Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces
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English

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Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces

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In this work, we consider a general algorithm for a countable family of nonexpansive mappings in Banach spaces. We proved that the proposed algorithm converges strongly to a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. It is worth pointing out that our proofs contain some new techniques. Our results improve and extend the corresponding ones announced by many others. MSC: 47H05, 47H09, 47H10.

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Publié le 01 janvier 2012
Nombre de lectures 9
Langue English

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WangFixed Point Theory and Applications2012,2012:207 http://www.fixedpointtheoryandapplications.com/content/2012/1/207
R E S E A R C HOpen Access Strong convergence of a general algorithm for nonexpansive mappings in Banach spaces * Shuang Wang
* Correspondence: wangshuang19841119@163.com School of Mathematical Sciences, Yancheng Teachers University, Yancheng, Jiangsu 224051, P.R. China
Abstract In this work, we consider a general algorithm for a countable family of nonexpansive mappings in Banach spaces. We proved that the proposed algorithm converges strongly to a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. It is worth pointing out that our proofs contain some new techniques. Our results improve and extend the corresponding ones announced by many others. MSC:47H05; 47H09; 47H10 Keywords:strong convergence; variational inequality; fixed points;k-Lipschitzian; η-strongly accretive; Banach space
1 Introduction LetXbe a real Banach space and letCbe a nonempty closed convex subset ofX. Recall that a mappingT:CCis said to be nonexpansive ifTxTy ≤ xy,x,yC. We denote byFix(T) the set of fixed points ofT. In , Yaoet al.[] considered the following algorithm in a Hilbert space. For an arbitrary pointxC, yn=PC[( –αn)xn], (.) xn+= ( –βn)xn+βnTyn,n.
They proved if{αn}and{βn}satisfy appropriate conditions, the{xn}defined by (.) con-verges strongly to a fixed point ofT. Recently, motivated and inspired by the above results, Wang and Hu [] introduced the following algorithm in a Hilbert space. For an arbitrary pointxC, yn=PC[(IαnF)xn], (.) xn+= ( –βn)xn+βnTnyn,n,
wherePC:XCis a metric projection,F:CXis aβ-Lipschitzian andη-strongly * monotone operators. They proved that the proposed algorithm converges strongly tox  ∞ ∞ * * T). Fix(Tn), which solves the variational inequalityFx,xu ≤,un=Fix(n n=
©2012 Wang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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