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We discuss the existence of weak solutions for a nonlinear boundary value problem of fractional differential equations in Banach space. Our analysis relies on the Mönch's fixed point theorem combined with the technique of measures of weak noncompactness.

This paper is mainly concerned with the existence results for the following fractional differential equation:

Recently, fractional differential equations have found numerous applications in various fields of physics and engineering [

To investigate the existence of solutions of the problem above, we use Mönch's fixed point theorem combined with the technique of measures of weak noncompactness, which is an important method for seeking solutions of differential equations. This technique was mainly initiated in the monograph of Banaś and Goebel [

The remainder of this is organized as follows. In Section

In this section, we present some basic notations, definitions, and preliminary results which will be used throughout this paper.

Let

Let

Moreover, for a given set

A function

The function

Let

If

Let

The De sBlasi measure of noncompactness satisfies the following properties:

Let

For completeness, we recall the definitions of the Pettis-integral and the Caputo derivative of fractional order.

Let

For a function

Let

Let us start by defining what we mean by a solution of the problem (

A function

For the existence results on the problem (

For

Assume that

We derive the corresponding Green's function for boundary value problem (

Let

By the Lemma

Applying the boundary conditions (

Therefore, the unique solution of problem (

From the expression of

Letting

To prove the main results, we need the following assumptions:

for each

for each

there exists

there exists

for each bounded set

there exists a constant

where

Let

Let the operator

First notice that, for

Let

Take

Let

Let

In the Theorem

Let

Let

Assume that the operator

First notice that, for

Let

Take

Let

Thus

Let

This means that

In this section we give an example to illustrate the usefulness of our main result.

Let us consider the following fractional boundary value problem:

Set

Clearly conditions (H1), (H2), and (H3) hold with

We have

A simple computation gives

We shall check that condition (

Wen-Xue Zhou’s work was supported by NNSF of China (11161027), NNSF of China (10901075), and the Key Project of Chinese Ministry of Education (210226).