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 Approximation of fixed points of strongly pseudocontractive mappings in uniformly smooth Banach spaces

2006 ◽  
Vol 2006 ◽  
pp. 1-6
Author(s):  
Xue Zhiqun
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Unique Fixed Point ◽  
Nonempty Closed Convex Subset ◽  
Iterative Sequence ◽  
Pseudocontractive Mappings ◽  
Uniformly Smooth ◽  
Ishikawa Iterative Sequence ◽  
Uniformly Smooth Banach Spaces ◽  
Strongly Pseudocontractive Mapping

LetEbe a real uniformly smooth Banach space, andKa nonempty closed convex subset ofE. Assume thatT1+T2:K→Kis a continuous and strongly pseudocontractive mapping, whereT1:K→Kis Lipschitz andT2:K→Khas the bounded range mapping. Then the Ishikawa iterative sequence converges strongly to the unique fixed point ofT1+T2.

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.

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 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.

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.

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 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 Mann Iteration Processes on Uniform Convex n-Banach Space

2019 ◽  
pp. 1063-1608
Author(s):  
Mustafa Mohamed Hamed ◽  
Zeana Zaki Jamil
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Set ◽  
Nonempty Closed Convex Subset ◽  
Nonlinear Mapping ◽  
Uniformly Convex ◽  
Mann Iteration ◽  
Iteration Processes

Let Y be a"uniformly convex n-Banach space, M be a nonempty closed convex subset of Y, and S:M→M be adnonexpansive mapping. The purpose of this paper is to study some properties of uniform convex set that help us to develop iteration techniques for1approximationjof"fixed point of nonlinear mapping by using the Mann iteration processes in n-Banachlspace.

scholarly journals Existence of common fixed points of non-linear semigroups of Enriched Kannan contractive mappings

2021 ◽  
Vol 38 (1) ◽  
pp. 169-178
Author(s):  
SAYANTAN PANJA ◽  
◽  
MANTU SAHA ◽  
RAVINDRA K. BISHT ◽  
◽  
...  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Banach Space ◽  
Normal Structure ◽  
Contractive Mapping ◽  
Common Fixed Points ◽  
Contractive Mappings ◽  
Non Linear

In this article, we consider the non-linear semigroup of \textit{enriched Kannan} contractive mapping and prove the existence of common fixed point on a non-empty closed convex subset $\mathcal C$ of a real Banach space $\mathscr X$, having uniformly normal structure.

Fixed points of nearly weak uniformly L-Lipschitzian mappings in real Banach spaces

2018 ◽  
Vol 27 (1) ◽  
pp. 63-70
Author(s):  
Adesanmi Alao Mogbademu ◽  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Unique Fixed Point ◽  
Stability Result ◽  
Iteration Scheme ◽  
Lipschitzian Mapping ◽  
Nonempty Convex Subset ◽  
Lipschitzian Mappings ◽  
Nonempty Convex

Let K be a nonempty convex subset of a real Banach space X. Let T be a nearly weak uniformly L-Lipschitzian mapping. A modified Mann-type iteration scheme is proved to converge strongly to the unique fixed point of T. Our result is a significant improvement and generalization of several known results in this area of research. We give a specific example to support our result. Furthermore, an interesting equivalence of T-stability result between the convergence of modified Mann-type and modified Mann iterations is included.

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