Let be a nonempty set, , be two metrics on , and be a self-map. We say that is a Picard point (modulo ) if (i) is -convergent; (ii) is in , i.e. . If this happens for each , and is a singleton, then we say that is a Picard operator (modulo ). is said to be -contractive if there exists some such that for all . We say that is subordinated to if for all .
M. G. Maia [Rend. Sem. Mat. Univ. Padova 40, 139–143 (1968; Zbl 0188.45603)] proved:
(A) Assume that is complete, is -continuous and -contractive, and is subordinated to . Then is a Picard operator (modulo ).
In particular, when , (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 be a partially ordered set such that every pair has a lower bound and an upper bound. Let be a metric on such that is complete. If is a continuous monotone self-map on such that
(i) there exists for all ;
(ii) there exists or ,
then is a Picard operator (modulo ).
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.