Functional Differential Equations: Advances and Applications

PREFACE xi

ACKNOWLEDGMENTS xv

1 Introduction, Classification, Short History, Auxiliary Results, and Methods 1

1.1 Classical and New Types of FEs 2

1.2 Main Directions in the Study of FDE 4

1.3 Metric Spaces and Related Concepts 11

1.4 Functions Spaces 15

1.5 Some Nonlinear Auxiliary Tools 21

1.6 Further Types of FEs 25

2 Existence Theory for Functional Equations 37

2.1 Local Existence for Continuous or Measurable Solutions 38

2.2 Global Existence for Some Classes of Functional Differential Equations 43

2.3 Existence for a Second-Order Functional Differential Equation 50

2.4 The Comparison Method in Obtaining Global Existence Results 55

2.5 A Functional Differential Equation with Bounded Solutions on the Positive Semiaxis 59

2.6 An Existence Result for Functional Differential Equations with Retarded Argument 64

2.7 A Second Order Functional Differential Equation with Bounded Solutions on the Positive Semiaxis 68

2.8 A Global Existence Result for a Class of First-Order Functional Differential Equations 72

2.9 A Global Existence Result in a Special Function Space and a Positivity Result 76

2.10 Solution Sets for Causal Functional Differential Equations 81

2.11 An Application to Optimal Control Theory 87

2.12 Flow Invariance 92

2.13 Further Examples/Applications/Comments 95

2.14 Bibliographical Notes 98

3 Stability Theory of Functional Differential Equations 105

3.1 Some Preliminary Considerations and Definitions 106

3.2 Comparison Method in Stability Theory of Ordinary Differential Equations 111

3.3 Stability under Permanent Perturbations 115

3.4 Stability for Some Functional Differential Equations 126

3.5 Partial Stability 133

3.6 Stability and Partial Stability of Finite Delay Systems 139

3.7 Stability of Invariant Sets 147

3.8 Another Type of Stability 155

3.9 Vector and Matrix Liapunov Functions 160

3.10 A Functional Differential Equation 163

3.11 Brief Comments on the Start and Evolution of the Comparison Method in Stability 168

3.12 Bibliographical Notes 169

4 Oscillatory Motion, with Special Regard to the Almost Periodic Case 175

4.1 Trigonometric Polynomials and APr-Spaces 176

4.2 Some Properties of the Spaces APr(R,C) 183

4.3 APr-Solutions to Ordinary Differential Equations 190

4.4 APr-Solutions to Convolution Equations 196

4.5 Oscillatory Solutions Involving the Space B 202

4.6 Oscillatory Motions Described by Classical Almost Periodic Functions 207

4.7 Dynamical Systems and Almost Periodicity 217

4.8 Brief Comments on the Definition of APr(R,C) Spaces and Related Topics 221

4.9 Bibliographical Notes 224

5 Neutral Functional Differential Equations 231

5.1 Some Generalities and Examples Related to Neutral Functional Equations 232

5.2 Further Existence Results Concerning Neutral First-Order Equations 240

5.3 Some Auxiliary Results 243

5.4 A Case Study, I 248

5.5 Another Case Study, II 256

5.6 Second-Order Causal Neutral Functional Differential Equations, I 261

5.7 Second-Order Causal Neutral Functional Differential Equations, II 268

5.8 A Neutral Functional Equation with Convolution 276

5.9 Bibliographical Notes 278

Appendix A On the Third Stage of Fourier Analysis 281

A.1 Introduction 281

A.2 Reconstruction of Some Classical Spaces 282

A.3 Construction of Another Classical Space 288

A.4 Constructing Spaces of Oscillatory Functions: Examples and Methods 290

A.5 Construction of Another Space of Oscillatory Functions 295

A.6 Searching Functional Exponents for Generalized Fourier Series 297

A.7 Some Compactness Problems 304

BIBLIOGRAPHY 307

INDEX 341