Fractional-Order Equations and Inclusions

J. Wang, Guizhou University, China; M. Fečkan, Comenius University, Slovakia; M. Pospíšil, Slovak Academy of Sciences, Slovakia.

Table of Content:

Introduction

1 Fractional Difference Equations
1.1 Fractional difference Gronwall inequalities
1.1.1 Introduction
1.1.2 Caputo like fractional difference
1.1.3 Linear fractional difference equation
1.1.4 Fractional difference inequalities
1.2 S-asymptotically periodic solutions
1.2.1 Introduction
1.2.2 Preliminaries
1.2.3 Non-existence of periodic solutions
1.2.4 Existence and uniqueness results

2 Fractional Integral Equations
2.1 Abel-type nonlinear integral equations
2.1.1 Introduction
2.1.2 Preliminaries
2.1.3 Existence and uniqueness of non-trivial solution in an order interval
2.1.4 General solutions of Erdélyi-Kober type integral equations
2.1.5 Illustrative examples
2.2 Quadratic Erdélyi-Kober type integral equations of fractional order
2.2.1 Introduction
2.2.2 Preliminaries
2.2.3 Existence and limit property of solutions
2.2.4 Uniqueness and another existence results
2.2.5 Applications
2.3 Fully nonlinear Erdélyi-Kober fractional integral equations
2.3.1 Introduction
2.3.2 Main result
2.3.3 Example
2.4 Quadratic Weyl fractional integral equations
2.4.1 Introduction
2.4.2 Preliminaries
2.4.3 Some basic properties of Weyl kernel
2.4.4 Existence and uniform local attractivity of 2pi-periodic solutions
2.4.5 Example

3 Fractional Differential Equations
3.1 Asymptotically periodic solutions
3.1.1 Introduction
3.1.2 Preliminaries
3.1.3 Non-existence results for periodic solutions
3.1.4 Existence results for asymptotically periodic solutions
3.1.5 Further extensions
3.2 Modified fractional iterative functional differential equations
3.2.1 Introduction
3.2.2 Notation, definitions and auxiliary facts
3.2.3 Existence
3.2.4 Data dependence
3.2.5 Examples
3.3 Ulam-Hyers-Rassias stability for semilinear equations
3.3.1 Introduction
3.3.2 Ulam-Hyers-Rassias stability for surjective linear equations on Banach spaces
3.3.3 Ulam-Hyers-Rassias stability for linear equations on Banach spaces with closed ranges
3.3.4 Ulam-Hyers-Rassias stability for surjective semilinear equations between Banach spaces
3.4 Practical Ulam-Hyers-Rassias stability for nonlinear equations
3.4.1 Introduction
3.4.2 Main results
3.4.3 Examples
3.5 Ulam-Hyers-Mittag-Leffler stability of fractional delay differential equations
3.5.1 Introduction
3.5.2 Preliminaries
3.5.3 Main results
3.5.4 Examples
3.6 Nonlinear impulsive fractional differential equations
3.6.1 Introduction
3.6.2 Preliminaries
3.6.3 Existence results for impulsive Cauchy problems
3.6.4 Ulam stability results for impulsive fractional differential equations
3.6.5 Existence results for impulsive boundary value problems
3.6.6 Applications
3.7 Fractional differential switched systems with coupled nonlocal initial and impulsive conditions
3.7.1 Introduction
3.7.2 Preliminaries
3.7.3 Existence and uniqueness result via Banach fixed point theorem
3.7.4 Existence result via Krasnoselskii fixed point theorem
3.7.5 Existence result via Leray-Schauder fixed point theorem
3.7.6 Existence result for the resonant case: Landesman-Lazer conditions
3.7.7 Ulam type stability results
3.8 Not instantaneous impulsive fractional differential equations
3.8.1 Introduction
3.8.2 Framework of linear impulsive fractional Cauchy problem
3.8.3 Generalized Ulam-Hyers-Rassias stability concept
3.8.4 Main results via fixed point methods
3.9 Center stable manifold result for planar fractional damped equations
3.9.1 Introduction
3.9.2 Asymptotic behavior of Mittag-Leffler functions E ,beta
3.9.3 Planar fractional Cauchy problems
3.9.4 Center stable manifold result
3.10 Periodic fractional differential equations with impulses
3.10.1 Introduction
3.10.2 FDE with Caputo derivatives with varying lower limits
3.10.3 FDE with Caputo derivatives with fixed lower limits
3.10.4 Conclusions

4 Fractional Evolution Equations: Continued
4.1 Fractional evolution equations with periodic boundary conditions
4.1.1 Introduction
4.1.2 Homogeneous periodic boundary value problem
4.1.3 Non-homogeneous periodic boundary value problem
4.1.4 Parameter perturbation methods for robustness
4.1.5 Example
4.2 Abstract Cauchy problems for fractional evolution equations
4.2.1 Introduction
4.2.2 Preliminaries
4.2.3 Existence and uniqueness theorems of solutions for Problem (I)
4.2.4 Existence and uniqueness theorems of solutions for Problem (II)
4.2.5 Existence and uniqueness theorems of solutions for Problem (III)
4.3 Nonlocal Cauchy problems for Volterra-Fredholm type fractional evolution equations
4.3.1 Introduction
4.3.2 Preliminaries
4.3.3 Existence of mild solutions
4.3.4 Example
4.4 Controllability of Sobolev type fractional functional evolution equations
4.4.1 Introduction
4.4.2 Preliminaries
4.4.3 Characteristic solution operators and their properties
4.4.4 Main results
4.4.5 Example
4.5 Relaxed controls for nonlinear impulsive fractional evolution equations
4.5.1 Introduction
4.5.2 Problem statement
4.5.3 Original and relaxed fractional impulsive control systems
4.5.4 Properties of relaxed trajectories for fractional impulsive control systems
4.5.5 Example

5 Fractional Differential Inclusions
5.1 Fractional differential inclusions with anti-periodic conditions
5.1.1 Introduction
5.1.2 Preliminaries
5.1.3 Existence results for (5.1)
5.1.4 Existence results for (5.2)
5.1.5 Applications to fractional lattice inclusions
5.2 Nonlocal Cauchy problems for semilinear fractional differential inclusions
5.2.1 Introduction
5.2.2 Preliminaries and notation
5.2.3 Existence results
5.2.4 Examples
5.3 Nonlocal impulsive fractional differential inclusions
5.3.1 Introduction
5.3.2 Preliminaries and notation
5.3.3 Existence results for convex case
5.3.4 Existence results for non-convex case

A Appendix
A.1 Functional analysis
A.1.1 Basic notation and results
A.1.2 Banach functional spaces
A.1.3 Linear operators
A.1.4 Semigroup of linear operators
A.1.5 Metric spaces
A.2 Fractional differential calculus
A.3 Henry-Gronwall's inequality
A.4 Measures of noncompactness
A.5 Multifunctions

Bibliography
Index