Iterative methods for monotone nonexpansive mappings in uniformly convex spaces

Advances in Nonlinear Analysis

article

Rahul Shukla¹  Andrzej Wiśnicki² 

[1] Department of Mathematics & Applied Mathematics, University of Johannesburg, Kingsway Campus;Department of Mathematics, Pedagogical University of Krakow

关键词: Monotone mapping;    nonexpansive mapping;    fixed point;    ergodic theorem;    Picard iteration;    ordered Banach space;   

DOI  :  10.1515/anona-2020-0170

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space ( X , ∣·∣, ⪯), T : C  →  C a monotone 1-Lipschitz mapping and x  ⪯  T ( x ), then the sequence of averages 1n∑i=0n−1Ti(x) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T . As a consequence, it is shown that the sequence of Picard’s iteration { T n ( x )} also converges weakly to a fixed point of T . The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

【 授权许可】

CC BY