Almost all about Rus-Hicks-Rhoades maps in quasi-metric spaces

Abstract

Let $(X, d)$ be a quasi-metric space. A Rus-Hicks-Rhoades (RHR) map $f : X \to X$ is the one satisfying $d(fx, f^2x) \le \alpha d(x, fx)$ for every $x\in X$, where $\alpha \in [0,1)$. In our previous work [37], we collected various fixed-point theorems closely related to RHR maps. In the present article, we collect almost all the things we know about RHR maps and their examples. Moreover, we derive new classes of generalized RHR maps and fixed point theorems on them. Consequently, many of the known results in metric fixed point theory are improved and reproved in an easy way.