Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces.
type: Item
instance of: scholarly article
title: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces.
zbMATH Open document ID: 1052.47049
DOI: 10.1016/S0362-546X(03)00193-7
author: John O'Hara
author: Paranjothi Pillay
author: Hong-Kun Xu
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-09-07T00:00:00Z
review text: In this paper, the authors establish the result that the iteration scheme \(x_{n+1}=\lambda_{n+1}y+(1-\lambda_{n+1})T_{n+1}x_n\) for infinitely/finitely many nonexpansive mappings \(T_i\) in a Hilbert space converges to \(Py\), where \(P\) is the projection onto the intersection of the fixed point sets of the \(T_i\)'s. This generalizes the result of \textit{T. Shimizu} and \textit{W. Takahashi} [J. Math. Anal. Appl. 211, 71--83 (1997; Zbl 0883.47075)], a complementary result to a result of \textit{H.~H. Bauschke} [J. Math. Anal. Appl. 202, 150--159 (1996; Zbl 0956.47024)], by introducing a condition on the sequence of parameters (\(\lambda_n\)). This condition improves \textit{P.-L. Lions}' condition [C. R. Acad. Sci., Paris, Sér. A 284, 1357--1359 (1977; Zbl 0349.47046)].
Mathematics Subject Classification ID: 47H09
Mathematics Subject Classification ID: 65J15
zbMATH DE Number: 1975873
zbMATH Keywords: iterative approach
zbMATH Keywords: convex feasibility problem
zbMATH Keywords: common fixed point
zbMATH Keywords: nearest point projection
zbMATH Keywords: nonexpansive mapping
reviewed by: Palaniappan Vijayaraju
MaRDI profile type: Publication
cites work: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space
cites work: A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces
cites work: On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces
cites work: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings
cites work: Q3998720
cites work: Fixed points of nonexpanding maps
cites work: Q4119693
cites work: Some problems and results in fixed point theory
cites work: Q4881791
cites work: Strong convergence to common fixed points of families of nonexpansive mappings
cites work: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces
cites work: Approximation of fixed points of nonexpansive mappings
cites work: An iterative approach to quadratic optimization
cites work: Quadratic optimization of fixed points of nonexpansive mappings in hubert space
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description: scientific article; zbMATH DE number 1975873