Metrical almost periodicity and applications

TL;DR

This paper studies various classes of multi-dimensional almost periodic functions in metric spaces, clarifies their structural properties, and applies these findings to abstract Volterra integro-differential equations.

Contribution

It introduces and analyzes new classes of multi-almost periodic functions and explores their structural properties and applications in differential equations.

Findings

Characterization of multi-almost periodic functions in metric spaces

Structural properties of the introduced function classes

Applications to Volterra integro-differential equations

Abstract

In this paper, we analyze various classes of multi-dimensional almost periodic type functions in general metric. The main classes of functions under our consideration are $({\mathrm R}, {\mathcal B},{\mathcal P},L)$-multi-almost periodic functions, $({\mathrm R}_{X}, {\mathcal B},{\mathcal P},L)$-multi-almost periodic functions, Bohr $({\mathcal B}, I',\rho,{\mathcal P})$-almost periodic functions and $({\mathcal B}, I',\rho,{\mathcal P})$-uniformly recurrent functions. We clarify the main structural properties for the introduced classes of almost periodic type functions and provide some applications of our results to the abstract Volterra integro-differential equations.