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Ran-Reurings theorems in ordered metric spaces. (English) Zbl 1251.54057
Let $X$ be a nonempty set, $d(\cdot,\cdot)$, $e(\cdot,\cdot)$ be two metrics on $X$, and $T: X\to X$ be a self-map. We say that $x\in X$ is a Picard point (modulo $(d,T)$) if (i) $T^nx: n\ge 0)$ is $d$-convergent; (ii) $z:= \lim_n T^nx$ is in $\text{Fix}(T)$, i.e. $z\in Tz$. If this happens for each $x\in X$, and $\text{Fix}(T)$ is a singleton, then we say that $T$ is a Picard operator (modulo $d$). $T$ is said to be $e$-contractive if there exists some $\alpha\in]0,1[$ such that $e(Tx,Ty,\le \alpha e(x,y)$ for all $x,y\in X$. We say that $d$ is subordinated to $e$ if $d(x,y)\le e(x,y)$ for all $x,y\in X$. {\it M. G. Maia} [Rend. Sem. Mat. Univ. Padova 40, 139--143 (1968; Zbl 0188.45603)] proved: (A) Assume that $d$ is complete, $T$ is $d$-continuous and $e$-contractive, and $d$ is subordinated to $e$. Then $T$ is a Picard operator (modulo $d$). In particular, when $d=e$, (A) is just the Banach contraction principle. The following analogue of the Banach contraction principle in partially ordered metric spaces was proved by {\it A. C. M. Ran} and {\it M. C. B. Reurings} [Proc. Am. Math. Soc. 132, No. 5, 1435--1443 (2004; Zbl 1060.47056)]: (B) Let $X$ be a partially ordered set such that every pair $x,y\in X$ has a lower bound and an upper bound. Let $d$ be a metric on $X$ such that $(X,d)$ is complete. If $T$ is a continuous monotone self-map on $X$ such that (i) there exists $0< c< 1: d(Tx,Ty)\le cd(x,y)$ for all $x\ge y$; (ii) there exists $x_0\in X: x_0\le Tx_0$ or $x_0\ge Tx_0$, then $T$ is a Picard operator (modulo $d$). In this paper, it is shown that the Ran-Reurings find point theorem is a particular case of Maia’s result. A “functional” version of this fast result in a convergence setting has been provided in the paper.

54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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