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A four-point coupled boundary value problem of fractional differential equations is studied. Based on Mawhin’s coincidence degree theory, some existence theorems are obtained in the case of resonance.

In this paper, we are concerned with the following four-point coupled boundary value problem for nonlinear fractional differential equation. Consider

The subject of fractional calculus has gained considerable popularity and importance because of its intensive development of the theory of fractional calculus itself and its varied applications in many fields of science and engineering. As a result, the subject of fractional differential equations has attracted much attention; see [

At the same time, we notice that coupled boundary value problems, which arise in the study of reaction-diffusion equations and Sturm-Liouville problems, have wide applications in various fields of sciences and engineering, for example, the heat equation [

In [

Recently, Cui and Sun [

A key assumption in the above papers is that the case studied is not at resonance; that is, the associated fractional (or ordinary) linear differential operators are invertible. In this paper, instead, we are interested in the resonance case due to the critical condition (

Now, we briefly recall some notations and an abstract existence result.

Let

Let

For convenience, let us set the following notations:

In this section, first we provide recall of some basic definitions and lemmas of the fractional calculus, 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

We use the classical Banach space

Let

In the following lemma, we use the unified notation of both for fractional integrals and fractional derivatives assuming that

Assume that

let

if

satisfies at any point on

let

note that, for

If

We also use the following two Banach spaces

Let the linear operator

Let the nonlinear operator

Let

Let

Let

If (

Define operator

Let

Define

In fact, if

In this section, we will use Theorem

There exist functions

There exists a constant

There exists a constant

or, for each

Suppose (

Set

We define the isomorphism

If the second part of (H3) holds, then define the set

In the following, we will prove that all conditions of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the presentation of the paper. The project was supported by the National Natural Science Foundation of China (11371221, 61304074), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province, the Postdoctoral Science Foundation of Shandong Province (201303074), and Foundation of SDUST.