Existence and Uniqueness of Positive Solutions for a New System of Fractional Differential Equations

Abstract

By virtue of a recent existing fixed point theorem of increasing -concave operator by Zhai and Wang, we consider the existence and uniqueness of positive solutions for a new system of Caputo-type fractional differential equations with Riemann–Stieltjes integral boundary conditions.

1. Introduction

In this paper, we consider the following nonlinear Caputo-type fractional system:where , ; and are the Caputo fractional derivative; and are continuous; are continuous; and and are bounded variation functions with positive measures with , .

In recent decades, fractional-order calculus has been widely used in engineering, biology, physics, and so on. Based on it, many scholars have been interested in the study of the existence of nontrivial solutions for various fractional boundary value problems. For some recent works, we can refer to [1–30] and the references therein. For example, in [10], by virtue of Guo–Krasnosel’skii fixed point theorem, Ma and Cui studied the following fractional boundary value problem:where , is the Caputo fractional derivative, , and is a parameter. In the case that the parameter satisfied some conditions, the existence of positive solutions for the boundary value problem (2) was proved. Meanwhile, many scholars considered some various fractional systems, such as [16–30] and the reference therein. For instance, in [16], the authors obtained the existence of two positive solutions for a nonlinear Caputo-type fractional system by virtue of fixed point index theory. In [17], by using monotone iterative approach, the authors investigated the iterative positive solutions of a system of fractional Riemann–Liouville-type equations with four-point boundary conditions. In [19], by using Banach’s contraction principle, the authors studied the uniqueness of solution for a system of Hadamard-type fractional differential equations with integral boundary conditions. In [21], by using Guo–Krasnosel’skii fixed point theorem, the authors studied the existence of positive solutions for an infinite system of fractional Caputo-type differential equations.

In [22], Zhai and Wang introduced a new concept of -concave operator and obtained fixed point theorems of increasing -concave operator. In recent years, by using the fixed point theorems of -concave operator, Zhai and Jiang investigated the uniqueness of positive solutions for a Riemann–Liouville-type fractional system with integral boundary conditions in [23]; Zhai and Wang considered the uniqueness of positive solutions for a system of Hadamard fractional differential equations with integral equations in [24]; Zhai and Zhu considered the uniqueness of positive solutions for a system of Riemann–Liouville fractional differential equations in [25].

Inspired by [10, 22–25], we introduce a new system of nonlinear Caputo-type fractional differential equations (1). There are few papers about the application of -concave operator in nonlinear Caputo-type fractional boundary value problems. So, in this paper, we use the recent fixed point theorems of -concave operator by Zhai and Wang to study system (1). The result of the existence and uniqueness of positive solutions for system (1) is obtained.

2. Preliminaries

In this section, we briefly introduce Caputo’s fractional derivative and the fixed point theorem of -concave operator. For details, we can refer to the literature [1, 22]. And we give some lemmas about the relevant Green’s functions.

Definition 1. (see [1]). For a function , we define Caputo’s fractional derivative of order as follows:where is the smallest integer greater than or equal to .
By [10], the following lemmas are listed.

Lemma 1 (see [10]). Let and . Then, the linear Caputo fractional differential equationhas a unique solutionwhereand .

Lemma 2. The above Green’s function has the following properties:(i) and is continuous on .(ii).

Proof. By (6), we know that and is continuous on .
By (6), we easily know thatFrom (6), when , we haveWhen , we haveBy (8) and (9), we haveLet be a real Banach space and be a cone. A partial order on is induced by . For any , the notation denotes that there are and such that . Take (i.e., and ); let ; then obviously, . Choose with , and let .

Definition 2. (see [22]). Let be an operator. If for any and , there exists such thatThen, is called a -concave operator.

Lemma 3 (see [22]). Let be a normal cone. Suppose that is an increasing -concave operator and ; then, has a unique fixed point . For any , the sequence , and then as .

3. Main Results

Let with the norm . Let with the norm . Let . Then, is a normal cone. Let . It is obvious that is a normal cone of . We define the following partial order on space : . For the detailed knowledge about the cone, we can refer to [31].

Define the following operators , and :where is defined by (6).

By [10], we easily know that fixed points of the operator are solutions of system (1).

Letwhere

Let

Set and . By [29], we have

Let .

For convenience, the following conditions are given.(i) is increasing about the second and third variables; is increasing about the second and third variables.(ii) For , there exists such that where(iii) , and , , .

Theorem 1. Suppose that conditions , and are satisfied. Then, system (1) has a unique solution , and for any given , we have , , where the sequences:where .

Proof. By Lemma 2, we know that . So,By Lemma 2, we haveFrom (21), for , we haveThus, we obtain thatIn the following, we divide three parts to prove this theorem. Firstly, we prove that is a concave operator. By , for and , we haveSimilarly, by , for and , we obtainBy (24) and (25), for , , and , we haveNamely, for and , we obtain thatSecondly, we show that is an increasing operator. By the definition of , for , we get that , i.e., . By the definitions of and , we obtain that there exist and such thatSo,From , we easily know that the operators and are increasing, so it is obvious that is increasing.
In the end, we prove . By [22], we know that . Since , we need to prove . In the following, by the definitions of and , we prove , respectively. From and , combine (21), and we haveLetObviously, . So, we have . And . Similarly, we have . So, is proved.
Therefore, by Lemma 3, the operator has a unique fixed point . For any given , define the sequences:where .

4. Application

Example 1. We study the following fractional system with integral boundary conditions:where ; ; andObviously,and , .
Green’s functions are as follows:Then, we haveLet , whereWe haveBy (34) and (35), we haveTake ; then, . For , , , and , we haveTherefore, from Theorem 1, we know that system (33) has a unique solution , whereFor any given , we obtain the following sequences:where , is defined by (37), .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (11801322) and Shandong Natural Science Foundation (ZR2018MA011).