Abstract and Applied Analysis
Volume 2003 (2003), Issue 8, Pages 449-477
Mann iterates of directionally nonexpansive mappings in hyperbolic spaces
1Department of Computer Science, University of Aarhus, Ny Munkegade, Aarhus C DK-8000, Denmark
2National Institute for Research and Development in Informatics, 8-10 Maresal Al. Averescu Avenue, Sector 1, Bucharest 71316, Romania
Received 2 July 2002
Copyright © 2003 Ulrich Kohlenbach and Laurenţiu Leuştean. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 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 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.