Existence and Global Uniform Asymptotic Stability of Pseudo Almost Periodic Solutions for Cohen-Grossberg Neural Networks with Discrete and Distributed Delays

Since the model of Cohen-Grossberg neural networks (CGNNs) was first proposed and studied by Cohen and Grossberg [1], it has been widely investigated because of the theoretical interest as well as the application considerations such as optimization, pattern recognition, automatic control, image processing, and associative memories. In recent years, there are many important results on dynamic behaviors of CGNNs. For instance, many sufficient conditions have been successively obtained to ensure the existence and stability of equilibrium point of CGNNs [1–10]. Some attractivity and asymptotic stability results have also been published [3, 11–14]. Many authors specially devote themselves to study the existence and global exponential stability of periodic or almost periodic solution to CGNNs [15–30]; for the other dynamic properties, see also the literatures [31, 32]. However, to the best of our knowledge, few authors have discussed the existence and the global uniform asymptotic stability of pseudo almost periodic solutions to CGNNs. In this paper, we discuss the existence and the global uniform asymptotic stability of pseudo almost periodic solutions to the following CGNNs: x 󸀠

In this paper, we discuss the existence and the global uniform asymptotic stability of pseudo almost periodic solutions to the following CGNNs: where   (),   (),   (),   (), Φ  () ∈ (, ),   () ∈ (,  + ) are pseudo almost periodic functions.The organization of this paper is as follows.In Section 2, some basic definitions, marks, and lemmas are given.In Section 3, some results are given to ascertain the existence of pseudo almost periodic solution to the system (1) by 2 Mathematical Problems in Engineering applying Schauder fixed point theorem.In Section 4, the global uniform asymptotic stability of pseudo almost periodic solutions to the system (1) is obtained.In Section 5, an example is provided to demonstrate the effectiveness of the main results.In Section 6, the final conclusions are drawn.

Preliminaries
In this section, some basic definitions, lemmas, and assumptions are introduced.
Definition 7. Assume that  * () is a pseudo almost periodic solution of system (1).By a translation transformation () = ()− * (), system ( 1) is transformed into a new system.If the zero solution of new system is globally uniformly asymptotically stable, then the pseudo almost periodic solution of system ( 1) is said to be globally uniformly asymptotically stable.As for the uniform asymptotical stability, see [35].

The Existence of Pseudo Almost Periodic Solution
In this section, we study the existence of pseudo almost periodic solution to system (1).It follows from (2.1) that the antiderivative of 1/  (  ) exists.Then we choose an antiderivative   (  ) of 1/  (  ) that satisfies   (0) = 0. Clearly,    (  ) = 1/  (  ).Because   (  ) > 0,   (  ) is increasing about   and the inverse function . Substituting these equations into system (1), we get the following equivalent equation: From ( 14), we get , where 0 ≤   ≤ 1. Putting it into (14), we obtain Thus, system (1) has at least one pseudo almost periodic solution if and only if the system (15) has at least one pseudo almost periodic solution.So we only consider the pseudo almost periodic solution of system (15).By Lagrange theorem, we have Again by (2.1),we get Combined with (2.2),we have In order to prove the main results, we give the following lemma.
Let  = {() ∈  : ‖‖ ≤ }, where  > 0 to be any constant.We denote  = max 1≤≤ {( where Therefore,  is equicontinuous.By the Ascoli-Arzela theorem, the operator  is compact; then it is completely continuous.By the Schauder fixed point theorem, the system (1) has at least one pseudo almost periodic solution.

The Global Uniform Asymptotic Stability of Pseudo Almost Periodic Solution
In order to discuss the global uniform asymptotic stability of pseudo almost periodic solution to system (1), we give the following assumptions: Theorem 11.Assume that (2.1)-(2.5)and (4.1)-(4.2) hold; then the pseudo almost periodic solution of system (1) is globally uniformly asymptotically stable.
By Lemma 8, the pseudo almost periodic solutions of system (1) are globally uniformly asymptotically stable.This completes the proof.