Revisiting jointly firmly nonexpansive families of mappings

Abstract

Recently, the author, together with L. Leuştean and A. Nicolae, introduced the notion of jointly firmly nonexpansive families of mappings in order to investigate in an abstract manner the convergence of proximal methods. Here, we further the study of this concept, by giving a characterization in terms of the classical resolvent identity, by improving on the rate of convergence previously obtained for the uniform case, and by giving a treatment of the asymptotic behaviour at infinity of such families.

Abstract strongly convergent variants of the proximal point algorithm

“…Moreover, from this formulation given by (3) one immediately obtains, using (2), that a self-mapping of a CAT(0) space satisfying property (P 2 ) is nonexpansive. Following [34,48], if T and U are self-mappings of X and λ, µ > 0, we say that T and U are (λ, µ)-mutually firmly nonexpansive if for all x, y ∈ X and all α, β ∈…”

“…The main result of that paper showed that this condition suffices for the working of the proximal point algorithm, namely that if X is complete, (T n ) n∈N is a family of self-mappings of X with a common fixed point and (γ n ) n∈N ⊆ (0, ∞) with ∞ n=0 γ 2 n = ∞, then, assuming that (T n ) is jointly (P 2 ) with respect to (γ n ), any sequence (x n ) ⊆ X such that for all n, x n+1 = T n x n , ∆-converges (a generalization of weak convergence to arbitrary metric spaces, due to Lim [36]) to a common fixed point of the family. Moreover, in [48], the reason for the effectiveness of this sort of condition was further elucidated: Theorem 3.3 of that paper shows that a family of self-mappings is jointly firmly nonexpansive if and only if each mapping in it is nonexpansive and the family as a whole satisfies the well-known resolvent identity.…”

“…The following is the quantitative version of Theorem 2.2, as obtained in [28]. An abstract version of it using the concept of jointly firmly nonexpansive families of mappings may be found in [48,Section 5], but here we shall only need the rate of metastability for the resolvents of nonexpansive mappings. (1).…”

“…(It was not accidental that we have already presented resolvents above in a quite abstract way.) We have recently revisited [48] the concept, providing a conceptual characterization of it and showing how it may be used to prove other kinds of results usually associated with resolvent-type mappings.…”