Mann iterates of directionally nonexpansive mappings in hyperbolic spaces

By Ulrich Kohlenbach, Laurentiu Leustean. 2002

67 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/366993edf23a6f2c654fee3e62f35063d44d1193

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In Applied Mathematics

In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Some logical metatheorems with applications in functional analysis (2003, methodology): “The proof in [12](Theorem 1) of the (non-uniform) convergence d(xn, f(xn))→ 0 easily extends to the larger class of directionally nonexpansive mappings introduced in [23] (see [23],[37]).”
A quadratic rate of asymptotic regularity for CAT(0)-spaces (2005, background): “The following result (which is a corollary of the main theorem in [21]) is proved there for hyperbolic spaces in the sense of [26], but the proof goes through for the setting used in the present paper.”
Proof Mining: A Systematic Way of Analysing Proofs in Mathematics (2002, background): “Very recently ([ 59 ]) the rst author (together with Laurent iu Leu stean) showed in this way that the results (as well as the basic structure of their proofs) presented above extend to hyperbolic spaces in the sense of Reich and Shafrir [78] (including the Hilbert ball with the hyperbolic metric) a”