Existence of Positive Solutions for Two-Point Boundary Value Problems of Nonlinear Finite Discrete Fractional Differential Equations and Its Application

Abstract

This paper is concerned with the two-point boundary value problems of nonlinear finite discrete fractional differential equations. On one hand, we discuss some new properties of the Green function. On the other hand, by using the main properties of Green function and the Krasnoselskii fixed point theorem on cones, some sufficient conditions for the existence of at least one or two positive solutions for the boundary value problem are established.

1. Introduction

Since 1974, the development of fractional differential equations is driven by an extremely wide application background. In those thirty years, a lot of papers and monographs have been produced and several international conferences were held about fractional calculus and fractional differential equation theory and application. At the same time, due to its application in physics, it also caused the fierce debate to classical laws of physics (see [1–11] and the references therein). However, it was basically confined to the fractional order differential equations and there is less literature available on paper concerned with discrete fractional differential equations.

In this paper, we will be interested in the nonlinear finite discrete FBVP given by where , , is continuous, . In Section 2, we will deduce some new properties of the Green function. In Section 3, by means of using the properties of the Green function and Krasnoselskii fixed point theorem on cones, the existence of two positive solutions and one positive solution of problem (1) is established. At last, an example is given to illustrate the main results of this paper.

The difference with other literatures is that we get some new results about Green’s function firstly, and then with the help of the nature of Green’s function we introduce a new function, thus establishing the new operator equation. By discussing the fixed point of the new operator equation, we gain some existence results of solution in (1).

2. Preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory.

Definition 1. The th fractional sum of a function defined on , for , is defined to be where . We also define the th fractional difference, where and with to be where .

Definition 2. We define for any and for which the right-hand side is defined. One appeals to the convention that if is a pole of the Gamma function and is not a pole, then .

Lemma 3. Let and be any numbers for which and are defined. Then,

Lemma 4. Let be a real-valued function, and let . Then, for all such that , we have where .

Lemma 5. Let . Then, for some with .

Definition 6. Let be a real Banach space. A nonempty convex closed set is called a cone provided that (1) , for all ; and (2) implies .

Lemma 7. Let be a Banach space and let be a cone. Assume that and are open sets contained in such that and . Assume, further, that is a completely continuous operator. If either   , and , or   , and , then has at least one fixed point in .

Moreover, there is one paper [12] in which the following two statements have been shown.

Lemma 8 (see [13]). The unique solution of the FBVP (1) is given by where is Green’s function for the problem where , which is given by

Lemma 9 (see [13]). Let be given as in the statement of Lemma 8. Then, we find that(1) for each ;(2), ;(3)there exists a number such that for .

Lemma 10. For given , is decreasing and is increasing with respect to for , where .

Proof. For , we find that Similar to the proof of [14], we have So, it shows that Then, is decreasing with respect to for .
This implies that Similar to the proof of [14], we have So, since , we have Then, is increasing with respect to for .

Lemma 11. For given , is decreasing with respect to for .

Proof. By Lemma 10, for any given , , we have Becausewe getNamely, for given , is decreasing with respect to for .

Theorem 12. The Green function defined by Lemma 8 has the following: and for each ,

Proof. By the definition of , it was obvious that for . By Lemma 11, for , we get For , we get soHence, for each ,

Theorem 13. The function has the following properties:

Proof. By Theorem 12, we get Let . It is clear that It is worth noting that two inequalities of Theorem 13 are very useful in the research of positive solutions to Dirichlet-type boundary value problems of discrete fractional differential equation (see the proofs of Theorems 15 and 17). Theorems 12 and 13 extend the results of integer-order Dirichlet boundary value problems.

3. Main Results and Proofs

In this section, we will give the existence results of positive solution to the boundary value problem (1) on the basis of Theorem 13 and Lemma 7 and make the following assumptions: () There exist , such that here For convenience, we introduce the following notations:  () , . () , . () There exists such that for , where . () There exists such that for , where .

We note that is a solution of the boundary value problem (1) if and only if is a fixed point of the operator where is Green’s function derived in this paper and and is the Banach space equipped with the norm

Define the cone by

Lemma 14. Assume is the operator defined by Then, the mapping is completely continuous.

Proof (). Hence, . On one hand, On the other hand, Then, we get And by means of the expression of , there are clearly whenever , whence .
Next, we show that is uniformly bounded.
Let be bounded; that is, there exists a positive real number such that for all . Let ; then for , by Theorem 13 and (), we have Thus, is bounded.
Finally, we prove that is equicontinuous.
Note that is uniformly continuous on ; then, for , each , , , there exists such that for . Then, we get Hence, we get In view of the Arzela-Ascoli theorem; it is easy to see that is completely continuous.

Theorem 15. Suppose that (), (), and () are satisfied. Then, problem (1) has two positive solutions.

Proof. Assume that () holds. Since , there exists such that for , where satisfies Let for ; we note that , where is the usual so-called floor function. So we getThen, this implies for .
Now, let for . It follows from () that we get Then, it implies that for .
Therefore, by Lemma 7, it follows that has a fixed point in , such that .
Further, since , there exists such that for , where is chosen so that .
Let and , for ; we get and so Hence, for ,
Thus, by Lemma 7, it follows that has a fixed point in such that .
Then, has a fixed point in and a fixed point in such that , and satisfy and . It is clear that are two solutions of (1); that is, Next, we will prove By means of (), we get Therefore , , and is a solution of problem (1).

Theorem 16. Suppose that (), (), and () are satisfied. Then, problem (1) has two positive solutions.

Proof. Assume that () holds. Since , there exists such that for , where satisfy Let for ; we get Then, this implies that for .
Now, let for ; then, , ; by (), we get Then, we see that for .
As a result, by Lemma 7, we obtain that has a fixed point in , such that .
Furthermore, because , there exists such that for , where is chosen so that .
Let + and , for ; we get Therefore, for we get .
So it suffices to obtain that has a fixed point in such that in view of Lemma 7.
Then, has fixed points in and in such that .
Therefore, problem (1) gets two positive solutions and .

Theorem 17. Suppose that () is satisfied. Furthermore, eitherorThen, problem (1) has one positive solution.

The proof of Theorem 17 is similar to the proof of Theorems 15 and 16.

4. Example

Consider the following boundary value problem:

Proof. Let ; it is clear that we have .
Let , and ; it follows that we get and , . Hence, this implies () and () in Theorem 15 hold.
Because .
Let , where . So, for , we get , Therefore, () holds. According to Theorem 15, problem (57) has two positive solutions.

Competing Interests

The authors declare that they have no competing interests.

Authors’ Contributions

All four authors read and approved the final paper.

Acknowledgments

The research was supported by a grant from of the National Natural Science Foundation of China (no. 11271235) and the Foundation of Datong University (2010-B-01, 2014Q10, and XJY-2012211).