Uniform asymptotic regularity for Mann iterates

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:279

Article

Kohlenbach, U

关键词: nonexpansive mappings;    fixed point theory;    Krasnoselski-Mann iteration;    asymptotic regularity;    proof mining;   

DOI  :  10.1016/S0022-247X(03)00028-3

来源: Elsevier

【 摘 要 】

In Numer. Funct. Anal. Optim. 22 (2001) 641-656, 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 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 (x(k)) 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. (C) 2003 Elsevier Science (USA). All rights reserved.

【 授权许可】

Free