## 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.

### MSC:

 54H25 Fixed-point and coincidence theorems (topological aspects) 47H10 Fixed-point theorems

### Citations:

Zbl 0188.45603; Zbl 1060.47056
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