Almost periodic functions and generalizations on complete-closed time scales

This Chapter mainly deals with almost periodic functions and its generalizations on complete-closed time scales under translations and it is divided into six sections and is organized as follows. In Sect. 3.1, some basic results of almost periodic functions are established which include their Bochner and Bohr form. In Sect. 3.2, the definitions of Bohr-Transform and Mean-Value of uniformly almost periodic functions are introduced and their corresponding results are presented. In Sect. 3.3, under the complete closedness of time scales, generalized pseudo almost periodic functions are introduced and some basic properties are investigated. In Sect. 3.4, we introduce the concept of Π-semigroup and moving operators and provide some of their fundamental properties. In Sect. 3.5, two equivalent definitions of relatively dense set on a group are introduced and discussed. Section 3.6 establishes properties of abstract weighted pseudo almost periodic functions. In Sect. 3.7, we introduce almost periodic functions on changing-periodic time scales and establish some basic properties under which almost periodic problems on an arbitrary time scales without complete closedness can be considered.