# Attracting and quasi-invariant sets for a class of impulsive stochastic difference equations

- Dingshi Li
^{1}Email author and - Shujun Long
^{1}

**2011**:3

https://doi.org/10.1186/1687-1847-2011-3

© Li and Long; licensee Springer. 2011

**Received: **25 December 2010

**Accepted: **17 May 2011

**Published: **17 May 2011

## Abstract

The aim of this article is to study the attracting and quasi-invariant sets for a class of impulsive stochastic difference equations. By establishing a difference inequality, we obtain the attracting and quasi-invariant sets of systems under consideration. An example is given to illustrate the theory.

## Keywords

## Introduction

Difference equations usually appear in the investigation of systems with discrete time or in the numerical solution of systems with continuous time [1]. A lot of difference systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc. In recent years, the stability investigation of stochastic difference equations has been interesting to many investigators, and various advanced results on this problem have been reported [2–5].

However, besides the stochastic effect, an impulsive effect likewise exists in a wide variety of evolutionary processes in which states are changed abruptly at certain moments of time, involving such fields as medicine and biology, economics, mechanics, electronics and telecommunications. Recently, the asymptotic behaviors of impulsive difference equations have attracted considerable attention. Many interesting results on impulsive effect have been obtained [6–8]. In [9], some stability conditions on impulsive stochastic difference equations are given. As is well known, stability is one of the major problems encountered in applications, and has attracted considerable attention due to its important role in applications. However, under impulsive perturbation, an equilibrium point sometimes does not exist in many physical systems, especially, in non-linear and nonautonomous dynamical systems. Therefore, an interesting subject is to discuss the invariant sets and the attracting sets of impulsive systems. Some significant progress has been made in the techniques and methods of determining the invariant sets and attracting sets for delay difference equations, delay differential equations, and impulsive functional differential equations [10–12]. Unfortunately, the corresponding problems for impulsive stochastic difference equations have not been considered.

Motivated by the above discussion, we here make a first attempt to arrive at results on the invariant sets and attracting sets of impulsive stochastic difference equations.

## Model description and preliminaries

Let *R*
^{
n
} be the space of *n*-dimensional real column vectors and *R*
_{+} = [0, +∞). *N*[*a*, *b*] {*a*, *a* + 1,..., *b*}, where *a < b* and *a*, *b* are integral numbers. *C* denotes the set of all functions *ϕ* : *N*[-*h*, 0] *→* *R*
^{
n
} , *h* is a nonnegative integer. For any *φ* ∈ *C*, we define
. *Z* denotes the integer set. Let {Ω, *P*, Σ} be a basic probability space, Σ_{
i-1}⊂ Σ_{
i
}⊂ Σ, *i* ∈ *Z* be a sequence of Σ-algebras *E* be the mathematical expectation, ξ_{0}, ξ_{1},... be a sequence of mutually independent random variables, ξ_{
i
}∈ *R*, ξ _{
i
} be Σ_{
i
}-adapted and independent on. Σ_{
i-1}, *Eξ*
_{
i
} = 0,
, *i* ∈ *Z*. Let *C*
_{Ω} denote the family of *C*-valued random variables on {Ω, *P*, Σ}.

where *F*, *G* : *Z* × *R*
^{
h+ 1}→ *R*, *H*
_{
i
}: *R* → *R*. *φ* (*i*) ∈ *C*
_{Ω}. The fixed moments of time *i*
_{
k
}∈ *Z*, and satisfy
. *x*
_{
i
}is an element of *C*
_{Ω} defined by *x*
_{
i
} = *x* (*i* + *s*), *s* ∈ *N* [-*h*, 0].

Throughout this article, we assume that for any *φ* (*i*) ∈ *C*
_{Ω}, there exists at least one solution of (1), which is denoted by *x*(*i*, 0, *φ*) or *x*
_{
i
} (0, *φ*) (simply *x*(*i*) and *x*
_{
i
} if no confusion should occur).

**Definition 2.1**. The set *S* ⊂ *C*
_{Ω} is called a quasi-invariant set of (1), if there exists a constant *k* such that for any initial value *φ* ∈ *S*, the solution *kx*
_{
i
} (0, *φ*) ∈ *S*, *i* ∈ *Z*. Especially, if *k* = 1, S is called a invariant set.

