δ-Almost Periodic Functions and Applications to Dynamic Equations

Chao Wang,Ravi P Agarwal,Donal O’Regan

doi:10.3390/math7060525

Abstract

In this paper, by employing matched spaces for time scales, we introduce a δ -almost periodic function and obtain some related properties. Also the hull equation for homogeneous dynamic equation is introduced and results of the existence are presented. In the sense of admitting exponential dichotomy for the homogeneous equation, the expression of a δ -almost periodic solution for a type of nonhomogeneous dynamic equation is obtained and the existence of δ -almost periodic solutions for new delay dynamic equations is considered. The results in this paper are valid for delay q-difference equations and delay dynamic equations whose delays may be completely separated from the time scale T .

Highlights

Almost periodic phenomena are common in real world phenomena and the concept of almost periodic functions was first introduced by H
It was difficult to combine almost periodic problems of q-difference dynamic equations on time scales in the past because qZ has no translation invariance since the graininess function μ is unbounded and there was no concept of relatively dense set defined on it
By employing the algebraic structure of matched spaces for time scales, some basic results of the shift closedness including non-translational shift closedness of time scales are established. This progress combines a larger scope of time scales without translation invariance

Summary

Introduction

Almost periodic phenomena are common in real world phenomena and the concept of almost periodic functions was first introduced by H. To consider more general time scales without translation periodicity, the authors in [22,23] studied a type of dynamic equation under a new concept of periodic time scale whose period set is contained in T. Wang and R.P. Agarwal for the first time proposed a type of delay dynamic equation whose delay function range is from the period set of the time scale and completely separated from the time scale T.

Matched Spaces for Time Scales

Almost Periodic Dynamic Equations under Matched Spaces

Conclusions