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

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