Metrical almost periodicity: Levitan and Bebutov concepts

TL;DR

This paper explores advanced concepts of almost periodic functions in multi-dimensional settings, focusing on Levitan and Bebutov approaches, with applications to differential equations.

Contribution

It introduces new analyses of metrical approximations and classes of multi-dimensional almost periodic functions, extending classical theories.

Findings

Established new approximation results for functions using trigonometric polynomials.

Characterized classes of multi-dimensional Levitan almost periodic functions.

Applied theoretical results to Volterra integro-differential and partial differential equations.

Abstract

In this paper, we analyze Levitan and Bebutov metrical approximations of functions $F :\Lambda \times X \rightarrow Y$ by trigonometric polynomials and $\rho$-periodic type functions, where $\emptyset \neq \Lambda \subseteq {\mathbb R}^{n}$, $X$ and $Y$ are complex Banach spaces, and $\rho$ is a general binary relation on $Y$. We also analyze various classes of multi-dimensional Levitan almost periodic functions in general metric and multi-dimensional Bebutov uniformly recurrent functions in general metric. We provide several applications of our theoretical results to the abstract Volterra integro-differential equations and the partial differential equations.