Research Article | Open Access

# Asymptotic Stability of Impulsive Cellular Neural Networks with Infinite Delays via Fixed Point Theory

**Academic Editor:**Qi Luo

#### Abstract

We employ the new method of fixed point theory to study the stability of a class of impulsive cellular neural networks with infinite delays. Some novel and concise sufficient conditions are presented ensuring the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. These conditions are easily checked and do not require the boundedness and differentiability of delays.

#### 1. Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have become a hot topic for their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.

Due to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out that the time delays should not be neglected, and therefore, the model of delayed cellular neural networks (DCNNs) is put forward, which is naturally of better realistic significances. In fact, besides delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. Accordingly many experts are showing a growing interest in the research on the dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs, with a result of many achievements [3–9] obtained.

Synthesizing the reported results about complex CNNs, we find that the existing research methods for dealing with stability are mainly based on Lyapunov theory. However, we also notice that there are still lots of difficulties in the applications of corresponding results to specific problems; correspondingly it is necessary to seek some new techniques to overcome those difficulties.

Encouragingly, in recent few years, Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results; for example, see the monograph [10] and papers [11–22]. In addition, more recently, there have been a few publications where the fixed point theory is employed to deal with the stability of stochastic (delayed) differential equations; see [23–29]. Particularly, in [24–26], Luo used the fixed point theory to study the exponential stability of mild solutions to stochastic partial differential equations with bounded delays and with infinite delays. In [27, 28], Sakthivel used the fixed point theory to investigate the asymptotic stability in th moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [29], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations.

Naturally, for complex CNNs which have high application values, we wonder if we can utilize the fixed point theory to investigate their stability, not just the existence and uniqueness of solution. With this motivation, in the present paper, we aim to discuss the stability of impulsive CNNs with infinite delays via the fixed point theory. It is worth noting that our research skill is the contraction mapping theory which is different from the usual method of Lyapunov theory. We employ the fixed point theorem to prove the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium all at once. Some new and concise algebraic criteria are provided, and these conditions are easy to verify and, moreover, do not require the boundedness and differentiability of delays.

#### 2. Preliminaries

Let denote the -dimensional Euclidean space and let represent the Euclidean norm. . . corresponds to the space of continuous mappings from the topological space to the topological space .

In this paper, we consider the following impulsive cellular neural network with infinite delays: where and is the number of neurons in the neural network. corresponds to the state of the th neuron at time . denote the activation functions, respectively. corresponds to the known transmission delay satisfying and as . Denote . The constant represents the connection weight of the th neuron on the th neuron at time . The constant denotes the connection strength of the th neuron on the th neuron at time . The constant represents the rate with which the ith neuron will reset its potential to the resting state when disconnected from the network and external inputs. The fixed impulsive moments () satisfy and . and stand for the right-hand and left-hand limits of at time , respectively. shows the abrupt change of at the impulsive moment and .

Throughout this paper, we always assume that for and . Thereby, problem (1) and (2) admits a trivial equilibrium .

Denote by the solution to (1) and (2) with the initial condition where and . Denote .

The solution of (1)–(3) is, for the time variable , a piecewise continuous vector-valued function with the first kind discontinuity at the points (), where it is left continuous; that is, the following relations are valid:

*Definition 1. * The trivial equilibrium is said to be stable, if, for any , there exists such that for any initial condition satisfying:

*Definition 2. * The trivial equilibrium is said to be asymptotically stable if the trivial equilibrium is stable, and for any initial condition , holds.

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

Theorem 3 (see [30]). * 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 will consider the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium by means of the contraction mapping principle. Before proceeding, we introduce some assumptions listed 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 such that, for any ,

Let , and let () be the space consisting of functions , where satisfies the following: (1) is continuous on ();(2) and exist; furthermore, for ; (3) on ;(4) as ;here () and () are defined as shown in Section 2. Also is a complete metric space when it is equipped with the following metric: where and .

In what follows, we will give the main result of this paper.

Theorem 4. *Assume that conditions (A1)–(A3) hold. Provided that *(i)*there exists a constant such that ,*(ii)*there exist constants such that for and , *(iii)*, *(iv)*, where ,**then the trivial equilibrium is asymptotically stable. *

*Proof. * Multiplying both sides of (1) with gives, for and ,
which yields after integrating from () to ()

Letting in (11), we have
for (). Setting () in (12), we get
which generates by letting

Noting , (14) can be rearranged as

Combining (12) and (15), we reach that
is true for (). Further,
holds for (). Hence,
which produces, for ,

Note in (19). We then define the following operator acting on , for :
where obeys the rules as follows:
on and on .

The subsequent part is the application of the contraction mapping principle, which can be divided into two steps. *Step **1*. We need to prove . Choosing (), it is necessary to testify .

First, since on and , we know is continuous on . For a fixed time , it follows from (21) that
where

Owing to , we see that is continuous on (); moreover, and exist, and .

