Effective results on compositions of nonexpansive mappings

Abstract

This paper provides uniform bounds on the asymptotic regularity for iterations associated to a finite family of nonexpansive mappings. We obtain our quantitative results in the setting of (r, δ)-convex spaces, a class of geodesic spaces which generalizes metric spaces with a convex geodesic bicombing.MSC: 47J25; 47H09; 53C23; 03F10.

Quantitative results for Halpern iterations of nonexpansive mappings

“…This alternative iterative method was introduced for Banach spaces in [13] as a discretization of an approximating curve considered in [2]. A more general version of this iteration is studied in [1], while a cyclic version of (x n ) associated to a finite family of nonexpansive mappings is considered in [7]. Define for n ≥ 0,…”

Linear rates of asymptotic regularity for Halpern-type iterations

“…One can easily see that the assumptions of Lemma 1.1 are met, so we can apply it to get (13). ( 14) follows easily: 10) and ( 12)…”

“…By letting f (x) = u ∈ C in (18), one obtains, as a particular case, the alternative regularization method, introduced by Xu [23]. The second author and Nicolae remarked in [13] that results on the asymptotic regularity and the strong convergence of the alternative regularization method can be immediately obtained from the ones for the Halpern iteration.…”

A computational study of a class of recursive inequalities

“…An early use of kind of recursive inequality was to extract rates of asymptotic regularity for the Halpern iterations of nonexpansive self-mappings by Leus ¸tean [41], and a particularly detailed discussion of variants of ( 14) is given by Kohlenbach and Leus ¸tean [31]. There have been many further examples since: the last years have seen numerous instances of ( 14) with combined conditions by several different authors, including Cheval, Dinis, Kohlenbach, Leus ¸tean, Nicolae, Pinto, and Sipos ¸ [9,10,13,11,12,24,33,42,43,44,46,53].…”

“…Many papers in the applied proof theory literature explicitly deal with the computational content of convergence results for sequences of reals satisfying recursive inequalities of some kind, from authors including Cheval, Dinis, Kohlenbach, Körnlein, Lambov, Leus ¸tean, López-Acedo, Nicolae, Pinto, Sipos ¸, Wiesnet, and the second author [9,10,13,11,12,24,26,29,30,31,33,34,41,42,43,44,46,53,50,54]. It is these are most relevant to the work presented below.…”