Existence results for a coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain

This paper deals with the existence results for solutions of coupled system of nonlinear fractional differential equations with boundary value problems on an unbounded domain. Also, we give an illustrative example in order to indicate the validity of our assumptions.


Introduction
Recently, fractional differential calculus has attracted a lot of attention by many researchers of different fields, such as: physics, chemistry, biology, economics, control theory and biophysics, etc. [11,15,16].In particular, study of coupled systems involving fractional differential equations is also important in several problems.
Many authors have investigated sufficient conditions for the existence of solutions for the following coupled systems of nonlinear fractional differential equations with different boundary conditions on finite domain.
In [13], Liang and Zhang investigated the existence of three positive solutions for the following m-point fractional boundary value problem where Wang et al. [20] by using Schauder's fixed point theorem investigated the existence and uniqueness of solutions for the following coupled system of nonlinear fractional differential equations on an unbounded domain where and D q denote Riemann-Liouville fractional derivatives of order p and q, respectively, as well as < Γ(q).Our aim in this paper is to generalize the above works on an infinite interval and more general boundary conditions, so we discuss the existence of the solutions of a coupled system of nonlinear fractional differential equations on an unbounded domain EJQTDE, 2013 No. 73, p. 2 ) and D is the Riemann-Liouville fractional derivative.This paper is organized as follows: in Section 2, some facts and results about fractional calculus are given, while inspired by [19] we prove the main result and some corollaries in Section 3, and we conclude this paper by considering an example in Section 4.

Preliminaries
In this section, we present some definitions and results which will be needed later.
Definition 2.1.[11] The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, ∞) → R is defined by provided that the right-hand side is pointwise defined.Definition 2.2.[11] The Riemann-Liouville fractional derivative of order α > 0 of a continuous function f : (0, ∞) → R is defined by where n = [α] + 1, provided that the right-hand side is pointwise defined.In particular, for α = n, Remark 1.The following properties are well known: The following two lemmas can be found in [5,11].
1+t α−1 and D α−1 u(t) are equicontinuous on any compact interval of J. (ii) Given > 0, there exists a constant T = T ( ) > 0 such that for any t 1 , t 2 ≥ T and u(t) ∈ Z.

Main result
In this section, we investigate sufficient conditions for the existence and uniqueness solutions for the boundary value problem (6).Before we state our main result, for the convenience, we introduce the following notations: has a unique solution Proof.We apply Lemma (2.2) to convert the boundary value problem (7) into the integral equation Since u(0) = 0, so c 2 = 0 and Now using the second boundary condition we obtain c 1 .Since EJQTDE, 2013 No. 73, p. 5 Therefore and and the proof is completed.
Define the operator T : where Proof.Take

Note that by conditions (H
First, we prove that T : B R → B R .In view of together with the definition of Av(t) and continuity of f we have Av(t), D α−1 Av(t) and similarly Bu(t), D β−1 Bu(t) are continuous on J.
For any (u, v) ∈ B R , we have In a similar way, we can get To do it note that, Similarly, we can get Similar process can be repeated for B and then Lebesgue's dominated convergence theorem asserts that T is continuous.Now we show that T maps bounded sets of X × Y to relatively compact sets of X × Y .It suffices to prove that both A and B map bounded sets to relatively compact sets.Now, for a bounded subset V of Y and U of X, by Lemma (2.3), we show that AV , BU are relatively compact.Let I ⊆ J be a compact interval, t 1 , t 2 ∈ I and t 1 < t 2 ; then for any v(t) ∈ V , we have ) is bounded on I. Then it is easy to see that Av(t) 1+t α−1 and D α−1 Av(t) are equicontinuous on I.
Next we show that for any v(t) ∈ V , functions Av(t) 1+t α−1 and D α−1 Av(t) satisfy the condition (ii) of Lemma (2.3).Based on condition (H 1 ) we know that for given > 0, there exists a constant L > 0 such that On the other hand, since lim t→∞ Similarly, lim t→∞ 1+t α−1 = 1 and thus there exists a constant T 2 > L > 0 such that for any t 1 , t 2 ≥ T 2 and 0 ≤ s ≤ L EJQTDE, 2013 No. 73, p. 10 Now choose T > max{T 1 , T 2 }; then for t 1 , t 2 ≥ T , we can obtain , EJQTDE, 2013 No. 73, p. 11 and Similar process can be repeated for B, thus T is relatively compact.Therefore, by Schauder's fixed point theorem the boundary value problem ( 6) has at least one solution.
Similarly, we can show that for the second equation condition (H 4 ) holds.Thus all the conditions of theorem (3.1) are satisfied and the problem ( 12) has at least one solution.