Fixed points and asymptotic stability of nonlinear fractional difference equations

In this paper, we discuss nonlinear fractional difference equations with the Caputo like difference operator. Some asymptotic stability results of equations under investigated are obtained by employing Schauder fixed point theorem and discrete Arzela-Ascoli's theorem. Three examples are also provided to illustrate our main results.


Introduction
This paper investigates the asymptotic stability of solutions for a class of nonlinear fractional difference equations where ∆ α * is a Caputo like discrete fractional difference, f : [0, +∞) × R → R is continuous with respect to t and x, N t = {t, t + 1, t + 2, • • •}.
Fractional differential equations have received increasing attention during recent years since these equations have been proved to be valuable tools in EJQTDE, 2011 No. 39, p. 1 the modeling of many phenomena in various fields of science and engineering, see the monographs [18,20,23,25] and the papers [1,7,11,19,24,28,30,31] and the references therein.
Fractional difference equations have also been studied more intensively of late [2][3][4][5][6]12].In particular, Atici and Eloe [3] investigated the commutativity properties of the fractional sum and the fractional difference operators, Atici and Sengül [6] developed Leibniz rule and summation by parts formula, Anastassiou [2] defined a Caputo like discrete fractional difference and compared it to the Riemann-Liouville fractional discrete analog, and Chen et al. [12] gave global and local existence results of solutions for nonlinear fractional difference equations with the Caputo like difference operator.
However, 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 differential equations, and there are few work on the stability of solutions for either fractional differential equations or fractional difference equations.Some local asymptotical stability, Mittag-Leffler stability and linear matrix inequality (LMI) stability are discussed in [13,15,21,22,27], Chen and Zhou [13] considered the attractivity of fractional functional differential equations by Schauder fixed point theorem, Deng [15] discussed the attractivity of nonlinear fractional differential equations by means of the principle of contraction mappings, but there's no work on asymptotic stability of fractional difference equations via fixed point theorems.
To study stability properties of differential equations, Burton [10] pointed out that many difficulties of Liapunov's direct method, such as constructing Liapunov functions and functionals, ascertaining limit sets when the equation becomes unbounded or the derivative is not definite, vanish when fixed point theory is used.
Motivated by applying fixed point theory to research stability of integerorder differential equations [8-10, 16, 17, 26], in this paper, we discuss asymptotic stability of nonlinear fractional difference equations by using Schauder fixed point theorem and discrete Arzela-Ascoli's theorem.
The rest of the paper is organized as follows.In section 2, we introduce some useful preliminaries.In section 3, we prove some sufficient conditions of asymptotic stability of IVP (1).Finally, three examples are given to illustrate our main results.
In this section, we introduce preliminary facts which are used throughout this paper.
Definition 2.1 [3,4] Let ν > 0. The ν−th fractional sum x is defined by where x is defined for s = a mod (1) and ∆ −ν x is defined for t = (a + ν) mod (1), and t (ν) = Γ(t+1) Γ(t−ν+1) .In (2), the fractional sum ∆ −ν maps functions defined on N a to functions defined on N a+ν .Atici and Eloe [3] pointed out that this definition is the development of the theory of the fractional calculus on time scales.Definition 2.2 [2] Let µ > 0 and m − 1 < µ < m, where m denotes a positive integer, m = ⌈µ⌉, ⌈•⌉ ceiling of number.Set ν = m − µ.The µ−th fractional Caputo like difference is defined as where ∆ m is the m−th order forward difference operator, the fractional Caputo like difference ∆ µ * maps functions defined on N a to functions defined on N a−µ .
Conversely, if x(t) is a solution of ( 5), comparing between with ( 4) and ( 5) we have EJQTDE, 2011 No. 39, p. 4 For t = 2, form (6) we have 1), we have It follows that ( 5) is equivalent to the following equation Lemma 2.3 Assume that β > 1 and γ > 0, then . Thus, holds for t ∈ N 1 .This completes the proof.Definition 2.3 The solution x = ϕ(t) of IVP ( 1) is said to be (i) stable, if for any ε > 0 and (iii) asymptotically stable if it is stable and attractive.The space l ∞ n 0 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 know that under the supremum norm l ∞ n 0 is a Banach space [29].
Definition 2.4 [14] A set Ω of sequences in l ∞ n 0 is uniformly Cauchy (or equi-Cauchy) if for every ε > 0, there exists an integer N such that |x(i) − x(j)| < ε whenever i, j > N for any x = {x(n)} in Ω.
Theorem 2.1 [14] (Discrete Arzela-Ascoli's Theorem) A bounded, uniformly Cauchy subset Ω of l ∞ n 0 is relatively compact.The following fixed point theorem are classical, which can be seen from many books.
Theorem 2.2 (Schauder fixed point theorem) Let Ω be a closed, convex and nonempty subset of a Banach space X.Let T : Ω → Ω be a continuous mapping such that T Ω is a relatively compact subset of X.Then T has at least one fixed point in Ω.That is, there exists an x ∈ Ω such that T x = x.EJQTDE, 2011 No. 39, p. 6

