Doss ρ-Almost Periodic Type Functions in Rn

Marko Kostić,Wei-Shih Du,Vladimir E Fedorov

doi:10.3390/math9212825

Abstract

In this paper, we investigate various classes of multi-dimensional Doss ρ-almost periodic type functions of the form F:Λ×X→Y, where n∈N,∅≠Λ⊆Rn, X and Y are complex Banach spaces, and ρ is a binary relation on Y. We work in the general setting of Lebesgue spaces with variable exponents. The main structural properties of multi-dimensional Doss ρ-almost periodic type functions, like the translation invariance, the convolution invariance and the invariance under the actions of convolution products, are clarified. We examine connections of Doss ρ-almost periodic type functions with (ω,c)-periodic functions and Weyl-ρ-almost periodic type functions in the multi-dimensional setting. Certain applications of our results to the abstract Volterra integro-differential equations and the partial differential equations are given.

Highlights

Introduction and PreliminariesAcademic Editor: Christopher GoodrichReceived: 12 October 2021Accepted: 5 November 2021Published: 7 November 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Licensee MDPI, Basel, Switzerland.This article is an open access articleThe notion of almost periodicity was introduced by H
We investigate the main structural properties of Doss ρ-almost periodic functions; in particular, we analyze the convolution invariance of Doss ρ-almost periodicity, the invariance of Doss ρ-almost periodicity under the actions of convolution products, and provide certain applications to the abstract Volterra integro-differential equations and the partial differential equations
We have described how the considered classes of Doss ρ-almost periodic functions can be further generalized and applied in the study of second-order partial differential equations whose solutions are governed by the Newtonian potential

Summary

Introduction and Preliminaries

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Regarding multi-dimensional Stepanov, Weyl and Besicovitch classes of almost periodic functions, the reader may consult the above-mentioned monographs [7,9]. The introduced class of functions retains, in a certain sense, many important structural properties of the corresponding classes of Stepanov, Weyl and Besicovitch almost periodic functions We continue, in such a way, our previous research studies [14,17,18,19,20] and revisit some known structural characterizations of one-dimensional Doss almost periodic functions [6]. Doss ρ-almost periodic type functions of the form F : Λ × X → Y, where Y is a Banach space equipped with the norm k · kY , ρ ⊆ Y × Y is a binary relation, Λ is a general nonempty subset of Rn , and p ∈ P (Λ); see Section 1 for the notion. Ruzicka and the lists of references quoted in this monograph and the forthcoming monograph [7]

Multi-Dimensional Doss ρ-Almost Periodic Type Functions

Relationship between Weyl Almost Periodicity and Doss Almost Periodicity

Applications to Abstract Volterra Integro-Differential Equations and Partial

Conclusions and Final Remarks