Journal of Applied MathematicsVolume 2012 (2012), Article ID 879657, 14 pagesdoi:10.1155/2012/879657

Research Article

Asymptotic Stability Results for Nonlinear Fractional Difference Equations

Fulai Chen and Zhigang Liu

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Received 1 August 2011; Revised 27 December 2011; Accepted 2 January 2012

Academic Editor: Michela Redivo-Zaglia

Copyright © 2012 Fulai Chen and Zhigang Liu. 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 present some results for the asymptotic stability of solutions for nonlinear fractional difference equations involvingRiemann-Liouville-likedifference operator. The results are obtained by using Krasnoselskii's fixed point theorem and discrete Arzela-Ascoli's theorem. Three examples are also provided to illustrate our main results.

1. Introduction

In this paper we consider the asymptotic stability of solutions for nonlinear fractional difference equations: where is aRiemann-Liouville-likediscrete fractional difference, is continuous with respect to and , .

Fractional differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Most of the present works were focused on fractional differential equations, see [1–12] and the references therein. However, very little progress has been made to develop the theory of the analogous fractional finite difference equation [13–19].

Due to the lack of geometry interpretation of the fractional derivatives, it is difficult to find a valid tool to analyze the stability of fractional difference equations. In the case that it is difficult to employ Liapunov’s direct method, fixed point theorems are usually considered in stability [20–25]. Motivated by this idea, in this paper, we discuss asymptotic stability of nonlinear fractional difference equations by using Krasnoselskii’s fixed point theorem and discrete Arzela-Ascoli’s theorem. Different from our previous work [18], in this paper, the sufficient conditions of attractivity are irrelevant to the initial value .

2. Preliminaries

In this section, we introduce preliminary facts of discrete fractional calculus. For more details, see [14].

Definition 2.1 (see [14]). Let . The th fractional sum is defined by where is defined for  mod  and is defined for  mod , and . The fractional sum maps functions defined on to functions defined on .

Definition 2.2 (see [14]). Let and , where denotes a positive integer, , ceiling of number. Set . The th fractional difference is defined as

Theorem 2.3 (see [15]). Let be a real-value function defined on and , then the following equalities hold: (i); (ii).

Lemma 2.4 (see [15]). Let and assume is not a nonpositive integer, then

Lemma 2.5 (see [15]). Assume that the following factorial functions are well defined: (i)If , then ;(ii).

Lemma 2.6 (see [13]). Let be noninteger, , , , thus one has

Lemma 2.7. The equivalent fractional Taylor’s difference formula of (1.1) is

Proof. Apply the operator to each side of the first formula of (1.1) to obtain Apply Theorem 2.3 to the left-hand side of (2.6) to obtain So, applying Definition 2.1 to the right-hand side of (2.6), for we obtain (2.5). The recursive iteration to this Taylor’s difference formula implies that (2.5) represents the unique solution of the IVP (1.1). This completes the proof.

Lemma 2.8 (see [4, (1.5.15)]). The quotient expansion of two gamma functions at infinityis

Corollary 2.9. One has

Proof. According to Lemma 2.8, Then, for . This completes the proof.

Definition 2.10. The solution of the IVP (1.1) is said to be(i) stable if for any and , there exists a such that for and all ;(ii) attractive if there exists such that implies(iii) asymptotically stable if it is stable and attractive.The space is the set of real sequences defined on the set of positive integers where any individual sequence is bounded with respect to the usual supremum norm. It is well known that under the supremum norm is a Banach space [26].

Definition 2.11 (see [27]). A set of sequences in is uniformly Cauchy (or equi-Cauchy), if for every , there exists an integer such that , whenever for any in .

Theorem 2.12 (see [27, (discrete Arzela-Ascoli’s theorem)]). A bounded, uniformly Cauchy subset of is relatively compact.

Theorem 2.13 (see [20, (Krasnoselskii’s fixed point theorem)]). Let be a nonempty, closed, convex, and bounded subset of the Banach space and let and be two operators such that (a)A is a contraction with constant ,(b)B is continuous, resides in a compact subset of ,(c).Then the operator equation has a solution in .

3. Main Results

Let be the set ofall real sequences with norm , then is a Banach space.

