Abstract:
In a previous paper we obtained an effective quantitative
analysis of a theorem due to Borwein, Reich and Shafrir on the asymptotic
behavior of general Krasnoselski-Mann iterations for nonexpansive
self-mappings of convex sets 
in arbitrary normed spaces. We used this
result to obtain a new strong uniform version of Ishikawa's theorem for
bounded . 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 contains a point 
whose Krasnoselski-Mann
iteration 
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
in arbitrary normed spaces. We used this
result to obtain a new strong uniform version of Ishikawa's theorem for
bounded . 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 contains a point whose Krasnoselski-Mann
iteration 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
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