Complete-closed time scales under shifts and related functions

Abstract

Abstract In this paper, we introduce the concept of complete-closed time scales under translational and non-translational shifts and propose new definitions of special functions arising from dynamic equations on time scales including the concepts of almost periodic functions, almost automorphic functions and Stepanov almost automorphic functions. All the functions introduced in the paper are not only effective on periodic time scales under translations but are also valid on irregular time scales like qZ‾ $\overline{q^{\mathbb{Z}}}$, (−q)Z‾ $\overline{(-q)^{\mathbb{Z}}}$ and N±12 $\mathbb{N}^{\frac{1}{2}}_{\pm }$, etc.

Keywords

• Time scales
• Complete-closed
• Shift functions