Fixed points for fuzzy mappings

Abstract In this paper the problem of the existence and computation of fixed points for fuzzy mappings is approached. A fuzzy mapping R over a set X is defined to be a function attaching to each x in X a fuzzy subset Rχ of X. An element x of X is called fixed point of R iff its membership degree to Rχ is at least equal to the membership degree to Rχ of any y ϵ X, i.e. Rχ(χ)⩾ Rχ(y)(∀y ϵ X). Two existence theorems for fixed points of a fuzzy mapping are proved and an algorithm for computing approximations of such a fixed point is described. The convergence theorem of our algorithm is proved under the restrictive assumption that for any x in X, the membership function of Rχ has a ‘complementary function’. Examples of fuzzy mappings having this property are given, but the problem of proving general criteria for a function to have a complementary remain open.