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

TL;DR

This paper explores the relationships between various types of almost periodic functions in multi-dimensional spaces, introduces new classes of such functions, and applies these concepts to differential equations.

Contribution

It introduces and analyzes new classes of remotely $c$-almost periodic functions and their relation to quasi-asymptotically almost periodic functions in ${ m I extbf{R}}^{n}$.

Findings

Remotely almost periodic functions are equivalent to bounded, uniformly continuous quasi-asymptotically almost periodic functions.

New classes of remotely $c$-almost periodic and slowly oscillating functions are defined and studied.

Applications to Volterra integro-differential and ordinary differential equations are provided.

Abstract

In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely $c$-almost periodic functions in ${\mathbb R}^{n},$ slowly oscillating functions in ${\mathbb R}^{n},$ and further analyze the recently introduced class of quasi-asymptotically $c$-almost periodic functions in ${\mathbb R}^{n}.$ We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations and the ordinary differential equations.