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

摘要

We study the existence of fixed points for multivalued mappings f : S → S $f: S o S$ , where ( S , F , T ) $(S,F,T)$ 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 ) ) $G=(V(G),E(G))$ such that V ( G ) = S $V(G)=S$ and Δ = { ( x , x ) : x ∈ S } ⊂ E ( G ) $Delta= { (x,x): x in S } subset E(G)$ . 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 ] ) $C([0,1])$ .