Open Access Research

On pth moment exponential stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses

Peiying Xiong12* and Lihong Huang1

Author Affiliations

1 College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, P.R. China

2 College of Mathematics and Computer Science, Hunan City University, Yiyang, Hunan, 413000, P.R. China

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Advances in Difference Equations 2013, 2013:172  doi:10.1186/1687-1847-2013-172


The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/172


Received:13 March 2013
Accepted:28 May 2013
Published:17 June 2013

© 2013 Xiong and Huang; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with the pth moment exponential stability of fuzzy cellular neural networks with time-varying delays under impulsive perturbations and stochastic noises. Based on the Lyapunov function, stochastic analysis and differential inequality technique, a set of novel sufficient conditions on pth moment exponential stability of the system are derived. These results generalize and improve some of the existing ones. Moreover, an illustrative example is given to demonstrate the effectiveness of the results obtained.

1 Introduction

In the last decades, cellular neural networks [1,2] have been extensively studied and applied in many different fields such as associative memory, signal processing and some optimization problems. In such applications, it is of prime importance to ensure that the designed neural networks are stable. In practice, due to the finite speeds of the switching and transmission of signals, time delays do exist in a working network and thus should be incorporated into the model equation. In recent years, the dynamical behaviors of cellular neural networks with constant delays or time-varying delays or distributed delays have been studied; see, for example, [3-11] and the references therein.

In addition to the delay effects, recently, studies have been intensively focused on stochastic models. It has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks or dynamical system described by stochastic functional differential equations (see [12-23]). On the other hand, most neural networks can be classified as either continuous or discrete. Therefore most of the investigations focused on the continuous or discrete systems, respectively. However, there are many real-world systems and neural processes that behave in piecewise continuous style interlaced with instantaneous and abrupt change (impulses). Motivated by this fact, several new neural networks with impulses have been recently proposed and studied (see [24-33]).

In this paper, we would like to integrate fuzzy operations into cellular neural networks. Speaking of fuzzy operations, Yang and Yang [34] first introduced fuzzy cellular neural networks (FCNNs) combining those operations with cellular neural networks. So far researchers have found that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs [34-40]. However, to the best of our knowledge, few author investigated the stability of stochastic fuzzy cellular neural networks with time-varying delays and impulses.

Motivated by the above discussions, in this paper, we consider the following stochastic fuzzy cellular neural networks with time-varying delays and impulses

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M1">View MathML</a>

(1)

where n corresponds to the number of units in the neural networks. For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M3">View MathML</a> corresponds to the state of the ith neuron. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M5">View MathML</a> are signal transmission functions. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M6">View MathML</a> denotes the rate at which a cell i resets its potential to the resting state when isolated from other cells and inputs; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M7">View MathML</a> corresponds to the transmission delay. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M8">View MathML</a> represents the elements of the feedback template. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M9">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M10">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M11">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M12">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M13">View MathML</a> are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operation, respectively; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M14">View MathML</a> denotes the external input of the ith neurons. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M15">View MathML</a> is the external bias of the ith unit. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M16">View MathML</a> is a transmission delay satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M17">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M18">View MathML</a> is the diffusion coefficient matrix and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M19">View MathML</a> is the ith row vector of <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M20">View MathML</a>: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M21">View MathML</a> is an n-dimensional Brownian motion defined on a complete probability space <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M22">View MathML</a> with a filtration <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M23">View MathML</a> satisfying the usual conditions (i.e., it is right continuous and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24">View MathML</a> contains all P-null sets). <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M25">View MathML</a> is the impulses at moment <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26">View MathML</a>, the fixed moments of time <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26">View MathML</a> satisfy <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M28">View MathML</a> , <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M29">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M30">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M31">View MathML</a>.

System (1) is supplemented with the initial condition given by

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M32">View MathML</a>

(2)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M33">View MathML</a> is <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24">View MathML</a>-measurable and continuous everywhere except at a finite number of points <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M26">View MathML</a>, at which <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M36">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M37">View MathML</a> exist and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M38">View MathML</a>.

