Abstract and Applied Analysis

Volume 2013 (2013), Article ID 325310, 10 pages

http://dx.doi.org/10.1155/2013/325310

Research Article

Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Qi Luo,¹ Xinjie Miao,¹ Qian Wei,¹ and Zhengxin Zhou²

¹School of Information & Control, Nanjing University of Information Science & Technology, Nanjing, Jiangsu 210044, China

²School of Mathematical Science, Yangzhou University, Yangzhou, Jiangsu 225009, China

Received 26 December 2012; Accepted 21 March 2013

Academic Editor: Xuerong Mao

Copyright © 2013 Qi Luo 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.

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.

1. Introduction

Since cellular neural networks (CNNs) were proposed by Chua and Yang in 1988 [1, 2], many researchers have put great effort into this subject due to their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.

Owing to the finite switching speed of amplifiers, there is no doubt that time delays exist in the communication and response of neurons. Moreover, as neural networks usually have a spatial extent due to the presences of a multitude of parallel pathways with a variety of axon sizes and lengths, there is a distribution of conduction velocities along these pathways and a distribution of propagation designed with discrete delays. Therefore, a more appropriate and ideal way is to incorporate continuously distributed delays with a result that a more effective model of cellular neural networks with time-varying and distributed delays proposed.

In fact, beside delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. So far, there have been many results [3–11] on the study of dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs. Summing up the existing researches on the stability of complex CNNs, we see that the primary method is Lyapunov theory. However, there are also lots of difficulties in the applications of corresponding theories to specific problems. It is therefore necessary to seek some new methods to deal with the stability in order to overcome those difficulties.

Recently, it is inspiring that Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable conclusions, for example, see the monograph [12] and the work in [13–24]. In addition, more recently, there have been a few papers where the fixed point theory is employed to investigate the stability of stochastic (delayed) differential equations, for instance, see [25–31]. Precisely, in [26–28], Luo used the fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [29, 30], Sakthivel used the fixed point theory to discuss the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [31], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations. We wonder if we can obtain some new and more applicable stability criteria of complex CNNs by applying the fixed point theory.

With this motivation, in this paper, we aim to discuss the global exponential stability of impulsive CNNs with time-varying and distributed delays. It is worth noting that our research technique is based on the contraction mapping principle rather than the usual method of Lyapunov theory. We deal with, by employing the fixed point theorem, the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time, for which Lyapunov method feels helpless. The obtained stability criteria are easily checked and do not require the differentiability of delays.

2. Preliminaries

Let denote the n-dimensional Euclidean space and represent the Euclidean norm and . 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 networks with time-varying and distributed delays:

where and is the number of neurons in the neural network. corresponds to the state of the th neuron at time . , , and denote the activation functions, respectively. The constant represents the rate with which the th neuron will reset its potential to the resting state when disconnected from the network and external inputs. The constants , , and represent the connection weights of the jth neuron to the th neuron, respectively. and correspond to the transmission delays meeting () and (). The fixed impulsive moments () satisfy and . and stand for the right-hand and left-hand limits of at time , respectively. shows the impulsive perturbation of the th neuron at the impulsive moment .

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

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

where , and .

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

Definition 1. The trivial equilibrium is said to be globally exponentially stable if for any initial condition , there exists a pair of positive constants and such that

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

Theorem 2 (see [32]). 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, for (1)–(3), use the contraction mapping principle to prove the existence and uniqueness of the solution and the global exponential stability of trivial equilibrium all at once. Before proceeding, we firstly introduce some assumptions 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 ,

(A4) There exist nonnegative constants such that for any ,

Let , and let () be the space embracing functions , wherein satisfies the following:

(1) is continuous on (),(2) and exist; moreover, for ,(3) on ,(4) as , where and ,

where and are defined as shown in Section 2. Also is a complete metric space when it is equipped with a metric defined by

where and .

Theorem 3. Assume that conditions (A1)–(A4) hold provided that

(i)there exists a constant such that ,(ii)there exist constants such that for and , (iii) ,

and then the trivial equilibrium is globally exponentially stable.

Proof. Multiplying both sides of (1) with gives, for and ,

which yields after integrating from () to () that

Letting in (12), we have, for (),

Setting () in (13), we get

which generates by letting

Noting , (15) can be rearranged as

Combining (13) and (16), we derive that

is true for (). Hence, we get, for (),

which results in

We therefore conclude, for ,

Note that in (20). We then define the following operator acting on , for :

where () obeys the rule as follows:

on and on .

In what follows, we will apply the contraction mapping principle to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time. The subsequent proof can be divided into two steps.

Step 1. We need to prove that . For (), it is necessary to show that . As defined above, we see that on . Owing to the continuity of on , we immediately know that is continuous on .

Choose a fixed time , and it is then derived from (22) that

where,

Since , we know that is continuous on (); moreover, and exist, in addition, .

Letting () in (23), it is easy to see that as for . Thus, as holds on and ().

Letting () in (23), it is not difficult to find that as for . Letting be small enough, we compute

which implies . Letting be small enough, we have

which implies .

According to the above discussion, we see that is continuous on , while for (), and exist; moreover, .

Next, we will prove that as for . To begin with, we give the expression of as follows:

Where

 , , , , and .

First, it is obvious that as . Furthermore, for (), we see . Then, for any , there exists a such that implies . Choose . It is derived form (A1) that

which leads to as .

Similarly, for the given above, there also exists a such that implies . Select . It follows from (A2) that

which results in as . In addition, it is derived from (A4) that

Since as , we know that, for any , there exists a such that implies . Selecting , it follows from (30) that

which yields as .

Furthermore, from (A3), we see 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 means that as .

Now, we can derive from (27) that as for . It is therefore concluded that which results in .

Step 2. We need to prove that is contractive. For and , we estimate

where

Note that

It is then derived from (36) that

which means that

where

In view of condition (iii), we know that is a contraction mapping, and hence, there exists a unique fixed point of in which means that is the solution to (1)–(3) and as . This completes the proof.

Lemma 4. Assume conditions (A1)–(A4) hold. Provided that

(i), (ii)there exist constants such that for and , (iii) ,

then the trivial equilibrium is globally exponentially stable.

Proof. Lemma 4 is a direct conclusion by letting in Theorem 3.

Remark 5. In Theorem 3, we use the fixed point theorem to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium all at once, while Lyapunov method fails to do this.

Remark 6. The presented sufficient conditions in Theorem 3 and Lemma 4 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

4. Example

Consider the following two-dimensional impulsive cellular neural network with time-varying and distributed delays.

with the initial conditions , on , where , , is defined as shown in (3), , , , , , , , , , , , and .

It is easily to find that , , and . Let and compute

From Theorem 3, we conclude that the trivial equilibrium of this two-dimensional impulsive cellular neural network with time-varying and distributed delays is globally exponentially stable.

5. Conclusions

This article is a new attempt of applying the fixed point theory to the stability analysis of impulsive neural networks with time-varying and distributed delays, which is different from the existing relevant publications where Lyapunov theory is the main technique. From what have been discussed above, we see that the contraction mapping principle is effective for not only the investigation of the existence and uniqueness of solution but also for the stability analysis of trivial equilibrium. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of complex neural networks.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant no. 61174077 and Natural Science Foundation of Jiangsu of China under Grant no. BK2012682.

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