The Ran-Reurings fixed point theorem without partial order: a simple proof

TL;DR

This paper generalizes the Ran-Reurings fixed point theorem to spaces with a transitive relation instead of a partial order, providing a simpler proof and connecting it to other fixed point results.

Contribution

It introduces a simplified proof of the Ran-Reurings theorem in a more general setting with transitive relations and relates it to existing fixed point theorems.

Findings

The theorem applies to spaces with transitive relations, not partial orders.

Connections established with Rakotch and Hu-Kirk extensions.

Simplified proof technique for fixed point theorems in relation-based spaces.

Abstract

The purpose of this note is to generalize the celebrated Ran and 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 Hu-Kirk of Edelstein's generalization of the Banach contraction principle to local contractions on chainable complete metric spaces derive from the theorem of Ran-Reurings.