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The problem of stability for nonlinear impulsive stochastic functional differential equations with delayed impulses is addressed in this paper. Based on the comparison principle and an impulsive delay differential inequality, some exponential stability and asymptotical stability criteria are derived, which show that the system will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the continuous stochastic flows. The obtained results complement ones from some recent works. Two examples are discussed to illustrate the effectiveness and advantages of our results.

Impulsive dynamical equations have received considerable attention during the recent decades since they provide a natural framework for mathematical modeling of many real world evolutionary processes where the states undergo abrupt changes at certain instants (see [

In the current literature concerning IFDEs, the impulses are assumed to take the form

On the other hand, stochastic perturbations are unavoidable in real equations (see [

Motivated by the above discussion, in this paper, we will further investigate the stability of ISFDEs-DI. By using the comparison principle and an impulsive delay differential inequality, some exponential and asymptotical stability criteria are derived, which are more convenient to be applied than those Razumikhin-type conditions. Our results complement ones from some recent works and show that the ISFDE-ID will be stable if the impulses’ frequency and amplitude are suitably related to the increase or decrease of the corresponding continuous stochastic flows. The rest of the paper is organized as follows. In Section

Throughout this paper, unless otherwise specified, we let

Let

For

Consider the following ISFDE-DI:

As a standing hypothesis, we assume that for any

We introduce the following scalar IFDE-DI as the comparison system:

For convenience, we introduce the following function classes:

=

At the end of this section, let us introduce the following definitions.

The trivial solution of (

A function

If

In this section, we will develop an impulsive delay differential inequality and comparison principles and establish some criteria on

Assume that

Then any solution

Set

On the other hand, for any

Let

Assume that there exists a function

For any

Write

Consider the system

In fact, if this is not true, then from the continuity of

Noting that

Assume that there exist functions

there exist positive constants

Firstly, assume that the trivial solution of IFDE-DI (

Let

If

Next, let us suppose that the trivial solution of IFDE-DI (

Thirdly, let us suppose that the trivial solution of IFDE-DI (

Assume that there exist a function

Let

Furthermore, let

An impulsive stochastic dynamical system can be viewed as a hybrid one comprised of two components: a continuous stochastic dynamic and a discrete dynamic. Theorem

It is noted that the exponential stability analysis in [

In this section, the effectiveness and advantages of the results derived in the preceding section will be illustrated by two examples.

Consider the two-dimensional nonlinear impulsive stochastic delay equation in the form

Denote

Take

Consider the following impulsive stochastic delayed neural network:

It is noted that (

The solution of system (

The mean square of the solution of system (

In the following, applying Theorem

Denote

Thus, the comparison system is

The solution of system (

The mean square of the solution of system (

This paper has investigated the exponential stability of ISFDEs-DI based on the comparison approach and an impulsive delay differential inequality. Some criteria on the

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (11226247, 11301004), the 211 Project of Anhui University (32030018/33010205), the Anhui Provincial Nature Science Foundation (1308085QA15/1308085MA01), the Natural Science Foundation of Jiangsu Province (BK20130239), the Research Fund for the Doctoral Program of Higher Education of China (20130094120015), the Key Foundation of Anhui Education Bureau (KJ2012A019), and the Research Fund for the Doctoral Program of Higher Education (20103401120002).