*(English)*Zbl 1251.54057

Let $X$ be a nonempty set, $d(\xb7,\xb7)$, $e(\xb7,\xb7)$ 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) ${T}^{n}x:n\ge 0)$ is $d$-convergent; (ii) $z:={lim}_{n}{T}^{n}x$ is in $\text{Fix}\left(T\right)$, i.e. $z\in Tz$. If this happens for each $x\in X$, and $\text{Fix}\left(T\right)$ 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 $x,y\in X$.

*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 *A. C. M. Ran* and *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 ${x}_{0}\in X:{x}_{0}\le T{x}_{0}$ or ${x}_{0}\ge T{x}_{0}$,

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.