Compactness criteria and new impulsive functional dynamic equations on time scales

In this paper, we introduce the concept of Δ-sub-derivative on time scales to define ε-equivalent impulsive functional dynamic equations on almost periodic time scales. To obtain the existence of solutions for this type of dynamic equation, we establish some new theorems to characterize the compact sets in regulated function space on noncompact intervals of time scales. Also, by introducing and studying a square bracket function [x(⋅),y(⋅)]:T→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[x(\cdot),y(\cdot) ]:\mathbb{T}\rightarrow\mathbb{R}$\end{document} on time scales, we establish some new sufficient conditions for the existence of almost periodic solutions for ε-equivalent impulsive functional dynamic equations on almost periodic time scales. The final section presents our conclusion and further discussion of this topic.