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21 April 2003 Mann iterates of directionally nonexpansive mappings in hyperbolic spaces
Ulrich Kohlenbach, Laurenţiu Leuştean
Abstr. Appl. Anal. 2003(8): 449-477 (21 April 2003). DOI: 10.1155/S1085337503212021


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.


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Ulrich Kohlenbach. Laurenţiu Leuştean. "Mann iterates of directionally nonexpansive mappings in hyperbolic spaces." Abstr. Appl. Anal. 2003 (8) 449 - 477, 21 April 2003.


Published: 21 April 2003
First available in Project Euclid: 27 April 2003

zbMATH: 1038.47037
MathSciNet: MR1983075
Digital Object Identifier: 10.1155/S1085337503212021

Primary: 47H09 , 47H10
Secondary: 03F10 , 03F60

Rights: Copyright © 2003 Hindawi

Vol.2003 • No. 8 • 21 April 2003
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