Impulsive Differential Inclusions: A Fixed Point Approach

by John R. Graef, Johnny Henderson, Abdelghani Ouahab

Overview

Differential equations with impulses arise as models of many evolving processes that are subject to abrupt changes, such as shocks, harvesting, and natural disasters. These phenomena involve short-term perturbations from continuous and smooth dynamics, whose duration is negligible in comparison with the duration of an entire evolution. In models involving such perturbations, it is natural to assume these perturbations act instantaneously or in the form of impulses. As a consequence, impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, ecology, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. There are also many different studies in biology and medicine for which impulsive differential equations provide good models.

During the last 10 years, the authors have been responsible for extensive contributions to the literature on impulsive differential inclusions via fixed point methods. This book is motivated by that research as the authors endeavor to bring under one cover much of those results along with results by other researchers either affecting or affected by the authors' work. The questions of existence and stability of solutions for different classes of initial value problems for impulsive differential equations and inclusions with fixed and variable moments are considered in detail. Attention is also given to boundary value problems. In addition, since differential equations can be viewed as special cases of differential inclusions, significant attention is also given to relative questions concerning differential equations. This monograph addresses a variety of side issues that arise from its simpler beginnings as well.

Product Details

ISBN-13:	9783110293616
Publisher:	De Gruyter
Publication date:	07/17/2013
Series:	De Gruyter Series in Nonlinear Analysis and Applications , #20
Pages:	410
Product dimensions:	6.69(w) x 9.45(h) x 0.04(d)
Age Range:	18 Years

About the Author

John R. Graef, University of Tennessee at Chattanooga, Tennessee, USA; Johnny Henderson, Baylor University, Waco, Texas, USA; Abdelghani Ouahab, University of Sidi Bel Abbes, Sidi Bel Abbes, Algeria.

