Remarks on Caristi’s fixed point theorem in metric spaces with a graph

Abstract

We discuss some extensions of Caristi’s fixed point theorem for mappings defined on a metric space endowed with a graph. This work should be seen as a generalization of the classical Caristi’s fixed point theorem where the assumptions in Caristi’s theorem can, a priori, be weakened. It extends some recent works on Caristi’s fixed point theorem for mappings defined on metric spaces with a graph. MSC: Primary 47H09; secondary 46B20; 47H10; 47E10.