Almost Periodic Functions on the Quantum Time Scale and Applications

Yongkun Li

doi:10.1155/2019/4529159

Abstract

In this paper, we first propose two types of concepts of almost periodic functions on the quantum time scale. Secondly, we study some basic properties of almost periodic functions on the quantum time scale. Thirdly, based on these, we study the existence and uniqueness of almost periodic solutions of dynamic equations on the quantum time scale by Lyapunov method. Then, we give an equivalent definition of almost periodic functions on the quantum time scale. Finally, as an application, we propose a class of high-order Hopfield neural networks on the quantum time scale and establish the existence and global exponential stability of almost periodic solutions of this class of neural networks.

Highlights

The concept of almost periodicity was initiated by the Bohr during the period 1923-1925 [1, 2]
By Theorems 34 and 35 and Remark 33, all of the properties of almost periodic functions on the quantum time scale can be directly obtained from the corresponding properties of the ordinary almost periodic functions defined on Z or Z × X
We proposed two types of concepts of almost periodic functions on the quantum time scale and investigated some their basic properties

Summary

Introduction

The concept of almost periodicity was initiated by the Bohr during the period 1923-1925 [1, 2]. Since the questions of the theory of almost periodic functions and almost periodic solutions of differential equations have been very interesting and challenge problems of great importance. Define for V ∈ Crd[qZ × Rn, R], Dq+VΔ(t, x(t)) to mean that, given ε > 0, there exists a right neighborhood Nε ⊂ N such that

Preliminaries

Almost Periodic Functions

Almost Periodic Dynamic Equations on the Quantum Time Scale

An Application

Conclusion