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Hybrid iteration method for common fixed points of an infinite family of nonexpansive mappings in Banach spaces

De
7 pages
Let E be a real uniformly convex Banach space, and let K be a nonempty closed convex subset of E . Let { T i } i = 1 ∞ be a sequence of nonexpansive mappings from K to itself with F :={ x ∈ K : T i x = x , ∀ i ≥ 1}≠ ∅. For an arbitrary initial point x 1 ∈ K , the modified hybrid iteration scheme { x n } is defined as follows: x n + 1 = α n x n + ( 1 - α n ) T n * x n - λ n + 1 μ A ( T n * x n ) , n ≥ 1 , where A : K → K is an L -Lipschitzian mapping, T n * = T i with i satisfying: n = [( k - i +1)( i + k )/2]+[1+( i -1)(i+2)/2], k ≥ i -1(i = 1,2,.),{ λ n } ⊂ [0,1), and { α n } is a sequence in [ a , 1 - a ] for some a ∈ (0,1). Under some suitable conditions, the strong and weak convergence theorems of { x n } to a common fixed point of the nonexpansive mappings { T i } i = 1 ∞ are obtained. The results in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000: 47H09; 47J25.
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DengFixed Point Theory and Applications2012,2012:22 http://www.fixedpointtheoryandapplications.com/content/2012/1/22
R E S E A R C H
Hybrid iteration method for common of an infinite family of nonexpansive Banach spaces
WeiQi Deng
Correspondence: dwq1273@126. com College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P. R. China
Open Access
fixed points mappings in
Abstract LetEbe a real uniformly convex Banach space, and letKbe a nonempty closed convex subset ofE. Let{Ti}be a sequence of nonexpansive mappings fromKto i=1 itself withF:={xÎK:Tix=x,i1}. For an arbitrary initial pointx1ÎK, the modified hybrid iteration scheme {xn} is defined as follows:   ∗ ∗ µA(T xn+1=αnxn+ (1αn)Tnxnλn+1nxn) ,n1, whereA:K®Kis anL=Tiwithisatisfyi +2)/2],kLipschitzian mapping,Tnng:n= [(ki+1)(i+k)/2]+[1+(i1)(i i1(i = 1,2,...),{ln}[0,1), and {an} is a sequence in [a, 1 a] for someaÎ(0,1). Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of the nonexpansive mappings{Ti}are obtained. The results i=1 in this article extend those of the authors whose related researches are restricted to the situation of finite families of nonexpansive mappings. Mathematics Subject Classifications 2000:47H09; 47J25. Keywords:infinite family of nonexpansive mappings, strong and weak convergence, common fixed point, hybrid iteration
1 Introduction LetKbe a nonempty closed convex subset of a real uniformly convex Banach spaceE. A selfmappingT:K®Kis said to be nonexpansive if ||TxTy||||xy|| for allx,yÎ K.F:K®Kis said to beLLipschitzian if there exists a constantL> 0 such that ||Fx Fy||L||xy|| for allx,yÎK. Iterative techniques for approximating fixed points of nonexpansive mappings have been studied by various authors (see, e.g., [19]). In 2007, Wang [10] introduced an explicit hybrid iteration method for nonexpansive mappings in Hilbert space; and then Osilike et al. [11] extended Wangs results to arbitrary Banach spaces without the strong monotonicity assumption imposed on the hybrid operator. In 2009, Wang et al. [12] obtained the following strong and weak convergence theorems for a finite family of nonexpansive mappings in uniformly convex Banach space by using hybrid iteration method, which further extended and improved his results and partially improved those of Osilikes. Theorem 1.1. [12]Let E be a real uniformly convex Banach space endowed with the norm||∙||.Let I= {1,2,...,N},{Ti:iÎI}be N nonexpansive mappings from E into itself
© 2012 Deng; 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.