Multiple Solutions to Fractional Difference Boundary Value Problems

Abstract

The following fractional difference boundary value problems , are considered, where is a real number and is the standard Riemann-Liouville fractional difference. Based on the Krasnosel’skiǐ theorem and the Schauder fixed point theorem, we establish some conditions on which are able to guarantee that this FBVP has at least two positive solutions and one solution, respectively. Our results significantly improve and generalize those in the literature. A number of examples are given to illustrate our main results.

1. 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 [1–8]. 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 [3–12]. Atici and Eloe developed some of the basic theory of both discrete fractional IVPs and BVPs with delta derivative on the time scale . In particular, Atıcı and Şengül [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 [7–12] 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 continuous and is not identically zero, , 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.

2. Preliminaries

In this section, we first review some basic notations and lemmas about fractional sums and differences in [6–8, 13].

For any and , we define for which the right-hand side is defined. We appeal to the convention that if is a pole of the Gamma function and is not a pole, then .

The th fractional sum of a function is for and . We also define the th fractional difference for by , where , and is chosen such that .

Let . Then

In order to prove our results, we now provide some properties on Green’s function associated with the problem (1).

Lemma 1 (see [6, Theorem 3.1]). Let and be given. Then the solution of the FBVP (1) is given by where Green’s function is defined by

Remark 2. It is easy to see that could be extended to ; we only discuss .

Lemma 3 (see [6, Theorem 3.2]). The Green function satisfies the following conditions.(i) . (ii), .(iii)There exists a number such that

Denote It is clear that is a Banach space with the norm . We choose a cone

Now consider the operator defined by

Referring to Lemma 3.1 of [8], we have the following.

Lemma 4.

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 be a Banach space and let be a cone. Assume that and are bounded open sets contained in such that and . Assume is a completely continuous operator. If either(i) for and for ; or(ii) for and for .Then the operator has at least one fixed point in .

3. 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 , for and . For convenience in what follows, we state these conditions of this section below.There is a such that for and .There is a such that for and ., ., .

Lemma 6 (see [8]). Suppose that there exist two different positive numbers and such that satisfies condition at and condition at . Then FBVP (1) has at least one positive solution satisfying .

Theorem 7. Assume that satisfies conditions and . Then FBVP (1) has at least two positive solutions and with .

Proof. Suppose that holds. Since , there exist and such that ,, . Let and note that . Thus for , we get that is, there is for .
On the other hand, since , there exist and such that , , . Choose . If , then for . So it follows that from which we see that for .
For any , from , we have , .
Consider that is, there is for .
Consequently, Lemma 5 implies that there are two fixed points and of operator such that . And this completes the proof.

Remark 8. By the proof of Theorem 7, we know that the conclusion of Theorem 7 is valid if is replaced by and . Namely, our result in this paper improve Theorem 3.4 in [8].

Theorem 9. Suppose that conditions and hold, for . Then FBVP (1) has at least two positive solutions and with .

Proof. From the assumption , one can find and such that , . Let ; then for , we have from which we see that for .
On the other hand, since , there exist and such that Denote ; then , . Let . For , we have Therefore, we have for .
Finally, for any , since for , we estimate Hence for .
By Lemma 5, the proof is complete.

Remark 10. From the proof of Theorem 9, we know that the conclusion of Theorem 9 is valid if the condition is replaced by and .

Remark 11. Theorem 9 is not included in [6, 8].
From the proof of Theorems 7 and 9, we have the following.

Theorem 12. Suppose that , . Then FBVP (1) has at least one positive solution.

Theorem 13. Suppose that , . Then FBVP (1) has at least one positive solution.

Remark 14. Theorems 12 and 13 in this paper significantly generalize Theorems 4.1 and 4.2 in [6].

Example 15. Consider the following boundary value problems: where and , and . A simple computation shows that , ( is the constant in Lemma 3(iii)), and . Taking , we get ,. All conditions in Theorem 7 are satisfied. Therefore FBVP (20) has at least two positive solutions and such that .

Example 16. Consider the following boundary value problems: where and , and ( is the constant in Lemma 3(iii)); it is easy to compute that which yields the condition of Theorem 12. By Theorem 12, FBVP (22) has at least one positive solution.

4. 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.

Theorem 17. Let be a continuous function. Suppose that one of the following conditions is satisfied.There exist a nonnegative function and a constant such that , where , ., where , .Then problem (1) has at least one solution.

Proof. First, suppose the condition can be satisfied. Let where Obviously, is a closed ball in the Banach space .
Now we prove that . For any , then Notice that Thus we get .
Second, suppose the condition can be satisfied; we choose With a similar argument as the above, we obtain . Consequently, we get .
Note that is a summation operator on a discrete finite set. Hence, is trivially completely continuous. Therefore, according to the Schauder fixed point theorem, has a fixed point . Namely, is a solution of problem (1). The theorem is proved.

Remark 18. In this section, is only a continuous function, without nonnegative assumptions on function .

Remark 19. If in , we need the condition . Then, choose If in , we only need the condition . Then the conclusion of Theorem 17 remains true.

Example 20. Consider the fractional difference equation where , or , , and , . By using Theorem 17, the existence of solutions is obvious for or .

Conflict of Interests

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

Acknowledgments

The authors are very grateful to the reviewers for their valuable suggestions and useful comments, which led to the improvement of this paper. Project supported by the National Natural Science Foundation of China (Grant no. 11271235) and Shanxi Datong University Institute (2009-Y-15, 2010-B-01, 2013K5) and the Development Foundation of Higher Education Department of Shanxi Province (20111020).