Relatives of a Theorem of Rus-Hicks-Rhoades
Sehie Park · Letters in Nonlinear Analysis and its Applications · 2023
Let $(X, d)$ be a complete metric space and $f : X \to X$ a mapsatisfying $d(fx, f^2x) \le \alpha d(x, fx)$ 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 byPark (1980), Harder-Hicks-Saliga (1993), Suzuki (2001), andJachymski (2003). Moreover, fixed point theorems of Zermelo (1904),Banach (1922), Caristi (1976), extended versions for multimapsdue to Nadler (1969) and Covitz-Nadler (1970) are also closelyrelated to the Rus-Hicks-Rhoades theorem. Finally, we unify these theoremsbased on a particular form of our 2023 Metatheorem.