In this paper, we analyze the Bohr $\rho$-almost periodic type sequences and the generalized $\rho$-almost periodic type sequences of the form $F:I\times X\to Y$, where $\varnothing \ne I\subseteq {\mathbb{Z}}^n$, X and Y are complex Banach spaces and $\rho$ is a general binary relation on $Y.$ We provide many structural results, observations and open problems about the introduced classes of $\rho$-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given.

The aim of this work is the resolution of a non-autonomous abstract differential equation of elliptic type set on unbounded domain. The study is performed in the framework of Hölder spaces. An example for a concrete elliptic problem in nonsmooth cylindrical domains will illustrate the theory.