Multiple Solutions to Fractional Difference Boundary Value Problems

and Applied Analysis 3 need to suppose thatf : []−1, ]+b]N]−1×[0, +∞) → [0, +∞) is continuous and f is not identically zero. Denote f 0 = lim inf y→0 min t∈[]−2,]+b]N]−2 f (t, y)


Introduction
Fractional difference equations have been of great interest recently.It is caused by the intensive development of the theory of discrete fractional calculus itself; see [1][2][3][4][5][6][7][8].Diaz and Osler [1] introduced a fractional difference defined as an infinite series, a generalization of the binomial formula for the th order difference   .Gray and Zhang [2] developed a special case for one composition rule and Leibniz formula.They worked exclusively with the nabla operator.A recent interest in discrete fractional calculus has been shown by Atici et al.; see [3][4][5][6][7][8][9][10][11][12].Atici and Eloe developed some of the basic theory of both discrete fractional IVPs and BVPs with delta derivative on the time scale Z.In particular, Atıcı and S ¸engül [5] provided some analysis of discrete fractional variational problems.Their paper also provided some initial attempts at using the discrete fractional calculus to model biological processes.Similarly, Goodrich [7][8][9][10][11][12] has established some results on both discrete fractional IVPs and BVPs.Holm [13] introduced fractional sum and difference operators and presented a complete and precise theory for composing fractional sums and differences.In addition, Wu and Baleanu [14] mainly concentrated on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem of -fractional difference equations.Following this trend, in [15,16], the authors discussed the boundary value problems of fractional difference equations depending on parameters.
In this paper, we consider the following boundary value problems for a fractional difference equation (FBVP): where is an integer, and  ] () is the standard Riemann-Liouville fractional difference.In this paper, we will use properties of Green's function of the FBVP (1) and the Krasnosel'skiǐ fixed point theorem to show that the FBVP (1) has at least one or two positive solutions.Our results significantly improve and generalize the results in [6,8].The plan of this paper is as follows.In Section 2, we will present some necessary lemmas.By using the Krasnosel'skiǐ theorem, Section 3 proves the existence of two positive solutions for the FBVP (1).Section 4 deduces the existence of one solution by using Schauder's fixed point theorem.

Preliminaries
In this section, we first review some basic notations and lemmas about fractional sums and differences in [6][7][8]13].
For any  and ], we define for which the right-hand side is defined.We appeal to the convention that if  + 1 − ] is a pole of the Gamma function and  + 1 is not a pole, then  ] = 0.The ]th fractional sum of a function  is for ] > 0 and  ∈ {+], +]+1, . ..} = N +] .We also define the ]th fractional difference for ] > 0 by  ] () =    −(−]) (), where  ∈ N +−] , and In order to prove our results, we now provide some properties on Green's function associated with the problem (1).
Then the solution of the FBVP (1) is given by where Green's function Lemma 3 (see [6,Theorem 3.2]).The Green function (, ) satisfies the following conditions.
(iii) There exists a number  ∈ (0, 1) such that Denote It is clear that B is a Banach space with the norm ‖‖ Now consider the operator  defined by Referring to Lemma 3.1 of [8], we have the following.
We notice that  is a summation operator on a discrete finite set.Hence,  is trivially completely continuous.And a fixed point of  is equivalent to a solution of the FBVP (1).We will obtain sufficient conditions on the nonlinear  for the existence of fixed points of .In order to prove our results, we need the following well-known Krasnosel'skiǐ fixed point theorem.
Lemma 5 (see [17]).Let B be a Banach space and let K ⊆ B be a cone.Assume that Ω 1 and Ω 2 are bounded open sets contained in B such that 0 ∈ Ω 1 and Then the operator  has at least one fixed point in K∩(Ω 2 \Ω 1 ).

Existence of Positive Solutions
In this section, we state and prove the multiplicity of positive solutions regarding FBVP (1).Then, we conclude this section with two examples to illustrate our main results.For this, we need to suppose that is continuous and  is not identically zero.Denote where  is the constant in Lemma 3. In the sequel, let Ω  = { ∈ K : ‖‖ < }, for  > 0, and Ω  = { ∈ K : ‖‖ = }.
For convenience in what follows, we state these conditions of this section below.(4)  0 < ,  ∞ < .
On the other hand, since  ∞ > , there exist  > 0 and from which we see that ‖‖ > ‖‖ for  ∈ K ∩ Ω  1 .
For any  ∈ Ω  , from (1), we have that is, there is ‖‖ ⩽ ‖‖ for  ∈ K ∩ Ω  .Consequently, Lemma 5 implies that there are two fixed points  1 and  2 of operator  such that 0 < ‖ 1 ‖ <  < ‖ 2 ‖.And this completes the proof.Remark 8.By the proof of Theorem 7, we know that the conclusion of Theorem 7 is valid if (3) is replaced by  0 = +∞ and  ∞ = +∞.Namely, our result in this paper improve Theorem 3.4 in [8].
Remark 10.From the proof of Theorem 9, we know that the conclusion of Theorem 9 is valid if the condition (4) is replaced by  0 = 0 and  ∞ = 0.
From the proof of Theorems 7 and 9, we have the following.

Existence of Solutions
In this section, we give the existence of solutions for problem (1).We will prove this result by using Schauder's fixed point theorem and provide an example to illustrate our results.Then problem (1) has at least one solution.