Almost automorphic functions on semigroups induced by complete-closed time scales and application to dynamic equations

In this paper, we introduce the concepts of Bochner and Bohr almost automorphic functions on the semigroup induced by complete-closed time scales and their equivalence is proved. Particularly, when \begin{document}$ \Pi = \mathbb{R}^{+} $\end{document} (or \begin{document}$ \Pi = \mathbb{R}^{-} $\end{document} ), we can obtain the Bochner and Bohr almost automorphic functions on continuous semigroup, which is the new almost automorphic case on time scales compared with the literature [ 20 ] (W.A. Veech, Almost automorphic functions on groups, Am. J. Math., Vol. 87, No. 3 (1965), pp 719-751) since there may not exist inverse element in a semigroup. Moreover, when \begin{document}$ \Pi = h\mathbb{Z}^{+},\,h>0 $\end{document} (or \begin{document}$ \Pi = h\mathbb{Z}^{-},\,h>0 $\end{document} ), the corresponding automorphic functions on discrete semigroup can be obtained. Finally, we establish a theorem to guarantee the existence of Bochner (or Bohr) almost automorphic mild solutions of dynamic equations on semigroups induced by time scales.

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