A classification of time scales and analysis of the general delays on time scales with applications

Abstract

In this paper, we present five classes/categories of time scales. Then, on each class, we introduce and analyze delays that not only lead to new types of delay systems on time scales but also reveal the limitations of the known results in the literature. To show the importance and significance of our analysis, several examples are illustrated. We conclude our paper with some interesting open problems.

Weighted piecewise pseudo double-almost periodic solution for impulsive evolution equations

“…Further, if let T be a bi-direction periodic time scale, i.e., ε 1 → 0, then Definition 3.4 will become the concept of piecewise almost periodic functions on periodic time scales from [33]. For more properties and applications related to these functions and dynamic systems, one should consult [30,31].…”

“…A time scale with such a type of "complete closedness" assists in conquering the difficulties of defining and studying functions on time scales. However, there exists a type of time scales that has "almost-complete closedness" but not "complete closedness" (see [28,30]). This type of time scales can describe the "double almost periodicity" of an object's status with almost periodic dynamic behavior in the real world and it is more general and comprehensive than periodic time scales.…”

“…Next, we introduce the concept of almost-complete closedness time scales which is more general than the concept of almost periodic time scales proposed in [28,30,31].…”

“…If ±τ ∈ E{T, ε 1 } in Definition 2.6 from [33], then there exists A 1 , which implies that almost periodic time scales introduced in [30,31] are bi-direction ACCTS with inf T = −∞, sup T = +∞. For more important comments and remarks of almost periodic time scales, one may consult [3,33].…”

Periodicity, almost periodicity for time scales and related functions

“…Recently, many papers were devoted to almost periodic issues on time scales (see for examples [1][2][3][4][5]). In 2014, to study the approximation property of time scales, Wang and Agarwal introduced some new concepts of almost periodic time scales in [6] and the results show that this type of time scales can not only unify the continuous and discrete situations but can also strictly include the concept of periodic time scales.…”