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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 701087, 10 pages
http://dx.doi.org/10.1155/2013/701087
Research Article

Periodic Solution for Impulsive Cellar Neural Networks with Time-Varying Delays in the Leakage Terms

1College of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, Hunan 415000, China
2College of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, China

Received 29 January 2013; Accepted 12 March 2013

Academic Editor: Chuangxia Huang

Copyright © 2013 Bingwen Liu and Shuhua Gong. 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

This paper is concerned with impulsive cellular neural networks with time-varying delays in leakage terms. Without assuming bounded and monotone conditions on activation functions, we establish sufficient conditions on existence and exponential stability of periodic solutions by using Lyapunov functional method and differential inequality techniques. Our results are complement to some recent ones.

1. Introduction

It is well known that impulsive differential equations are mathematical apparatus for simulation of process and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth [13]. Thus, many neural networks with impulses have been studied extensively, and a great deal of literature is focused on the existence and stability of an equilibrium point [47]. In [810], the authors discussed the existence and global exponential stability of periodic solution of a class of cellular neural networks (CNNs) with impulse. Recently, Wang et al. [11] considered the following CNNs with impulses and leakage delays: where are the impulses at moments and is a strictly increasing sequence such that ; and are constants, and , , and are continuous periodic functions with period . Suppose that the following conditions are satisfied. There exist constants , , such that, for any , and for , there exists a constant , such that By using the continuation theorem of coincidence degree theory and a suitable degenerate Lyapunov-Krasovskii functional together with model transformation technique, some results were obtained in [11] to guarantee that all solutions of system (1) converge exponentially to a periodic function. However, to the best of our knowledge, few authors have considered the existence and stability of periodic solutions of system (1) without the assumptions and . Thus, it is worthwhile to continue to investigate the convergence behavior of system (1) in this case. In view of the fact that the coefficients and delays in neural networks are usually time varying in the real world, motivated by the above discussions, in this paper, we will consider the problem on periodic solution of the following impulsive CNNs with time-varying delays in the leakage terms: in which corresponds to the number of units in a neural network, corresponds to the state vector of the th unit at the time , and 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 at the time . and are the connection weights at the time , and denote the transmission delays, denotes the external bias on the th unit at the time , and are activation functions of signal transmission, are the impulses at moments , and is a strictly increasing sequence such that , and . It is obvious that when and is a constant function, (1) is a special case of (4).

The main purpose of this paper is to give the conditions for the existence and exponential stability of the periodic solutions for system (4). By applying Lyapunov functional method and differential inequality techniques, without assuming and , we derive some new sufficient conditions ensuring the existence, uniqueness, and exponential stability of the periodic solution for system (4), which are new and complement previously known results. Moreover, an example is also provided to illustrate the effectiveness of our results.

Throughout this paper, we assume that the following conditions hold.

For , and are continuous periodic functions with period , and for all . In addition, there exist constants , , , , , , and such that

The sequence of times satisfies and , and satisfies for and , where denotes the set of all positive integers.

There exists a such that

For each , there exist nonnegative constants and such that, for all ,

For all and , there exist constants and such that

For convenience, let , in which “” denotes the transposition. We define and . As usual in the theory of impulsive differential equations, at the points of discontinuity of the solution , we assume that . It is clearly that, in general, the derivative does not exist. On the other hand, according to system (4), there exists the limit . In view of the above convention, we assume that .

The initial conditions associated with (4) are assumed to be of the form where denotes a real-valued continuous function defined on .

2. Preliminary Results

The following lemmas will be used to prove our main results in Section 3.

Lemma 1. Let hold. Suppose that is a solution of system (1) with the initial conditions where , . Then

Proof. Assume that (11) does not hold. From , we have So, if , then . Thus, we may assume that there exist and such that According to (4), we get
Calculating the upper left derivative of , together with (13), (14), , and we obtain It is a contradiction and shows that (11) holds. The proof is now completed.

Remark 2. After the conditions , the solution of system (4) always exists (see [1, 2]). In view of the boundedness of this solution, from the theory of impulsive differential equations in [1], it follows that the solution of system (4) with initial conditions (10) can be defined on .

Lemma 3. Suppose that are true. Let be the solution of system (4) with initial value , and let be the solution of system (4) with initial value . Then, there exists a positive constant such that

Proof. Let . Then, for , it is followed by
Define continuous functions by setting Then which, together with the continuity of , implies that we can choose two positive constants and such that
Let Then
We define a positive constant as follows: Let be a positive number such that We claim that Obviously, (27) holds for . We first prove that (27) is true for . Otherwise, there exist and such that one of the following two cases must occur;
Now, we distinguish two cases to finish the proof.
Case (i). If (28) holds. Then, from (21), (23), and , we have
Case (ii). If (29) holds. Then, from (21), (23), and , we get Therefore, (27) holds for . From (24) and (27), we know that Thus, for , we may repeat the above procedure and obtain Further, we have That is,

Remark 4. If is the -periodic solution of system (4), it follows from Lemma 3 that is globally exponentially stable.

3. Main Results

In this section, we will study existence and exponential stability for periodic solutions of system (4).

Theorem 5. Suppose that all conditions in Lemma 3 are satisfied. Then system (4) has exactly one -periodic solution . Moreover, is globally exponentially stable.

Proof. Let be a solution of system (4) with initial conditions (10). By Remark 2, the solution can be defined for all . By hypothesis , we have, for any natural number , Further, by hypothesis of , we obtain Thus, for any natural number , we obtain that is a solution of system (4) for all . Hence, is also a solution of (4) with initial values Then, by the proof of Lemma 3, there exists a constant such that for any natural number , Moreover, for any natural number , we can obtain
Combining (39) with (40), we know that will converge uniformly to a piecewise continuous function on any compact set of .
Now we are in the position of proving that is a -periodic solution of system (4). It is easily known that is -periodic since where . Noting that the right side of (4) is piecewise continuous, together with (36) and (37), we know that converges uniformly to a piecewise continuous function on any compact set of . Therefore, letting on both sides of (36) and (37), we get Thus, is a -periodic solution of system (4).
Finally, by Lemma 3, we can prove that is globally exponentially stable. This completes the proof.

4. An Example

In this section, we give an example to demonstrate the results obtained in the previous sections.

Example 6. Consider the following impulsive cellar neural network consisting of two neurons with time-varying delays in the leakage terms, which is described by Here, it is assumed that the activation functions are
Noting that then we obtain This yields that system (43) satisfies . Hence, from Theorem 5, system (43) has exactly one 2-periodic solution. Moreover, the 2-periodic solution is globally exponentially stable.

Remark 7. Since and CNNs (43) is a very simple form of CNNs with time-varying delays in the leakage terms, it is clear that the conditions and are not satisfied. Therefore, all the results in [1119] and the references therein cannot be applicable to system (43) to obtain the existence and exponential stability of the 2-periodic solutions.

Acknowledgments

The authors would like to express their sincere appreciation to the reviewers for their helpful comments in improving the presentation and quality of the paper. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Hunan Provincial of China (Grant no. 11JJ6006), the Construct Program of the Key Discipline in Hunan Province (Mechanical Design and Theory), the Natural Scientific Research Fund of Hunan Provincial Education Department of China (Grants nos. 11C0916 and 11C0915), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).

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