov functions.
This book will be of value to researchers and advanced graduate students in mathematics, electrical engineering and biomathematics.
Author(s): V. Lakshmikantham, Lizhi Wen, Binggen Zhang
Series: Mathematics and Its Applications
Edition: Reprint
Publisher: Springer
Year: 2013,1994
Language: English
Pages: 386
Tags: Математика;Дифференциальные уравнения;
CONTENTS
PREFACE ix
CHAPTER 1 PRELIMINARIES 1
1.0 Introduction 1
1.1 Classification and Description 2
1.2 Notes and Comments 13
CHAPTER 2 EXISTENCE AND THEORY FOR $p$-TYPE NFDE 15
2.0 Introduction 15
2.1 E xistence and Uniqueness 15
2.2 Continuous Dependence 33
2.3 Differentiability Relative to Initial Data 38
2.4 Continuation of Solutions 41
2.5 Notes and Comments 45
CHAPTER 3 EXISTENCE THEORY OF NFDE WITH INFINITE DELAY 47
3.0 Introduction 47
3.1 Description of Phase Spaces 48
3.2 Existence and Uniqueness 50
3.3 Continuation of Solutions 56
3.4 Implication to Admissible Phase Spaces 61
3.5 Notes and Comments 72
CHAPTER 4 STABILITY AND BOUNDEDNESS FOR RFDE WITH BOUNDED DELAY 75
4.0 Introduction 75
4.1 Definitions 76
4.2 C lassical Results for Stability 79
4.3 Razumikhin-type Theorems with Lyapunov f unctionals 82
4.4 Stability in Terms of Two Measures 101
4.5 Weak Exponential Asymptotical Stability 111
4.6 Boundedness C riteria 123
4.7 Notes and Comments 125
CHAPTER 5 STABILITY CRITERIA FOR $p$-TYPE NFDE 127
5.0 Introduction 127
5.1 Stability Criteria 127
5.2 Boundedness Results 149
5.3 Notes and C omments 174
CHAPTER 6 STABILITY AND BOUNDEDNESS FOR EQUATIONS WITH INFINITE DELAY 177
6.0 Introduction 177
6.1 Notation and Definitions 177
6.2 Stability in Terms of Two Measures 180
6.3 Boundedness in Terms of Two Measures 194
6.4 Razumikhin's Method for NFD by Comparison Method 211
6.5 Notes and C omments 243
CHAPTER 7 ASYMPTOTIC BEHAVIOR 245
7.0 Introduction 245
7.1 Invariance Principle 245
7.2 Convergence of Solutions 257
7.3 Asymptotic Behavior 268
7.4 Notes and C omments 273
CHAPTER 8 OSCILLATION THEORY 275
8.0 Introduction 275
8.1 Oscillation of Systems 275
8.2 Oscillation of Scalar Equations 283
8.3 Comparison Theorems 289
8.4 Nonoscillatory Solutions 296
8.5 Notes and Comments 304
CHAPTER 9 PERIODIC SOLUTIONS 307
9.0 Introduction 307
9.1 Extension of ODE Methods 308
9.2 Periodic Solutions Generated by ODE s 317
9.3 Nussbaum's f ixed Point Theorem 328
9.4 Periodic Delay Logistic Equations 332
9.5 Volterra E quations with Infinite Delay 341
9.6 Method of Lyapunov functionals 349
9.7 Notes and Comments 354
REFERENCES 355
INDEX 383
