Local pseudo almost automorphic functions with applications to semilinear dynamic equations on changing-periodic time scales

Chao Wang,Rathinasamy Sakthivel,Donal O’Regan,Ravi P Agarwal

doi:10.1186/s13661-019-1247-4

Abstract

In this paper, based on the concepts of changing-periodic time scales, we introduce a notion of local pseudo almost automorphic functions on an arbitrary time scale with a bounded graininess function μ. Then, some properties of local pseudo almost automorphic functions are investigated. As applications, by adopting the concept of a Π-semigroup on time scales, we obtain some new sufficient conditions for the existence of local pseudo almost automorphic mild solutions for a class of semilinear dynamic equations on arbitrary time scales.

Highlights

Almost periodic and almost automorphic functions are widely studied and applied in the literature, and there have been several natural and powerful generalizations of classical almost automorphic functions
In 2016, Wang, Agarwal, and O’Regan attached time scales under translations with “translation direction” and introduced several new concepts of periodic time scales to improve the results from [31, 32], Agarwal and O’Regan provided some key notes and comments on changing-periodic time scales. Based on these fundamental works, we find a feasible and plausible way to solve the closedness under translations for arbitrary time scales with bounded graininess function μ, which paves the way for introducing some new well-defined functions on periodic sub-timescales with “complete closedness”
Remark 2.1 Note that local Π -semigroups on changing-periodic time scales are operator semigroups on the sub-timescale Tτt, so the semigroups induced by time scales and Π -semigroups on time scales are completely different concepts through comparing Definitions 2.3 and 2.6

Summary

Introduction

Almost periodic and almost automorphic functions are widely studied and applied in the literature, and there have been several natural and powerful generalizations of classical almost automorphic functions (see [1,2,3,4]). As applications, using the concept of a Π -semigroup that was introduced in 2016 for time scales (see [36]), we obtain some new sufficient conditions for the existence of local pseudo almost automorphic solutions for a class of semilinear dynamic equations on time scales. 4, as applications of our results, by using the concept of a Π -semigroup that was introduced in 2016 for time scales, we establish the existence of the local pseudo almost automorphic solutions for a class of semilinear dynamic equations on arbitrary time scales.

Results

Conclusion