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.

scholarly journals Strong Convergence for Hybrid ImplicitS-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Shin Min Kang ◽  
Arif Rafiq ◽  
Faisal Ali ◽  
Young Chel Kwun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Iteration Scheme ◽  
Nonempty Closed Convex Subset ◽  
Pseudocontractive Mappings

LetKbe a nonempty closed convex subset of a real Banach spaceE, letS:K→Kbe nonexpansive, and let  T:K→Kbe Lipschitz strongly pseudocontractive mappings such thatp∈FS∩FT=x∈K:Sx=Tx=xandx-Sy≤Sx-Sy and x-Ty≤Tx-Tyfor allx, y∈K. Letβnbe a sequence in0, 1satisfying (i)∑n=1∞βn=∞; (ii)limn→∞⁡βn=0.For arbitraryx0∈K, letxnbe a sequence iteratively defined byxn=Syn, yn=1-βnxn-1+βnTxn, n≥1.Then the sequencexnconverges strongly to a common fixed pointpofSandT.

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.

The Equivalence of Mann and Implicit Mann Iterations for Uniformly Pseudocontractive Mappings in Uniformly Smooth Banach Spaces

2011 ◽  
Vol 50-51 ◽  
pp. 718-722
Author(s):  
Cheng Wang ◽  
Zhi Ming Wang
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Banach Spaces ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonempty Closed Convex Subset ◽  
Mann Iteration ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Uniformly Smooth Banach Spaces

In this paper, suppose is an arbitrary uniformly smooth real Banach space, and is a nonempty closed convex subset of . Let be a generalized Lipschitzian and uniformly pseudocontractive self-map with . Suppose that , are defined by Mann iteration and implicit Mann iteration respectively, with the iterative parameter satisfying certain conditions. Then the above two iterations that converge strongly to fixed point of are equivalent.

scholarly journals Iterative approximation of fixed point forΦ-hemicontractive mapping without Lipschitz assumption

2005 ◽  
Vol 2005 (17) ◽  
pp. 2711-2718
Author(s):  
Xue Zhiqun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Unique Fixed Point ◽  
Nonempty Closed Convex Subset ◽  
Iterative Sequence ◽  
Iterative Approximation ◽  
Ishikawa Iterative Sequence ◽  
Uniformly Continuous

LetEbe an arbitrary real Banach space and letKbe a nonempty closed convex subset ofEsuch thatK+K⊂K. Assume thatT:K→Kis a uniformly continuous andΦ-hemicontractive mapping. It is shown that the Ishikawa iterative sequence with errors converges strongly to the unique fixed point ofT.

scholarly journals Strong convergence theorem for two asymptotically quasi-nonexpansive mappings with errors in Banach space

2007 ◽  
Vol 38 (1) ◽  
pp. 85-92 ◽  
Author(s):  
G. S. Saluja
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Fixed Points ◽  
Convex Subset ◽  
Convergence Theorem ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Nonempty Closed Convex Subset ◽  
Strong Convergence Theorem

In this paper, we study strong convergence of common fixed points of two asymptotically quasi-nonexpansive mappings and prove that if $K$ is a nonempty closed convex subset of a real Banach space $E$ and let $ S, T\colon K\to K $ be two asymptotically quasi-nonexpansive mappings with sequences $ \{u_n\}$, $\{v_n\}\subset [0,\infty) $ such that $ \sum_{n=1}^{\infty}u_n

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.

Fixed point theorems for uniformly generalized Kannan type semigroup of self-mappings

2017 ◽  
Vol 26 (2) ◽  
pp. 231-240
Author(s):  
AHMED H. SOLIMAN ◽  
MOHAMMAD IMDAD ◽  
MD AHMADULLAH
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Real Banach Space ◽  
Normal Structure ◽  
Uniform Normal Structure

In this paper, we consider a new uniformly generalized Kannan type semigroup of self-mappings defined on a closed convex subset of a real Banach space equipped with uniform normal structure and employ the same to show that such semigroup of self-mappings admits a common fixed point provided the underlying semigroup of self-mappings has a bounded orbit.

scholarly journals Convergence Theorems for Finite Family of Multivalued Maps in Uniformly Convex Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Yekini Shehu
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Iterative Process ◽  
Real Banach Space ◽  
Finite Family ◽  
Uniformly Convex ◽  
Multivalued Maps ◽  
Convergence Theorems ◽  
Uniformly Convex Banach Spaces

We introduce a new iterative process to approximate a common fixed point of a finite family of multivalued maps in a uniformly convex real Banach space and establish strong convergence theorems for the proposed process. Furthermore, strong convergence theorems for finite family of quasi-nonexpansive multivalued maps are obtained. Our results extend important recent results.

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.

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