Basic Theory of Fractional Differential Equations

The following sections are included:

• Preface to the Third Edition
• About the Author
• Contents

The following sections are included:

• Introduction
• Some Notations, Concepts and Lemmas
• Fractional Calculus
  • Definitions
  • Properties
  • Mittag-Leffler Functions
• Some Results from Nonlinear Analysis
  • Laplace and Fourier Transforms
  • Sobolev Spaces
  • Measure of Noncompactness
  • Topological Degree
  • Picard Operator
  • Fixed Point Theorems
  • Critical Point Theorems
• Semigroups
  • C₀-Semigroup
  • Almost Sectorial Operators

The following sections are included:

• Introduction
• Neutral Equations with Bounded Delay
  • Introduction
  • Existence and Uniqueness
  • Extremal Solutions
• p-Type Neutral Equations
  • Introduction
  • Existence and Uniqueness
  • Continuous Dependence
• Neutral Equations with Infinite Delay
  • Introduction
  • Existence and Uniqueness
  • Continuation of Solutions
• Iterative Functional Differential Equations
  • Introduction
  • Existence
  • Data Dependence
  • Examples and General Cases
• Oscillations and Nonoscillations
  • Introduction
  • Preliminaries
  • Oscillation of Neutral Differential Systems
  • Existence of Nonoscillatory Solutions
  • Fractional Partial Functional Differential Equations
• Notes and Remarks

The following sections are included:

• Introduction
• Cauchy Problems via Measure of Noncompactness Method
  • Introduction
  • Existence
• Cauchy Problems via Topological Degree Method
  • Introduction
  • Qualitative Analysis
• Cauchy Problems via Picard Operators Technique
  • Introduction
  • Results via Picard Operators
  • Results via Weakly Picard Operators
• Notes and Remarks

The following sections are included:

• Introduction
• Evolution Equations with Riemann-Liouville Derivative
  • Introduction
  • Definition of Mild Solutions
  • Preliminary Lemmas
  • Compact Semigroup Case
  • Noncompact Semigroup Case
• Evolution Equations with Caputo Derivative
  • Introduction
  • Definition of Mild Solutions
  • Preliminary Lemmas
  • Compact Semigroup Case
  • Noncompact Semigroup Case
• Nonlocal Problems for Evolution Equations
  • Introduction
  • Definition of Mild Solutions
  • Existence
• Optimal Controls of Fractional Evolution Equations
  • Introduction
  • Preliminaries
  • Existence of α-Mild Solutions
  • Existence of Fractional Optimal Controls
• Abstract Cauchy Problems with Almost Sectorial Operators
  • Introduction
  • Properties of Operators
  • Linear Problems
  • Nonlinear Problems
  • Applications
• Evolution Equations with Hilfer Derivative
  • Introduction
  • Preliminaries
  • Some Lemmas
  • Existence Results
• Infinite Interval Problems with Hilfer Derivative
  • Introduction
  • Preliminaries
  • Lemmas
  • Existence on Infinite Interval
• Notes and Remarks

The following sections are included:

• Introduction
• Impulsive Initial Value Problems
  • Introduction
  • Formula of Solutions
  • Existence
• Impulsive Boundary Value Problems
  • Introduction
  • Formula of Solutions
  • Existence
• Impulsive Langevin Equations
  • Introduction
  • Formula of Solutions
  • Existence
• Impulsive Evolution Equations
  • Introduction
  • Cauchy Problems
  • Nonlocal Problems
• Notes and Remarks

The following sections are included:

• Introduction
• Solutions for BVP with Left and Right Fractional Integrals
  • Introduction
  • Fractional Derivative Space
  • Variational Structure
  • Existence under Ambrosetti-Rabinowitz Condition
  • Superquadratic Case
  • Asymptotically Quadratic Case
• Multiple Solutions for BVP with Parameters
  • Introduction
  • Existence
• Infinite Solutions for BVP with Left and Right Fractional Integrals
  • Introduction
  • Existence
• Solutions for BVP with Left and Right Fractional Derivatives
  • Introduction
  • Variational Structure
  • Existence of Weak Solutions
  • Existence of Solutions
• Notes and Remarks

The following sections are included:

• Introduction
• Existence and Multiplicity of Homoclinic Solutions (I)
  • Fractional Derivative Space
  • Some Lemmas
  • Existence of Homoclinic Solutions
• Existence and Multiplicity of Homoclinic Solutions (II)
  • Introduction
  • Some Lemmas
  • Existence and Multiplicity
• Notes and Remarks

The following sections are included:

• Introduction
• Fractional Navier-Stokes Equations
  • Introduction
  • Preliminaries
  • Global Existence
  • Local Existence
  • Regularity
• Fractional Rayleigh-Stokes Equations
  • Introduction
  • Preliminaries
  • Globally Lispchitz Source Term
  • Locally Lispchitz Source Term
• Fractional Euler-Lagrange Equations
  • Introduction
  • Functional Spaces
  • Variational Structure
  • Existence of Weak Solution
• Fractional Diffusion Equations
  • Introduction
  • Regularity and Unique Existence
• Fractional Wave Equations
  • Introduction
  • Preliminaries
  • Approximation Solution
  • Energy Estimates
  • Well-Posedness and Regularity
• Notes and Remarks

The following sections are included:

• Bibliography
• Index