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M39">View MathML</a> denote the family of all nonnegative functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M40">View MathML</a> on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M41">View MathML</a> which are continuous once differentiable in t and twice differentiable in x. If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M42">View MathML</a>, define an operator LV associated with (1) as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M43">View MathML</a>

(3)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M44">View MathML</a>

For convenience, we introduce several notations. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M45">View MathML</a> denotes a column vector. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M46">View MathML</a> denotes a vector norm defined by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M47">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M48">View MathML</a> denotes the space of continuous mappings from topological space X to topological space Y. Denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M49">View MathML</a> is the family of all bounded <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M24">View MathML</a>-measurable, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M51">View MathML</a>-valued random variables ϕ, satisfying <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M52">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M53">View MathML</a> denotes the expectation of a stochastic process.

Throughout the paper, we give the following assumptions.

(A1) The signal transmission functions <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M4">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M5">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M56">View MathML</a>) are Lipschitz continuous on R with Lipschitz constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M57">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M58">View MathML</a>, namely, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M59">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M60">View MathML</a>

(A2) There exist non-negative numbers <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M61">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M62">View MathML</a> such that for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M63">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M64">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M65">View MathML</a>

(A3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M66">View MathML</a>, where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M67">View MathML</a> is the equilibrium point of (1) with the initial condition (2), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M68">View MathML</a> satisfies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M69">View MathML</a>.

Definition 1.1 The equilibrium point <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M70">View MathML</a> of system (1) is said to be pth moment exponentially stable if there exist positive constants <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M71">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72">View MathML</a> such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M73">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M74">View MathML</a> is any solution of system (1) with initial value <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M75">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2">View MathML</a>.

When <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M77">View MathML</a>, it is usually said to be exponentially stable in mean square.

Lemma 1.1[34]

Supposexandyare two states of system (1), then we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M78">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M79">View MathML</a>

Lemma 1.2If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M80">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M81">View MathML</a>) denotepnonnegative real numbers, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M82">View MathML</a>

(4)

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M83">View MathML</a>denotes an integer. A particular form of (4), namely

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M84">View MathML</a>

2 Main results

In this section, we consider the existence and global pth moment exponential stability of system (1).

Lemma 2.1For two positive real numbersaandb, assume that there exists a constant number<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M85">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M86">View MathML</a>. Then the equation

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M87">View MathML</a>

(5)

has a unique solution<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72">View MathML</a>.

Proof Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M89">View MathML</a>. It is easy to see <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M90">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M91">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M92">View MathML</a>. Thus, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M93">View MathML</a> is strictly increasing on <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M94">View MathML</a>. Therefore, Eq. (5) has a unique positive solution <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M72">View MathML</a>. □

Lemma 2.2[18]

For two positive real numbersaandb, assume that there exists a constant number<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M85">View MathML</a>such that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M86">View MathML</a>. Assume that<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M98">View MathML</a>is a nonnegative continuous function on<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M99">View MathML</a>and satisfies the following inequality:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M100">View MathML</a>

(6)

then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M101">View MathML</a>, whereλis a solution of (5) and the upper right Dini derivative of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M98">View MathML</a>is defined as

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M103">View MathML</a>

Theorem 2.1Under conditions (A1)-(A3), if there exist constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M104">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M106">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M107">View MathML</a>

(7)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M108">View MathML</a>

Then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M109">View MathML</a>is a unique equilibrium which is globallypth moment exponential stable.

Proof The proof of existence and uniqueness of equilibrium for the system is similar to that of [39]. So we omit it. Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M110">View MathML</a> is the unique equilibrium of system (1).

