scholarly journals Strong Convergence of Viscosity Iteration Methods for Nonexpansive Mappings

2009 ◽  
Vol 2009 ◽  
pp. 1-17
Author(s):  
Jong Soo Jung
Keyword(s):  
Strong Convergence ◽  
Reflexive Banach Space ◽  
Iterative Scheme ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Point Property ◽  
Weakly Compact ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Differentiable Norm ◽  
Iteration Methods

We propose a new viscosity iterative scheme for finding fixed points of nonexpansive mappings in a reflexive Banach space having a uniformly Gâteaux differentiable norm and satisfying that every weakly compact convex subset of the space has the fixed point property for nonexpansive mappings. Certain different control conditions for viscosity iterative scheme are given and strong convergence of viscosity iterative scheme to a solution of a ceratin variational inequality is established.

scholarly journals An Iterative Algorithm on Approximating Fixed Points of Pseudocontractive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Youli Yu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Reflexive Banach Space ◽  
Contractive Mapping ◽  
Bounded Subset ◽  
Point Property ◽  
Uniformly Gâteaux Differentiable Norm ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm ◽  
Uniformly Continuous

LetEbe a real reflexive Banach space with a uniformly Gâteaux differentiable norm. LetKbe a nonempty bounded closed convex subset ofE,and every nonempty closed convex bounded subset ofKhas the fixed point property for non-expansive self-mappings. Letf:K→Ka contractive mapping andT:K→Kbe a uniformly continuous pseudocontractive mapping withF(T)≠∅. Let{λn}⊂(0,1/2)be a sequence satisfying the following conditions: (i)limn→∞λn=0; (ii)∑n=0∞λn=∞. Define the sequence{xn}inKbyx0∈K,xn+1=λnf(xn)+(1−2λn)xn+λnTxn, for alln≥0. Under some appropriate assumptions, we prove that the sequence{xn}converges strongly to a fixed pointp∈F(T)which is the unique solution of the following variational inequality:〈f(p)−p,j(z−p)〉≤0, for allz∈F(T).

scholarly journals Iterative Methods for Pseudocontractive Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Jong Soo Jung
Keyword(s):  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Initial Value ◽  
Weakly Compact ◽  
Pseudocontractive Mapping ◽  
Pseudocontractive Mappings ◽  
Differentiable Norm

LetEa reflexive Banach space having a uniformly Gâteaux differentiable norm. LetCbe a nonempty closed convex subset ofE,T:C→Ca continuous pseudocontractive mapping withF(T)≠∅, andA:C→Ca continuous bounded strongly pseudocontractive mapping with a pseudocontractive constantk∈(0,1). Let{αn}and{βn}be sequences in(0,1)satisfying suitable conditions and for arbitrary initial valuex0∈C, let the sequence{xn}be generated byxn=αnAxn+βnxn-1+(1-αn-βn)Txn,  n≥1.If either every weakly compact convex subset ofEhas the fixed point property for nonexpansive mappings orEis strictly convex, then{xn}converges strongly to a fixed point ofT, which solves a certain variational inequality related toA.

A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

2019 ◽  
Vol 28 (2) ◽  
pp. 191-198
Author(s):  
T. M. M. SOW
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Mann Iteration ◽  
Infinite Dimensional ◽  
Mann Algorithm

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

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.

scholarly journals Strong Convergence Theorems by Modified Four Step Iterative Scheme with Errors for Three Nonexpansive Mappings

2015 ◽  
Vol 55 (3) ◽  
pp. 667-678
Author(s):  
PANKAJ KUMAR JHADE ◽  
AMARJEET SINGH SALUJA
Keyword(s):  
Strong Convergence ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Convergence Theorems

scholarly journals Convergence of two-step iterative scheme with errors for two asymptotically nonexpansive mappings

2004 ◽  
Vol 2004 (37) ◽  
pp. 1965-1971 ◽  
Author(s):  
Hafiz Fukhar-ud-din ◽  
Safeer Hussain Khan
Keyword(s):  
Strong Convergence ◽  
Fixed Points ◽  
Banach Spaces ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Weak And Strong Convergence ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
The Common

A two-step iterative scheme with errors has been studied to approximate the common fixed points of two asymptotically nonexpansive mappings through weak and strong convergence in Banach spaces.

scholarly journals Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces

2002 ◽  
Vol 31 (4) ◽  
pp. 251-257 ◽  
Author(s):  
Wei-Shih Du ◽  
Young-Ye Huang ◽  
Chi-Lin Yen
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorems ◽  
Compact Convex ◽  
Nonexpansive Mappings ◽  
Finite Union ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Nonconvex Sets ◽  
Firmly Nonexpansive ◽  
Asymptotically Regular

It is shown that every asymptotically regular orλ-firmly nonexpansive mappingT:C→Chas a fixed point wheneverCis a finite union of nonempty weakly compact convex subsets of a Banach spaceXwhich is uniformly convex in every direction. Furthermore, if{T i}i∈Iis any compatible family of strongly nonexpansive self-mappings on such aCand the graphs ofT i,i∈I, have a nonempty intersection, thenT i,i∈I, have a common fixed point inC.

scholarly journals Strong convergence theorems for a sequence of nonexpansive mappings with gauge functions

2013 ◽  
Vol 21 (1) ◽  
pp. 183-200
Author(s):  
Prasit Cholamjiak ◽  
Yeol Je Cho ◽  
Suthep Suantai
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Convex Banach Space ◽  
Duality Mapping ◽  
Convergence Theorems ◽  
Strictly Convex Banach Space ◽  
Differentiable Norm ◽  
Gauge Functions

Abstract In this paper, we first prove a path convergence theorem for a nonexpansive mapping in a reflexive and strictly convex Banach space which has a uniformly Gˆateaux differentiable norm and admits the duality mapping jφ, where φ is a gauge function on [0,∞). Using this result, strong convergence theorems for common fixed points of a countable family of nonexpansive mappings are established.

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.

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