Existence and Uniqueness Results for a Coupled System of Nonlinear Fractional Differential Equations with Antiperiodic Boundary Conditions

and Applied Analysis 3 let ‖(x, y)‖ X×X = ‖x‖ X + ‖y‖ X . Obviously, (X, ‖ ⋅ ‖ X ) is a Banach space, and the product space (X × X, ‖(⋅, ⋅)‖ X×X ) is also a Banach space. Consider the following coupled system of the integral equations:


Introduction
In this paper, we consider the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations: where 4 < , ≤ 5, and denotes the Caputo fractional derivative of order . Here our nonlinearity , : [0, ] × R × R → R 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][2][3][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][6][7][8][9][10][11][12][13][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][16][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][19][20][21][22]. In [5], Alsaedi et al. study an Antiperiodic boundary value problem of nonlinear fractional differential equations of order ∈ (4,5]. 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.

Background Materials
For the convenience of the readers, we present here some necessary definitions and lemmas which are used throughout this paper.
Lemma 3 (see [19]). Consider ( ) = ( ) + 0 + 1 + Lemma 4 (see [5]). For any ∈ [0, ], the unique solution of the boundary value problem is where ( , ) is the Green function given by Let We call ( 1 , 2 ) Green's function for problem (1). We define the space Abstract and Applied Analysis 3 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. Proof. The proof is immediate from the discussion above, and we omit the details here.
It is obvious that a fixed point of the operator is a solution of problem (1).

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 1 ( , ), 2 ( , ) satisfy the following estimates: Proof. For any ∈ [0, ], On the other hand, Abstract and Applied Analysis 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.

(17)
In addition, it is assumed that Then problem (1) has at least one solution.
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: (H 1 ) there exist two constants > 0 and > 0, = 1, 2 such that Then problem (1) has a unique solution.

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