Table of Contents

Notations xi

1 Introduction and Motivations 1

1.1 Introduction 1

1.2 Motivational Models 8

1.2.1 Kruger-Thiemer Model 8

1.2.2 Lotka-Volterra Model 8

1.2.3 Pulse Vaccination Model 9

1.2.4 Management Model 9

1.2.5 Some Examples in Economics and Biomathematics 10

2 Preliminaries 11

2.1 Some Definitions 11

2.2 Some Properties in Fréchet Spaces 12

2.3 Some Properties of Set-valued Maps 13

2.3.1 Hausdorff Metric Topology 15

2.3.2 Vietoris Topology 18

2.3.3 Continuity Concepts and Their Relations 20

2.3.4 Selection Functions and Selection Theorems 28

2.3.5 Hausdorff Continuity 30

2.3.6 Measurable Multifunctions 32

2.3.7 Decomposable Selection 35

2.4 Fixed Point Theorems 36

2.5 Measures of Noncompactness: MNC 37

2.6 Semigroups 40

2.6.1 C₀-semigroups 40

2.6.2 Integrated Semigroups 42

2.6.3 Examples 44

2.7 Extrapolation Spaces 45

3 FDEs with Infinite Delay 47

3.1 First Order FDEs 47

3.1.1 Examples of Phase Spaces 48

3.1.2 Existence and Uniqueness on Compact Intervals 50

3.1.3 An Example 57

3.2 FDEs with Multiple Delays 58

3.2.1 Existence and Uniqueness Result on a Compact Interval 58

3.2.2 Global Existence and Uniqueness Result 65

3.3 Stability 66

3.3.1 Stability Result 67

3.4 Second Order Impulsive FDEs 69

3.4.1 Existence and Uniqueness Results 71

3.5 Global Existence and Uniqueness Result 76

3.5.1 Uniqueness Result 77

3.5.2 Example 82

3.5.3 Stability 83

4 Boundary Value Problems on Infinite Intervals 86

4.1 Introduction 86

4.1.1 Existence Result 87

4.1.2 Uniqueness Result 92

4.1.3 Example 96

5 Differential Inclusions 98

5.1 Introduction 98

5.1.1 Filippov's Theorem 98

5.1.2 Relaxation Theorem 111

5.2 Functional Differential Inclusions 113

5.2.1 Filippov's Theorem for FDIs 114

5.2.2 Some Properties of Solution Sets 123

5.3 Upper Semicontinuity without Convexity 125

5.3.1 Nonconvex Theorem and Upper Semicontinuity 126

5.3.2 An Application 130

5.4 Inclusions with Dissipative Right Hand Side 131

5.4.1 Existence and Uniqueness Result 131

5.5 Directionally Continuous Selection and IDIs 136

5.5.1 Directional Continuity 136

6 Differential Inclusions with Infinite Delay 140

6.1 Existence Results 140

6.2 Boundary Differential Inclusions 150

7 Impulsive FDEs with Variable Times 154

7.1 Introduction 154

7.1.1 Existence Results 154

7.1.2 Neutral Functional Differential Equations 155

7.2 Impulsive Hyperbolic Differential Inclusions with Infinite Delay 156

7.3 Existence Results 157

7.3.1 Phase Spaces 157

7.3.2 The Nonconvex Case 168

8 Neutral Differential Inclusions 171

8.1 Filippov's Theorem 171

8.2 The Relaxed Problem 182

8.2.1 Existence and Compactness Result: an MNC Approach 189

9 Topology and Geometry of Solution Sets 199

9.1 Background in Geometric Topology 199

9.2 Aronszajn Type Results 201

9.2.1 Solution Sets for Impulsive Differential Equations 206

9.3 Solution Sets of Differential Inclusions 208

9.4 σ-selectionable Multivalued Maps 208

9.4.1 Contractible and Rδ-contractible 212

9.4.2 Rδ-sets 218

9.5 Impulsive DIs on Proximate Retracts 219

9.5.1 Viable Solution 220

9.6 Periodic Problems 226

9.6.1 Poincaré Translation Operator 226

9.6.2 Existence Result 227

9.7 Solution Set for Nonconvex Case 231

9.7.1 Continuous Selection and AR of Solution Sets 232

9.8 The Terminal Problem 245

9.8.1 Existence and Solution Set 245

10 Impulsive Semilinear Differential Inclusions 254

10.1 Nondensely Defined Operators 254

10.2 Integral Solutions 255

10.3 Exact Controllability 267

10.3.1 Controllability of Impulsive FDIs 267

10.3.2 Controllability of Impulsive Neutral FDIs 276

10.4 Controllability in Extrapolation Spaces 282

10.5 Second Order Impulsive Semilinear FDIs 290

10.5.1 Mild Solutions 291

10.5.2 Filippov's Theorem 292

10.5.3 Filippov-Wazewski's Theorem 303

11 Selected Topics 306

11.1 Stochastic Differential Equations 306

11.1.1 Itô Integral 307

11.1.2 Definition of a Mild Solution 308

11.1.3 Existence and Uniqueness 311

11.1.4 Global Existence and Uniqueness 321

11.2 Impulsive Sweeping Processes 327

11.2.1 Preliminaries in Nonsmooth Analysis 327

11.2.2 Uniqueness Result 328

11.3 Integral Inclusions of Volterra Type in Banach Spaces 331

11.3.1 Resolvent Family 332

11.3.2 Existence results 334

11.3.3 The Convex Case: an MNC Approach 339

11.3.4 The Nonconvex Case 342

11.4 Filippov's Theorem 346

11.4.1 Filippov's Theorem on a Bounded Interval 346

11.5 The Relaxed Problem 351

Appendix 357

A.1 Cech Homology Functor with Compact Carriers 357

A.2 The Bochner Integral 359

A.3 Absolutely Continuous Functions 361

A.4 Compactness Criteria in C([a, b], E), C_b([0, ∞), E), and PC([a, b], E) 363

A.5 Weak-compactness in L¹ 365

A.6 Proper Maps and Vector Fields 367

A.7 Fundamental Theorems in Functional Analysis 367

Bibliography 369

Index 399