Metrical almost periodicity: Levitan and Bebutov concepts

Abstract

In this paper, we analyze Levitan and Bebutov metrical approximations of functions $F:\Lambda \times X\to Y$ by trigonometric polynomials and $\rho$-periodic type functions, where $\varnothing \ne \Lambda \subseteq {\mathbb{R}}^n$, $X$ 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.