1. Introduction

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 [1]). 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 [7] and Wang et al. [8] 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

Proof. Let be a solution of system (1.1) with any initial value ,and let In view of (1.1), we have Let 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: where .
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 .