Discontinuous generalized double-almost-periodic functions on almost-complete-closed time scales

In this paper, we introduce the concept of almost-complete-closed time scales (ACCTS) that allows independent variables of functions to possess almost-periodicity under translations. For this new type of time scale, a class of piecewise functions with double-almost-periodicity is proposed and studied. Based on these, concepts of weighted pseudo-double-almost-periodic functions (WPDAP) in Banach spaces and a translation-almost-closed set are introduced. Further, we prove that the function space WPDAP₀ affiliated to WPDAP is a translation-almost-closed set. Then, by introducing the concept of almost-uniform convergence for piecewise functions on ACCTS and using measure theory on time scales, some composition theorems of WPDAP and the completeness of the function space are proved.