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We discuss the existence of solutions, under the Pettis integrability assumption, for a class of boundary value problems for fractional differential inclusions involving nonlinear nonseparated boundary conditions. Our analysis relies on the Mönch 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 inclusion with non-separated boundary conditions:

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 [

In 2007, Ouahab [

The remainder of this paper is organized as follows. In Section

In this section, we introduce notation, definitions, and preliminary facts that will be used in the remainder of this paper. Let

Let

Moreover, for a given set

For any

A function

A function

The function

Let

If

Let

The De Blasi measure of noncompactness satisfies the following properties:

The following result follows directly from the Hahn-Banach theorem.

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 problem (

A function

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

for each continuous

there exist

for each bounded set

there exists a constant

Let

Let

From the expression of

We transform the problem (

Let

Indeed, if

To see this, take

Therefore, by (H5), we have

Next suppose

Step

Let

We must show that there exists

Since

Since function

This means that

In the sequel we present an example which illustrates Theorem

We consider the following partial hyperbolic fractional differential inclusion of the form

Set

Let

Hence conditions

The first author’s work was supported by NNSF of China (11161027), NNSF of China (10901075), and the Key Project of Chinese Ministry of Education (210226). The authors are grateful to the referees for their comments according to which the paper has been revised.