Almost periodic functions on Hausdorff almost periodic time scales

This paper is devoted to generalizing the notion of almost periodic functions on time scales. We introduce a new class of almost periodic time scales called Hausdorff almost periodic time scales by using the Hausdorff distance and propose a more general notion of almost periodic functions on these new time scales. Then we explore some properties of Hausdorff almost periodic time scales and prove that the family of almost periodic functions on Hausdorff almost periodic time scales is a Banach space. Especially, our analysis also indicates that a function on a Hausdorff almost periodic time scale is almost periodic if and only if its affine extension is Bohr almost periodic on the real numbers R. As an application, we establish the existence of almost periodic solutions for a single species model on Hausdorff almost periodic time scales.