Asymptotic regularity, fixed points and successive approximations

Abstract

Let $(M,d)$ be a metric space. In this paper we survey some of the most relevant results which relate the three concepts involved in the title: a) the asymptotic regularity; b) the existence (and uniqueness) of fixed points and c) the convergence of the sequence of successive approximations  to the fixed point(s), for a given operator $f:M\rightarrow M$ or for two operators $f,g:M\rightarrow M$ connected to each other in some sense.