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Discrete Dynamics in Nature and Society
Volume 2017, Article ID 1526478, 9 pages
https://doi.org/10.1155/2017/1526478
Research Article

Almost Automorphic Functions on the Quantum Time Scale and Applications

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Yongkun Li; nc.ude.uny@eilky

Received 28 September 2017; Accepted 23 November 2017; Published 14 December 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Yongkun Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers; by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.