Uniform asymptotic regularity for Mann iterates

In [16] we obtained an effective quantitative analysis of a theorem due to Borwein, Reich and Shafrir on the asymptotic behavior of general KrasnoselskiMann iterations for nonexpansive self-mappings of convex sets C in arbitrary normed spaces. We used this result to obtain a new strong uniform version of Ishikawa’s theorem for bounded C. In this paper we give a qualitative improvement of our result in the unbounded case and prove the uniformity result for the bounded case under the weaker assumption that C contains a point x whose Krasnoselski-Mann iteration (xk) is bounded. We also consider more general iterations for which asymptotic regularity is known only for uniformly convex spaces (Groetsch). We give uniform effective bounds for (an extension of) Groetsch’s theorem which generalize previous results by Kirk/Martinez-Yanez and the author. ∗Basic Research in Computer Science, funded by the Danish National Research Foundation.