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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) \(T^nx: n\geq 0)\) 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 \(e(Tx,Ty,\leq \alpha e(x,y)\) for all \(x,y\in X\). We say that \(d\) is subordinated to \(e\) if \(d(x,y)\leq e(x,y)\) 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 \(0< c< 1: d(Tx,Ty)\leq cd(x,y)\) for all \(x\geq y\);

(ii) there exists \(x_0\in X: x_0\leq Tx_0\) or \(x_0\geq Tx_0\),

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.

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 \(0< c< 1: d(Tx,Ty)\leq cd(x,y)\) for all \(x\geq y\);

(ii) there exists \(x_0\in X: x_0\leq Tx_0\) or \(x_0\geq Tx_0\),

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.

Reviewer: T. D. Narang (Amritsar)