Uniformly Almost Periodic Functions and Almost Periodic Solutions to Dynamic Equations on Time Scales

Yongkun Li,Chao Wang

doi:10.1155/2011/341520

Abstract

Firstly, we propose a concept of uniformly almost periodic functions on almost periodic time scales and investigate some basic properties of them. When time scaleT=ℝorℤ, our definition of the uniformly almost periodic functions is equivalent to the classical definitions of uniformly almost periodic functions and the uniformly almost periodic sequences, respectively. Then, based on these, we study the existence and uniqueness of almost periodic solutions and derive some fundamental conditions of admitting an exponential dichotomy to linear dynamic equations. Finally, as an application of our results, we study the existence of almost periodic solutions for an almost periodic nonlinear dynamic equations on time scales.

Highlights

In recent years, researches in many fields on time scales have received much attention
A function f : T → R is right-dense continuous provided it is continuous at rightdense point in T and its left-side limits exist at left-dense points in T
The set of all regressive and rd-continuous functions p : T → R will be denoted by RRTRT, R

Summary

Introduction

Researches in many fields on time scales have received much attention. Our main purpose of this paper is firstly to propose a concept of uniformly almost periodic functions on time scales and investigate some basic properties of them. We study the existence and uniqueness of almost periodic solutions to linear dynamic equations on almost time scales. As an application of our results, we study the existence of almost periodic solutions for almost periodic nonlinear dynamic equations on time scales.

Preliminaries

Uniformly Almost Periodic Functions

Almost Periodic Dynamic Equations on Almost Periodic Time Scales

Almost Periodic Solutions to a Nonlinear Dynamic Equation on Time Scales