Define the operator Obviously, , the operator is a contraction with the constant 0, which implies that condition of Theorem 2.13 holds, and is a solution of (1.1) if it is a fixed point of the operator .

Lemma 3.1. Assume that the following condition is satisfied: there exist constants and such that Then the operator is continuous and is a compact subset of for , where , and satisfies that

Proof. For , apply Lemma 2.8 and , and we have that as , then there exists a such that inequality (3.4) holds, which implies that the set exists.We firstly show that maps in .It is easy to know that is a closed, bounded, and convex subset of .Apply condition , Lemma 2.5, Corollary 2.9 and (3.4), for , we have which implies that for .Nextly, we show that is continuous on .Let be given then there exist and such that implies that Let be a sequence such that . For , applying the continuity of and Lemma 2.6, we have For , Thus, for all , we have which implies that is continuous.Lastly, we show that is relatively compact.Let and , thus we have Thus, is a bounded and uniformly Cauchy subset by Definition 2.11, and is relatively compact by means of Theorem 2.12. This completes the proof.

Lemma 3.2. Assume that condition holds, then a solution of (1.1) is in for .

Proof. Notice if that is a fixed point of , then it is a solution of (1.1). To prove this, it remains to show that, for fixed , holds.If , applying condition and (3.4), for , we have Thus, for . According to Theorem 2.13 and Lemma 3.1, there exists a such that , that is, has a fixed point in which is a solution of (1.1) for . This completes the proof.

Theorem 3.3. Assume that condition holds, then the solutions of (1.1) is attractive.

Proof. By Lemma 3.2, the solutions of (1.1) exist and are in . All functions in tend to 0 as . Then the solutions of (1.1) tend to zero as . This completes the proof.

Theorem 3.4. Assume that the following condition is satisfied: there exist constants and such that Then the solutions of (1.1) are stable provided that

Proof. Let be a solution of (1.1), and let be a solution of (1.1) satisfying the initial value condition . For , applying condition , we have which implies that For any given , let , follows that , which yields that the solutions of (1.1) are stable. This completes the proof.

Theorem 3.5. Assume that conditions and hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds. Theorem 3.5 is the simple consequence of Theorems 3.3 and 3.4.

Theorem 3.6. Assume that the following condition is satisfied: there exist constants , , and such that Then the solutions of (1.1) is attractive.

Proof. Set where satisfies that We first prove condition of Theorem 2.13, that is, for fixed and for all , holds.If , applying condition and (3.19), for , we have Thus, condition of Theorem 2.13 holds.The proof of condition of Theorem 2.13 is similar to that of Lemma 3.1, and we omit it. Therefore, has a fixed point in by using Theorem 2.13, that is, the IVP (1.1) has a solution in . Moreover, all functions in tend to 0 as , then the solution of (1.1) tends to zero as , which shows that the zero solution of (1.1) is attractive. This completes the proof.

Theorem 3.7. Assume that conditions and hold, then the solutions of (1.1) are asymptotically stable provided that (3.14) holds.

Theorem 3.8. Assume that the following condition is satisfied: there exist constants , and such that Then the solutions of (1.1) is attractive.

Proof. Set where , and satisfies that Here we only prove that condition of Theorem 2.13 holds, and the remaining part of the proof is similar to that of Theorem 3.6.Since , and , then .If , applying condition , Lemma 2.5 and (3.23), for , we have Thus, condition of Theorem 2.13 holds. This completes the proof.

4. Examples

Example 4.1. Consider where , .Since thisimplies that condition holds.In addition, Thus, condition is satisfied.Moreover, from , , and , we have which implies that inequality (3.14) holds.Thus the solutions of (4.1) are asymptotically stable by Theorem 3.5.

Example 4.2. Consider where , .Since , we have that , and which implies that condition is satisfied.Meanwhile, which implies that condition is satisfied.From , , and , we have which implies that inequality (3.14) holds.Thus the solutions of (4.5) are asymptotically stable by Theorem 3.7.

Example 4.3. Consider where , .Since , we have that and then condition is satisfied.The solutions of (4.9) are attractive by Theorem 3.8.

Acknowledgments

Thisresearch was supported by the NSF of Hunan Province (10JJ6007, 2011FJ3013), the Scientific Research Foundation of Hunan Provincial Education Department, and the Construct Program of the Key Discipline in Hunan Province.

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