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In this paper, using the algebraic structure of the Abelian group, we introduce the concept of a matched space for time scales, and we construct the algebraic structure of matched spaces to solve the closedness of time scales under non-translational shifts. Using a matched space for time scales, a new concept of periodic time scales is introduced. Based on it, new concepts of periodic functions, almost periodic functions and almost automorphic functions whose concepts were defined on translations of their arguments are proposed through non-translational shifts. The results in this paper provide new methods to consider periodic solution, almost periodic solution and almost automorphic solutions for q-difference equations and others on irregular time scales via the background of the algebraic structure.

A matched space for time scales and applications to the study on functions