Ulrich Kohlenbach — Page 2

arXiv2008

Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces

This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the Krasnoselski-Mann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from prima-facie ineffective proofs in the fixed point theory of such mappings.

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arXiv2005

The approximate fixed point property in product spaces

In this paper we generalize to unbounded convex subsets C of hyperbolic spaces results obtained by W.A. Kirk and R. Espinola on approximate fixed points of nonexpansive mappings in product spaces $(C\times M)_\infty$, where M is a metric space and C is a nonempty, convex, closed and bounded subset of a normed or a CAT(0)-space. We extend the results further, to families $(C_u)_{u\in M}$ of unbounded convex subsets of a hyperbolic space. The key ingredient in obtaining these generalizations is a uniform quantitative version of a theorem due to Borwein, Reich and Shafrir, obtained by the authors in a previous paper using techniques from mathematical logic. Inspired by that, we introduce in the last section the notion of uniform approximate fixed point property for sets C and classes of self-mappings of C. The paper ends with an open problem.

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