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This work addresses the stability study for stochastic cellular neural networks with time-varying delays. By utilizing the new research technique of the fixed point theory, we find some new and concise sufficient conditions ensuring the existence and uniqueness as well as mean-square global exponential stability of the solution. The presented algebraic stability criteria are easily checked and do not require the differentiability of delays. The paper is finally ended with an example to show the effectiveness of the obtained results.

Cellular neural networks (CNNs), firstly proposed by Chua and Yang in 1988 [

On the other hand, it is noteworthy that, besides delay effects, stochastic and impulsive as well as diffusion effects are also likely to exist in the neural networks. Up to now, there have been a mass of works [

Referring to the current publications of complex CNNs, we note that Lyapunov theory is always the primary method for the stability analysis. However the unavoidable reality is that there also exist lots of difficulties in the application of corresponding results to specific problems. So it does seem that some new methods are needed to resolve those difficulties.

Encouragingly, the fixed point theory is successfully applied by Burton and other authors to investigate the stability of deterministic systems, followed by some valid conclusions presented; for example, see the monograph [

The motivation of this paper is discussing the feasibility of using the fixed point theory to tackle the stability research of complex CNNs and thereupon enlarging the applications of the fixed point theory as well as enriching the stability theory of complex CNNs. In detail, via Banach contraction mapping principle, studied in this paper is the mean-square global exponential stability of stochastic delayed CNNs. Remarkably, Banach contraction mapping principle is far different from Lyapunov method. By establishing a new inequality, we first construct a proper Banach space and thereby investigate, in mean-square sense, the existence and uniqueness as well as global exponential stability of the solution simultaneously. The obtained results show that, in regard to the stability research of complex CNNs, the fixed point theory does work and has its own advantage; namely, it works with no need for Lyapunov functions. Some algebraic stability criteria are finally presented, which are easily checked and do not require the differentiability of delays, let alone the monotone decreasing behavior of delays.

Let

Consider the following stochastic cellular neural network with time-varying delays:

Throughout this paper, we always assume that

Equation (

Assume that

As

Assume that

In Lemma

The consideration of this paper is based on the following fixed point theorem.

Let

In this section, we discuss, by means of the contraction mapping principle stated in Theorem

There exist nonnegative constants

There exist nonnegative constants

There exist nonnegative constants

Let

here

Assume that conditions (A1)–(A3) hold. If there exist constants

By Ito formula, we compute the differential of

Note

Now we will, by applying the contraction mapping principle, prove the existence and uniqueness as well as global exponential stability of solution to (

Next, we will prove

In addition, from (A2), we deduce that

Therefore,

As

The main idea of this proof is based on the fixed point theory rather than Lyapunov method. By using Banach contraction mapping principle with no need for Lyapunov functions, we simultaneously explore the existence and uniqueness as well as global exponential stability of solution to (

Assume that conditions (A1)–(A3) hold. If

Lemma

The obtained algebraic stability criteria are easily checked and do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

Consider the following two-dimensional stochastic cellular neural network with time-varying delays:

The main contribution of this work is confirming the feasibility of utilizing the fixed point theory to address the stability research of complex CNNs and thereby enlarging the applications of the fixed point theory as well as enriching the stability theory of complex CNNs. Specifically, by Banach contraction mapping principle with no need for Lyapunov functions, we complete the proof of the existence and uniqueness as well as global exponential stability of solution to stochastic delayed neural networks simultaneously, whereas Lyapunov method cannot do this. The derived algebraic stability criteria are novel and easily checked and do not require the differentiability of delays. As we all know, the fixed point theory has various forms, for example, Krasnosleskii’s fixed point theorem. Considering many mathematical models can be transformed into a linear part and other nonlinear parts, our future work is trying to explore the application of Krasnosleskii’s fixed point theorem to the stability analysis of complex CNNs.

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

This work is supported by the National Natural Science Foundation of China under Grant nos. 71171116 and 60904028, Humanities and Social Sciences Foundation of Ministry of Education of China under Grant no. 09YJC630129, Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, and “China’s Manufacturing Industry Development Academy”—Key Philosophy and Social Science Research Center of University in Jiangsu Province.