The Ran-Reurings fixed point theorem without partial order: a simple   proof
  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. 
Hichem Ben-El-Mechaiekh
math.GN