Metrical almost periodicity, metrical approximations of functions and applications

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