Table of Contents
Preface v
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 8
1.4 Some Results from Nonlinear Analysis 11
1.4.1 Sobolev Spaces 11
1.4.2 Measure of Noncompactness 12
1.4.3 Topological Degree 13
1.4.4 Picard Operator 15
1.4.5 Fixed Point Theorems 16
1.4.6 Critical Point Theorems 17
1.5 Semigroups 20
1.5.1 C0-Semigroup 20
1.5.2 Almost Sectorial Operators 21
2 Fractional Functional Differential Equations 23
2.1 Introduction 23
2.2 Neutral Equations with Bounded Delay 24
2.2.1 Introduction 24
2.2.2 Existence and Uniqueness 24
2.2.3 Extremal Solutions 29
2.3 p-Type Neutral Equations 38
2.3.1 Introduction 38
2.3.2 Existence and Uniqueness 40
2.3.3 Continuous Dependence 50
2.4 Neutral Equations with Infinite Delay 53
2.4.1 Introduction 53
2.4.2 Existence and Uniqueness 55
2.4.3 Continuation of Solutions 62
2.5 Iterative Functional Differential Equations 66
2.5.1 Introduction 66
2.5.2 Existence 66
2.5.3 Data Dependence 72
2.5.4 Examples and General Cases 74
2.6 Notes and Remarks 80
3 Fractional Ordinary Differential Equations in Banach Spaces 81
3.1 Introduction 81
3.2 Cauchy Problems via Measure of Noncompactness Method 83
3.2.1 Introduction 83
3.2.2 Existence 83
3.3 Cauchy Problems via Topological Degree Method 92
3.3.1 Introduction 92
3.3.2 Qualitative Analysis 92
3.4 Cauchy Problems via Picard Operators Technique 96
3.4.1 Introduction 96
3.4.2 Results via Picard Operators 96
3.4.3 Results via Weakly Picard Operators 102
3.5 Notes and Remarks 107
4 Fractional Abstract Evolution Equations 109
4.1 Introduction 109
4.2 Evolution Equations with Riemann-Liouville Derivative 110
4.2.1 Introduction 110
4.2.2 Definition of Mild Solutions 111
4.2.3 Preliminary Lemmas 114
4.2.4 Compact Semigroup Case 120
4.2.5 Noncompact Semigroup Case 124
4.3 Evolution Equations with Caputo Derivative 127
4.3.1 Introduction 127
4.3.2 Definition of Mild Solutions 128
4.3.3 Preliminary Lemmas 130
4.3.4 Compact Semigroup Case 133
4.3.5 Noncompact Semigroup Case 136
4.4 Nonlocal Cauchy Problems for Evolution Equations 138
4.4.1 Intorduction 138
4.4.2 Definition of Mild Solutions 139
4.4.3 Existence 140
4.5 Abstract Cauchy Problems with Almost Sectorial Operators 146
4.5.1 Introduction 146
4.5.2 Preliminaries 150
4.5.3 Properties of Operators 154
4.5.4 Linear Problems 160
4.5.5 Nonlinear Problems 164
4.5.6 Applications 172
4.6 Notes and Remarks 175
5 Fractional Boundary Value Problems via Critical Point Theory 177
5.1 Introduction 177
5.2 Existence of Solution for BVP with Left and Right Fractional Integrals 177
5.2.1 Introduction 177
5.2.2 Fractional Derivative Space 180
5.2.3 Variational Structure 185
5.2.4 Existence under Ambrosetti-Rabinowitz Condition 192
5.2.5 Superquadratic Case 196
5.2.6 Asymptotically Quadratic Case 200
5.3 Multiple Solutions for BVP with Parameters 203
5.3.1 Introduction 203
5.3.2 Existence 204
5.4 Infinite Solutions for BVP with Left and Right Fractional Integrals 214
5.4.1 Introduction 214
5.4.2 Existence 215
5.5 Existence of Solutions for BVP with Left and Right Fractional Derivatives 223
5.5.1 Introduction 223
5.5.2 Variational Structure 224
5.5.3 Existence of Weak Solutions 227
5.5.4 Existence of Solutions 231
5.6 Notes and Remarks 235
6 Fractional Partial Differential Equations 237
6.1 Introduction 237
6.2 Fractional Euler-Lagrange Equations 237
6.2.1 Introduction 237
6.2.2 Functional Spaces 239
6.2.3 Variational Structure 242
6.2.4 Existence of Weak Solution 245
6.3 Time-Fractional Diffusion Equations 249
6.3.1 Introduction 249
6.3.2 Regularity and Unique Existence 250
6.4 Fractional Hamiltonian Systems 257
6.4.1 Introduction 257
6.4.2 Fractional Derivative Space 257
6.4.3 Existence and Multiplicity 263
6.5 Fractional Schrödinger Equations 271
6.5.1 Inroduction 271
6.5.2 Existence and Uniqueness 273
6.6 Notes and Remarks 278
Bibliography 279
