Existence Results for a Coupled System of Nonlinear Fractional Differential Equations in Banach Spaces

Abstract

We investigate boundary value problems for a coupled system of nonlinear fractional differential equations involving Caputo derivative in Banach spaces. A generalized singular type coupled Gronwall inequality system is given to obtain an important a priori bound. Existence results are obtained by using fixed point theorems and an example is given to illustrate the results.

1. Introduction

Fractional differential equations involving the Riemann-Liouville fractional derivative or the Caputo fractional derivative have been shown to be very useful in the study of models of many phenomena in various fields of science and engineering, such as physics, chemistry, biology, signal and image processing, biophysics, blood flow phenomena, control theory, economics, aerodynamics and fitting of experimental data. For more details, see, for example, [1–6].

In recent years, many researchers paid much attention to the coupled system of fractional differential equations due to its applications in differential fields. The reader is referred to the papers [7–11] and the references cited therein.

Up to now, there are fewer results of fractional differential equations with boundary conditions in infinite dimensional spaces than in finite dimensional spaces. Recently, Wang et al. [12] investigated the existence and uniqueness of solutions for a fractional boundary value problem involving the Caputo derivative in Banach space as follows: which extended the earlier work [13], where is the Caputo fractional derivative of order , , where is a Banach spaces and , and are real constants with .

To the best of our knowledge, there is no effort being made in the literature to study the existence of solutions for a coupled system of fractional boundary value problems involving the Caputo derivative in Banach space. Motivated by the above-mentioned works, in this paper, we study a coupled system of fractional differential equations with boundary conditions of the type where and are the Caputo fractional derivatives of order and , respectively, , where is a Banach spaces and , , , , , and are real constants with and . We will apply Schaefer fixed point theorem, nonlinear alternative of Leray-Schauder type, and a new singular coupled Gronwall inequality system given by us to establish the existence of solutions for BVP (2).

This paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel, and we give a generalized singular type coupled Gronwall inequality system which can be used to obtain an important a priori bound. In Section 3, we give two existence results of the problem (2) which is based on two fixed point theorems, respectively. Finally, an example is given to illustrate the results in Section 4.

2. Preliminaries

For the convenience of the reader, we first briefly recall some definitions of fractional calculus; for more details, see [1, 2, 5], for example.

Definition 1. The Riemann-Liouville fractional integral of order of a function is given by provided that the right side is pointwise defined on , where is the Gamma function.

Definition 2. The Caputo fractional derivative of order of a function can be written as

Definition 3. The Mittag-Leffler function in two parameters is defined as where ,  , and ; denotes the complex plane. In particular, for , one has
The Laplace transform of Mittag-Leffler function is where and are, respectively, the variables in the time domain and Laplace domain; stands for the Laplace transform.

Throughout this paper, let be the Banach space of all continuous functions from into with the norm . Let be the Banach space endowed with the norm as follows:

Now, we give the definition of the solution for problem (2).

Definition 4. A is said to be a solution of a coupled system of fractional BVP (2) if satisfies the system , on and the conditions , .

By Lemma 3.2 in [14], we have the following.

Lemma 5. is a solution of the fractional integral system

Wang et al. in [15] gave a generalized Gronwall inequality as follows.

Lemma 6. Let satisfy the following inequality: where , , are constants, and . Then there exists a constant such that

By using the above generalized Gronwall inequality, we now give the following generalized singular type coupled Gronwall inequality system.

Lemma 7. Let satisfy the following inequality system: where , , , , and () are constants. Then there exists a constant such that

Proof. Let By (12), we have where , , , , and . It is easy to know that Adding (15) to (16), we get by Cauchy inequality and (17) that () From Lemma 6, we obtain that there exists such that . Thus,

Theorem 8 (Schaefer’s fixed point theorem [16]). Let completely continuous operator. If the set is bounded, then has fixed points.

Theorem 9 (Nonlinear alternative of Leray-Schauder type [17]). Let be a Banach space, a closed, convex subset of , an open subset of , and . Assume that is a continuous and compact map. Then either(i)has fixed points or(ii)there exists and with .

3. Main Results

In order to obtain main result, we make the following assumptions.(H1)The functions are continuous.(H2)There exist constants such that (H3)For each , the sets  are relatively compact.

Define the operator as follows: where

It is easy to know that the existence of solution of the coupled fractional BVP (2) is equivalent to the operator having a fixed point on .

Theorem 10. Suppose that (H1)–(H3) hold. Then the coupled fractional BVP (2) has at least one solution on .

Proof. We will use Schaefer’s fixed point theorem to prove that has a fixed point. The proof is divided into several steps.
Firstly, is continuous. Let be a sequence such that in . For each , we have Similarly, we obtain Since , are continuous ((H1)), we have by (25) and (26) that Thus, we get
Secondly, we will prove that maps bounded sets into bounded sets in .
By (H2), for any , we have for each and that which implies that . Similarly, we can obtain that , where Thus, we get
Thirdly, maps bounded sets into equicontinuous sets of . Let , . By (H2), we have Similarly, we obtain Hence, is equicontinuous.
Let , , be a sequence on , and where
According to the condition (H3) and Mazur Lemma [18], we know that is compact. For any , where Since is convex and compact, we have that . Thus, for any , the set is relatively compact. From Ascoli-Arzela theorem [17], every contains a uniformly convergent subsequence , , on . Hence, the set is relatively compact. Similarly, one can obtain that contains a uniformly convergent subsequence , . Thus, the set is relatively compact. Similar to the above process, we can get that the set is relatively compact. Thus, the set is relatively compact.
From the above three steps, we can conclude that is continuous and completely compact.
Finally, we will show that the set is bounded.
Let ; then for some . Hence, for any , we obtain For each , we obtain By Lemma 7, there exists a such that Hence for any , we obtain which implies that the set is bounded. From Theorem 8 (Schaefer’s fixed point theorem), we have that has a fixed point which is a solution of the fractional BVP (2).

Next, we give the second result of this paper, which applies Theorem 9. We firstly introduce the following assumption.(H4)There exist functions and nondecreasing functions () such that

For convenience, let

Theorem 11. Let (H1), (H3), and (H4) hold. Assume that there exists , with where Then problem (2) has at least one solution.

Proof. Firstly, we prove that maps sets into bounded sets in . Let be a bounded subset of . For each , and with (), we have which implies that where is as in (45). Similarly, we have where is as in (46). Combining (47), (51), and (52), we obtain where and are as in (49). This implies that is bounded in .
Secondly, we claim that is continuous and completely continuous. The proof of this claim is the same as the corresponding part in the proof of Theorem 10 by the conditions (H1), (H3), and (H4).
Finally, let for some . Then for any , we have by (53) that By (48), we know that there exists such that . Let From the choice of , there is no such that for some . Therefore, Theorem 9 guarantees that has a fixed point which is a solution of (2). This completes the proof.

4. An Example

In this section, we give an example to illustrate the main results.

Example 1. Consider the following fractional boundary value problem:
Set For each and , we have On the other hand, we easily see that Thus the sets are bounded and closed which implies that and are compact. Hence, all the assumptions in Theorem 10 are satisfied. By Theorem 10, the fractional boundary value problem (56) has at least one solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the editor and reviewer for their valuable comments. This work is supported by the Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212).