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.
Kim and TuyenFixed Point Theory and Applications2011,2011:52 http://www.fixedpointtheoryandapplications.com/content/2011/1/52
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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, 631701, 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 write〈x,x*〉instead ofx*(x) forx*ÎE* andxÎE. We denote as⇀and®, 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:
0∈A 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)
A− whereJ= (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, Güler’s example [2] shows that proximal fies the condition=en<. Howe point algorithm (1.2) has only weak convergence in an infinitedimensional Hilbert space. Recently, several authors proposed modifications of Rockafellar’s proximal point algorithm (1.2) for the strong convergence. For example, Solodov and Svaiter [3] and