Functional Differential Equations: Advances and Applications

Features new results and up–to–date advances in modeling and solving differential equations

Introducing the various classes of functional differential equations, Functional Differential Equations: Advances and Applications presents the needed tools and topics to study the various classes of functional differential equations and is primarily concerned with the existence, uniqueness, and estimates of solutions to specific problems. The book focuses on the general theory of functional differential equations, provides the requisite mathematical background, and details the qualitative behavior of solutions to functional differential equations.

The book addresses problems of stability, particularly for ordinary differential equations in which the theory can provide models for other classes of functional differential equations, and the stability of solutions is useful for the application of results within various fields of science, engineering, and economics. Functional Differential Equations: Advances and Applications also features: 

Discussions on the classes of equations that cannot be solved to the highest order derivative, and in turn, addresses existence results and behavior types Oscillatory motion and solutions that occur in many real–world phenomena as well as in man–made machines Numerous examples and applications with a specific focus on ordinary differential equations and functional differential equations  with finite delay An appendix that introduces generalized Fourier series and Fourier analysis after periodicity and almost periodicity An extensive bibliography with over 550 references that connects the presented concepts to further topical exploration 

Functional Differential Equations: Advances and Applications is an ideal reference for academics and practitioners in applied mathematics, engineering, economics, and physics. The book is also an appropriate textbook for graduate– and PhD–level courses in applied mathematics, differential and difference equations, differential analysis, and dynamics processes.

Constantin Corduneanu, PhD, is Emeritus Professor in the Department of Mathematics at The University of Texas at Arlington.  The author of six books and over 200 journal articles, he is currently Associate Editor for seven journals; a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Romanian Academy; and past president of the American Romanian Academy of Arts and Sciences. 

Yizeng Li, PhD, is Professor in the Department of Mathematics at Tarrant County College. He is a member of the Society for Industrial and Applied Mathematics.

Mehran Mahdavi, PhD, is Professor in the Department of Mathematics at Bowie State University.  The author of numerous journal articles, he is a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Mathematical Association of America.

Preface v

1 Introduction 1

1.1 Classical and New Types of Functional Equations 2

1.2 Main Directions in the Study of FDE 5

1.3 Metric Spaces and Related Concepts 11

1.4 Functions Spaces 16

1.5 Some Nonlinear Auxiliary Tools 21

1.6 Further Types of Functional Equations 26

2 Existence Theory 37

2.1 Local Existence 38

2.2 Global Existence 43

2.3 Second Order FDE 50

2.4 Comparison Method 55

2.5 Bounded Solutions on Semi–Axis 59

2.6 FDE with Retarded Argument 64

2.7 Another Second Order Equation 68

2.8 Global Existence for FDE 72

2.9 Global Existence in Some Function Spaces 76

2.10 Solution Sets for Causal FDE 81

2.11 An Application to Optimal Control Theory 87

2.12 Flow Invariance 92

2.13 Further Examples/Applications 95

2.14 Bibliographical Notes 98

3 Stability 107

3.1 Preliminaries 108

3.2 Comparison Method in Stability 113

3.3 Permanent Perturbations 117

3.4 Stability for Some FDE 129

3.5 Partial Stability 136

3.6 Finite Delay Systems 142

3.7 Stability of Invariant Sets 150

3.8 Another Type of Stability 158

3.9 Vector and Matrix Liapunov Functions 163

3.10 A Functional Differential Equation 166

3.11 Brief Comments 171

3.12 Bibliographical Notes 173

4 Oscillatory Motion 179

4.1 Trigonometric Polynomials 180

4.2 Properties of APr(R, C) Spaces 187

4.3 APr–Solutions for ODE 193

4.4 APr–Solutions to Convolution Equations 200

4.5 Oscillatory Solutions in the Space B 206

4.6 Oscillatory Motion and Almost Periodicity 211

4.7 Dynamical Systems and Almost Periodicity 220

4.8 Brief Comments 225

4.9 Bibliographical Notes 228

5 Neutral Functional Equations 235

5.1 Generalities and Examples 236

5.2 Neutral First Order Equations 244

5.3 Some Auxiliary Results 247

5.4 A Case Study, I 252

5.5 Another Case Study, II 260

5.6 Second Order Causal FDE, I 264

5.7 Second Order Causal FDE, II 271

5.8 A Neutral FE with Convolution 279

5.9 Bibliographical Notes 281

A Appendix 285

A.1 Reconstruction of Some Classical Spaces 286

A.2 Construction of Another Classical Space 292

A.3 Constructing Spaces of Oscillatory Functions 294

A.4 Another Space of Oscillatory Functions 299

A.5 Functional Exponents for Generalized Series 302

A.6 Some Compactness Problems 309

Bibliography 311

Index 356