Asymptotic Behavior of Impulsive Infinite Delay Difference Equations with Continuous Variables
 Zhixia Ma^{1} and
 Liguang Xu^{2}Email author
DOI: 10.1155/2009/495972
© Z. Ma and L. Xu 2009
Received: 3 June 2009
Accepted: 2 August 2009
Published: 19 August 2009
Abstract
A class of impulsive infinite delay difference equations with continuous variables is considered. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of "ϱcone," we obtain the attracting and invariant sets of the equations.
1. Introduction
Difference equations with continuous variables are difference equations in which the unknown function is a function of a continuous variable [1]. These equations appear as natural descriptions of observed evolution phenomena in many branches of the natural sciences (see, e.g., [2, 3]). The book mentioned in [3] presents an exposition of some unusual properties of difference equations, specially, of difference equations with continuous variables. In the recent years, the asymptotic behavior and other behavior of delay difference equations with continuous variables have received much attention due to its potential application in various fields such as numerical analysis, control theory, finite mathematics, and computer science. Many results have appeared in the literatures; see, for example, [1, 4–7].
However, besides the delay effect, an impulsive effect likewise exists in a wide variety of evolutionary process, in which states are changed abruptly at certain moments of time. Recently, impulsive difference equations with discrete variable have attracted considerable attention. In particular, delay effect on the asymptotic behavior and other behaviors of impulsive difference equations with discrete variable has been extensively studied by many authors and various results are reported [8–12]. However, to the best of our knowledge, very little has been done with the corresponding problems for impulsive delay difference equations with continuous variables. Motivated by the above discussions, the main aim of this paper is to study the asymptotic behavior of impulsive infinite delay difference equations with continuous variables. By establishing an infinite delay difference inequality with impulsive initial conditions and using the properties of " cone," we obtain the attracting and invariant sets of the equations.
2. Preliminaries
where , , , and are real constants, (here, and will be defined later), and are positive real numbers. is an impulsive sequence such that . , , , and : are realvalued functions.
By a solution of (2.1), we mean a piecewise continuous realvalued function defined on the interval which satisfies (2.1) for all .
this function will be called the solution of the initial problem (2.1)–(2.3).
where , , , , , , , , , , and , in which .
In what follows, we introduce some notations and recall some basic definitions. Let be the space of dimensional (nonnegative) real column vectors, be the set of (nonnegative) real matrices, be the dimensional unit matrix, and be the Euclidean norm of . For or , means that each pair of corresponding elements of and satisfies the inequality " ( )."Especially, is called a nonnegative matrix if , and is called a positive vector if . and .
and denotes the spectral radius of .
Definition 2.1.
The set is called a positive invariant set of (2.4), if for any initial value , the solution , .
Definition 2.2.
where dist , , for .
Definition 2.3.
If and , then .
Lemma 2.5 (La Salle [14]).
Suppose that and , then there exists a positive vector such that .
which is a nonempty set by Lemma 2.5, satisfying that for any scalars , , and vectors . So is a cone without vertex in , we call it a " cone" [12].
3. Main Results
In this section, we will first establish an infinite delay difference inequality with impulsive initial conditions and then give the attracting and invariant sets of (2.4).
Theorem 3.1.
 (a)Then(3.2)
 (b)Then(3.5)
Proof.
 (b)For any given initial function: , , where , there is a constant such that . To prove (3.5), we first prove that(3.12)
where ( small enough), provided that the initial conditions satisfies .
which contradicts the first equality of (3.13), and so (3.12) holds for all . Letting , then (3.5) holds, and the proof of part (b) is completed.
Remark 3.2.
Suppose that in part (a) of Theorem 3.1, then we get [15, Lemma 3].

(A_{1}) For any , there exist nonnegative diagonal matrices such that(3.15)

(A_{2}) For any , there exist nonnegative matrices such that(3.16)

(A_{3}) Let , where(3.17)

(A_{4}) There exists a constant such that(3.18)where the scalar satisfies and is determined by the following inequality(3.19)where , and(3.20)

(A_{5}) Let(3.21)where satisfy(3.22)
Theorem 3.3.
If ( )–( ) hold, then is a global attracting set of (2.4).
Proof.
Since and , then, by Lemma 2.5, there exists a positive vector such that . Using continuity and noting , we obtain that inequality (3.19) has at least one positive solution .
where
Since and , then, by Lemma 2.4, we can get , and so .
This implies that the conclusion of the theorem holds and the proof is complete.
Theorem 3.4.
If ( )–( ) with hold, then is a positive invariant set and also a global attracting set of (2.4).
Proof.
Therefore, is a positive invariant set. Since , a direct calculation shows that and in Theorem 3.3. It follows from Theorem 3.3 that the set is also a global attracting set of (2.4). The proof is complete.
For the case , we easily observe that is a solution of (2.4) from and . In the following, we give the attractivity of the zero solution and the proof is similar to that of Theorem 3.3.
Corollary 3.5.
If hold with , then the zero solution of (2.4) is globally exponentially stable.
Remark 3.6.
If , that is, they have no impulses in (2.4), then by Theorem 3.4, we can obtain the following result.
Corollary 3.7.
If and hold, then is a positive invariant set and also a global attracting set of (2.4).
4. Illustrative Example
The following illustrative example will demonstrate the effectiveness of our results.
Example 4.1.
where and are nonnegative constants, and the impulsive sequence satisfies: . For System (4.1), we have , . So, it is easy to check that , , provided that . In this example, we may let .
Let , then satisfy ,
Case 1.
Moreover, , . Clearly, all conditions of Theorem 3.3 are satisfied. So is a global attracting set of (4.1).
Case 2.
Let and , then . Therefore, by Theorem 3.4, is a positive invariant set and also a global attracting set of (4.1).
Case 3.
Clearly, all conditions of Corollary 3.5 are satisfied. Therefore, by Corollary 3.5, the zero solution of (4.1) is globally exponentially stable.
Declarations
Acknowledgment
The work is supported by the National Natural Science Foundation of China under Grant 10671133.
Authors’ Affiliations
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