Open AccessMann Iterates of Directionally Nonexpansive Mappings in Hyperbolic Spaces
Author(s) -
Ulrich Kohlenbach,
Laurenţiu Leuştean
Publication year - 2002
Publication title -
brics report series
Language(s) - English
Resource type - Journals
eISSN - 1601-5355
pISSN - 0909-0878
DOI - 10.7146/brics.v9i28.21744
Subject(s) - mathematics , iterated function , regular polygon , bounded function , pure mathematics , convex function , functional analysis , discrete mathematics , mathematical analysis , geometry , biochemistry , chemistry , gene
In a previous paper, the first author derived an explicit quantitative version of a theorem due to Borwein, Reich and Shafrir on the asymptotic behaviour of Mann iterations of nonexpansive mappings of convex sets C in normed linear spaces. This quantitative version, which was obtained by a logical analysis of the ineffective proof given by Borwein, Reich and Shafrir, could be used to obtain strong uniform bounds on the asymptotic regularity of such iterations in the case of bounded C and even weaker conditions. In this paper we extend these results to hyperbolic spaces and directionally nonexpansive mappings. In particular, we obtain significantly stronger and more general forms of the main results of a recent paper by W.A. Kirk with explicit bounds. As a special feature of our approach, which is based on logical analysis instead of functional analysis, no functional analytic embeddings are needed to obtain our uniformity results which contain all previously known results of this kind as special cases.