Relatives of a Theorem of Rus-Hicks-Rhoades
Keywords:
complete metric space, fixed point, poset, fixed point theorem, preorder, stationary point, maximal elementAbstract
Let $(X, d)$ be a complete metric space and $f : X \to X$ a map
satisfying $d(fx, f^2x) \le \alpha d(x, fx)$ for every $x\in X$,
where $0 < \alpha < 1$. The fixed point theorems due to Rus (1973)
and Hicks-Rhoades (1979) on such maps were extended or improved by
Park (1980), Harder-Hicks-Saliga (1993), Suzuki (2001), and
Jachymski (2003). Moreover, fixed point theorems of Zermelo (1904),
Banach (1922), Caristi (1976), extended versions for multimaps
due to Nadler (1969) and Covitz-Nadler (1970) are also closely
related to the Rus-Hicks-Rhoades theorem. Finally, we unify these theorems
based on a particular form of our 2023 Metatheorem.
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