A new periodicity concept for time scales

TL;DR

This paper introduces a novel periodicity concept on time scales using shift operators, allowing analysis of broader classes of time scales, including those relevant for q-difference equations, thus expanding the scope of periodicity studies.

Contribution

It proposes a new periodicity definition on time scales that does not require the traditional uniform shift condition, enabling investigation of periodic solutions in more general contexts.

Findings

Defines a new periodicity concept using shift operators.

Allows analysis of non-uniform time scales.

Facilitates study of periodic solutions in q-difference equations.

Abstract

By means of the shift operators we introduce a new periodicity concept on time scales. This new approach will enable researchers to investigate periodicity notion on a large class of time scales whose members may not satisfy the condition: "there exists a $P>0$ such that $t\pm P\in\mathbb{T}$ for all $t\in\mathbb{T}$", unlike the existing literature. Therefore, the results of this paper open an avenue for the investigation of periodic solutions of $q$-difference equations.