Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 459161, 7 pages

http://dx.doi.org/10.1155/2013/459161

Research Article

Existence and Ulam Stability of Solutions for Discrete Fractional Boundary Value Problem

Fulai Chen¹ and Yong Zhou²

¹Department of Mathematics, Xiangnan University, Chenzhou 423000, China

²School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411005, China

Received 28 May 2013; Accepted 9 July 2013

Academic Editor: Shurong Sun

Copyright © 2013 Fulai Chen and Yong Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We discuss the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations. Two examples are also provided to illustrate our main results.

1. Introduction

This paper investigates the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations:

where is a Caputo fractional difference operator, and for any number and each interval of , and is a continuous function with respect to the second variable.

Accompanied with the development of the theory on fractional differential equations, fractional difference equations have also been studied more intensively of late. In particular, some properties and inequalities of the fractional difference calculus are discussed in [1–7], the existence and asymptotic stability of the solutions for fractional difference equations are investigated in [8–10], and the boundary value problems of fractional difference equations are considered in [11–13]. But there are a lot of works to do in the future, and to the best of my knowledge, there is no work on the existence of solutions for antiperiodic boundary value problem and the Ulam stability for nonlinear fractional difference equations.

To research the boundary value problem of fractional difference equations, we need to select a suitable fixed-point theorem because of the discrete property of the difference operator; here, we choose the Banach contraction mapping principle and the Brower fixed-point theorem. Motivated by the work of the Ulam stability for fractional differential equations [14], in this paper, we also introduce four types of the Ulam stability definitions for fractional difference equations and study the Ulam-Hyers stable and the Ulam-Hyers-Rassias stable.

The rest of the paper is organized as follows. In Section 2, we introduce some useful preliminaries. In Section 3, we consider the existence of solutions for antiperiodic boundary value problem of fractional difference equations. In Section 4, we discuss the Ulam stability for fractional difference equations. Finally, two examples are given to illustrate our main results.

2. Preliminaries

In this section, we introduce preliminary facts which are used throughout this paper.

Definition 1 (see [3, 4]). Let . The th fractional sum of is defined by

where .

In (2), the fractional sum maps functions defined on to functions defined on . Atici and Eloe [3] pointed out that this definition is the development of the theory of the fractional calculus on time scales.

Definition 2 (see [1]). Let and , where denotes a positive integer and , ceiling of number. Set . The th fractional Caputo difference operator is defined as

where is the th order forward difference operator; the fractional Caputo like difference maps functions defined on to functions defined on .

Lemma 3 (see [2, 13]). Assume that and is defined on . Then,

where is the smallest integer greater than or equal to , .

Lemma 4. One has

Proof. For ,  ,   , we have [1]

that is,

Then,

The following result is an immediate consequence of Lemma 4.

Corollary 5. One has

(i), (ii).

3. Antiperiodic Boundary Value Problem

In this section, we consider the following antiperiodic boundary value problem:

where is a forward difference operator.

Let be the set of all real sequences with norm . Then, is a Banach space.

Lemma 6. A solution is a solution for antiperiodic boundary value problem

if and only if is a solution of the the following fractional Taylor’s difference formula:

Proof. Suppose that defined on is a solution of (10). Using Lemma 3, for some constants , we have

Then, we obtain [3]

In view of and , we have

Then,

Substituting the values of and into (12), we obtain (11).

Conversely, if is a solution of (11), by a direct computation, it follows that the solution given by (11) satisfies (10). The proof is completed.

The following fixed-point theorems are needed to prove the existence and uniqueness of solutions for the BVP (9).

Lemma 7 (see [15] (Banach contraction mapping principle)). A contraction mapping on a complete metric space has exactly one fixed point.

Lemma 8 (see [16] (Brower fixed-point theorem)). Let be a continuous mapping, where is a nonempty, bounded, close, and convex set. Then, has a fixed point.

Define the operator

Obviously, is a solution of (9) if it is a fixed point of the operator .

Theorem 9. Assume that.

There exists a constant such that  for each and all .

Then, the BVP (9) has a unique solution on provided that

Proof. Let ; then for each , we have

According to (17), we obtain

Then,

which implies that is a contraction. Therefore, the Banach fixed-point theorem (Lemma 7) guarantees that has a unique fixed point which is a unique solution of the BVP (9). This completes the proof.

Theorem 10. Assume that.

There exists a bounded function such that for all .

Then, the BVP (9) has at least a solution on provided that

where .

Proof. Let ; define the set

To prove this theorem, we only need to show that maps in .

For , we have

From (21), we have ; then, which implies that maps in . has at least a fixed point which is a solution of the BVP (9) according to Brower fixed-point theorem (Lemma 8). This completes the proof.

4. The Ulam Stability

Similar to the definitions of the Ulam stability for fractional differential equation [14], we introduce four types of the Ulam stability definitions for fractional difference equation.

Consider (1) and the following inequalities:

Definition 11. Equation (1) is the Ulam-Hyers stable if there exists a real number such that for each and for each solution of inequality (24), there exists a solution of (1) with

Equation (1) is the generalized Ulam-Hyers stable if we substitute the function for the constant on inequality (26), where and .

Definition 12. Equation (1) is the Ulam-Hyers-Rassias stable with respect to if there exists a real number such that for each and for each solution of inequality (25), there exists a solution of (1) with

Equation (1) is the generalized Ulam-Hyers-Rassias stable if we substitute the function for the function on inequalities (25) and (27).

Remark 13. If is a constant function in Definition 12, we say that the integral equation (25) has also the Hyers-Ulam stability.

Remark 14. A function is a solution of inequality (24) if and only if there exists a function such that

(i), (ii),

similar remark for inequality (25).

Theorem 15. Assume that holds. Let be a solution of inequality (24) and let be a solution of the following boundary value problem:

Then, (1) is the Ulam-Hyers stable provided that

Proof. By Lemma 3, the solution of the BVP (28) is given by

From inequality (24), for , it follows that

Combining (30) and (31), for , we have

Then,

Applying (29) to the previous inequality yields that

where , thus; (1) is the Ulam-Hyers stable.

Theorem 16. Assume that and the following condition hold.

Let be an increasing function. There exists a constant such that

Let be a solution of inequality (25) and let be a solution of the following boundary value problem (28). Then, (1) is the Ulam-Hyers-Rassias stable provided that (29) holds.

The proof of Theorem 16 is similar to that of Theorem 15, and we omit it.

5. Examples

As the applications of our main results, we consider the following examples.

Example 1. Consider the fractional difference BVP

where for and conditions and are satisfied. Since

inequalities (17) and (21) are satisfied if . According to Theorem 9, the BVP (36) has a unique solution. At the same time, the BVP (36) has at least a solution by Theorem 10.

Example 2. Consider the fractional difference equation

with the boundary conditions

Since

if and the inequality

hold, then (38) is the Ulam-Hyers stable by Theorem 15.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China (10971173), the National Natural Science Foundation of Hunan Province under Grant 13JJ3120, and the Construct Program of the Key Discipline in Hunan Province.

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