On set-valued nonlinear equations in Menger probabilistic normed spaces

Abstract

We introduce the concept of probabilistic ( ϕ , ψ ) -contractor couple, which allows us to consider set-valued nonlinear operator equations in a large class of probabilistic normed spaces. Our results complete those from Cho et al. [Nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems 110 (2000) 115–122] and Fang [On nonlinear equations for fuzzy mappings in probabilistic normed spaces, Fuzzy Sets and Systems 131 (2002) 357–364].