## Abstract and Applied Analysis

### Mann iterates of directionally nonexpansive mappings in hyperbolic spaces

#### Abstract

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

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

Dates
First available in Project Euclid: 27 April 2003

https://projecteuclid.org/euclid.aaa/1051453543

Digital Object Identifier
doi:10.1155/S1085337503212021

Mathematical Reviews number (MathSciNet)
MR1983075

Zentralblatt MATH identifier
1038.47037

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

#### Citation

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