- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2012 (2012), Article ID 358362, 14 pages

http://dx.doi.org/10.1155/2012/358362

## Globally Exponential Stability of Periodic Solutions to Impulsive Neural Networks with Time-Varying Delays

^{1}College of Science, Guilin University of Technology, Guangxi, Guilin 541004, China^{2}Guizhou Key Laboratory of Economics System, Guizhou College of Finance and Economics, Guizhou, Guiyang 550004, China

Received 12 January 2012; Accepted 26 February 2012

Academic Editor: Josef Diblík

Copyright © 2012 Yuanfu Shao et al. 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.

#### Abstract

By using Schaeffer's theorem and Lyapunov functional, sufficient conditions of the existence and globally exponential stability of positive periodic solution to an impulsive neural network with time-varying delays are established. Applications, examples, and numerical analysis are given to illustrate the effectiveness of the main results.

#### 1. Introduction

It is well known that in implementation of neural networks, time delays are inevitably encountered because of the finite switching speed of amplifiers. Specially in electronic neural networks, delays are usually time-varying and often become sources of instability. So it is important to investigate the dynamics of neural networks with delays [1–7]. Recently, the study of the existence of periodic solutions of neural networks has received much attention. The common approaches are based on using Mawhin continuation theorem [1, 2, 8–10], Banach’s fixed point theorem [11–13], fixed point theorem in a cone [14], Schaeffer’s theorem [15, 16], and so on. On the other hand, studies on neural dynamical systems not only involve the existence of periodic solutions, but also involve other dynamical behaviors such as stability of periodic solutions, bifurcations, and chaos. In recent years, the stability of solutions of neural networks has attracted attention of many researchers and many nice results have been obtained [1–3, 5–13, 16–25]. For example, M. Tan and Y. Tan [1] considered the following neural network with variable coefficients and time-varying delays: By using the Mawhin continuation theorem, they discussed the existence and globally exponential stability of periodic solutions.

However, in real world, many physical systems often undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. In fact, impulsive differential equation represents a more natural framework for mathematical modelling of many real world phenomena such as population dynamic and neural networks. The theory of impulsive differential equations is now being recognized to be richer than the corresponding theory of differential equations without impulse, and various kinds of impulsive differential equations have been extensively studied, see [8–16, 18, 19, 21–25] and references therein. Then, considering impulsive effects, it is necessary and interesting for us to study further the dynamics of system (1.1). Furthermore, as pointed by Gopalsamy and Sariyasa [4], it would be of great interest to study neural networks in periodic environment. On the other hand, to the best of our knowledge, few authors considered the existence of periodic solutions by using Schaeffer’s theorem. Hence, in this paper, by using Schaeffer’s theorem and Lyapunov functional, we aim to discuss the existence and exponential stability of periodic solutions to a class of impulsive neural networks with periodic coefficients and time-varying delays. The model is as follows: with initial conditions where corresponds to the state of the th unit, represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, denotes the output of the th unit, and denote the strength of the th unit on the th unit, respectively, is the external bias on the th unit, corresponds to the transmission delay along the axon of the th unit, denotes the impulsive moment, and is a strictly increasing sequence such that , denotes the Banach space of continuous mapping from to equipped with the norm for all , where .

Throughout this paper, we always assume the following.() are all continuous -periodic functions for .() is continuous and there exists positive constant such that for any and .()There exists positive integer such that . Then For convenience, we use the following notations: where is continuous and -periodic function.

The rest of this paper is organized as follows. In Section 2, by using Schaeffer’s theorem, sufficient conditions of the existence of -periodic solution to system (1.2) with initial conditions (1.3) are established. In Section 3, by using Lyapunov functional, we derive the conditions under which the periodic solution is globally exponentially stable. In Section 4, applications, illustrative examples, and simulations are given to show the effectiveness of the main results. Finally, some conclusions are drawn in Section 5.

#### 2. Existence of Periodic Solution

First we make some preparations. As usual in the theory of impulsive differential equation, by a solution of model (1.2), it means the following.(i) is piecewise continuous such that exists, and is differentiable on for .(ii) satisfies (1.2) for .

*Definition 2.1. *The set is said to be quasi-equicontinuous in if for any , there exists such that, if and , then .

Lemma 2.2 (see [26, Compactness criterion]). * The set is relatively compact if and only if*(i)* is bounded, that is, for each and some ,*(ii)* is quasi-equicontinuous in .*

The following lemma is fundamental to our discussion. The method is similar to that of [13, 16], so the proof is omitted here.

Lemma 2.3. * is an -periodic solution of system (1.2) which is equivalent to is an -periodic solution of the following equation:
**
where , and
**It is easy to show that and**
where and .*

Lemma 2.4 (see [27, Schaeffer’s theorem]). *Let be a normed space and be a compact operator. Define
**
Then either*(i)*set is unbounded, or*(ii)*operator has a fixed point in .**In order to use Lemma 2.4, let**
with the norm , then is a Banach space.**Define a mapping by , where and**
By Lemma 2.3, it is easy to see that the existence of -periodic solution of (1.2) is equivalent to the existence of fixed point of the mapping in .*

Theorem 2.5. *Suppose that – hold. Further,*()*. **Then system (1.2) admits an -periodic solution.*

*Proof. *By Lemma 2.3, it suffices to prove that the mapping admits a fixed point in .

For any constant , let . For , from (2.3) and , we have
It implies that is uniformly bounded.

For any , we have
If , it is obvious that . Hence, from (2.7) and (2.8), we have
Therefore, is a family of uniformly bounded and equicontinuous subset. By Lemma 2.2, the mapping is compact.

Let , and considering the following operator equation:
If is a solution of (2.10), then
According to , we deduce that
It implies that is bounded, which is independent of . By Lemma 2.4, we obtain that the mapping admits a fixed point in . Hence system (1.2) admits an -periodic solution such that . This completes the proof.

#### 3. Globally Exponentially Stable

In this section, the sufficient conditions ensuring that (1.2) admits a unique -periodic solution and all solutions of (1.2) exponentially converge to the unique -periodic solution are to be established.

*Definition 3.1. *Let be an -periodic solution of system (1.2) with initial value . If there exist constants , for every solution of (1.2) with initial , such that
then is said to be globally exponentially stable.

Theorem 3.2. *Suppose that – hold. Further,*()* ,*()* ,**where is a constant, is a constant determined in (3.5).**Then system (1.2) admits a unique -periodic solution, which is globally exponentially stable.*

*Proof. *By Theorem 2.5, system (1.2) admits an -periodic solution with initial value . Let be an arbitrary solution of (1.2) with initial value . Define and , then we have
By , we have for . Let
It is clear that is continuous on and . In addition,
and , then is strictly monotone increasing. Therefore, there exists a unique such that for . Let
then
Obviously, for and the above , we have
where .

Define by
In view of (3.2) and (3.8), for , we have
We claim that
If not, then there exist and such that
for . Then, it follows from (3.9) and (3.11) that
Equation (3.12) leads to
which contradicts (3.6). Thus (3.10) holds, that is,
If , we have
for . Similar to the steps of (3.10)–(3.14), we can derive that
If , then
By repeating the same procedure, then
It follows from that , which leads to
for any . So the combination (3.18) and (3.19) gives
In addition, it is clear that
Therefore, from (3.20) and (3.21), for any , we have
It implies that the -periodic solution of (1.2) is globally exponentially stable. Hence, (1.2) admits a unique -periodic solution, which is globally exponentially stable. This completes the proof.

*Remark 3.3. *Theorem 3.2 implies that the impulse affects the existence and exponential stability of the periodic solution of system (1.2). It shows the dynamics of impulsive differential system (1.2) is richer than the corresponding system (1.1) without impulse.

#### 4. Applications and Examples

In (1.2), if , then (1.2) reads: For system (4.1), we have the following result.

Proposition 4.1. *Suppose that – hold. Further,*()*,*()*,*()*,
**where is a constant, is determined in Theorem 3.2.**Then system (4.1) admits a unique -periodic solution, which is globally exponentially stable.*

If the impulses are absent in system (1.2), that is, , then (1.2) leads to (1.1). Similarly we have the following.

