sunny nonexpansive retraction
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2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Shuo Sun

We introduce a new regularization iterative algorithm for equilibrium and fixed point problems of nonexpansive mapping. Then, we prove a strong convergence theorem for nonexpansive mappings to solve a unique solution of the variational inequality and the unique sunny nonexpansive retraction. Our results extend beyond the results of S. Takahashi and W. Takahashi (2007), and many others.


2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Shunhou Fan ◽  
Shin Min Kang ◽  
Muhammad Aslam Noor ◽  
Yonghong Yao

Motivated and inspired by Korpelevich's and Noor's extragradient methods, we suggest an extragradient method by using the sunny nonexpansive retraction which has strong convergence for solving the generalized variational inequalities in Banach spaces.


2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Esref Turkmen ◽  
Safeer Hussain Khan ◽  
Murat Ozdemir

Suppose thatKis nonempty closed convex subset of a uniformly convex and smooth Banach spaceEwithPas a sunny nonexpansive retraction andF:=F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅. LetT1,T2:K→Ebe two weakly inward nonself asymptotically nonexpansive mappings with respect toPwith two sequences{kn(i)}⊂[1,∞)satisfying∑n=1∞(kn(i)-1)<∞(i=1,2), respectively. For any givenx1∈K, suppose that{xn}is a sequence generated iteratively byxn+1=(1-αn)(PT1)nyn+αn(PT2)nyn,yn=(1-βn)xn+βn(PT1)nxn,n∈N, where{αn}and{βn}are sequences in[a,1-a]for somea∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of{xn}to a common fixed point ofT1andT2are obtained.


2010 ◽  
Vol 2010 ◽  
pp. 1-14 ◽  
Author(s):  
Yekini Shehu ◽  
Jerry N. Ezeora

LetEbe a real Banach space which is uniformly smooth and uniformly convex. LetKbe a nonempty, closed, and convex sunny nonexpansive retract ofE, whereQis the sunny nonexpansive retraction. IfEadmits weakly sequentially continuous duality mappingj, path convergence is proved for a nonexpansive mappingT:K→K. As an application, we prove strong convergence theorem for common zeroes of a finite family ofm-accretive mappings ofKtoE. As a consequence, an iterative scheme is constructed to converge to a common fixed point (assuming existence) of a finite family of pseudocontractive mappings fromKtoEunder certain mild conditions.


Author(s):  
Rudong Chen ◽  
Zhichuan Zhu

LetCbe a closed convex subset of a uniformly smooth Banach spaceE, andT:C→Ea nonexpansive nonself-mapping satisfying the weakly inwardness condition such thatF(T)≠∅, andf:C→Ca fixed contractive mapping. Fort∈(0,1), the implicit iterative sequence{xt}is defined byxt=P(tf(xt)+(1−t)Txt), the explicit iterative sequence{xn}is given byxn+1=P(αnf(xn)+(1−αn)Txn), whereαn∈(0,1)andPis a sunny nonexpansive retraction ofEontoC. We prove that{xt}strongly converges to a fixed point ofTast→0, and{xn}strongly converges to a fixed point ofTasαnsatisfying appropriate conditions. The results presented extend and improve the corresponding results of Hong-Kun Xu (2004) and Yisheng Song and Rudong Chen (2006).


2002 ◽  
Vol 66 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Tomas Dominguez Benavides ◽  
Genaro López Acedo ◽  
Hong-Kun Xu

Let  be a commutative family of nonexpansive self-mappings of a closed convex subset C of a uniformly smooth Banach space X such that the set of common fixed points is nonempty. It is shown that if a certain regularity condition is satisfied, then the sunny nonexpansive retraction from C to F can be constructed in an iterative way.


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