# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Ran-Reurings theorems in ordered metric spaces. (English) Zbl 1251.54057

Let $X$ be a nonempty set, $d\left(·,·\right)$, $e\left(·,·\right)$ 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 $\left(d,T\right)$) if (i) ${T}^{n}x:n\ge 0\right)$ 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 $\alpha \in \right]0,1\left[$ such that $e\left(Tx,Ty,\le \alpha e\left(x,y\right)$ for all $x,y\in X$. We say that $d$ is subordinated to $e$ if $d\left(x,y\right)\le e\left(x,y\right)$ 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 $\left(X,d\right)$ is complete. If $T$ is a continuous monotone self-map on $X$ such that

(i) there exists $0 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.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 47H10 Fixed point theorems for nonlinear operators on topological linear spaces