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By establishing a comparison result and using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence of solutions for nonlinear fractional differential systems with coupled four-point boundary value problems.

This paper discusses the coupled four-point boundary value problems

It is well known that

Fractional differential equation’s modeling capabilities in physics, chemistry, economics, and other fields, over the last few decades, have resulted in the rapid development of the theory of fractional differential equations; we refer the reader to the books [

Motivated by the above papers, in this paper, we will investigate the existence of a solution of problem (

In the following, we denote

In this section, we introduce the definition of the lower and upper solutions and present some existence and uniqueness results for linear problems together with comparison results for differential systems (

Throughout this paper, we always assume that the following condition is satisfied:

If

Let

We have the following important result.

Let

Suppose the contrary. By Lemma

Let

Let

It follows from [

Now we enunciate the following existence and uniqueness results for differential system:

Let

Indeed, by Lemma

In this section, we prove the existence of extremal solutions of differential system (

Assume that

Then differential system (

Let us define two sequences

Now we show that

By mathematical induction we can show that

We need to show now that

Consider the following problems:

On the other hand, it is easy to verify that conditions

By Theorem

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

The authors sincerely thank the reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original paper. The Project is 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.