Research Article | Open Access

# Mean-Square Exponential Stability Analysis of Stochastic Neural Networks with Time-Varying Delays via Fixed Point Method

**Academic Editor:**Naseer Shahzad

#### Abstract

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.

#### 1. Introduction

Cellular neural networks (CNNs), firstly proposed by Chua and Yang in 1988 [1, 2], have become a research focus owing to their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision. Taking into account the finite switching speed of amplifiers in the implementation of neural networks, we see that the time delays are inevitable and therefore a new important model, namely, delayed cellular neural networks (DCNNs), is put forward.

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 [3–12] on the dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs.

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 [13] and the papers [14–25]. Furthermore, this new idea is developed to discuss the stability of stochastic (delayed) differential equations, turning out to be effective for the stability analysis of dynamical systems with delays and stochastic effects; see [26–32]. Specifically, in [27–29], Luo used the fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [30, 31, 33–35], Sakthivel et al. used the fixed point theory to investigate the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [32], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations.

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.

#### 2. Preliminaries

Let be a complete probability space equipped with some filtration satisfying the usual conditions; that is, the filtration is right continuous and contains all -null sets. Let denote a standard Brownian motion defined on . stands for the -dimensional Euclidean space and represents the Euclidean norm. . . corresponds to the space of continuous mappings from the topological space to the topological space .

Consider the following stochastic cellular neural network with time-varying delays: where and is the number of the neurons in the neural network. corresponds to the state of the th neuron at time . ; moreover, is the activation function of the th neuron at time and is the activation function of the th neuron at time . The constant represents the connection weight of the th neuron on the th neuron at time and the constant represents the rate with which the th neuron will reset its potential to the resting state when disconnected from the network and external inputs. The constant represents the connection strength of the th neuron on the th neuron at time , where corresponds to the transmission delay along the axon of the th neuron and satisfies . denotes the diffusion coefficient. and .

Throughout this paper, we always assume that (). Denote by the solution of (1) with the initial condition (2).

*Definition 1. *Equation (1) is said to be globally exponentially stable in mean square if, for any , there exists a pair of positive constants and such that

Lemma 2. *Assume that , where denotes the space consisting of functions satisfying ; then
**
holds for and .*

*Proof. *As
we derive . Hence, it is easy to see

Lemma 3 (see [36]). *Assume that , where denotes the space consisting of functions satisfying , ; then
*

*Remark 4. *In Lemma 3, it is derived from letting that

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

Theorem 5 (see [37]). *Let be a contraction operator on a complete metric space ; then there exists a unique point for which .*

#### 3. Main Results

In this section, we discuss, by means of the contraction mapping principle stated in Theorem 5, the existence and uniqueness as well as global exponential stability of the solution to (1)-(2) in mean-square sense. Before proceeding, we introduce some assumptions as follows.(A1)There exist nonnegative constants such that, for any , (A2)There exist nonnegative constants such that, for any , (A3)There exist nonnegative constants and such that, for any ,

Let , and let () be the space consisting of -adapted processes which satisfy, for fixed , the following:(1) is continuous in mean-square;(2) on ;(3), where is a positive constant satisfying ; here is defined as shown in (2). From Lemma 2, we equip with the norm , where ; thereby is also a complete metric space.

Theorem 6. *Assume that conditions (A1)–(A3) hold. If there exist constants , such that , where
**
then (1) is globally exponentially stable in mean square.*

*Proof. *By Ito formula, we compute the differential of along the solution of (1)-(2):
which yields after integrating from 0 to

Note in (14); hence we define the following operator acting on , for ,
where obeys the following rules:
on and on , .

Now we will, by applying the contraction mapping principle, prove the existence and uniqueness as well as global exponential stability of solution to (1)-(2) in mean-square sense. The subsequent proof can be divided into two steps.*Step 1*. We need to prove . To testify , it is necessary to show the mean-square continuity of on and for . First, in light of the expression of , we have, for a fixed time ,
where
It is obvious that holds. In addition, it is easy to see that for . Furthermore, by Lemma 3, we get
Thus, we know for , which means is continuous in mean square on . Moreover, owing to , we conclude is indeed continuous in mean square on for .