Let

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M111">View MathML</a>

then system (1) can be transformed into the following equation, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M2">View MathML</a>:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M113">View MathML</a>

(8)

We define a Lyapunov function <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M114">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M115">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M116">View MathML</a>, then we can get the operator <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M117">View MathML</a> associated with system (8) of the following form:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M118">View MathML</a>

(9)

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M119">View MathML</a>

Firstly, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M120">View MathML</a>, applying the Ito formula, we obtain that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M121">View MathML</a>

(10)

Since <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M122">View MathML</a>, taking expectations on both sides of equality (10) and applying the inequality (9) yields

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M123">View MathML</a>

(11)

Since the Dini derivative <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M124">View MathML</a> is

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M125">View MathML</a>

(12)

denote <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M126">View MathML</a>, the preceding result (11) leads directly to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M127">View MathML</a>

(13)

Hence, from Lemma 2.2, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M128">View MathML</a>

Namely,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M129">View MathML</a>

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M130">View MathML</a>, λ is the unique positive solution of the following equation:

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M131">View MathML</a>

Next, suppose that for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M132">View MathML</a>, the inequality

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M133">View MathML</a>

(14)

holds. From (A3), we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M134">View MathML</a>

This, together with (14), leads to

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M135">View MathML</a>

(15)

On the other hand, for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M136">View MathML</a>, applying the Ito formula, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M137">View MathML</a>

Then, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M138">View MathML</a>

(16)

So, for small enough <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M139">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M140">View MathML</a>

(17)

From (16) and (17), we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M141">View MathML</a>

(18)

Similarly, we obtain

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M142">View MathML</a>

(19)

Hence, by the mathematical induction, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M143">View MathML</a> , we conclude that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M144">View MathML</a>

which implies that the equilibrium point of the impulsive system (1) is pth moment exponentially stable. This completes the proof of the theorem. □

3 Comparisons and remarks

It can be easily seen that many neural networks are special cases of system (1). Thus, in this section, we give some comparisons and remarks.

Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M145">View MathML</a>, system (1) becomes the stochastic fuzzy cellular neural networks with time-varying delays.

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M146">View MathML</a>

(20)

For (20), we have the following corollary by Theorem 2.1.

Corollary 3.1If (A1)-(A2) hold, if there exist constants<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M104">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105">View MathML</a>), <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M106">View MathML</a>such that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M150">View MathML</a>

where

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M151">View MathML</a>

then the unique equilibrium<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M109">View MathML</a>of system (20) is globallypth moment exponential stable.

If we do not consider fuzzy AND and fuzzy OR operations and when <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M145">View MathML</a> in system (1), then system (1) becomes impulsive stochastic cellular neural networks with time-varying delays

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M154">View MathML</a>

(21)

Remark 3.1 The stability of system (21) has been investigated in [31]. In [31], authors required the differentiability and monotonicity of a time delays function which satisfied <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M155">View MathML</a>. Hence, this assumption may impose a very strict constraint on the model because time delays sometimes vary dramatically with time in real circuits. Obviously, Theorem 2.1 does not require these conditions.

Remark 3.2 In Theorem 2.1, if we do not consider fuzzy AND and OR operation, it becomes traditional cellular neural networks. The results in [32] are the corollary of Theorem 2.1. Therefore the results of this paper extend the previous known publication.

4 An example

Example 4.1 Consider the following stochastic fuzzy neural networks with time-varying delays and time-varying delays and impulses

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M156">View MathML</a>

(22)

where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M157">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M158">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M143">View MathML</a> , and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M160">View MathML</a>

and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M161">View MathML</a> (<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M162">View MathML</a>).

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M77">View MathML</a>, obviously, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M164">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M105">View MathML</a>. By simple computation, we can easily get that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M166">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M167">View MathML</a>. Letting <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M168">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/172/mathml/M169">View MathML</a>

(23)

Thus, system (22) satisfies assumptions (A1)-(A3). It follows from Theorem 2.1 that system (22) is exponentially stable in mean square.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors indicated in parentheses made substantial contributions to the following tasks of research: drafting the manuscript (PX); participating in the design of the study (LH); writing of paper (PX).

Acknowledgements

The authors would like to thank the editor and the referees for their helpful comments and valuable suggestions regarding this paper. This work is partially supported by 973 program (2009CB326202), the National Natural Science Foundation of China (11071060, 60835004, 11101133), Fundamental Research Fund for the Central Universities of China (531107040222) and Scientific Research Fund of Hunan Provincial Education Department (Grant No. 12A206).

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