Proposition 4.2. *Suppose that – hold. Further,*()* ,*()* ,**then system (1.1) admits a unique -periodic solution, which is globally exponentially stable.*

*Remark 4.3. *Proposition 4.2 implies that the sufficient conditions of the existence and globally exponential stability of periodic solution to (1.1) are independent of the time-varying delays, while the corresponding results obtained by authors [5] are dependent on delays. Without effect from time-varying delays, our results are better for people to keep the stability of system (1.1). Although the authors [1] also established similar conditions which are independent of delays, their employed tool and analysis techniques are very different so that their main results are different from ours. Particularly, (1.1) is the special case of (1.2) without impulse. Hence, in this sense, results of this paper complement or improve some previously known results [1, 5].

Finally, two examples and numerical analysis are given to show the usefulness of the main results.

*Example 4.4. *Let
where , , . Then .

By easy computation, , and , which implies holds. On the other hand, it is easy to verify that holds. By verification, , namely, holds too. From Theorems 2.5 and 3.2, we obtain that (4.2) has a unique 2-periodic solution, which is globally exponentially stable, see Figure 1.

*Example 4.5. *Let
where for , ,,, , , , . Then .

By computation, , which implies that holds. It is easy to verify that holds too. From Proposition 4.2, system (4.3) has a unique 1-periodic solution, which is globally exponentially stable, see Figure 2. However, by calculation, conditions of the results of [1] fail, then one cannot obtain the existence of periodic solution of system (4.3) by results of reference [1], which further shows that the results complement or improve previously known results.

#### 5. Conclusions

In this paper, the existence and globally exponential stability of the periodic solution of system (1.2) are studied. Model (1.2) is very general, including such models as continuous bidirectional associative memory networks, cellular neural networks, and Hopfield-type neural networks (see, e.g., [6, 7, 28]). The main methods employed here are Schaeffer’ theorem, differential inequality techniques, and Lyapunov functional, which are very different from [1]. The sufficient conditions obtained here are new and complement or improve the previously known results [1, 5–7]. Finally, applications, two illustrative examples and simulations, are given to show the effectiveness of the main results.

#### Acknowledgments

The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which are very useful for improving the quality of this paper. This paper is supported by National Natural Science Foundation of China (11161015) and Doctoral Foundation of Guilin University of Technology (2010).

