Abstract
We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided. Furthermore, the concepts of almost periodic functions are redefined under the sense of this new timescale concept. Finally, almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. Besides, as an application, we apply our results to a class of high-order Hopfield neural networks.
Highlights
Almost periodicity is a recent concept in the literature of time scales
We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided
Of [7], all invariant under translations time scales are periodic time scales, that is, from Example 3.9 to Example 3.11, which indicate that we investigated almost periodic problems of dynamic equations under the periodic time scales in the past, and all the obtained results are valid for all periodic time scales, for two special periodic time scales: T = R and T = Z. This method can unify the continuous and discrete situations effectively, whether or not there exists a time scale which is almost periodic but not periodic if we introduce a new concept of almost periodic time scales
Summary
Received 2 June 2014; Revised 10 July 2014; Accepted 13 July 2014; Published 7 August 2014. We introduce three equivalent concepts of almost periodic time scales as a further study of the corresponding concept proposed in Li and Wang (2011) and several examples of almost periodic time scales which are not periodic are provided. The concepts of almost periodic functions are redefined under the sense of this new timescale concept. Almost periodicity of Cauchy matrix for dynamic equations is proved under these new definitions. Based on these results, the existence of almost periodic solutions to a class of nonlinear dynamic equations is investigated by the almost periodicity of Cauchy matrix on almost periodic time scales. As an application, we apply our results to a class of high-order Hopfield neural networks
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