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 hooked right arrow 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) is an element of X x X}.