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 
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
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
casesAvailable as PostScript, PDF, DVI.