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

Let .X; d/ be a metric space and F V 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/ D f.x; y/ V .x; y/ 2 X Xg.