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We employ the new method of fixed point theory to study the stability of a class of impulsive cellular neural networks with infinite delays. Some novel and concise sufficient conditions are presented ensuring the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium at the same time. These conditions are easily checked and do not require the boundedness and differentiability of delays.

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [

Due to the finite switching speed of neurons and amplifiers in the implementation of neural networks, it turns out that the time delays should not be neglected, and therefore, the model of delayed cellular neural networks (DCNNs) is put forward, which is naturally of better realistic significances. In fact, besides delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. Accordingly many experts are showing a growing interest in the research on the dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs, with a result of many achievements [

Synthesizing the reported results about complex CNNs, we find that the existing research methods for dealing with stability are mainly based on Lyapunov theory. However, we also notice that there are still lots of difficulties in the applications of corresponding results to specific problems; correspondingly it is necessary to seek some new techniques to overcome those difficulties.

Encouragingly, in recent few years, Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable results; for example, see the monograph [

Naturally, for complex CNNs which have high application values, we wonder if we can utilize the fixed point theory to investigate their stability, not just the existence and uniqueness of solution. With this motivation, in the present paper, we aim to discuss the stability of impulsive CNNs with infinite delays via the fixed point theory. It is worth noting that our research skill is the contraction mapping theory which is different from the usual method of Lyapunov theory. We employ the fixed point theorem to prove the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium all at once. Some new and concise algebraic criteria are provided, and these conditions are easy to verify and, moreover, do not require the boundedness and differentiability of delays.

Let

In this paper, we consider the following impulsive cellular neural network with infinite delays:

Throughout this paper, we always assume that

Denote by

The solution

The trivial equilibrium

The trivial equilibrium

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

Let

In this section, we will consider the existence and uniqueness of solution and the asymptotic stability of trivial equilibrium by means of the contraction mapping principle. Before proceeding, we introduce some assumptions listed as follows.

There exist nonnegative constants

There exist nonnegative constants

There exist nonnegative constants

Let

In what follows, we will give the main result of this paper.

Assume that conditions (A1)–(A3) hold. Provided that

there exists a constant

there exist constants

Multiplying both sides of (

Letting

Noting

Combining (

Note

The subsequent part is the application of the contraction mapping principle, which can be divided into two steps.

First, since

Owing to

Consequently, when

On the other hand, as

According to the above discussion, we find that

Next, we will prove

Due to

On the other hand, since

Furthermore, from (A3), we know that

As

From (

Note

It hence follows from (

Therefore,

In view of condition (iii), we see

To obtain the asymptotic stability, we still need to prove that the trivial equilibrium

Suppose there exists

So

Assume that conditions (A1)–(A3) hold. Provided that

there exist constants

Corollary

In Theorem

The presented sufficient conditions in Theorems

Provided that

Assume that conditions (A1)-(A2) hold. Provided that

Consider the following two-dimensional impulsive cellular neural network with infinite delays:

It is easy to see that

This work is devoted to seeking new methods to investigate the stability of complex neural networks. From what has been discussed above, we find that the fixed point theory is feasible. With regard to a class of impulsive cellular neural networks with infinite delays, we utilize the contraction mapping principle to deal with the existence and uniqueness of solution and the asymptotic analysis of trivial equilibrium at the same time, for which Lyapunov method feels helpless. Now that there are different kinds of fixed point theorems and complex neural networks, our future work is to continue the study on the application of fixed point theory to the stability analysis of complex neural networks.

This work is supported by the National Natural Science Foundation of China under Grants 60904028, 61174077, and 41105057.