Existence and uniqueness solution for integral boundary value problem of fractional differential equation

In this work, we will study the existence and uniqueness results of a class of nonlinear fractional differential equations with integral boundary value conditions. By using Leray-Schauder nonlinear alternative and the Banach contraction mapping principle. As an application, an example is given to prove our conclusions.


Introduction
Fractional calculus describe various phenomena in diverse areas of natural science such as physics, aerodynamics, biology, control theory, and chemistry. Boundary value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems. In recent years, many researchers focused on the solutions for boundary value problems of fractional differential equations for details, see [1 − 3, 5 − 7, 10, ...] and references therein. Motivated by the above work, we investigate the following integral boundary value problems of fractional differential equations The organization of the paper is as follows. In section 2, we introduce some definitions and lemmas needed in our proofs. In section 3, we establish the existence and uniqueness of the solution by using Leray-Schauder nonlinear alternative and the Banach contraction theorem. Last, we give an example illustrating the previous results.

Preliminaries
In this section, we present some definitions and lemmas.from fractional calculus theory. Let E be the Banach space of continuous functions C [0, 1] , endowed with the norm ∥u∥ E = max 1] |u (t)| .
almost everywhere on [a, b], where n is the smallest integer greater than or equal to α.

Lemma 5. [7]
The function G (t, s) defined by (2.2) satisfies the following properties To use the fixed point theorem, according to Lemma 6, we define the operator T as The operator T : E −→ E, is completely continuous.

Existence and uniqueness results
In this section, we prove the existence and uniqueness solution for the boundary value problem (1.1) by Leray-Schauder nonlinear alternative and the Banach contraction mapping principle.
Theorem 1. Assume that there exists L 1 , L 2 > 0 such that Then the boundary value broblem (1.1) , has a unique solution in E Proof. We have, Then T is a contraction mapping. Therefore, by the Banach contraction mapping principle, it has a unique fixed point which is the unique solution of the boundary value broblem (1.1) .