Multi-dimensional $\rho$-almost periodic type functions and applications

Abstract

In this paper, we analyze various classes of multi-dimensional $\rho$-almost periodic type functions $F:I\times X\to Y$ and multi-dimensional $(\omega ,\rho )$-almost periodic type functions $F:I\times X\to Y,$ where $n\in \mathbb{N},$ $\varnothing \ne I\subseteq {\mathbb{R}}^n,$ $X$ and $Y$ are complex Banach spaces and $\rho$ is a binary relation on $Y.$ The proposed notion is new even in the one-dimensional setting, for the functions of the form $F:I\to Y.$ The main structural properties and characterizations for the introduced classes of functions are presented. We provide certain applications of our theoretical results to the abstract Volterra integro-differential equations, as well.