Asymptotic Stability Results for Nonlinear
Fractional Difference Equations
Fulai Chen1and Zhigang Liu1
Academic Editor: Michela Redivo-Zaglia
Received01 Aug 2011
Revised27 Dec 2011
Accepted02 Jan 2012
Published08 Mar 2012
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 [1β12] and the references therein. However, very little progress has been made to develop the theory of the analogous fractional finite difference equation [13β19].
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 Liapunovβs direct method, fixed point theorems are usually considered in stability [20β25]. Motivated by this idea, in this paper, we discuss asymptotic stability of nonlinear fractional difference equations by using Krasnoselskiiβs fixed point theorem and discrete Arzela-Ascoliβs 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 βmodβ and is defined for βmodβ, 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 Taylorβs 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 Taylorβs 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-Ascoliβs theorem)]). A bounded, uniformly Cauchy subset of is relatively compact.
Theorem 2.13 (see [20, (Krasnoselskiiβs 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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