Construction d’un point fixe commun à une famille de contractions fermes

Abridged English Version – Let (Ti)i∈I be a countable family of firmly nonexpansive mappings on a real Hilbert space H [i.e., (1) holds] and let F denote their set of common fixed points, which is assumed to be nonempty. In this Note we consider the problem (P) of finding a point in F . This generic problem arises in many branches of applied mathematics, e.g., solving systems of variational inequalities, solving convex feasibility problems, or finding a common zero of maximal monotone mappings [3], [9]. In addition, since Ti is firmly nonexpansive if and only if it can be written as Ti = (Id+Ci)/2, where Ci is nonexpansive [9], (P) is closely related to the problem of finding a common fixed point of nonexpansive mappings.