Almost automorphic funtions on time scales and almost automorphic solutions to shunting inhibitory cellular neural networks on time scales

Abstract

In this paper, we first propose a new concept of almost periodic time scales, a new definition of almost automorphic functions on almost periodic time scales, and study some their basic properties. Then we prove a result ensuring the existence of an almost automorphic solution for both the linear nonhomogeneous dynamic equation on time scales and its associated homogeneous equation, assuming that the associated homogeneous equation admits an exponential dichotomy. Finally, as an application of our results, we establish the existence and global exponential stability of almost automorphic solutions to a class of shunting inhibitory cellular neural networks with time-varying delays on time scales. Our results about the shunting inhibitory cellular neural network with time-varying delays on time scales are new even for the both cases of differential equations(the time scale $\mathbb{T}=\mathbb{R})$ and difference equations(the time scale $\mathbb{T}=\mathbb{Z})$.