Multiple Positive Solutions for a System of Nonlinear Caputo-Type Fractional Differential Equations

Li, Hongyu; Chen, Yang

doi:https://doi.org/10.1155/2020/2437530

Journal of Function Spaces

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Abstract Introduction Preliminaries Applications Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Research Article | Open Access

Volume 2020 | Article ID 2437530 | https://doi.org/10.1155/2020/2437530

Multiple Positive Solutions for a System of Nonlinear Caputo-Type Fractional Differential Equations

Hongyu Li¹and Yang Chen¹

Academic Editor: Liguang Wang

Received20 Mar 2020

Accepted18 Apr 2020

Published19 May 2020

Abstract

By using fixed-point index theory, we consider the existence of multiple positive solutions for a system of nonlinear Caputo-type fractional differential equations with the Riemann-Stieltjes boundary conditions.

1. Introduction

Fractional order calculus is more widely used than integer order calculus (see [1]). Based on it, many researchers have been focused on the study of the existence of positive solutions for various fractional differential equations with some boundary conditions. We can refer to [2–35] for some recent works about fractional boundary value problems.

In [14], Ma and Cui discussed the following fractional boundary value problem: where , is the Caputo fractional derivative, is a parameter. By using the Guo-Krasnosel’skii fixed-point theorem, they proved that the fractional differential equation has at least a positive solution when satisfies some conditions.

A few researchers have studied the existence of solutions for the systems of nonlinear fractional differential equations (see [23–35]). For example, in [24], by monotone iterative technique and cone theory, Zhang et al. have studied the uniqueness of a solution for fractional systems with the Riemann-Stieltjes integral boundary condition. Hao et al. [32] have studied a system of fractional boundary value problems with two parameters; by using the Guo-Krasnosel’skii fixed-point theorem, they obtained the existence of positive solutions for the system in terms of different values of parameters. Meanwhile, a few researchers have been considering the systems for nonlinear integer order boundary value problems, such as in [33], where the researchers considered the existence of multiple positive solutions for a system of nonlinear third-order differential equations. In [34], by Banach’s contraction principle, the researchers considered the existence of a unique solution for a system of second order differential equations with coupled integral boundary conditions.

Inspired by [14, 23–35], in this paper, we want to study the following system of fractional differential equations: where is continuous; is the Caputo fractional derivative; is a bounded variation function with positive measure with .

The main purpose of this paper is that we prove that system (2) has one and multiple positive solutions by the fixed-point index theory.

2. Preliminaries

In this section, we only give the definition of Caputo’s fractional derivative. For some of the other definitions and properties of Caputo’s fractional derivative, we can refer to [2]. We mainly give the relevant Green functions and the used lemmas.

Definition 1 (see [2, 14]). For a function , we define Caputo’s fractional derivative of order as follows: where is the smallest integer greater than or equal to .
From [14], we have the following lemmas.

Lemma 2 (see [14]). Let and . Then is the solution of the linear Caputo fractional differential equation if and only if is the solution of the integral equation where and .

Lemma 3 (see [14]). The above Green’s function has the following properties: (i)(ii)where

Lemma 4 (see [36]). Let be a Banach space and be a cone. Define , where . Suppose that is completely continuous, and , for . (i)(ii)

3. Main Results

Let be a Banach space with the norm on , where

Define the cone where

and are defined by (7), and the nonlinear operators , , and are defined by where is defined by (6).

It is known that fixed points of the operator in are positive solutions of the system (2).

Lemma 5. The operator is completely continuous.

Proof. For , when , we have .
When by Lemma 3, we obtain Similar to the proof of (12), for we have By (12) and (14), we get By (15), . Similar to the proof of Lemma 6 in [14], we know that are completely continuous. So is completely continuous.

For convenience, some marks and assumptions are given.

Some of the marks are as follows: where is defined by (9).

Some of the assumptions are as follows:

S₁. There exist and such that (i)(ii)where also satisfies , and are defined by (17).

S₂. There exists such that where is defined by (9) and is defined by (17).

Theorem 6. Suppose that and uniformly on . Then, system (2) has a positive solution.

Proof. By , we know that there exist and such that and , .
Set . By Lemma 3 and (19), for , we have Similar to the proof of (20), we have So by (20) and (21), we have By Lemma 4 and (22), we have From , we know that there exist and such that and . In this part, we divide two cases. One case is that and are bounded. Namely, there exists such that Take . Set . For and , we get The other case is that and are unbounded. By the continuity of , there exists such that Hence, set . For and we get So in either case, there always exists such that Similarly, we have So by (29) and (30), we have By Lemma 4 and (31), we have From (23) and (32), we obtain So has a fixed point in . It is obvious that is a positive solution of system (2).

Theorem 7. Suppose that S₁ holds and . Then, the system (2) has two positive solutions.

Proof. Set . By Lemma 3 and S₁ (i), for and , we get So Similar to the proof of (35), we have So by (35) and (36), we have By Lemma 4 and (37), we have Set . By Lemma 3 and S₁ (ii), for and , we get So Similarly, So by (40) and (41), we have By Lemma 4 and (42), we have By there exist and such that and .
Choose . Set . For Then, by (44) and Lemma 3, we get So Similarly, So by (47) and (48), we have By Lemma 4 and (49), we have Since by (38), (43), and (50), we get From (51), we know that has a fixed point . From (52), we know that has another fixed point. So the system (2) has two positive solutions and , with .

Theorem 8. Let and . In addition, suppose that S₂ holds. Then, system (2) has two positive solutions.

Proof. By , we know that there exist and such that and . Set .
By Lemma 3 and (53), for and , we get So Similarly, So by (55) and (56), we have By Lemma 4 and (57), we have From , we know that there exist and such that and . In this part, we divide two cases. One case is that and are bounded. Namely, there exists such that Take . Set . For and , we get Another case is that and are unbounded. By the continuity of , there exists such that Hence, set . For , we get So in either case, there always exists such that Similarly, we have Thus So by Lemma 4 and (66), we get Set . For Then, by S₂ and Lemma 3, we get So Similarly, So by (70) and (71), we have By Lemma 4 and (72), we have Since , by (58), (67), and (73), we get . So has a fixed point .. So has another fixed point . Therefore, system (2) has at least two positive solutions.

4. Applications

Example 9. We study the following Caputo-type fractional system: where , , , , and . Obviously, and . From Theorem 6, system (74) has a positive solution.

Example 10. We study the following Caputo-type fractional system: where , , and .
Take and By simple computation, we get , , and .
Choose , , and , then .
When , , (1); (2); When , , (1); (2); (3); Obviously, . So by Theorem 7, system (74) has at least two positive solutions.

Example 11. We study the following Caputo-type fractional system: where , , and .
Take and Take , then , . When , . Obviously, and . So, by Theorem 8, system (78) has at least two positive solutions.

Data Availability

The data set supporting the conclusions is included within this article.

Conflicts of Interest

The authors declare that they have no competing interests regarding the publication of this paper.

Acknowledgments

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

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Copyright

Copyright © 2020 Hongyu Li and Yang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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