Journal of Applied Mathematics

Volume 2013 (2013), Article ID 516293, 6 pages

http://dx.doi.org/10.1155/2013/516293

Research Article

Existence and Global Uniform Asymptotic Stability of Almost Periodic Solutions for Cellular Neural Networks with Discrete and Distributed Delays

Zongyi Hou,¹ Hongying Zhu,¹ and Chunhua Feng²

¹Department of Mathematics, Hechi University, Yizhou 546300, China

²Department of Mathematics and Statistics, Guangxi Normal University, Guiling 541004, China

Received 29 September 2013; Revised 26 November 2013; Accepted 14 December 2013

Academic Editor: Samir Saker

Copyright © 2013 Zongyi Hou 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 paper discusses the existence and global uniform asymptotic stability of almost periodic solutions for cellular neural networks (CNNS). By utilizing the theory of the almost periodic differential equation and the Lyapunov functionals method, some sufficient conditions are obtained to ensure the existence and global uniform asymptotic stability. An example is given to illustrate the effectiveness of the main results.

1. Introduction

Cellular neural networks (CNNS) are composed of a large number of simple processing units (called neurons), widely interconnected to form a complex network system. It reflects many basic features of the human brain functions. It is a highly complicated nonlinear dynamics system and has successful applications in many fields such as associative, signal, and image processing, pattern recognition, and optimization.

In 1984, Hopfield proposed that the dynamic behavior of neurons should be described with a set of ordinary differential equations or functional differential equations. Since then, a lot of research achievements have been published in the world.

Recently, many scholars have paid much attention to the research on the dynamics and applications of CNNS. Specially, some scholars have studied the existence and stability of almost periodic solution for neural networks, which can be seen from [1–10] and therein references.

In [4], without product systems, by utilizing the generalized Halanay inequality technique and combining the theory of exponential dichotomy with fixed point method, Huang et al. study the existence and global exponential stability of almost periodic solutions for recurrent neural network with continuously distributed delays as follows:

where are incentive functions, which satisfy .   are all almost periodic functions.

In [5], Xiang and Cao discuss the following system:

Without product systems, by using the Lyapunov functionals method and analytical skills, the results about the existence, attractivity, and exponential stability of almost periodic solutions for the system (2) are obtained.

However, a more general system than the systems above is discussed in this paper. We consider the existence and global uniform asymptotic stability of almost periodic solutions to the CNNS with discrete and continuously distributed delays. The system is as follows:

By using Lemmas 3 and 4 in the next section and under the less restrictive conditions, some sufficient conditions are obtained to ensure the existence and global uniform asymptotic stability of almost periodic solutions to the system (3). An example is given to illustrate the effectiveness of the main results at last.

2. Preliminaries

In order to facilitate the following section description, we introduce some marks and basic definitions in this section.

If is an almost periodic in uniformly for , where is an open set, then the equation

is called lagging-type almost periodic differential equation. The following system is defined as the product systems of (4):

Definition 1 (see [11]). If there is a constant for such that the solution of (5) through when satisfies , then the solution is bound.

Definition 2 (see [11]). Lyapunov functionals . Suppose that the solution of (5) through is , is defined as . The total derivative is defined as follows:

Then is the right derivative of functionals along (5).

Lemma 3 (see [11]). Lagging-type almost periodic differential equation (5) has an asymptotically almost periodic solution , which satisfies or for all defined in ; then (5) has an almost periodic solution.

Lemma 4 (see [11]). There is a continuous functional of for such that

(Ha),(Hb),(Hc),

where is a positive constant, and are continuous and non-decreasing, when , , and is a positive constant. At this time, if (5) has a bounded solution such that , where , then (5) in has a unique almost periodic solution which is global uniform asymptotic stability.

Throughout this paper, we make the following assumptions.

(H2.1) is uniform almost periodic continuous function to about . for all , and we denote . In addition, also satisfies the Lipschitz condition as follows:

 where .(H2.2), , and are almost periodic continuous functions. we denote constants, respectively, as follows:

(H2.3)Functions , , and are bounded continuous functions, and they satisfy the following Lipschitz conditions:

(H2.4)Delay kernel functions satisfying

3. Main Results

Theorem 5. Assume that (H2.1)–(H2.4) hold; then all solutions of system (3) are bounded.

Proof. Let , , and set . From system (3) and the assumption (H2.3), we get

for . By (11), we obtain

This shows that

This completes the proof of the theorem.

From Theorem 5, all solutions of system (3) are bounded. In order to investigate the globally uniform asymptotic stability of the almost periodic solutions, we assume that .

Theorem 6. Assume that (H2.1)–(H2.4) hold, and suppose further that

Then, in system (3), there exists an almost periodic solution, which is global uniform asymptotic stable.

Proof. From the condition (H2.1), we rewrite system (3) as follows:

The product system of the system (15) is the following form:

In order to apply Lemma 4, we construct the Lyapunov functionals about the product system (16) as follows:

For convenience sake, we denote

where

Using (H2.1) and , by the triangle inequality, we have

Calculating the upright derivative of along system (16) as follows:

Similarly, we calculate the upright derivatives of and along system (16), respectively, as follows:

Note that

Combining with (21) and (22) and the assumptions of Theorem 5, we get

By Theorem 5 and Lemmas 3 and 4, there exists an almost periodic solution of system (3), which is global uniform asymptotic stable. This completes the proof of Theorem 6.

4. An Example

Example 1. Consider the following cellular neural network which consists of two neurons:

where

We select the functions and the kernel functions . Then, . Because the periods of and are and , respectively. The quotient of and is irrational. Then system (25) is an almost periodic system. In addition, and

From , we have ; then we get .

It is easy for us to verify that the conditions (H2.1)–(H2.4) in Theorem 5 hold. Therefore, in the system (25), there exists an almost periodic solution, which is global uniform asymptotic stable.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11361010 and 11161018); the Institutions of Higher Learning in Guangxi of China, Scientific Research Fund Project (201204LX391); the Scientific Research Project in Guangxi of China, Department of Education (201106LX613).

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