Abstract

This paper is concerned with the problem of the existence, uniqueness, and global exponential stability for neutral-type cellular neural networks with distributed delays. Based on fixed point theory and Lyapunov functional, several sufficient conditions are established for the existence, uniqueness, and global exponential stability of almost periodic solution for the above system. Finally, a simple example is given to illustrate the feasibility and effectiveness of our main results.

1. Introduction

It is well known that the delayed cellular neural networks (DCNNs) have been successfully applied to many practical problems, such as signal and image processing, pattern recognition, and optimization. Hence, they have been investigated extensively by many researchers in recent years. In particular, the stability and the existence of almost periodic solutions are two important properties, which have close relation to the applications of neural networks, so they have been widely studied. Therefore, there have been extensive results on the existence and stability of periodic solutions and almost periodic solutions of delayed cellular neural networks in the literature. We refer the reader to [1–12] and the references cited therein for more details.

As pointed out in [13], in the biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process. Moreover, in many cases the existing neural network models cannot characterize the properties of a neural reaction process precisely. To solve this problem, it is naturally introducing the neutral networks. To the best of our knowledge, however, the problem of global exponential stability of almost periodic solution for neutral-type cellular neural networks has not been fully investigated in the literature.

In this paper, we consider the following neutral-type cellular neural networks with distributed delays: where denotes the number of units in a neural network, corresponds to the state of the th unit at time , represents the passive decay rates at time , , , and are the neuronal output signal functions, , , , and denote the strength of the th unit on the th unit at time , denotes the external inputs at time , and , , , , , and are all almost periodic functions, . The kernel functions are continuous with and satisfy where and are continuous functions in and , .

Let and be the space of continuous functions and continuously differential functions which map into , respectively. Especially, , . For any bounded function , , .

The initial conditions associated with system (1) are of the form

We list some assumptions which will be used in this paper.(H₁).(H₂)There exist some positive constants , , and such that for  all ,  .(H₃), where

The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Sections 3 and 4, by using a fixed point theorem and constructing suitable Lyapunov functional, we obtain some sufficient conditions ensuring existence, uniqueness, and global exponential stability of almost periodic solution of system (1). Finally, an example is given to illustrate that our results are feasible.

2. Preliminaries

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

Definition 1 (see [14]). is called almost periodic, if for any it is possible to find a real number and for any interval with length there exists a number in this interval such that , . The collection of those functions is denoted by .

Definition 2 (see [14]). Let and let be a continuous matrix defined on . The linear system is said to be an exponential dichotomy on if there exist constants , projection , and the fundamental matrix satisfying

Lemma 3 (see [14]). If the linear system has an exponential dichotomy, then almost periodic system has a unique almost periodic solution which can be expressed as follows:

Lemma 4 (see [14]). Let be an almost periodic function and
Then the linear system admits an exponential dichotomy, where .

Definition 5. The almost periodic solution of system (1) with the initial value is said to be globally exponentially stable, if there exist constants and , for any solution of system (1) with initial value such that where

Lemma 6 (see [15, 16]). Let be a Banach space. Assume that is an open bounded subset of ; is completely continuous satisfying then has a fixed point in .

Let and with the norm where ,  ,  ,  . Then is a Banach space with the norm .

Let , where

Define an open bounded subset in by

By Lemmas 3 and 4, system (1) has a unique almost periodic solution which can be expressed as follows: where , where , .

Let the map be defined by

Lemma 7. is completely continuous.

Proof. First, we show that is continuous. Assume that ; then which implies that where ,  . Since where ,  , so
For arbitrary , taking , when , one has which implies that is continuous.
Next, we show that maps bounded set onto itself. Assume is a positive constant and . By the almost periodicity of system (1), there exists a constant such that
From the above analysis, we get that which imply that is uniformly bounded. In addition, notice that
Using Arzela-Ascoli theorem, we obtain that is relatively compact. Hence, is completely continuous. The proof of this lemma is complete.

3. Existence of Almost Periodic Solution

In this section, we study the existence of almost periodic solutions of system (1).

Theorem 8. Assume that ()–() hold; then system (1) admits at least one almost periodic solution.

Proof. Consider the following nonlinear operator:
For we have . From Lemma 7, is completely continuous. Similar to the arguments as that in Lemma 7, it follows from the definition of and that which yields that where . Hence,
So
By Lemma 6, there exists at least a fixed point satisfying , which implies system (1) has at least one almost periodic solution. This completes the proof.