Existence and Uniqueness of Solutions for Fractional Integro-Differential Equations Involving the Hadamard Derivatives

Abstract

In this paper, we study the existence and uniqueness of solutions for the following fractional boundary value problem, consisting of the Hadamard fractional derivative: HDαx(t)=Af(t,x(t))+∑i=1kCiHIβigi(t,x(t)),t∈(1,e), supplemented with fractional Hadamard boundary conditions: HDξx(1)=0,HDξx(e)=aHDα−ξ−12(HDξx(t))|t=δ,δ∈(1,e), where 1<α≤2, 0<ξ≤12, a∈(0,∞), 1<α−ξ<2, 0<βi<1, A,Ci, 1≤i≤k, are real constants, HDα is the Hadamard fractional derivative of order α and HIβi is the Hadamard fractional integral of order βi. By using some fixed point theorems, existence and uniqueness results are obtained. Finally, an example is given for demonstration.