Abstract and Applied Analysis

Mann iterates of directionally nonexpansive mappings in hyperbolic spaces

Ulrich Kohlenbach and Laurenţiu Leuştean

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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.

Article information

Abstr. Appl. Anal., Volume 2003, Number 8 (2003), 449-477.

First available in Project Euclid: 27 April 2003

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H10
Secondary: 03F10 03F60


Kohlenbach, Ulrich; Leuştean, Laurenţiu. Mann iterates of directionally nonexpansive mappings in hyperbolic spaces. Abstr. Appl. Anal. 2003 (2003), no. 8, 449--477. doi:10.1155/S1085337503212021. https://projecteuclid.org/euclid.aaa/1051453543

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