Multiplicity Solutions for Integral Boundary Value Problem of Fractional Differential Systems

Song, Shiying; Cui, Yujun

doi:https://doi.org/10.1155/2020/2651845

Discrete Dynamics in Nature and Society

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

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Fractional Difference and Differential Operators and their Applications in Nonlinear Systems

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Research Article | Open Access

Volume 2020 | Article ID 2651845 | https://doi.org/10.1155/2020/2651845

Multiplicity Solutions for Integral Boundary Value Problem of Fractional Differential Systems

Shiying Song¹and Yujun Cui²

Guest Editor: Thabet Abdeljawad

Received24 Dec 2019

Accepted07 Feb 2020

Published12 Mar 2020

Abstract

This paper deals with the existence and multiplicity of solutions for the integral boundary value problem of fractional differential systems: , where is continuous and is the standard Riemann–Liouville’s fractional derivative of order α. Our result is based on an extension of the Krasnosel’skiĭ’s fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in 2019. The main results are explained by the help of an example in the end of the article.

1. Introduction

With the deepening of people's understanding of mathematics, the knowledge of mathematics is more and more closely related to the way of production and life of human beings. In recent years, fractional calculus is very active in the field of applied mathematics. It can be applied not only in biochemistry, mathematical physics equation, physical science experiment, and other academic fields but also in precision production [1–3].

In many recent papers are researched the fractional differential equations with the existence of the solutions [4–43]. For example, Zhang and Zhong [38] used the fixed-point theorem on cones to find the existence result of at least two positive solutions which are considered the nonlinear fractional differential equations of nonlocal boundary value problems as follows:where with

In [22], the authors obtained the uniqueness results of positive solution by the contraction map principle and obtained some existence results of positive solution through the fixed-point index theory, which is as follows:where , , is the standard Riemann–Liouville differentiation, and the function f is continuous on .

However, in recent years, many scholars have begun to use the fixed-point index to study the existence and multiplicity of operator equation and operator systems [44–51]. For example, in [46], the authors use the fixed-point index in cones to study the existence, localization, and multiplicity of positive solutions to operator systems of the following form:where for each is invertible, , and . It should be noted that each component of the fixed-point operator systems may have the same or different behaviors.

In particular, only few papers studied the existence and multiplicity of solutions to specific differential systems [52–58]. Therefore, in this paper, we will apply an extension of the Krasnosel’skiǐ’s fixed-point theorem to investigate the existence and multiplicity of solutions for a class of fractional differential systems. More precisely, the following fractional differential systems are studied:where , , and are nondecreasing on , left continuous at .

2. Preliminaries

In this part, we first give the basic definitions, lemmas, and theorems related to fractional calculus.

Definition 1 (see [2]). Define the Riemann–Liouville fractional derivative of order for function σ aswhere is the Euler gamma function.

Definition 2 (see [2]). Let σ define the Riemann–Liouville fractional integral of order for σ aswhere is the Euler gamma function.

Lemma 1 (see [2]). Let and ; then,where , .

For convenience, we first consider the following linear fractional differential equation:where , , , , and is nondecreasing on , left continuous at .

Lemma 2. Let and ; then, boundary value problem (8) has an unique solution where

Proof. It follows from Lemma 1 that . With consideration of the boundary value conditions , we can get . Consequently,Notice that ; we getSince and , we conclude that . Therefore, (10) reduces toTaking into account that and , we haveThis yieldsTaking the above equality into (12), we havewhere and are given in Lemma 2.

The following proposition of Green’s function will be used throughout the paper.

Lemma 3. The function and is nondecreasing on .

Proof. After direct computation, we will getThen, we conclude that is a nondecreasing function on , which implies that This proves the content of lemma.

Lemma 4. The function has the following properties:(i)(ii)(iii)

Proof. According to Lemma 2, we have learned that Green’s function is divided into two cases, and next, we will prove three properties of , respectively.(i)When then by a direct calculation, it is easy to get When (ii)Based on the property (i), it follows that is increasing with respect to t. Obviously, we have(iii)For, we discuss two cases.When , , it is easy to get that .
When ,This yields the desired result.

The main proof of this research uses the following theorem in [46].

Theorem 1 (see [46]). Let be a Banat space, two cones, and the corresponding cone of . Let with , . Assume that , is a compact map (where for ) and there exist , such that for each , the following condition is satisfied in :(i) for and (ii) for and Then,(1)T has at least one fixed point in K such that for if for (2)T has at least two fixed points located in and if and (3)T has at least two fixed points located in and if and (4)T has at least four fixed points located in if for

3. Main Results

Let , , , and . Then, X becomes a real Banach space with the norm and are cones on X. Also the product space is a Banach space endowed with norm and is a cone in .

For convenience, we present some basic conditions as follows which we will be used later: (H1) . (H2) There exist and such that for for for for where (H3) , .

Employing Lemma 2 and the condition of (H1), system (4) has the following integral representation:

Let us define two operators as follows:

Then, we can define an operator as follows:

Lemma 5. Assume that (H1), (H2), and (H3) hold. Then, is completely continuous.

Proof. Firstly, we prove that . In fact, for , by (H1), it is obvious that for and . In addition, if , thenSo,On the other hand, for any and any , it follows from Lemma 4 thatThus, from the above discussion, we conclude that , and then, it obviously shows that T is well defined. The complete continuity of operator T can be given by a standard argument with the help of the Arzela–Arscoli Theorem. We omit the details.

Theorem 2. Assume that (H1), (H2), and (H3) hold. Then, we have(i)If and , then (23) has at least a positive solution located in , where (ii)If and , then (23) has at least two positive solutions located in and (iii)If and , then (23) has at least two positive solutions located in and (iv)If and , then (23) has at least three positive solutions located in

Proof. It follows from Lemma 5 that the existence of a positive solution of problem (23) is equivalent to the existence of a nontrivial fixed point of T in . Let .
First, note that if with , then and by the definition of ,In the following, we conclude that for , the following properties hold:guaranteeing the validity of Theorem 1.
In fact, if and for a , then by (H2),whence, in particular, we conclude , a contradiction. Now, if for and , then by (H2), we obtainfor all . This yields the contradiction . Hence, (30) holds for . Similarly, (30) is true for .

Example 1. Consider the following integral boundary value problem of fractional differential systems:Then, (33) has at least two positive solutions with and .
To see this, we will apply Theorem 2 withClearly,Thus, (H3) holds.
TakeLet . By simple computation, we haveWe choose , , , and . Then, , ,Consequently, holds with and , and our conclusion follows from Theorem 2.

4. Conclusions

In this paper, we investigate the existence and multiplicity of positive solutions for the integral boundary value problem of higher-order fractional differential systems. This result is based on an extension of the Krasnosel’skiǐ’s fixed-point theorem due to Radu Precup and Jorge Rodriguez-Lopez in [46]. We rewrite the original fractional differential systems as equivalent fractional integral systems. With the help of properties of Green’s function, we obtain some sufficient conditions of existence and multiplicity of positive solutions. Finally, an example is presented to illustrate the effectiveness of the main result. The interesting point is that the integral boundary condition is dependent on the lower-order fractional derivative.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This project was supported by the National Natural Science Foundation of China (11571207 and 51774197), Shandong Natural Science Foundation (ZR2018MA011), SDUST Graduate Innovation Project (SDKDYC190238), and the Tai’shan Scholar Engineering Construction Fund of Shandong Province of China.

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Copyright © 2020 Shiying Song and Yujun Cui. 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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