Next, we will prove as for . It is derived from (A1) that
So, , which leads to

In addition, from (A2), we deduce that
Therefore, , which leads to
Moreover, from (A3) and Lemma 3, we get
So,
It then follows from (21)–(25) that as for . We therefore conclude .*Step 2*. We need to prove is contractive. For and , we know that, for ,
where
Note
which implies
which implies
In addition,
which implies

Therefore,
which results in

As , is a contraction mapping and hence there exists a unique fixed point of in which means is the solution of (1)-(2) and as . This completes the proof.

*Remark 7. *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 (1)–(14) in mean-square sense, whereas Lyapunov method fails to do this.

Lemma 8. *Assume that conditions (A1)–(A3) hold. If , where , then (1) is globally exponentially stable in mean square.*

*Proof. *Lemma 8 is the direct corollary of Theorem 6 by choosing and .

*Remark 9. *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.

#### 4. Example

Consider the following two-dimensional stochastic cellular neural network with time-varying delays: with the initial conditions , on , where , , , , , , , . It is easy to see that and . We compute, for , which yields . From Lemma 8, we conclude this two-dimensional stochastic cellular neural network with time-varying delays is mean-square globally exponentially stable.

#### 5. Conclusions

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.

#### Conflict of Interests

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

#### Acknowledgments

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.

