Remotely $c$-almost periodic type functions in ${\mathbb R}^{n}$

Abstract

In this paper, we relate the notions of remote almost periodicity and quasi-asymptotical almost periodicity; in actual fact, we observe that a remotely almost periodic function is nothing else but a bounded, uniformly continuous quasi-asymptotically almost periodic function. We introduce and analyze several new classes of remotely c-almost periodic functions in R n , slowly oscillating functions in R n , and further analyze the recently introduced class of quasi-asymptotically c-almost periodic functions in R n . We provide certain applications of our theoretical results to the abstract Volterra integrodifferential equations and the ordinary differential equations.2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. Remotely c-almost periodic functions in R n , slowly oscillating functions in R n , quasi-asymptotically c-almost periodic functions in R n , abstract Volterra integro-differential equations, Richard-Chapman ordinary differential equation with external perturbation.