Main Results
Let l ∞ 1 be the set of all real sequence x = {x(t)} ∞ t=1 with norm x = sup t∈N 1 |x(t)|.Then l ∞ 1 is a Banach space.Define the operator Obviously, x(t) is a solution of (1) if it is a fixed point of the operator T .Lemma 3.1 Assume that the following condition is satisfied: (H 1 ) there exist constants γ 1 , L 1 > 0 such that Then IVP (1) exists at least one solution x(t) for t ∈ N 1 .
Proof.Define the set It is easy to know that S 1 is a closed, bounded and convex subset of R. In addition, for t ∈ N 1 , we have → 0 for t → ∞.
To prove that T has a fixed point, we firstly show that T maps S 1 in S 1 .
Nextly, we show that T is continuous on S 1 .Let ε > 0 be given, there exists a ], Lemma 2.5), we EJQTDE, 2011 No. 39, p. 7 have Thus, for all t ∈ N 1 , we have which means that T is continuous.
Lastly, we show that T S 1 is relatively compact.Let t 1 , t 2 ∈ N 1 and t 2 > t 1 ≥ N 1 , we have EJQTDE, 2011 No. 39, p. 8 Therefore, {T x : x ∈ S 1 } is a bounded and uniformly Cauchy subset.Hence, by Theorem 2.1, T S 1 is relatively compact.
According to Theorem 2.2, we have that T has a fixed point in S 1 which is a solution of IVP (1).This completes the proof.
Theorem 3.1 Assume that condition (H 1 ) holds, then the solutions of ( 1) is attractive.
Proof.By Lemma 3.1, the solutions of (1) exist and are in S 1 .All functions x(t) in S 1 tend to 0 as t → ∞.Then the solutions of (1) tend to zero as t → ∞.This completes the proof.
Combining Theorem 3.1 and Theorem 3.2, we have Theorem 3.3 Assume that conditions (H 1 ) and (H 2 ) hold, then the solutions of IVP (1) are asymptotically stable provided that (10) holds.
Lemma 3.2 Assume that the following condition are satisfied: Then IVP (1) exists at least one solution x(t) on N 1 .
Proof.Define the set From the above assumption of S 2 , it is easy to know that S 2 is a closed, bounded and convex subset of R.
We firstly show that T maps S 2 in S 2 .For t ∈ N 1 , from condition (H 3 ) we have Nextly, we show that T is continuous on S 2 .Let ε > 0 be given, there exists a N 2 ∈ N 1 such that t > N 2 implies that Let {x n } be a sequence such that Thus, for all t ∈ N 1 , we have which means that T is continuous.
The proof of T S 2 be relatively compact is similar to that of Lemma 3.1, and we omit it.By Theorem 2.2, we have that T has a fixed point in S 2 which is a solution of IVP (1).This completes the proof.EJQTDE, 2011 No. 39, p. 11 Theorem 3.4 Assume that condition (H 3 ) holds, then the solutions of (1) are attractive.
Theorem 3.5 Assume that conditions (H 2 ) and (H 3 ) hold, then the solutions of IVP (1) are asymptotically stable provided that (10) holds.
Theorem 3.6 Assume that condition (H 4 ) and ( 11) hold, then the solutions of (1) are attractive.

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