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

  • You-Cai Zhang
  • Ke GuoEmail author
  • Tao Wang
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Abstract

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.

Keywords

Krasnoselskii–Mann-type iteration Nonexpansive mappings Weak convergence Accretive operator proximal point algorithm Banach spaces 

Mathematics Subject Classification

47H05 47H09 
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© Operations Research Society of China, Periodicals Agency of Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and InformationChina West Normal UniversityNanchongChina

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