The study of fixed points for multivalued mappings in a Menger probabilistic metric space endowed with a graph

We study the existence of fixed points for multivalued mappings f:S -> S,where ( S,F,T) is a complete Menger PM-space with a t-norm of H-type T and S is endowed with a directed graph G = (V(G), E(G)) such that V(G) = S and Delta = {( x, x) : x is an element of S} subset of E( G). The obtained results recover several existing fixed point theorems from the literature. As applications, we obtain a convergence result of successive approximations for certain nonlinear operators defined on a complete metric space. This last result allows us to establish a Kelisky-Rivlin type result for a class of modified q-Bernstein operators on the space C([0, 1]).