Generalized $c$-almost periodic type functions in ${\mathbb R}^n$

TL;DR

This paper extends the theory of almost periodic functions to multi-dimensional settings, introducing new classes and generalizations, and applies these results to Volterra integro-differential equations in Banach spaces.

Contribution

It introduces generalized multi-dimensional $c$-almost periodic functions, explores their subclasses, and applies the theory to abstract Volterra equations in Banach spaces.

Findings

Characterization of multi-dimensional quasi-asymptotically $c$-almost periodic functions

Reconsideration of semi-$c$-periodicity in variable exponent Lebesgue spaces

Applications to Volterra integro-differential equations

Abstract

In this paper, we analyze multi-dimensional quasi-asymptotically $c$-almost periodic functions and their Stepanov generalizations as well as multi-dimensional Weyl $c$-almost periodic type functions. We also analyze several important subclasses of the class of multi-dimensional quasi-asymptotically $c$-almost periodic functions and reconsider the notion of semi-$c$-periodicity in the multi-dimensional setting, working in the general framework of Lebesgue spaces with variable exponent. We provide certain applications of our results to the abstract Volterra integro-differential equations in Banach spaces.