S-Almost Automorphic Solutions for Impulsive Evolution Equations on Time Scales in Shift Operators

Chao Wang,Rathinasamy Sakthivel,Gaston M. N’Guérékata

doi:10.3390/math8061028

Chao Wang, Rathinasamy Sakthivel + Show 1 more

Open Access

https://doi.org/10.3390/math8061028

Copy DOI

Abstract

In this paper, based on the concept of complete-closed time scales attached with shift direction under non-translational shifts (or S-CCTS for short), as a first attempt, we develop the concepts of S-equipotentially almost automorphic sequences, discontinuous S-almost automorphic functions and weighted piecewise pseudo S-almost automorphic functions. More precisely, some novel results about their basic properties and some related theorems are obtained. Then, we apply the introduced new concepts to investigate the existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains. The obtained results are valid for q-difference partial dynamic equations and can also be extended to other dynamic equations on more general time scales. Finally, some heat dynamic equations on various hybrid domains are provided as applications to illustrate the obtained theory.

Highlights

Almost automorphic functions, which are more general than the almost periodic functions, were introduced by Bochner in relation to some aspects of differential geometry.Almost automorphic solutions in the context of differential equations have been studied by several researchers
In 2020, based on the concepts the authors introduced on translation time scales, Wang et al established a theory of closedness of translation time scales and their applications to evolution equations and dynamical models
We provide four types of impulsive evolution dynamic equations in the above, (1) will turn into other different types of dynamic equations on different types of complete-closed time scales attached with shift direction under translational or non-translational shifts

Summary

Introduction

Almost automorphic functions, which are more general than the almost periodic functions, were introduced by Bochner (see [1,2,3]) in relation to some aspects of differential geometry. In 2020, based on the concepts the authors introduced on translation time scales, Wang et al established a theory of closedness of translation time scales and their applications to evolution equations and dynamical models (see the monograph [33]). Some Lemmas are obtained and the exponential stability of weighted piecewise pseudo S-almost automorphic mild solutions is studied We apply these obtained results to study a class of ∆-partial differential equations on S-CCTS. We provide four types of impulsive evolution dynamic equations in the above, (1) will turn into other different types of dynamic equations on different types of complete-closed time scales attached with shift direction under translational or non-translational shifts. The existence of weighted piecewise pseudo S-almost automorphic mild solutions for the impulsive evolution equations on irregular hybrid domains is studied. The obtained results in this paper are effective for q-difference heat equations and other dynamic equations on more general hybrid domains

S-Equipotentially Almost Automorphic Sequence Under S-CCTS

Applications

Conclusions and Open Problems