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

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: 47H09, 46B20, 47H10, 47E10.