scholarly journals Convergence to Common Fixed Point for Generalized Asymptotically Nonexpansive Semigroup in Banach Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Yali Li ◽  
Jianjun Liu ◽  
Lei Deng
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Contractive Mapping ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Nonexpansive Semigroup ◽  
Iterative Schemes ◽  
Strictly Convex Banach Space ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm

LetKbe a nonempty closed convex subset of a reflexive and strictly convex Banach spaceEwith a uniformlyGâteauxdifferentiable norm,ℱ={T(h):h≥0}a generalized asymptotically nonexpansive self-mapping semigroup ofK, andf:K→Ka fixed contractive mapping with contractive coefficientβ∈(0,1). We prove that the following implicit and modified implicit viscosity iterative schemes{xn}defined byxn=αnf(xn)+(1−αn)T(tn)xnandxn=αnyn+(1−αn)T(tn)xn, yn=βnf(xn−1)+(1−βn)xn−1strongly converge top∈Fasn→∞andpis the unique solution to the following variational inequality:〈f(p)−p,j(y−p)〉≤0for ally∈F.

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 Fixed points of mappings satisfying semicontractivity conditions

1992 ◽  
Vol 53 (1) ◽  
pp. 25-38
Author(s):  
Jürgen Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Points ◽  
Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Opial’S Condition ◽  
Asymptotically Nonexpansive

AbstractLet A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.

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 Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

1991 ◽  
Vol 43 (1) ◽  
pp. 153-159 ◽  
Author(s):  
J. Schu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces ◽  
Asymptotically Nonexpansive ◽  
Fourth Kind

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

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.

scholarly journals Convergence theorems of the sequence of iterates for a finite family asymptotically nonexpansive mappings

2001 ◽  
Vol 27 (11) ◽  
pp. 653-662 ◽  
Author(s):  
Jui-Chi Huang
Keyword(s):  
Banach Space ◽  
Nonexpansive Mappings ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Bounded Sequences

LetEbe a uniformly convex Banach space,Ca nonempty closed convex subset ofE. In this paper, we introduce an iteration scheme with errors in the sense of Xu (1998) generated by{Tj:C→C}j=1ras follows:Un(j)=an(j)I+bn(j)TjnUn(j−1)+cn(j)un(j),j=1,2,…,r,x1∈C,xn+1=an(r)xn+bn(r)TrnUn(r−1)xn+cn(r)un(r),n≥1, whereUn(0):=I,Ithe identity map; and{un(j)}are bounded sequences inC; and{an(j)},{bn(j)}, and{cn(j)}are suitable sequences in[0,1]. We first consider the behaviour of iteration scheme above for a finite family of asymptotically nonexpansive mappings. Then we generalize theorems of Schu and Rhoades.

scholarly journals Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

2011 ◽  
Vol 2011 ◽  
pp. 1-19
Author(s):  
Tanakit Thianwan
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Iteration Scheme ◽  
Convex Banach Space ◽  
Uniformly Convex Banach Space ◽  
Uniformly Convex ◽  
Ishikawa Iteration ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Fréchet Differentiable

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach spaceEwhich has a Fréchet differentiable norm or its dualE*has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

scholarly journals Fixed point iteration for asymptotically quasi-nonexpansive mappings in Banach spaces

2005 ◽  
Vol 2005 (11) ◽  
pp. 1685-1692 ◽  
Author(s):  
Somyot Plubtieng ◽  
Rabian Wangkeeree
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Subset ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Completely Continuous

Suppose thatCis a nonempty closed convex subset of a real uniformly convex Banach spaceX. LetT:C→Cbe an asymptotically quasi-nonexpansive mapping. In this paper, we introduce the three-step iterative scheme for such map with error members. Moreover, we prove that ifTis uniformlyL-Lipschitzian and completely continuous, then the iterative scheme converges strongly to some fixed point ofT.

scholarly journals Asymptotic behavior of almost-orbits of reversible semigroups of non-Lipschitzian mappings in Banach spaces

1997 ◽  
Vol 20 (1) ◽  
pp. 51-60
Author(s):  
Jong Soo Jung ◽  
Jong Yeoul Park ◽  
Jong Seo Park
Keyword(s):  
Metric Projection ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Convex Banach Space ◽  
Semitopological Semigroup ◽  
Strong Limit ◽  
Lipschitzian Mappings ◽  
Asymptotically Nonexpansive ◽  
Differentiable Norm ◽  
Fréchet Differentiable

LetCbe a nonempty closed convex subset of a uniformly convex Banach spaceEwith a Fréchet differentiable norm,Ga right reversible semitopological semigroup, and𝒮={S(t):t∈G}a continuous representation ofGas mappings of asymptotically nonexpansive type ofCinto itself. The weak convergence of an almost-orbit{u(t):t∈G}of𝒮={S(t):t∈G}onCis established. Furthermore, it is shown that ifPis the metric projection ofEonto setF(S)of all common fixed points of𝒮={S(t):t∈G}, then the strong limit of the net{Pu(t):t∈G}exists.

Sign in / Sign up

Export Citation Format

Share Document