scholarly journals Strong Convergence Theorems for Common Fixed Points of an Infinite Family of Asymptotically Nonexpansive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Yuanheng Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In the framework of a real Banach space with uniformly Gateaux differentiable norm, some new viscosity iterative sequences{xn}are introduced for an infinite family of asymptotically nonexpansive mappingsTii=1∞in this paper. Under some appropriate conditions, we prove that the iterative sequences{xn}converge strongly to a common fixed point of the mappingsTii=1∞, which is also a solution of a variational inequality. Our results extend and improve some recent results of other authors.

The Strong Convergence of a New Iterative Algorithm for Asymptotically Nonexpansive Mappings

2013 ◽  
Vol 756-759 ◽  
pp. 3628-3633
Author(s):  
Yuan Heng Wang ◽  
Wei Wei Sun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Iterative Sequence ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

In a real Banach space E with a uniformly differentiable norm, we prove that a new iterative sequence converges strongly to a fixed point of an asymptotically nonexpansive mapping. The results in this paper improve and extend some recent results of other authors.

scholarly journals A New Iteration Process for Approximation of Common Fixed Points for Finite Families of Total Asymptotically Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
C. E. Chidume ◽  
E. U. Ofoedu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Recursion Formula ◽  
Iterative Sequence ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Total Asymptotically Nonexpansive Mappings

LetEbe a real Banach space, andKa closed convex nonempty subset ofE. LetT1,T2,…,Tm:K→Kbemtotal asymptotically nonexpansive mappings. A simple iterative sequence{xn}n≥1is constructed inEand necessary and sufficient conditions for this sequence to converge to a common fixed point of{Ti}i=1mare given. Furthermore, in the case thatEis a uniformly convex real Banach space, strong convergence of the sequence{xn}n=1∞to a common fixed point of the family{Ti}i=1mis proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.

scholarly journals Mixed Approximation for Nonexpansive Mappings in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Qing-Bang Zhang ◽  
Fu-Quan Xia ◽  
Ming-Jie Liu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Viscosity Approximation ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Type Approximation ◽  
Differentiable Norm ◽  
Convergence Of The Scheme

The mixed viscosity approximation is proposed for finding fixed points of nonexpansive mappings, and the strong convergence of the scheme to a fixed point of the nonexpansive mapping is proved in a real Banach space with uniformly Gâteaux differentiable norm. The theorem about Halpern type approximation for nonexpansive mappings is shown also. Our theorems extend and improve the correspondingly results shown recently.

scholarly journals Weak convergence theorem for Passty type asymptotically nonexpansive mappings

1999 ◽  
Vol 22 (1) ◽  
pp. 217-220
Author(s):  
B. K. Sharma ◽  
B. S. Thakur ◽  
Y. J. Cho
Keyword(s):  
Banach Space ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Weak Convergence Theorem ◽  
Fréchet Differentiable

In this paper, we prove a convergence theorem for Passty type asymptotically nonexpansive mappings in a uniformly convex Banach space with Fréchet-differentiable norm.

scholarly journals A New Iterative Scheme of Modified Mann Iteration in Banach Space

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jinzuo Chen ◽  
Dingping Wu ◽  
Caifen Zhang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Accretive Operators ◽  
Mann Iteration ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

We introduce the modified iterations of Mann's type for nonexpansive mappings and asymptotically nonexpansive mappings to have the strong convergence in a uniformly convex Banach space. We study approximation of common fixed point of asymptotically nonexpansive mappings in Banach space by using a new iterative scheme. Applications to the accretive operators are also included.

scholarly journals Common fixed points for generalized asymptotically nonexpansive mappings

2012 ◽  
Vol 44 (1) ◽  
pp. 23-29
Author(s):  
Sumit Chandok ◽  
T. D. Narang
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Metric Spaces ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Asymptotically Nonexpansive Mappings ◽  
Convex Metric Spaces ◽  
Asymptotically Nonexpansive ◽  
The Subject ◽  
Common Fixed Point Theorem

A common fixed point theorem for noncommuting generalized asymptotically nonexpansive mappings has been obtained in convex metric spaces. As an application, a result on the set of best approximation is also derived for such class of mappings. The proved results unify and extend some of the known results on the subject.

scholarly journals Strong Convergence Theorems for Two Total Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xi-bing He ◽  
Xiao-jian Hui ◽  
Hui Xing
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Mathematical Programming ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Hybrid Method ◽  
Nonexpansive Mappings ◽  
Convergence Theorems ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi-ϕ-asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

scholarly journals Existence ofixed points for pointwise eventually asymptotically nonexpansive mappings

2019 ◽  
Vol 20 (1) ◽  
pp. 119
Author(s):  
M. Radhakrishnan ◽  
S. Rajesh
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Fixed Point Property ◽  
Uniformly Convex ◽  
Point Property ◽  
Opial Condition ◽  
Nonexpansive Maps ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

<p>Kirk introduced the notion of pointwise eventually asymptotically non-expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically non expansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings when ever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X  has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) &lt; 1. Also, we study the asymptotic behavior of the sequence {T<sup>n</sup>x} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map.</p>

scholarly journals Strong convergence and control condition of modified Halpern iterations in Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-10
Author(s):  
Yonghong Yao ◽  
Rudong Chen ◽  
Haiyun Zhou
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonempty Closed Convex Subset ◽  
Unique Element ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
And Control

LetCbe a nonempty closed convex subset of a real Banach spaceXwhich has a uniformly Gâteaux differentiable norm. LetT∈ΓCandf∈ΠC. Assume that{xt}converges strongly to a fixed pointzofTast→0, wherextis the unique element ofCwhich satisfiesxt=tf(xt)+(1−t)Txt. Let{αn}and{βn}be two real sequences in(0,1)which satisfy the following conditions:(C1)lim⁡n→∞αn=0;(C2)∑n=0∞αn=∞;(C6)0<lim⁡inf⁡n→∞βn≤lim⁡sup⁡n→∞βn<1. For arbitraryx0∈C, let the sequence{xn}be defined iteratively byyn=αnf(xn)+(1−αn)Txn,n≥0,xn+1=βnxn+(1−βn)yn,n≥0. Then{xn}converges strongly to a fixed point ofT.

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