ON THE REALM OF WEAKLY NONEXPANSIVE MAPS ON QUASI-METRIC SPACES

Abstract

Let $(X,d)$ be a metric space. There have appeared thousands of works on nonexpansive maps $f: X\to X$ and the way to get their fixed points. Recently, many maps of the type satisfying $d(fx, f^2x) \le d(x, fx)$ on $x\in X$ appeared in the literature and they are called the weakly nonexpansive maps. There are a large number of fixed point theorems on weakly nonexpansive maps on metric spaces. Their proofs are different each other. In the present article, we trace the history of their applications to ordered fixed point theory. Certain new proofs are also given for known theorems.