# Almost automorphic functions on the quantum time scale and applications

## TL;DR

This paper introduces and studies almost automorphic functions on the quantum time scale, establishing their properties, transformations, and applications to dynamic equations, thus extending the theory to quantum calculus.

## Contribution

It defines two types of almost automorphic functions on the quantum time scale and explores their properties and applications to dynamic equations.

## Key findings

- Established properties of almost automorphic functions on the quantum time scale
- Provided a transformation linking quantum time scale functions to generalized integer functions
- Proved existence of almost automorphic solutions for certain dynamic equations

## Abstract

In this paper, we first propose two types of concepts of almost automorphic functions on the quantum time scale. Secondly, we study some basic properties of almost automorphic functions on the quantum time scale. Then, we introduce a transformation between functions defined on the quantum time scale and functions defined on the set of generalized integer numbers, by using this transformation we give equivalent definitions of almost automorphic functions on the quantum time scale. Finally, as an application of our results, we establish the existence of almost automorphic solutions of linear and semilinear dynamic equations on the quantum time scale.