Journal of Function Spaces
Volume 2020 (2020), Article ID 7319098, 7 pages
https://doi.org/10.1155/2020/7319098

Research Article

Monotone Iterative Method for Fractional Differential Equations with Integral Boundary Conditions

Shiying Song,¹ Hongyu Li,¹ and Yumei Zou²

¹Department of Applied Mathematics, Shandong University of Science and Technology, Qingdao 266590, China
²Department of Statistics and Finance, Shandong University of Science and Technology, Qingdao 266590, China

Correspondence should be addressed to Yumei Zou; sdzouym@126.com

Received 4 February 2020; Accepted 4 March 2020; Published 30 March 2020

Guest Editor: Chuanjun Chen

Copyright © 2020 Shiying Song et al. 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.

Abstract.

In this paper, the existence of extremal solutions for fractional differential equations with integral boundary conditions is obtained by using the monotone iteration technique and the method of upper and lower solutions. The main equations studied are as follows: where is the standard Riemann–Liouville fractional derivative of order and is a positive measure function. Moreover, an example is given to illustrate the main results.

1. Introduction

The purpose of this paper is to study the following differential equations with integral boundary conditions:

where is the standard Riemann–Liouville fractional derivative and and is a positive measure function, and it satisfies .

Fractional differential equations have been widely used in physics, chemistry, aerodynamics, electrodynamics of complex media, and rheology of polymers [1–7]. As a result, various nonlinear functional analysis methods have been used to study the existence of solutions for differential equations [8–38]. However, it is difficult to obtain the exact solutions of fractional differential equations, so the monotone iteration method and the upper and lower solutions are generally used to obtain the approximate solutions of fractional differential problems. This method is also applicable to both initial value problems and boundary value problems for integer-order differential equations and differential systems [39–43]. In recent years, many scholars have used this method to study various fractional differential equation problems [44–55]. For example, in [47, 48, 52], the authors paid attention to the Riemann–Liouville fractional differential equations of order , and in [44, 45, 49, 55], the authors considered the Riemann–Liouville fractional differential equation of order , while in [46], the authors considered the boundary value problem with Riemann–Liouville fractional order . Of course, some papers also use the monotone iterative method to deal with nonlinear Caputo fractional differential equations [56].

Based on the upper and lower solutions, this paper presents a method to prove the existence of solutions of Riemann–Liouville fractional differential equation (1). By using the monotone iteration technique coupled with the upper and lower solution method, a new comparison principle is established and the existence of the extremal solution of integral boundary value problems (1) is proved.

This paper is mainly divided into the following two parts: Section 2 mainly introduces the preparation of this article, and then in Section 3, the monotone sequence of solutions is constructed, and the main result of integral boundary value problems (1) is given.

2. Preliminaries

In this section, we will briefly introduce some of the necessary definitions and results that will be used in the main results.

Definition 1. (see [2, 5]). The fractional integral of order of a function is given by

provided that the right-hand side is point-wise defined on .

Definition 2. (see [2, 5]). The Riemann–Liouville fractional derivative of order of a function is given by

where , denotes the integer part of number , provided that the right-hand side is point-wise defined on .
Let be endowed with the norm in which , and then is a Banach space.

Definition 3. We say that is an upper solution of (1) if it satisfies

Definition 4. We say that is a lower solution of (1) if it satisfies

Denote

It is easy to check that (see [57, 58])

Therefore, there exists a unique number such that

Set , where is the Mittag-Leffler function (see [2, 5]).
In this article, we list the following assumption for convenience:

 : the parameter satisfies .  is a positive measure function and . : assume that are the upper and lower solutions of problem (1), respectively, and , .  and for .

Next, we will consider the following auxiliary linear boundary value problem:

Lemma 1. Suppose that (H1) and (H2) hold and . Then, fractional boundary value problem (9) has the following unique solution:

where

Proof. The main idea of Lemma 1 comes from [57]. By [2], we first find the solution of the fractional differential equations

with two-point boundary condition

which can be expressed by

Since , we have . Then, by the condition , we calculated that

Therefore, the solution of (12) and (13) is

Next, we consider problem (9). Integrating equality (16) with respect to , we have

Making use of the condition in the above equality yields

and then we get

Obviously,

Therefore,

The proof is completed.

Lemma 2. Suppose that (H1) and (H2) hold, and satisfies

Then, for , .

Proof. Let and . From (12), we have , , and

By Lemma 1, we obtain that problem (23) has unique solution , which can be expressed as follows:

From [57], it follows that for . This together with (H1) and (H2) yields

Hence, we conclude that

which completes the proof.

3. Main Results

For with for , we denote an ordered interval:

Theorem 1. Suppose (H1)–(H4) hold, and then there exist monotone iterative sequences such that as uniformly in , and are a minimal and a maximal solution of (1) in , respectively.

Proof. For , , we define two sequences satisfying the following fractional differential equation:

By consideration of Lemma 1, for any , problems (28) and (29) have a unique solution , respectively, which are well defined.
Firstly, we need to show that, for any , . Let , and the definition of together with yields

In the light of Lemma 2, we have , namely, . Similarly, it can be shown that , .
Secondly, we make . From , we get

Also, and . Thus, Lemma 2 implies that, for any , .
Thirdly, we prove that are upper and lower solutions of problem (1), respectively. Note that

And by assumption , and . This shows that is a lower solution of problem (1). Similarly, we can infer that is an upper solution of (1).
Using mathematical induction, it is easy to verify that

Clearly, it is easy to conclude that and are uniformly bounded in . Moreover, by Lemma 1, problems (28) and (29) are equivalent to the following integral equation:

respectively. Therefore, the continuity of the functions allows us to conclude that and are equicontinuous in . Using (28) and (29) again, we know that and are uniformly bounded and equicontinuous in . So, and are uniformly bounded and equicontinuous in . Using the standard arguments, we have and converging, say, to and , uniformly on , respectively. That is,

Furthermore, and are the solutions of problem (1), and on [0, 1].
Next, we need to prove that are extremal solutions of (1) in . Let be any solution of problem (1). We assume that for some . Take . Then, by assumption , we obtain

By Lemma 2, we have

Applying mathematical induction, one has on for any . Taking the limit, we conclude . The proof is complete.

Example 1. Consider the following problem:

where , , and .
Taking and , we have

which shows that and are a lower and an upper solution of (38), respectively, and . So, (H3) holds.
Using the strictly monotone increasing property of gamma function on , we have

By MATLAB, we obtain . Therefore, . In addition, we have

for . Hence, (H1) and (H4) hold.
Note that . Then,

It shows that (H2) holds. Thus, Theorem 1 ensures that problem (38) has extremal solutions in .

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 (51774197 and 11801322), the Shandong Natural Science Foundation (ZR2018MA011), and SDUST graduate innovation project (SDKDYC190238).

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