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By combining the techniques of fractional calculus with measure of weak noncompactness and fixed point theorem, we establish the existence of weak solutions of multipoint boundary value problem for fractional integrodifferential equations.

In recent years, fractional differential equations in Banach spaces have been studied and a few papers consider fractional differential equations in reflexive Banach spaces equipped with the weak topology. As long as the Banach space is reflexive, the weak compactness offers no problem since every bounded subset is relatively weakly compact and therefore the weak continuity suffices to prove nice existence results for differential and integral equations [

The theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored. There are many papers dealing with multipoint boundary value problems both on resonance case and on nonresonance case; for more details see [

Moreover, theory for boundary value problem of integrodifferential equations of fractional order in Banach spaces endowed with its weak topology has been few studied until now. In [

Motivated by the above works, in this paper, we use the techniques of measure of weak noncompactness combine with the fixed point theorem to discuss the existence theorem of weak solutions for a class of nonlinear fractional integrodifferential equations of the form

The problems of our research are different between this paper and paper [

The paper is organized as follows: In Section

Throughout this paper, we introduce notations, definitions, and preliminary results which will be used.

Let

Let

Now, for the convenience of the reader, we recall some useful definitions of integrals.

A function

A function

This function

A family

Let

for every

for every sequence

If

Let

Let

Let

We give the fixed point theorem, which play a key role in the proof of our main results.

Let

The following we recall the definition of the Caputo derivative of fractional order.

Let

In the above definition the sign “

The Riemann-Liouville derivative of order

The Caputo fractional derivative of order

In this section, we present the existence of solutions to problem (

A function

Let

From the lemma above, we deduce the following statement.

Assume that

The following we give the corresponding Greens function for problem (

Let

Based on the idea of paper [

On the other hand, by the relations

Let

Problem (

For each weakly continuous function

For any

for all

For each bounded set

where

For each

The family

From assumption

Now, we present the existence theorem for problem (

Assume that conditions (

Let

Defining the set

We will show that

Let

Similarly, we have

Moreover, because

In this paper, we use the techniques of measure of weak noncompactness and Henstock-Kurzweil-Pettis integrals to discuss the existence theorem of weak solutions for a class of the multipoint boundary value problem of fractional integrodifferential equations equipped with the weak topology. Our results generalized some classical results.

The authors declare that they have no conflicts of interest.

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

This work is supported by National Natural Science Foundation of China (Grant no. 11061031).