**Dedinition 2.2**. The set

*S*⊂

*C*

_{Ω}is called a global attracting set of (1), if for any initial value

*φ*∈

*C*

_{Ω}, the solution

*x*

_{ i }(0,

*φ*) satisfies

where *ρ*(·,·) is any distance in *C*
_{Ω}.

**Definition 2.3**. The zero solution of Equation (1) is called mean square exponential stable if there are positive constants

*λ*and

*M*such that for any initial condition

*φ*∈

*C*

_{Ω},

Here *λ* is called the exponential convergence rate. Of course, conditions are needed to ensure that the zero function is a solution of (1).

Based on discrete Halanay inequality [13] and its extension [9], we develop the following difference in-equality with the impulsive initial condition.

**Lemma 2.1**. Suppose

*c*

_{ j }(

*i*) ∈

*R*

_{+},

*i*∈

*Z*,

*j*∈

*N*[0,

*h*], and

*b*> 0. Let

*u*(

*i*) be a sequence of real numbers satisfying the following difference inequality:

- (a)

where *i'* ∈ *Z* and *γ* ≥ 1.

*i**≥

*i'*such that

which contradicts the first inequality of (10). So (6) holds. The proof of part (b) is complete.

## Main results

_{1}) For any

*i*∈

*Z*, there exist positive constants

*a*

_{ j }(

*i*),

*b*

_{ j }(

*i*),

*J*

_{1}and

*J*

_{2}such that

**Theorem 3.1**. If (A_{1})- (A_{4}) hold, then *S* = {*ϕ* ∈ *C*
_{Ω}|*E*||*ϕ*||^{2} ≤*e*
^{σ}(1-*μ*)^{-1}
*J*} is a global attracting set of (1), where
.

*Proof*. From (1), Condition (A

_{1}), (

*a*+

*b*)

^{2}≤ 2 (

*a*

^{2}+

*b*

^{2}) and the Hölder inequality, we have

*x*(

*s*) =

*φ*(

*s*),

*s*∈

*N*[-

*h*, 0], where

*φ*∈

*C*

_{Ω}, we have a positive constant

*K*such that

This implies that the conclusion holds and the proof is complete.

**Theorem 3.2**. If (A_{1})-(A_{4}) hold, then *S* = {*ϕ* ∈ *C*
_{Ω} |*E*||*ϕ*||^{2} ≤ *γ* (1-*μ*)^{-1}
*J*, *γ ≥* 1}is a quasi-invariant set of (1).

This implies that the conclusion holds and the proof is complete.

**Theorem 3.3**. If (A_{1})-(A_{2}) and (A_{5}) hold, then *S* = {*ϕ* ∈ *C*
_{Ω} |*E*||*ϕ*||^{2} ≤ *γ* (1-*μ*)^{-1}
*J*} is a invariant set and also a global attracting set of (1).

*Proof*. Since *d*
_{
k
} ≤ 1, a direct calculation shows that *α** = 0 and *σ* = 0 in Theorems 3.1 and 3.2. It follows from Theorem 3.1 the set *S* is a global attracting set of (1). It follows from Theorem 3.2 the set *S* is a invariant set of (1).

By Theorem 3.3, we can obtain the following result.

**Corollary 3.1**. If (A_{1}) and (A2) hold, then *S* = {*ϕ* ∈ *C*
_{Ω}|*E*||*ϕ*||^{2} ≤ (1-*μ*)^{-1}
*J*} is a invariant set and also a global attracting set of (21).

We easily observe *x*(*i*) = 0 is a solution of (1) from (A_{3}) and (A_{6}). In the following, we give the attractivity of the zero solution and the proof is similar to that of Theorem 3.1.

**Theorem 3.4**. If (A_{3}), (A_{4}), (A_{6}), and (A_{7}) hold, then the zero solution of Equation (1) is mean square exponential stable and the exponential convergence rate is equal to *λ** - *α**.

### Example

**Example 4.1**. Consider the following impulsive stochastic difference equation:

*α** = 0.016 and . Moreover, . Then the condition (A

_{4}) is satisfied. So, by Theorem 3.1, we can get that

is a global attracting set of (22). By Theorem 3.2, we can get that S is a quasi-invariant set of (22).

