The Existence Results of Solutions for System of Fractional Differential Equations with Integral Boundary Conditions

Abstract

In this paper, we investigated the system of fractional differential equations with integral boundary conditions. By using a fixed point theorem in the Banach spaces, we get the existence of solutions for the fractional differential system. By constructing iterative sequences for any given initial point in space, we can approximate this solution. As an application, an example is presented to illustrate our main results.

1. Introduction

The fractional differential equations attracted many people’s attention, and they have many applications in different fields of science and biology. In recent years, the research on the qualitative properties of fractional differential equations has become a hot topic, and many results have been obtained; see [1–29]. From the literature, there are many articles investing the existence of solutions or the solutions for some nonlinear systems; see [4–10, 21].

In [6], Su studied the following system of nonlinear fractional differential equations:where are given functions and are Riemann-Liouville fractional derivatives of order . The author gave sufficient conditions for the existence of solutions of system (1).

In [7], Wang et al. studied the existence of solutions for the following system of nonlinear fractional differential equations:where are given functions and are Riemann-Liouville fractional derivatives of order .

In [8], Yang studied the existence of solutions for the following system of nonlinear fractional differential equations:where are nonnegative and , and are also the standard Riemann-Liouville fractional derivatives of order .

In [9], Alsulami et al. studied the existence of solutions for the following system of nonlinear Caputo fractional integrodifferential equations:subject to the nonseparated coupled boundary conditions:where are Caputo fractional derivatives of order are functions, and are real constants with .

In [10], Yang et al. studied the existence of solutions for the following system of fractional differential equations with integral boundary conditions:where are nonnegative and , and are also the standard Riemann-Liouville fractional derivatives of order .

In this paper, we investigate the following system of fractional differential equations:where , and are also the standard Riemann-Liouville fractional derivatives of order . The system of (7) satisfies the following conditions: .  with and , whereWe know

2. Preliminaries and Basic Lemmas

In this section, we will present some definitions and lemmas, which will be used throughout this paper.

From [1], the definitions of Riemann-Liouville fractional integral and derivative are given as follows:where ,   and is the gamma function.where and

Lemma 1 (see [2]). If ,  , then, for any , the unique solution of the boundary value problem,is given bywhereThen is Green’s function of (7).

Lemma 2 (see [2]). If , then is continuous for all for all , where is given in (13).

Lemma 3 (see [3]). The function has the following properties:where and   is given in (14).

Lemma 4. Assume condition (2°) holds; then satisfies the following property:where is given in (13) and are defined in (8).

Proof. From (13)–(15), we haveFrom [10], we have

Throughout this paper, we list the following definitions and marks for our convenience. For details, see [11–14].

Let be a real Banach space, for any .(1)If , then cone is called partially ordered, i.e., .(2) is the zero element of .(3)There is a constant ; if is called normal, then when .(4)Let the mapping , if is an increasing operator, then when .(5)The notation is an equivalence relation. ; i.e., there exist and such that .(6)Set , where .(7) is the class of function , which satisfies the condition for .

Lemma 5 (see [14]). Let be a normal cone in a real Banach space ,   The operator satisfies the following conditions:
(i) is an increasing
(ii) There is such that
(iii) For any and , there exists such that
Then,
(1) has a unique fixed point
(2) for any initial value , constructing successively the sequence ,  , we have as .

3. Main Results

Set , the norm . For , we define the norm .

We define ,  .

We give the definition of the partially ordered in the space ; let , then .

We also define , where .

Lemma 6. Define ; there exist such that ,And then , where .

Proof. On one hand, for any , there exists such that . Then,From the definition of the partial order, we can get . So
That is to say,
Hence, .
On the other hand, for any , we haveThere exist , such thatLet .
From the definition of the partial order, we haveThat is, , and thus .
Hence holds.

Lemma 7. Assume conditions (1°) and (2°) hold; then is a solution of system (7) if and only ifFrom Lemma 1, we can get that (25) holds.

Theorem 8. Assume condition (2°) and the following conditions hold:
(1) are continuous and
(2) are increasing functions in , for any
(3) There exists such thatfor any and
Then system (7) has a unique positive solution , where .

Proof. For , we define operators , and byIf then , that is,To begin with, we prove that condition (i) of Lemma 5 holds. For any with , according to the definition of the partially order, we have .
From Lemma 6 and condition (2), we haveSo, we can get the following result:Hence, is an increasing operator.
Then, we prove that condition (ii) of Lemma 5 holds. Let
According to Lemma 7, we only need to prove
Applying (17) and Theorem 3.1 of [10], we haveBy computation we haveFrom [10], we know Then we haveLetSince ,
we can obtain
Hence, .
That is,
Therefore,
Last, we prove that condition (iii) of Lemma 5 holds. For any and , by condition (3), we can getThus,Let . Hence,
We have proved that the operator satisfies all the conditions of Lemma 5. Hence has a unique fixed point
Moreover, we give the sequences, whose limit is the solution of system (7).
For any , we can construct the sequences as follows:The limit of exists as , respectively.
From Lemma (16), we haveHence, is a positive solution of system (7).

4. An Example

Example 1. Consider the system of fractional differential equations:where are continuous with , then system (39) has a unique positive solution.

Proof. Compared with (7), we know andAfter calculation, we getWe find . The functions are continuous and In addition, are increasing in .
Let , then
For any , we haveSo, system (39) satisfies all conditions of Theorem 8.
Hence, (39) has a unique positive solution ,
where , and we can get this positive solution by the following way.
For any , we can construct the sequences as follows:whereThen, the limit of is the solution of system (39).
That is,From Theorem 8, system (39) has a unique positive solution.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by National Science Foundation of China (11671227).