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This paper deals with the integral boundary value problems of fractional differential equations at resonance. By Mawhin’s coincidence degree theory, we present some new results on the existence of solutions for a class of differential equations of fractional order with integral boundary conditions at resonance. An example is also included to illustrate the main results.

In this paper, we are concerned with the following integral boundary value problem for nonlinear fractional differential equation:

Recently, fractional differential equations have received considerable attentions not only because of a generalization of ordinary differential equations but also because they have played a significant role in science, engineering, economy, and other fields; see, for example, [

When

In present, many papers are devoted to the integral boundary value problem for fractional differential equation under nonresonance conditions; see [

Motivated by the above results, in this paper, we consider the existence of solutions for the resonance integral boundary value problem (

Now, we recall the essentials of the coincidence degree theory. Let

The abstract equation

Let

Then the equation

Throughout this paper, we always suppose that

In this section, first we provide recall some necessary basic definitions and lemmas of the fractional calculus theory, which will be used in this paper. For more details, we refer to books [

The Riemann-Liouville fractional integral of order

The Riemann-Liouville fractional derivative of order

Assume that

Assume that

Let

The function

We use the classical Banach space

Define

Then integral boundary value problems (

The operator

Firstly, we show that

Now we prove

Clearly,

Next, define the projections

The generalized inverse operator of

In fact, if

For

Using (

By a standard method, we obtain the following lemma.

For

It is easy to see that

In this section, we will use Theorem

There exist functions

There exists a constant

or for each

There exists a constant

There exists a constant

Suppose (

Set

Let

We define the isomorphism

If (

Next, we will prove that all the assumptions of Theorem

At last, we will prove that (iii) of Theorem

Then by Theorem

Suppose (

As in the proof of Theorem

In view of

Consider the IBVP

Taking

Hence

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The project was supported by the National Natural Science Foundation of China (11371221, 11571207, and 51774197).