Abstract
The purpose of this note is to generalize the celebrated Ran–Reurings fixed point theorem to the setting of a space with a binary relation that is only transitive (and not necessarily a partial order) and a relation-complete metric. The arguments presented here are simple and straightforward. It is also shown that extensions by Rakotch and by Hu and Kirk of Edelstein’s generalization of the Banach contraction principle to local contractions on chainable complete metric spaces are derived from the Ran–Reurings theorem.
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Ben-El-Mechaiekh, H. The Ran–Reurings fixed point theorem without partial order: A simple proof. J. Fixed Point Theory Appl. 16, 373–383 (2014). https://doi.org/10.1007/s11784-015-0218-3
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DOI: https://doi.org/10.1007/s11784-015-0218-3