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2012 Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces
Songnian He, Jun Guo
J. Appl. Math. 2012(SI03): 1-13 (2012). DOI: 10.1155/2012/787419

Abstract

Let $C$ be a nonempty closed convex subset of a real uniformly smooth Banach space $X$, $\{{T}_{k}{\}}_{k=1}^{\infty }:C\to C$ an infinite family of nonexpansive mappings with the nonempty set of common fixed points ${\bigcap }_{k=1}^{\infty }\mathrm{Fix}({T}_{k})$, and $f:C\to C$ a contraction. We introduce an explicit iterative algorithm ${x}_{n+1}={\alpha }_{n}f({x}_{n})+(1-{\alpha }_{n}){L}_{n}{x}_{n}$, where ${L}_{n}={\sum }_{k=1}^{n}\left({\omega }_{k}/{\text{s}}_{n}\right){T}_{k},{S}_{n}={\sum }_{k=1}^{n}{\omega }_{k},\mathrm{ }$ and ${w}_{k}>0$ with ${\sum }_{k=1}^{\infty }{\omega }_{k}=1$. Under certain appropriate conditions on $\{{\alpha }_{n}\}$, we prove that $\{{x}_{n}\}$ converges strongly to a common fixed point ${x}^{*}$ of $\{{T}_{k}{\}}_{k=1}^{\infty }$, which solves the following variational inequality: $〈{x}^{*}-f({x}^{*}),J({x}^{*}-p)〉\le 0,\mathrm{ }\mathrm{ }p\in {\bigcap }_{k=1}^{\infty }\text{Fix}({T}_{k})$, where $J$ is the (normalized) duality mapping of $X$. This algorithm is brief and needs less computational work, since it does not involve $W$-mapping.

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Songnian He. Jun Guo. "Iterative Algorithm for Common Fixed Points of Infinite Family of Nonexpansive Mappings in Banach Spaces." J. Appl. Math. 2012 (SI03) 1 - 13, 2012. https://doi.org/10.1155/2012/787419

Information

Published: 2012
First available in Project Euclid: 3 January 2013

zbMATH: 1366.47011
MathSciNet: MR2904524
Digital Object Identifier: 10.1155/2012/787419

Rights: Copyright © 2012 Hindawi

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Vol.2012 • No. SI03 • 2012
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