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

Volume 8, Issue 6, pp 1190--1211 http://dx.doi.org/10.22436/jnsa.008.06.27

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Authors

Yongkun Li - Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People's Republic of China. Bing Li - Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People's Republic of China. Xiaofang Meng - Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, People's Republic of China.

Abstract

In this paper, we first propose a new definition of almost automorphic functions on almost periodic time scales and study some of 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 latter 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 networks with time-varying delays on time scales are new both for the case of differential equations (the time scale \(\mathbb{T} = \mathbb{R}\)) and difference equations (the time scale \(\mathbb{T} = \mathbb{Z}\)).

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Keywords

• Time scales
• Almost automorphic functions
• Dynamic equations
• Exponential dichotomy.

MSC

•  34N05
•  34K14
•  43A60
•  92B20

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