#### References

- M. Tan and Y. Tan, “Global exponential stability of periodic solution of neural network with variable coefficients and time-varying delays,”
*Applied Mathematical Modelling*, vol. 33, no. 1, pp. 373–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Wu, “Exponential convergence of BAM neural networks with time-varying coefficients and distributed delays,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 1, pp. 562–573, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - I. A. Dzhalladova, J. Baštinec, J. Diblík, and D. Y. Khusainov, “Estimates of exponential stability for solutions of stochastic control systems with delay,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 920412, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Gopalsamy and Sariyasa, “Time delays and stimulus-dependent pattern formation in periodic environments in isolated neurons,”
*IEEE Transactions on Neural Networks*, vol. 13, no. 3, pp. 551–563, 2002. View at Publisher · View at Google Scholar · View at Scopus - L. Huang, C. Huang, and B. Liu, “Dynamics of a class of cellular neural networks with time-varying delays,”
*Physics Letters, Section A*, vol. 345, no. 4–6, pp. 330–344, 2005. View at Publisher · View at Google Scholar · View at Scopus - Y. Li, L. Zhu, and P. Liu, “Existence and stability of periodic solutions of delayed cellular neural networks,”
*Nonlinear Analysis: Real World Applications*, vol. 7, no. 2, pp. 225–234, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - B. Liu and L. Huang, “Existence and exponential stability of periodic solutions for cellular neural networks with time-varying delays,”
*Physics Letters, Section A*, vol. 349, no. 6, pp. 474–483, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. Shao and Y. Zhou, “Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 74, no. 4, pp. 1107–1118, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. Bai, “Global exponential stability and existence of periodic solution of Cohen-Grossberg type neural networks with delays and impulses,”
*Nonlinear Analysis: Real World Applications*, vol. 9, no. 3, pp. 747–761, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Shao and B. Dai, “The existence of exponential periodic attractor of impulsive BAM neural network with periodic coefficients and distributed delays,”
*Neurocomputing*, vol. 73, no. 16–18, pp. 3123–3131, 2010. View at Publisher · View at Google Scholar · View at Scopus - Q. Zhou, “Global exponential stability of BAM neural networks with distributed delays and impulses,”
*Nonlinear Analysis: Real World Applications*, vol. 10, no. 1, pp. 144–153, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Xia and P. J. Y. Wong, “Global exponential stability of a class of retarded impulsive differential equations with applications,”
*Chaos, Solitons and Fractals*, vol. 39, no. 1, pp. 440–453, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Yang and J. Cao, “Stability and periodicity in delayed cellular neural networks with impulsive effects,”
*Nonlinear Analysis: Real World Applications*, vol. 8, no. 1, pp. 362–374, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Li, X. Zhang, and D. Jiang, “A new existence theory for positive periodic solutions to functional differential equations with impulse effects,”
*Computers & Mathematics with Applications*, vol. 51, no. 12, pp. 1761–1772, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. J. Nieto, “Impulsive resonance periodic problems of first order,”
*Applied Mathematics Letters*, vol. 15, no. 4, pp. 489–493, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - X. Yang, “Existence and global exponential stability of periodic solution for Cohen-Grossberg shunting inhibitory cellular neural networks with delays and impulses,”
*Neurocomputing*, vol. 72, no. 10–12, pp. 2219–2226, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. Diblík, D. Ya. Khusainov, I. V. Grytsay, and Z. Šmarda, “Stability of nonlinear autonomous quadratic discrete systems in the critical case,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 539087, 23 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Asymptotic stability of stochastic delayed recurrent neural networks with impulsive effects,”
*Journal of Optimization Theory and Applications*, vol. 147, no. 3, pp. 583–596, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Sakthivel, R. Samidurai, and S. M. Anthoni, “Exponential stability for stochastic neural networks of neural type with impulsive effects,”
*Modern Physics Letters B*, vol. 24, no. 11, pp. 1099–1110, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Baštinec, J. Diblík, D. Ya. Khusainov, and A. Ryvolová, “Exponential stability and estimation of solutions of linear differential systems of neutral type with constant coefficients,”
*Boundary Value Problems*, vol. 2010, Article ID 956121, 20 pages, 2010. View at Zentralblatt MATH - R. Samidurai, R. Sakthivel, and S. M. Anthoni, “Global asymptotic stability of BAM neural networks with mixed delays and impulses,”
*Applied Mathematics and Computation*, vol. 212, no. 1, pp. 113–119, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Raja, R. Sakthivel, S. M. Anthoni, and H. Kim, “Stability of impulsive Hopfield neural networks with Markovian switching and time-varying delays,”
*International Journal of Applied Mathematics and Computer Science*, vol. 21, no. 1, pp. 127–135, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. Shao, “Exponential stability of periodic neural networks with impulsive effects and time-varying delays,”
*Applied Mathematics and Computation*, vol. 217, no. 16, pp. 6893–6899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. Chen and B. Cui, “Impulsive effects on global asymptotic stability of delay BAM neural networks,”
*Chaos, Solitons and Fractals*, vol. 38, no. 4, pp. 1115–1125, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Wu and C. Shan, “Stability analysis for periodic solution of BAM neural networks with discontinuous neuron activations and impulses,”
*Applied Mathematical Modelling*, vol. 33, no. 6, pp. 2564–2574, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - D. Bainov and P. Simeonov,
*Impulsive Defferential Equatios: Periodic Solution and Applications*, Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 66, 1993. - D. R. Smart,
*Fixed Point Theorems*, Cambridge University Press, London, UK, 1974. - S. Mohamad and K. Gopalsamy, “Dynamics of a class of discrete-time neural networks and their continuous-time counterparts,”
*Mathematics and Computers in Simulation*, vol. 53, no. 1-2, pp. 1–39, 2000. View at Publisher · View at Google Scholar