Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions

Phollakrit Thiramanus,Jessada Tariboon,Sotiris K Ntouyas

doi:10.1155/2014/902054

Abstract

We study the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard-type fractional differential equations and nonlocal fractional integral boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.

Highlights

Let C = C([1, e], R) denote the Banach space of all continuous functions from [1, e] to R endowed with the norm defined by ‖x‖ = supt∈[1,e]|x(t)|
It should be noticed that problem (1)-(2) has solutions if and only if the operator F has fixed points
We show that FBr ⊂ Br, where Br = {x ∈ C : ‖x‖ ≤ r}

Summary

Introduction

We investigate the following Hadamard boundary value problem: Dqx (t) = f (t, x (t)) , 1 < q ≤ 2, t ∈ (1, e) , (1). It has been noticed that most of the work on this topic is based on Riemann-Liouville and Caputo type fractional differential equations Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due Abstract and Applied Analysis to Hadamard introduced in 1892 [21], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains logarithmic function of arbitrary exponent. For some recent results on Hadamard boundary value problem we refer to [27, 28]. A second existence and uniqueness result is proved in Theorem 7, via nonlinear contractions and a fixed point theorem due to Boyd and Wong.

Preliminaries

Main Results

Examples