The contraction principle for set valued mappings on a metric space with a graph

Abstract

Let (X,d) be a metric space and F:X↝X be a set valued mapping. We obtain sufficient conditions for the existence of a fixed point of the mapping F in the metric space X endowed with a graph G such that the set V(G) of vertices of G coincides with X and the set of edges of G is E(G)={(x,y):(x,y)∈X×X}.