  In this paper, we analyze multi-dimensional quasi-asymptotically $c$-almost
periodic functions and their Stepanov generalizations as well as
multi-dimensional Weyl $c$-almost periodic type functions. We also analyze
several important subclasses of the class of multi-dimensional
quasi-asymptotically $c$-almost periodic functions and reconsider the notion of
semi-$c$-periodicity in the multi-dimensional setting, working in the general
framework of Lebesgue spaces with variable exponent. We provide certain
applications of our results to the abstract Volterra integro-differential
equations in Banach spaces.
