Existence of Solutions for Boundary Value Problem of a Caputo Fractional Difference Equation

The theory of fractional difference equations and their applications have been receiving intensive attention. In the last ten years, new research achievements kept emerging (see [1–16] and the references therein). In [1–3], the authors introduced fractional sum and difference operators, studied their behavior, and developed a complete theory governing their compositions. Abdeljawad [3] considered the initial value problem to Caputo fractional difference equation. The authors explored BVP of fractional difference equation, and they deduced the existence of one or more positive solutions in [4–7]. Abdeljawad and Baleanu [8] introduced the fractional differences and integration by parts. In [9] the authors studied the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference by using Lyapunov’s direct method. They discussed the conditions for uniform stability, uniform asymptotic stability, and uniform global stability. In [10, 11], the authors studied multiple solutions to fractional difference boundary value problems by means of Krasnosel’skii theorem and Schauder fixed point theorem. They obtained sufficient conditions of the existence of two positive solutions for the boundary value problem of fractional difference equations depending on parameters in [12]. Chen et al. [13] presented the existence of at least one positive solution for Caputo fractional boundary value problems. In [14], Kang et al. discussed existence of positive solutions for a system of Caputo fractional difference equations depending on parameters on the basis of [13]. Recently, Wu and Baleanu introduced some applications of the Caputo fractional difference to discrete chaoticmaps in [15, 16]. Thus, the fractional difference equation has recently attracted increasing attention from a growing number of researchers. Following this trend, we investigate the following boundary value problem of fractional difference equation (FBVP):


Introduction
The theory of fractional difference equations and their applications have been receiving intensive attention.In the last ten years, new research achievements kept emerging (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] and the references therein).In [1][2][3], the authors introduced fractional sum and difference operators, studied their behavior, and developed a complete theory governing their compositions.Abdeljawad [3] considered the initial value problem to Caputo fractional difference equation.The authors explored BVP of fractional difference equation, and they deduced the existence of one or more positive solutions in [4][5][6][7].Abdeljawad and Baleanu [8] introduced the fractional differences and integration by parts.In [9] the authors studied the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference by using Lyapunov's direct method.They discussed the conditions for uniform stability, uniform asymptotic stability, and uniform global stability.
In [10,11], the authors studied multiple solutions to fractional difference boundary value problems by means of Krasnosel'skii theorem and Schauder fixed point theorem.They obtained sufficient conditions of the existence of two positive solutions for the boundary value problem of fractional difference equations depending on parameters in [12].
Chen et al. [13] presented the existence of at least one positive solution for Caputo fractional boundary value problems.In [14], Kang et al. discussed existence of positive solutions for a system of Caputo fractional difference equations depending on parameters on the basis of [13].
Recently, Wu and Baleanu introduced some applications of the Caputo fractional difference to discrete chaotic maps in [15,16].Thus, the fractional difference equation has recently attracted increasing attention from a growing number of researchers.
Following this trend, we investigate the following boundary value problem of fractional difference equation (FBVP): where The reason for studying (1) is that, in [13], Chen et al. studied similar FBVP (1) by using the cone fixed point theorem.In this note, we consider existence of solutions to FBVP (1) by using Schauder theorem.Here, the nonnegative function  is not necessary, but  is nonnegative in [13].We extend the domain of .
This paper is organized as follows.Section 1 introduces the developing of fractional difference equations in a simple way.We present some necessary definitions and lemmas in Section 2. In Section 3, we prove existence of solution to FBVP (1).Finally, we provide some examples to illustrate our main results.

Preliminaries
We first introduce some theory about fractional sums and differences.The definitions and some basic results about fractional sums and differences can be seen in [1][2][3][4][5][6][7], so we omit their proof.
Definition 1 (see [3]).For any  and ], one defines for which the right-hand side is defined.One appeals to the convention that if  + 1 − ] is a pole of the Gamma function and  + 1 is not a pole, then  ] = 0.
is given by where Green's function Lemma 6 (see [13]).The Green function  has the following properties: where

Main Results
In this section, we give the main result of this paper.We will prove this result by using Schauder fixed point theorem and provide some examples to illustrate our main results.
For the sake of convenience, we write out the conditions as follows: Proof.First, suppose that condition ( 1 ) is satisfied.Let where Obviously B is the ball in the Banach space.Now we prove that  : B → B. For any  ∈ B, then As Hence,          ⩽ .Namely,  : B → B.
Repeating the course of the above, we obtain |()| ⩽ .Consequently, we get  : B → B. By means of the continuity of  and , it is easy to see that operator  is continuous.Next, we show that  is a completely continuous operator.For this, we take For any (] − 1) ( − ] + 3) (] − 1) (] − 1) Since functions  ] ,  are uniformly continuous on interval [] − 2,  + ]] N ]−2 , we conclude that (B) is an equicontinuous set.Obviously, it is uniformly bounded since (B) ⊂ B. Thus, we know  is completely continuous.
Consequently, it follows at once by Schauder fixed point theorem that  has a fixed point ; namely,  is a solution of (1).The theorem is proved.
Remark 9.In this paper,  is only continuous function, without nonnegative assumptions on function .