Almost Automorphic Functions on the Quantum Time Scale and Applications

Yongkun Li

doi:10.1155/2017/1526478

Abstract

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; following the idea of the transformation, we also give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and 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.

Highlights

IntroductionBecause the theory of quantum calculus has important applications in quantum theory (see Kac and Cheung [1]), it has received much attention
Because the theory of quantum calculus has important applications in quantum theory, it has received much attention
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; following the idea of the transformation, we give a concept of almost automorphic functions on more general time scales that can unify the concepts of almost automorphic functions on almost periodic time scales and on the quantum time scale

Summary

Introduction

Because the theory of quantum calculus has important applications in quantum theory (see Kac and Cheung [1]), it has received much attention. Since Bohner and Chieochan [2] introduced the concept of the periodicity for functions defined on the quantum time scale, quite a few authors have devoted themselves to the study of the periodicity for dynamic equations on the quantum time scale [3,4,5,6]. Our main purpose of this paper is to propose two types of definitions of almost automorphic functions on the quantum time scale, study some of their basic properties, and establish the existence of almost automorphic solutions of nonautonomous linear dynamic equations on the quantum time scale.

Preliminaries

Almost Automorphic Functions on the Quantum Time Scale

Conclusion