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Exponential convergence of Cohen-Grossberg neural networks with continuously distributed leakage delays
Journal of Inequalities and Applications volume 2014, Article number: 48 (2014)
Abstract
This paper is concerned with the global exponential convergence of Cohen-Grossberg neural networks with continuously distributed leakage delays. By using the Lyapunov functional method and differential inequality techniques, we propose a new approach to establishing some sufficient conditions ensuring that all solutions of the networks converge exponentially to the zero point. Our results complement some recent ones.
MSC: 34C25, 34K13, 34K25.
1 Introduction
It is well known that Cohen-Grossberg neural networks (CGNNs) have been successfully applied in many fields such as pattern recognition, parallel computing, associative memory, and combinatorial optimization (see [1–5]). Such applications heavily depend on the global exponential convergence behaviors, because the exponential convergent rate can be unveiled. Many good results on the problem of the global exponential convergence of the equilibriums and periodic solutions of for CGNNs are given in the literature. We refer the reader to [6–13] and the references cited therein. Recently, in real applications, a typical time delay called leakage (or ‘forgetting’) delay has been introduced in the negative feedback terms of the neural network system, and these terms are variously known as forgetting or leakage terms (see [14–16]). Subsequently, Gopalsamy [17] investigated the stability on the equilibrium for the bidirectional associative memory (BAM) neural networks with constant delay in the leakage term. Following this, the authors of [18–22] dealt with the existence and stability of equilibrium and periodic solutions for neuron networks model involving constant leakage delays. In particular, Peng [23] established some delay dependent criteria for the existence and global attractive periodic solutions of the bidirectional associative memory neural network with continuously distributed delays in the leakage terms. However, to the best of our knowledge, few authors have considered the exponential convergence behavior for all solutions of CGNNs with continuously distributed delays in the leakage terms. Motivated by the arguments above, in the present paper, we shall consider the following CGNNs with time-varying coefficients and continuously distributed delays in the leakage terms:
where and are continuous functions on , , , , , , and are continuous functions on R; n corresponds to the number of units in a neural network; denotes the potential (or voltage) of cell i at time t; represents an amplification function; is an appropriately behaved function; and denote the strengths of connectivity between cell i and j at time t, respectively; the activation functions and show how the i th neuron reacts to the input, corresponds to the transmission delays, and correspond to the transmission delay kernels, and denotes the i th component of an external input source introduced from outside the network to cell i at time t for .
Throughout this paper, for , it will be assumed that and are continuous functions, and there exist constants , , , and such that
We also make the following assumptions.
(H1) For each , there exist nonnegative constants β, α, and such that
(H2) For , there exist positive constants and such that
(H3) For , , and there exist positive constants and such that
(H4) For all and , there exist constants and such that
and
(H5) (), .
The initial conditions associated with system (1.1) are of the form
where denotes a real-valued bounded continuous function defined on .
The remaining part of this paper is organized as follows. In Section 2, we present some new sufficient conditions to ensure that all solutions of CGNNs (1.1) with initial conditions (1.4) converge exponentially to the zero point. In Section 3, we shall give some examples and remarks to illustrate our results obtained in the previous sections.
2 Main results
Theorem 2.1 Let (H1)-(H5) hold. Then, for every solution of CGNNs (1.1) with initial conditions (1.4), there exists a positive constant K such that
Proof Let be a solution of system (1.1) with initial conditions (1.4), and let
In view of (1.1), we have
Let
From (1.2) and (H5), we can choose a positive constant such that
Then it is easy to see that
We now claim that
Otherwise, one of the following two cases must occur.
-
(1)
There exist and such that
(2.5) -
(2)
There exist and such that
(2.6)
Now, we distinguish two cases to finish the proof.
Case (1). If (2.5) holds. Then, from (2.1), (2.3), and (H1)-(H4), we have
This contradiction implies that (2.5) does not hold.
Case (2). If (2.6) holds, then, from (2.1), (2.3), and (H1)-(H4), by using a similar argument as in Case (1), we can derive a contradiction, which shows that (2.6) does not hold.
Therefore, (2.4) is proved and
This implies that the proof of Theorem 2.1 is now completed. □
3 An example
Example 3.1 Consider the following CGNNs with time-varying delays in the leakage terms:
where , , .
It follows that
and
Define a continuous function by setting
According to the continuity of and , we can choose constants and such that
which implies that the CGNNs (3.1) satisfied (H1)-(H5). Hence, from Theorem 2.1, all solutions of the CGNNs (3.1) with initial value converge exponentially to the zero point .
Remark 3.1 It is easy to check that the results in [17–23] and [24–34] are invalid for the global exponential convergence of (3.1), since the leakage delays are continuously distributed.
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Acknowledgements
The authors would like to express the 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 nos. 51375160, 11201184), and the Scientific Research Fund of Hunan Provincial Natural Science Foundation of China (grant no. 12JJ3007).
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ZC gave the proof of Theorem 2.1 and drafted the manuscript. SG proved and gave the example to illustrate the effectiveness of the obtained results. All authors read and approved the final manuscript.
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Chen, Z., Gong, S. Exponential convergence of Cohen-Grossberg neural networks with continuously distributed leakage delays. J Inequal Appl 2014, 48 (2014). https://doi.org/10.1186/1029-242X-2014-48
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DOI: https://doi.org/10.1186/1029-242X-2014-48