## Conclusion

The aim of this article is to study the attracting and quasi-invariant sets for a class of impulsive stochastic difference equations. By establishing a difference inequality, we obtain the attracting and quasi-invariant sets of systems under consideration. As pointed out by the reviewer, when *F* and *G* do not depend on *i*, the solutions of (1) are time-homogeneous Markovian in character except that there is an impulse at predetermined times *i*
_{1}, *i*
_{2}, *i*
_{3}.... For time-homogeneous Markov chains there is a well-established stability theory most eloquently summarized by Meyn and Tweedie [14]. We will explore the relationship between our work and the established theory of stochastic stability for Markov chains in the next article.

## Declarations

### Acknowledgements

The work is supported by National Natural Science Foundation of China under Grant 10971147. The authors would like to thank the referees for their detailed comments and valuable suggestions which considerably improved the presentation of the paper.

## Authors’ Affiliations

## References

- Kolmanovskii VB, Shaikhet LE:
**Control of Systems with Aftereffect.**In*Translations of Mathematical Monographs*.*Volume 157*. American Mathematical Society, Providence, RI; 1996.Google Scholar - Kuchkina N, Shaikhet L:
**Optimal control of Volterra type stochastic difference equations.***Comput Math Appl*1998,**36**(10-12):251-259. 10.1016/S0898-1221(98)80026-6MathSciNetView ArticleMATHGoogle Scholar - Taniguchi T:
**Stability theorems of stochastic difference equations.***J Math Anal Appl*1990,**147**(1):81-96. 10.1016/0022-247X(90)90386-TMathSciNetView ArticleMATHGoogle Scholar - Ma F, Caughey TK:
**Mean stability of stochastic difference systems.***Int J Non-Linear Mech*1982,**17**(2):69-84. 10.1016/0020-7462(82)90040-3MathSciNetView ArticleMATHGoogle Scholar - Ahmadi G:
**On the mean square stability of linear difference equations.***Appl Math Comput*1979,**5**(3):233-241. 10.1016/0096-3003(79)90015-8MathSciNetView ArticleMATHGoogle Scholar - He ZM, Zhang XM:
**Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions.***Appl Math Comput*2004,**156:**605-620. 10.1016/j.amc.2003.08.013MathSciNetView ArticleMATHGoogle Scholar - Zhang QQ:
**On a linear delay difference equation with impulses.***Ann Diff Equ*2002,**18**(2):197-204.MathSciNetMATHGoogle Scholar - Zhu W, Xu DY, Yang ZC:
**Global exponential stability of impulsive delay difference equation.***Appl Math Comput*2006,**181:**65-72. 10.1016/j.amc.2006.01.015MathSciNetView ArticleMATHGoogle Scholar - Yang ZG, Xu DY:
**Mean square exponential stability of impulsive stochastic difference equations.***Appl Math Lett*2007,**20:**938-945. 10.1016/j.aml.2006.09.006MathSciNetView ArticleMATHGoogle Scholar - Xu DY:
**Asymptotic behavior of nonlinear difference equations with delays.***Comput Math Appl*2001,**42:**393-398. 10.1016/S0898-1221(01)00164-XMathSciNetView ArticleMATHGoogle Scholar - Seifert G:
**Positively invariant closed sets for systems of delay differential equations.***J Diff Equ*1976,**22:**292-304. 10.1016/0022-0396(76)90029-2MathSciNetView ArticleMATHGoogle Scholar - Xu DY, Yang ZC:
**Attracting and invariant sets for a class of impulsive functional differential equations.***J Math Anal Appl*2007,**329:**1036-1044. 10.1016/j.jmaa.2006.05.072MathSciNetView ArticleMATHGoogle Scholar - Liz E, Ferreiro JB:
**A note on the global stability of generalized difference equations.***Appl Math Lett*2002,**15:**655-659. 10.1016/S0893-9659(02)00024-1MathSciNetView ArticleMATHGoogle Scholar - Meyn SP, Tweedie R:
*Markov Chains and Stochastic Stability.*Springer-Verlag, London; 1993.View ArticleGoogle Scholar

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