We discuss 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. It extends some recent works on the extension of Banach contraction principle to metric spaces with graph.
"Caristi Fixed Point Theorem in Metric Spaces with a Graph." Abstr. Appl. Anal. 2014 (SI41) 1 - 5, 2014. https://doi.org/10.1155/2014/303484