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Ran-Reurings theorems in ordered metric spaces. (English) Zbl 1251.54057

Let X be a nonempty set, d(·,·), e(·,·) be two metrics on X, and T:XX be a self-map. We say that xX is a Picard point (modulo (d,T)) if (i) T n x:n0) is d-convergent; (ii) z:=lim n T n x is in Fix(T), i.e. zTz. If this happens for each xX, and 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 α]0,1[ such that e(Tx,Ty,αe(x,y) for all x,yX. We say that d is subordinated to e if d(x,y)e(x,y) for all x,yX.

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,yX 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)cd(x,y) for all xy;

(ii) there exists x 0 X:x 0 Tx 0 or x 0 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:
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces