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

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