$\rho$-Almost periodic type functions in ${\mathbb R}^{n}$

We investigate various classes of multi-dimensional $(S,{\mathbb D}, {\mathcal B})$-asymptotically $(\omega,\rho)$-periodic type functions, multi-dimensional quasi-asymptotically $\rho$-almost periodic type functions and multi-dimensional $\rho$-slowly oscillating type functions of the form $F : I \times X \rightarrow Y,$ where $n\in {\mathbb N},$ $\emptyset \neq I \subseteq {\mathbb R}^{n},$ $\omega \in {\mathbb R}^{n} \setminus \{0\},$ $X$ and $Y$ are complex Banach spaces and $\rho$ is a binary relation on $Y.$ The main structural properties of these classes of almost periodic type functions are deduced. We also provide certain applications of our results to the abstract Volterra integro-differential equations.