BASIC THEO FRACT DIFFER (2ND ED)

by Yong Zhou, Jinrong Wang, Lu Zhang

Overview

This invaluable monograph is devoted to a rapidly developing area on the research of qualitative theory of fractional ordinary and partial differential equations. It provides the readers the necessary background material required to go further into the subject and explore the rich research literature. The tools used include many classical and modern nonlinear analysis methods such as fixed point theory, measure of noncompactness method, topological degree method, the technique of Picard operators, critical point theory and semigroup theory. Based on the research work carried out by the authors and other experts during the past seven years, the contents are very recent and comprehensive.In this edition, two new topics have been added, that is, fractional impulsive differential equations, and fractional partial differential equations including fractional Navier-Stokes equations and fractional diffusion equations.

Product Details

ISBN-13:	9789813148185
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/20/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	380
File size:	59 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface to the Second Edition v

Preface to the First Edition vii

1 Preliminaries 1

1.1 Introduction 1

1.2 Some Notations, Concepts and Lemmas 1

1.3 Fractional Calculus 3

1.3.1 Definitions 4

1.3.2 Properties 9

1.3.3 Mittag-Leffler functions 12

1.1 Some Results from Nonlinear Analysis 13

1.4.1 Sobolev Spaces 13

1.4.2 Measure of Noncompactness 14

1.4.3 Topological Degree 15

1.4.4 Picard Operator 17

1.4.5 Fixed Point Theorems 18

1.4.6 Critical Point Theorems 20

1.5 Semigroups 22

1.5.1 C_o-semigroup 22

1.5.2 Almost Sectorial Operators 23

2 Fractional Functional Differential Equations 27

2.1 Introduction 27

2.2 Neutral Equations with Bounded Delay 28

2.2.1 Introduction 28

2.2.2 Existence and Uniqueness 28

2.2.3 Extremal Solutions 33

2.3 p-Type Neutral Equations 42

2.3.1 Introduction 42

2.3.2 Existence and Uniqueness 44

2.3.3 Continuous Dependence 55

2.4 Neutral Equations with Infinite Delay 58

2.4.1 Introduction 58

2.4.2 Existence and Uniqueness 60

2.4.3 Continuation of Solutions 67

2.5 Iterative Functional Differential Equations 71

2.5.1 Introduction 71

2.5.2 Existence 72

2.5.3 Data Dependence 78

2.5.4 Examples and General Cases 79

2.6 Notes and Remarks 86

3 Fractional Ordinary Differential Equations in Banach Spaces 87

3.1 Introduction 87

3.2 Cauchy Problems via Measure of Noncompactness Method 89

3.2.1 Introduction 89

3.2.2 Existence 89

3.3 Cauchy Problems via Topological Degree Method 98

3.3.1 Introduction 98

3.3.2 Qualitative Analysis 98

3.4 Cauchy Problems via Picard Operators Technique 102

3.4.1 Introduction 102

3.4.2 Results via Picard Operators 102

3.4.3 Results via Weakly Picard Operators 109

3.5 Notes and Remarks 113

4 Fractional Abstract Evolution Equations 115

4.1 Introduction 115

4.2 Evolution Equations with Riemann-Liouville Derivative 116

4.2.1 Introduction 116

4.2.2 Definition of Mild Solutions 117

4.2.3 Preliminary Lemmas 120

4.2.4 Compact Semigroup Case 126

4.2.5 Noncompact Semigroup Case 131

4.3 Evolution Equations with Caputo Derivative 134

4.3.1 Introduction 134

4.3.2 Definition of Mild Solutions 134

4.3.3 Preliminary Lemmas 136

4.3.4 Compact Semigroup Case 140

4.3.5 Noncompact Semigroup Case 143

4.4 Nonlocal Problems for Evolution Equations 145

4.4.1 Introduction 145

4.4.2 Definition of mild solutions 145

4.4.3 Existence 147

4.5 Abstract Cauchy Problems with Almost Sectorial Operators 153

4.5.1 Introduction 153

4.5.2 Properties of Operators 158

4.5.3 Linear Problems 164

4.5.4 Nonlinear Problems 169

4.5.5 Applications 177

4.6 Notes and Remarks 179

5 Fractional Impulsive Differential Equations 181

5.1 Introduction 181

5.2 Impulsive Initial Value Problems 182

5.2.1 Introduction 182

5.2.2 Formula of Solutions 182

5.2.3 Existence 185

5.3 Impulsive Boundary Value Problems 190

5.3.1 Introduction 190

5.3.2 Formula of Solutions 190

5.3.3 Existence 193

5.4 Impulsive Langevin Equations 197

5.4.1 Introduction 197

5.4.2 Formula of Solutions 198

5.4.3 Existence 206

5.5 Impulsive Evolution Equations 213

5.5.1 Introduction 213

5.5.2 Cauchy Problems 214

5.5.3 Nonlocal Problems 216

5.6 Notes and Remarks 222

6 Fractional Boundary Value Problems 223

6.1 Introduction 223

6.2 Solution for BVP with Left and Right Fractional Integrals 223

6.2.1 Introduction 223

6.2.2 Fractional Derivative Space 226

6.2.3 Variational Structure 231

6.2.4 Existence Under Ambrosetti-Rabinowitz Condition 238

6.2.5 Superquadratic Case 243

6.2.6 Asymptotically Quadratic Case 247

6.3 Multiple Solutions for BVP with Parameters 250

6.3.1 Introduction 250

6.3.2 Existence 251

6.4 Infinite Solutions for BVP with Left and Right Fractional Integrals 261

6.4.1 Introduction 261

6.4.2 Existence 262

6.5 Solutions for BVP with Left and Right Fractional Derivatives 271

6.5.1 Introduction 271

6.5.2 Variational Structure 272

6.5.3 Existence of Weak Solutions 275

6.5.4 Existence of Solutions 279

6.6 Notes and Remarks 283

7 Fractional Partial Differential Equations 285

7.1 Introduction 285

7.2 Fractional Navier-Stokes Equations 285

7.2.1 Introduction 285

7.2.2 Preliminaries 287

7.2.3 Global Existence 290

7.2.4 Local Existence 297

7.2.5 Regularity 301

7.3 Fractional Euler-Lagrange Equations 309

7.3.1 Introduction 309

7.3.2 Functional Spaces 311

7.3.3 Variational Structure 314

7.3.4 Existence of Weak Solution 317

7.4 Fractional Diffusion Equations 321

7.4.1 Introduction 321

7.4.2 Preliminaries 324

7.4.3 Existence and Regularity 327

7.5 Fractional Schrödinger Equations 336

7.5.1 Introduction 336

7.5.2 Preliminaries 337

7.5.3 Existence and Uniqueness 340

7.6 Notes and Remarks 342

Bibliography 343

Index 363