Consequently, when () in (22), it is easy to find that as for , and so is continuous on the fixed time ().

On the other hand, as () in (22), it is not difficult to find that as for . Furthermore, if letting be small enough, we derive
which implies as . While lettingtend to zero gives
which yields as .

According to the above discussion, we find that is continuous on (); moreover, and exist; in addition, .

Next, we will prove as . For convenience, denote
where , , , and.

Due to (), we know . Then for any , there exists a such that implies . Choose . It is derived from (A1) that, for ,
Moreover, as , we can find a for the given such that implies , which leads to
namely,

On the other hand, since as , we get . Then for any , there also exists a such that implies . Select . It follows from (A2) that
which results in

Furthermore, from (A3), we know that . So

As , we have . Then for any , there exists a nonimpulsive point such that implies . It then follows from conditions (i) and (ii) that
which produces

From (30), (32), and (35), we deduce as for . We therefore conclude that () which means .*Step **2*. We need to prove is contractive. For and , we estimate
where ,.

Note

It hence follows from (37) that
which implies

Therefore,

In view of condition (iii), we see is a contraction mapping, and, thus there exists a unique fixed point of in which means the transposition of is the vector-valued solution to (1)–(3) and its norm tends to zero as .

To obtain the asymptotic stability, we still need to prove that the trivial equilibrium is stable. For any , from condition (iv), we can find satisfying such that . Let . According to what has been discussed above, we know that there exists a unique solution to (1)–(3); moreover,
here , , , and .

Suppose there exists such that and as . It follows from (41) that
As
we obtain .

So . This contradicts the assumption of . Therefore, holds for all . This completes the proof.

Corollary 5. * Assume that conditions (A1)–(A3) hold. Provided that *(i)*, *(ii)*there exist constants such that for and , *(iii)*, *(iv)*, where ,**then the trivial equilibrium is asymptotically stable.*

*Proof. * Corollary 5 is a direct conclusion by letting in Theorem 4.

*Remark 6. *In Theorem 4, we can see it is the fixed point theory that deals with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, while Lyapunov method fails to do this.

*Remark 7. *The presented sufficient conditions in Theorems 4 and Corollary 5 do not require even the boundedness and differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

Provided that , (1) and (2) will become the following cellular neural network with infinite delays and without impulsive effects: where , , , , , , and are the same as defined in Section 2. Obviously, (44) also admits a trivial equilibrium . From Theorem 4, we reach the following.

Theorem 8. * Assume that conditions (A1)-(A2) hold. Provided that*(i)*, *(ii)*, where ,**then the trivial equilibrium is asymptotically stable. *

#### 4. Example

Consider the following two-dimensional impulsive cellular neural network with infinite delays: with the initial conditions , on , where , , , , , , , , and .

It is easy to see that , , and . Let and compute where . From Theorem 4, we conclude that the trivial equilibrium of this two-dimensional impulsive cellular neural network with infinite delays is asymptotically stable.

#### 5. Conclusions

This work is devoted to seeking new methods to investigate the stability of complex neural networks. From what has been discussed above, we find that the fixed point theory is feasible. With regard to a class of impulsive cellular neural networks with infinite delays, we utilize the contraction mapping principle to deal with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, for which Lyapunov method feels helpless. Now that there are different kinds of fixed point theorems and complex neural networks, our future work is to continue the study on the application of fixed point theory to the stability analysis of complex neural networks.

#### Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grants 60904028, 61174077, and 41105057.

#### 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 - 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 | Zentralblatt MATH - S. Ahmad and I. M. Stamova, “Global exponential stability for impulsive cellular neural networks with time-varying delays,”
*Nonlinear Analysis: Theory, Methods & Applications*, 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 and 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 and Fractals*, vol. 45, no. 8, pp. 1033–1040, 2012. View at: Publisher Site | Google Scholar | 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*, 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*, 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*, 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 & Applications*, vol. 63, no. 5–7, pp. e217–e224, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - 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 & Applications*, vol. 63, no. 5–7, pp. e233–e242, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Luo, “Fixed points and stability of neutral stochastic delay differential equations,”
*Journal of Mathematical Analysis & 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 J. Luo, “Asymptotic stability of impulsive stochastic partial differential equations with infinite delays,”
*Journal of Mathematical Analysis and Applications*, vol. 356, no. 1, pp. 1–6, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. Sakthivel and J. Luo, “Asymptotic stability of nonlinear impulsive stochastic differential equations,”
*Statistics & Probability Letters*, vol. 79, no. 9, pp. 1219–1223, 2009. 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 - D. R. Smart,
*Fixed Point Theorems*, Cambridge University Press, Cambridge, UK, 1980. View at: MathSciNet

#### Copyright

Copyright © 2013 Yutian Zhang and Yuanhong Guan. 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.