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Regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in Banach spaces

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We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. 2010 Mathematics Subject Classification : 47H09; 47J25; 47J30. We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. 2010 Mathematics Subject Classification : 47H09; 47J25; 47J30.
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Kim and TuyenFixed Point Theory and Applications2011,2011:52 http://www.fixedpointtheoryandapplications.com/content/2011/1/52
R E S E A R C H
Open Access
Regularization proximal point algorithm for finding a common fixed point of a finite family nonexpansive mappings in Banach spaces 1* 2 Jong Kyu Kim and Truong Minh Tuyen
* Correspondence: jongkyuk@kyungnam.ac.kr 1 Department of Mathematics Education, Kyungnam University, Masan, Kyungnam, 631701, Korea Full list of author information is available at the end of the article
Abstract We study the strong convergence of a regularization proximal point algorithm for finding a common fixed point of a finite family of nonexpansive mappings in a uniformly convex and uniformly smooth Banach space. 2010 Mathematics Subject Classification: 47H09; 47J25; 47J30. Keywords:accretive operators, uniformly smooth and uniformly convex, Banach space, sunny nonexpansive retraction, weak sequential continuous, mapping, regularization
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1 Introduction LetEbe a Banach space with its dual spaceE*. For the sake of simplicity, the norms of EandE* are denoted by the symbol || ∙ ||. We writex,x*instead ofx*(x) forx*ÎE* andxÎE. We denote asand®, the weak convergence and strong convergence, respectively. A Banach spaceEis reflexive ifE=E**. The problem of finding a fixed point of a nonexpansive mapping is equivalent to the problem of finding a zero of the following operator equation:
0A x
(1:1)
involving the accretive mappingA. One popular method of solving equation 0ÎA(x) is the proximal point algorithm of Rockafellar [1] which is recognized as a powerful and successful algorithm for finding a zero of monotone operators. Starting from any initial guessx0ÎH, this proximal point algorithm generates a sequence {xn} given by
A xn+1=J(xn+en) c
(1:2)
AwhereJ= (I+rA),r> 0 is the resolvent ofAin a Hilbert spaceH. Rockafellar [1] proved the weak convergence of the algorithm (1.2) provided that the regularization sequence {cn} remains bounded away from zero, and that the error sequence {en} satis ver, Gülers example [2] shows that proximal fies the condition=en<. Howe point algorithm (1.2) has only weak convergence in an infinitedimensional Hilbert space. Recently, several authors proposed modifications of Rockafellars proximal point algorithm (1.2) for the strong convergence. For example, Solodov and Svaiter [3] and
© 2011 Kim and Tuyen; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution 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.