Abstract

This paper studies the existence and uniqueness of solutions for a coupled system of nonlinear fractional differential equations of order with antiperiodic boundary conditions. Our results are based on the nonlinear alternative of Leray-Schauder type and the contraction mapping principle. Two illustrative examples are also presented.

1. Introduction

In this paper, we consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: where , , and denotes the Caputo fractional derivative of order . Here our nonlinearity are given continuous functions.

Fractional differential equations have recently been addressed by many researchers in various fields of science and engineering, such as rheology, porous media, fluid flows, chemical physics, and many other branches of science; see [1–4]. As a matter of fact, fractional-order models become more realistic and practical than the classical integer-order models; as a consequence, there are a large number of papers and books dealing with the existence and uniqueness of solutions to nonlinear fractional differential equations; see [5–14]. The study of a coupled system of fractional order is also very significant because this kind of system can often occur in applications; see [15–17].

Antiperiodic boundary value problems arise in the mathematical modeling of a variety of physical process; many authors have paid much attention to such problems; for examples and details of Antiperiodic boundary conditions, see [5, 18–22]. In [5], Alsaedi et al. study an Antiperiodic boundary value problem of nonlinear fractional differential equations of order .

It should be noted that, in [23], Ntouyas and Obaid have researched a coupled system of fractional differential equations with nonlocal integral boundary conditions, but this paper researches a coupled system of fractional differential equations with Antiperiodic boundary conditions. On the other hand, in [5, 19], the authors have discussed some existence results of solutions for Antiperiodic boundary value problems of fractional differential equation but not the coupled system. The rest of the papers above for the coupled systems have been devoted to the case of Riemann-Liouville fractional derivatives but not the Caputo fractional derivatives.

This paper is organized as follows. In Section 2, we recall some basic definitions and preliminary results. In Section 3, we give the existence results of (1) by means of the Leray-Schauder alternative; then we obtain the uniqueness of solutions for system (1) by the contraction mapping principle. We give two examples in Section 4 to illustrate the applicability of our results.

2. Background Materials

For the convenience of the readers, we present here some necessary definitions and lemmas which are used throughout this paper.

Definition 1 (see [2, 3]). The Riemann-Liouville fractional integral of order of a function is given by provided the right hand side is pointwise defined on .

Definition 2 (see [2, 3]). The Caputo fractional derivative of order of a continuous function is given by where and denotes the integer part of number , provided that the right side is pointwise defined on .

Lemma 3 (see [19]). Consider , for some , , where and denotes the integer part of number .

Lemma 4 (see [5]). For any , the unique solution of the boundary value problem is where is the Green function given by

Let

We call Green’s function for problem (1).

We define the space endowed with ; for , let . Obviously, is a Banach space, and the product space is also a Banach space.

Consider the following coupled system of the integral equations: As a result, differential problem (1) turns into integral problem (8), and here is a conclusion about the relationship between their solutions.

Lemma 5. Assume that are continuous functions. Then is a solution of (1) if and only if is a solution of system (8).

Proof. The proof is immediate from the discussion above, and we omit the details here.

Let be an operator defined as , where It is obvious that a fixed point of the operator is a solution of problem (1).

3. Main Results

In this section, we will discuss the existence and uniqueness of solutions for problem (1).

Lemma 6. One can conclude that the Green functions , satisfy the following estimates:

Proof. For any , On the other hand, Inequalities (11) and (13) can be proved in the same way.

The first result is based on Leray-Schauder alternative.

Lemma 7 (see [24]). Let be a completely continuous operator (i.e., a map that is restricted to any bounded set in is compact). Let Then either the set is unbounded or has at least one fixed point.

Theorem 8. Let and satisfy the following growth condition: In addition, it is assumed that Then problem (1) has at least one solution.

For sake of convenience, we set .

Proof. First, we show that operator is completely continuous.
Step 1. In view of the continuity of , , , and , , it is obvious that the operator is continuous.
Step 2. Let be bounded; then there exist positive constants and such that Then for any , according to the inequalities (10) and (11), we have Similarly, we get Thus, it follows from the above inequalities that the operator is uniformly bounded.
Step 3. We show that is equicontinuous: Hence, for , by the inequalities (12) and (13), we have Analogously, we can obtain Therefore, the operator is equicontinuous, and thus the operator is completely continuous.
Finally, it will be verified that the set is bounded.
Let , then . For any , we have , ; then, Hence, we have which imply that As a result, for any , which proves that is bounded; thus by Lemma 7, the operator has at least one fixed point. Hence the boundary value problem (1) has at least one solution; this completes the proof.

In the second result, we prove uniqueness of solutions of the boundary value problem (1) via contraction mapping principle.

Theorem 9. Let and satisfy the following growth conditions: (H1)there exist two constants and , such that (H2)Consider Then problem (1) has a unique solution.

Proof. Let ; then Analogously,
Thus,
Hence, we conclude that problem (1) has a unique solution by (H2) and the contraction mapping principle; this ends the proof.

4. Examples

In this section, two examples are given in order to verify the validity of Theorems 8 and 9.

Example 1. Consider the system where , , , , , , . Consider Then and ; by Theorem 8, the existence of the solution for the system (34) is obvious.

Example 2. Consider the system where , , . , .
Noting that we have
Obviously, then we can conclude from Theorem 9 that system (36) has a unique solution.

Conflict of Interests

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

Acknowledgments

The authors express their thanks to the referee for his/her valuable suggestions. The second author was supported by NSF of China (11271154) and the first author was supported by the Fundamental Research Funds for the Central Universities (13cx02015A).