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

We generalize the Caristi’s fixed point theorem for single valued as well as multivalued mappings defined on a metric space endowed with a graph and -distance. Particularly, we modify the concept of the (OSC)-property due to Alfuraidan and Khamsi (Alfuraidan M. R. and Khamsi, M. A.,  Caristi fixed point theorem in metric spaces with graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5.) which enable us to reformulated their stated graph theory version theorem (Theorem 3.2 in Alfuraidan M. R. and Khamsi, M. A.,  Caristi fixed point theorem in metric spaces with graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5. ) to the case of -distance. Consequently, we extend and improve some recent works concerning extension of Banach Contraction Theorem to -distance with graph e.g. (Jachymski, J., The contraction principle for mappings on a metric space with graph, Proc. Amer. Math. Soc.,  136 (2008), No. 4, 1359–1373; Nieto, J. J., Pouso, R. L. and Rodriguez-Lopez R.,  Fixed point theorems in ordered abstract spaces, Proc. Amer. Math. Soc., 135 (2007), 2505–2517 and Petrusel, A. and Rus, I., Fixed point theorems in ordered spaces endowed with graph, {Proc. Amer, Math. Soc.,} 134 (2006), 411–418).