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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 941063, 11 pages
Exponential Convergence for Cellular Neural Networks with Time-Varying Delays in the Leakage Terms
1School of Science, Hunan University of Technology, Hunan, Zhuzhou 412000, China
2College of Mathematics, Physics and Information Engineering, Jiaxing University, Zhejiang, Jiaxing 314001, China
Received 18 August 2012; Accepted 3 October 2012
Academic Editor: Narcisa C. Apreutesei
Copyright © 2012 Zhibin Chen and Junxia Meng. 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.
We consider a class of cellular neural networks with time-varying delays in the leakage terms. By applying Lyapunov functional method and differential inequality techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.
It is well known that the delayed cellular neural networks (CNNs) have been successfully applied to signal and image processing, pattern recognition, and optimization (see ). Hence, they have been the object of intensive analysis by numerous authors in the past decades. In particular, extensive results on the problem of the existence and stability of the equilibrium point for CNNs are given out in many works in the literature. We refer the reader to [2–6] and the references cited therein. Recently, to consider CNNs with the incorporation of time delays in the leakage terms, Gopalsamy  and Wang et al.  investigated a class of CNNS described by 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 leakage delay and transmission delay, respectively, denotes the external bias on the th unit at the time , and are activation functions of signal transmission, and .
Suppose that the following conditions and are constants, where , for each , there exists a nonnegative constant such that are satisfied. Avoiding the continuously distributed delay terms, the authors of [7, 8] obtained that all solutions of system (1.1) converge to the equilibrium point or the periodic solution. However, to the best of our knowledge, few authors have considered the convergence behavior for all solutions of system (1.1) without the assumptions and . Thus, it is worthwhile to continue to investigate the convergence behavior of system (1.1) in this case.
The main purpose of this paper is to give the new criteria for the convergence behavior for all solutions of system (1.1). By applying Lyapunov functional method and differential inequality techniques, without assuming and , we derive some new sufficient conditions ensuring that all solutions of system (1.1) converge exponentially to zero point. Moreover, an example is also provided to illustrate the effectiveness of our results.
Throughout this paper, for , it will be assumed that and are continuous functions, and there exist constants and such that We also assume that the following conditions , , and hold: for each , there exist nonnegative constants and such that for all and , there exist constants and such that .
The initial conditions associated with system (1.1) are of the form where denotes real-valued-bounded continuous function defined on .
2. Main Results
Theorem 2.1. Let , , and hold. Then, for every solution of system (1.1) with any initial value , there exists a positive constant such that
Let be a solution of system (1.1) with any initial value ,and let
In view of (1.1), we have
From (1.3), , and , we can choose a positive constant such that
Then, it is easy to see that
We now claim thatIf this is not valid, then, one of the following two cases must occur:(1)there exist and such that
(2)there exist and such that
Now, we consider two cases.
Case i. If (2.8) holds. Then, from (2.3), (2.5), and ()−(), we have This contradiction implies that (2.8) does not hold.
Case ii. If (2.9) holds. Then, from (2.3), (2.5), and ()−(), we get which is a contradiction and yields that (2.9) does not hold.
Consequently, we can obtain that (2.7) is true. Thus, This implies that the proof of Theorem 2.1 is now completed.
3. An Example
Example 3.1. Consider the following CNNs with time-varying delays in the leakage terms:
Noting that Define a continuous function by setting Then, we obtain Therefore, which, together with the continuity of , implies that we can choose positive constants and such that for all , there holds This yields that system (3.1) satisfied , , and . Hence, from Theorem 2.1, all solutions of system (3.1) converge exponentially to the zero point .
Remark 3.2. Since , and CNNs (3.1) are 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 [7–9] and the references therein cannot be applicable to prove that all solutions of system (3.1) converge exponentially to the zero point.
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 Hunan Provincial National Natural Science Foundation of China (12JJ3007), the Natural Scientific Research Fund of Zhejiang Provincial of China (Grants nos. Y6110436, LY2A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant no. Z201122436).
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