#### References

- L. O. Chua and L. Yang, “Cellular neural networks: theory,”
*IEEE Transactions on Circuits and Systems*, vol. 35, no. 10, pp. 1257–1272, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. O. Chua and L. Yang, “Cellular neural networks: applications,”
*IEEE Transactions on Circuits and Systems*, vol. 35, no. 10, pp. 1273–1290, 1988. View at: Publisher Site | Google Scholar | MathSciNet - J. Cao, “New results concerning exponential stability and periodic solutions of delayed cellular neural networks,”
*Physics Letters A*, vol. 307, no. 2-3, pp. 136–147, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. P. Civalleri, M. Gilli, and L. Pandolfi, “On stability of cellular neural networks with delay,”
*IEEE Transactions on Circuits and Systems I*, vol. 40, no. 3, pp. 157–165, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Cao, “A set of stability criteria for delayed cellular neural networks,”
*IEEE Transactions on Circuits and Systems I*, vol. 48, no. 4, pp. 494–498, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. T. Stamov and I. M. Stamova, “Almost periodic solutions for impulsive neural networks with delay,”
*Applied Mathematical Modelling*, vol. 31, no. 7, pp. 1263–1270, 2007. View at: Publisher Site | Google Scholar - S. Ahmad and I. M. Stamova, “Global exponential stability for impulsive cellular neural networks with time-varying delays,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 69, no. 3, pp. 786–795, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Li, X. Zhang, and Z. Li, “Global exponential stability of impulsive cellular neural networks with time-varying and distributed delay,”
*Chaos, Solitons and Fractals*, vol. 41, no. 3, pp. 1427–1434, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Qiu, “Exponential stability of impulsive neural networks with time-varying delays and reaction-diffusion terms,”
*Neurocomputing*, vol. 70, no. 4-6, pp. 1102–1108, 2007. View at: Publisher Site | Google Scholar - X. Wang and D. Xu, “Global exponential stability of impulsive fuzzy cellular neural networks with mixed delays and reaction-diffusion terms,”
*Chaos, Solitons & Fractals*, vol. 42, no. 5, pp. 2713–2721, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Zhang and Q. Luo, “Global exponential stability of impulsive delayed reaction-diffusion neural networks via Hardy-Poincarè inequality,”
*Neurocomputing*, vol. 83, pp. 198–204, 2012. View at: Publisher Site | Google Scholar - Y. Zhang and Q. Luo, “Novel stability criteria for impulsive delayed reaction-diffusion Cohen-Grossberg neural networks via Hardy-Poincarè inequality,”
*Chaos, Solitons & Fractals*, vol. 45, no. 8, pp. 1033–1040, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton,
*Stability by Fixed Point Theory for Functional Differential Equations*, Dover, New York, NY, USA, 2006. View at: MathSciNet - L. C. Becker and T. A. Burton, “Stability, fixed points and inverses of delays,”
*Proceedings of the Royal Society of Edinburgh A*, vol. 136, no. 2, pp. 245–275, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton, “Fixed points, stability, and exact linearization,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 61, no. 5, pp. 857–870, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton, “Fixed points, Volterra equations, and Becker's resolvent,”
*Acta Mathematica Hungarica*, vol. 108, no. 3, pp. 261–281, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton, “Fixed points and stability of a nonconvolution equation,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 12, pp. 3679–3687, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton, “Perron-type stability theorems for neutral equations,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 55, no. 3, pp. 285–297, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton, “Integral equations, implicit functions, and fixed points,”
*Proceedings of the American Mathematical Society*, vol. 124, no. 8, pp. 2383–2390, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton and T. Furumochi, “Krasnoselskii's fixed point theorem and stability,”
*Nonlinear Analysis. Theory, Methods & Applications A*, vol. 49, no. 4, pp. 445–454, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. A. Burton and B. Zhang, “Fixed points and stability of an integral equation: nonuniqueness,”
*Applied Mathematics Letters*, vol. 17, no. 7, pp. 839–846, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Furumochi, “Stabilities in FDEs by Schauder's theorem,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5-7, pp. e217–e224, 2005. View at: Publisher Site | Google Scholar - C. Jin and J. Luo, “Fixed points and stability in neutral differential equations with variable delays,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 3, pp. 909–918, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. N. Raffoul, “Stability in neutral nonlinear differential equations with functional delays using fixed-point theory,”
*Mathematical and Computer Modelling*, vol. 40, no. 7-8, pp. 691–700, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Zhang, “Fixed points and stability in differential equations with variable delays,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 63, no. 5-7, pp. e233–e242, 2005. View at: Publisher Site | Google Scholar - J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 334, no. 1, pp. 431–440, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Luo, “Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays,”
*Journal of Mathematical Analysis and Applications*, vol. 342, no. 2, pp. 753–760, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Luo, “Stability of stochastic partial differential equations with infinite delays,”
*Journal of Computational and Applied Mathematics*, vol. 222, no. 2, pp. 364–371, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Luo and T. Taniguchi, “Fixed points and stability of stochastic neutral partial differential equations with infinite delays,”
*Stochastic Analysis and Applications*, vol. 27, no. 6, pp. 1163–1173, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Sakthivel and Y. Ren, “Exponential stability of second-order stochastic evolution equations with Poisson jumps,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 12, pp. 4517–4523, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Sakthivel, P. Revathi, and N. I. Mahmudov, “Asymptotic stability of fractional stochastic neutral differential equations with infinite delays,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 769257, 9 pages, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Luo, “Fixed points and exponential stability for stochastic Volterra-Levin equations,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 3, pp. 934–940, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Mathiyalagan, R. Sakthivel, and S. Marshal Anthoni, “Exponential stability result for discrete-time stochastic fuzzy uncertain neural networks,”
*Physics Letters A*, vol. 376, no. 8-9, pp. 901–912, 2012. View at: Publisher Site | Google Scholar - K. Mathiyalagan, R. Sakthivel, and S. M. Anthoni, “New robust exponential stability results for discrete-time switched fuzzy neural networks with time delays,”
*Computers & Mathematics with Applications*, vol. 64, no. 9, pp. 2926–2938, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Sakthivel, R. Samidurai, and S. M. Anthoni, “New exponential stability criteria for stochastic BAM neural networks with impulses,”
*Physica Scripta*, vol. 82, no. 4, Article ID 045802, 2010. View at: Publisher Site | Google Scholar - S. Hu, C. Huang, and F. Wu,
*Stochastic Differential Equations*, Science Press, Beijing, China, 2008. - D. R. Smart,
*Fixed Point Theorems*, Cambridge University Press, Cambridge, UK, 1980.

#### Copyright

Copyright © 2014 Tianxiang Yao and Xianghong Lai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.