Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii–Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzo...

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Bibliographic Details
Published in:Journal of the Operations Research Society of China (Internet) Vol. 9; no. 1; pp. 195 - 206
Main Authors: Zhang, You-Cai, Guo, Ke, Wang, Tao
Format: Journal Article
Language:English
Published: Beijing Operations Research Society of China 01.03.2021
Springer Nature B.V
Subjects:
Algorithms
Banach spaces
Codes
Convergence
Hilbert space
Investigations
Iterative methods
Management Science
Mathematics
Mathematics and Statistics
Operations Research
Weak convergence
Krasnoselskii–Mann-type iteration
47H09
Banach spaces
47H05
Accretive operator
Nonexpansive mappings
proximal point algorithm
ISSN:2194-668X, 2194-6698
Online Access:Get full text
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Summary:The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii–Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzow and Shehu proposed a generalized Krasnoselskii–Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces. In this paper, we show that the generalized Krasnoselskii–Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces. As applications, we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.
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ISSN:2194-668X
2194-6698
DOI:10.1007/s40305-018-0235-1