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When an impulsive control is adopted for a stochastic delay difference system (SDDS), there are at least two situations that should be contemplated. If the SDDS is stable, then what kind of impulse can the original system tolerate to keep stable? If the SDDS is unstable, then what kind of impulsive strategy should be taken to make the system stable? Using the Lyapunov-Razumikhin technique, we establish criteria for the stability of impulsive stochastic delay difference equations and these criteria answer those questions. As for applications, we consider a kind of impulsive stochastic delay difference equation and present some corollaries to our main results.

In recent years, stochastic delay difference equations (SDDEs) have been studied by many researchers; a number of results have been reported [

For SDDEs, when we take impulsive effects into account, we have at least two problems to deal with.

As well known, Lyapunov-Razumikhin technique is one of main methods to investigate the stability of delay systems [

In the sequel,

Consider the impulsive stochastic delay difference equations of the form

Assume that

One calls the trivial solution of system (

If the trivial solution of system (

In this section, we will establish two theorems on

First, we present the theorem on impulsive stability. The technique adopted in the proof is motivated by [

Assume that there exist a positive function

For

For

Let

Next, we will show that, for any

For a given

Under this situation, we have

Making use of the definition of

When

Now, we are in position to state the theorem on impulsive stabilization. The method used in the proof is motivated by [

Assume that there exist a function

For

Choose

Write

First we will show that, for any

Note that there may not exist the natural number

For any

Assume that (

Now we will show that, when

If (

For

By induction, we know that (

Using condition (

In this section, we consider a kind of impulsive stochastic delay difference equation as follows:

Using the obtained results, we present three corollaries for system (

Assume that conditions

There exist constants

Let

Let

Now we assume that

From the above proof, we know that constant

Assume that conditions (

There exists a constant

Let

Since

From the above proof, we know that constant

Now, we present a corollary of Theorem

Assume that there exist positive constants

Let

Now we study some examples to illustrate our results.

We consider a linear impulsive stochastic delay difference equation as following:

First we take

Mean square exponential stability of (

Now we take

Instability without impulsive effects of (

Mean square exponential stability of (

It should be pointed that the conditions of Corollary

Stability of (

In this paper, we considered the

This work was supported by the Scientific Foundations of Harbin Institute of Technology at Weihai under Grants no.HIT(WH)20080008 and no.HIT(WH)2B200905. The authors would like to give their thanks to Professor Leonid Shaikhet for his valuable comments and to the referees who suggest that the authors give examples to illustrate the results.