scholarly journals A convergence theorem of multi-step iterative scheme for nonlinear maps

2015 ◽  
Vol 98 (112) ◽  
pp. 281-285
Author(s):  
Adesanmi Mogbademu
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Convergence Theorem ◽  
Real Banach Space ◽  
Iterative Scheme ◽  
Nonempty Closed Convex Subset ◽  
Nonlinear Maps

Let K be a nonempty closed convex subset of a real Banach space X,T:K ? K a nearly uniformly L-Lipschitzian (with sequence {rn}) asymptotically generalized ?-hemicontractive mapping (with sequence kn ? [1,?), lim n?? kn = 1) such that F(T) = {p?K:Tp=p}. Let {?n}n?0, {?kn}n?0 be real sequences in [0,1] satisfying the conditions: (i) ?n?0 ?n = 1 (ii) limn?? ?n, ?kn = 0, k = 1, 2,..., p?1. For arbitrary x0 ? K, let {xn}n?0 be a multi-step sequence iteratively defined by xn+1=(1??n)xn + ?nTny1n, n?0, ykn = (1 ? ?kn )xn + ?kn Tnyk+1n, k = 1,2,..., p?2 (0.1), yp?1n=(1? ?p?1n)xn + ?p?1n Tnxn, n ? 0, p ? 2. Then, {xn}n?0 converges strongly to p ? F(T). The result proved in this note significantly improve the results of Kim et al. [2].

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 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 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 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 The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Zhiqun Xue ◽  
Yaning Wang ◽  
Haiyun Zhou
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Real Banach Space ◽  
Nonempty Closed Convex Subset ◽  
Pseudocontractive Mapping ◽  
Uniformly Smooth ◽  
Convergence Results ◽  
Strongly Pseudocontractive Mapping ◽  
Bounded Sequences

LetEbe an arbitrary uniformly smooth real Banach space, letDbe a nonempty closed convex subset ofE, and letT:D→Dbe a uniformly generalized Lipschitz generalized asymptoticallyΦ-strongly pseudocontractive mapping withq∈F(T)≠∅. Let{an},{bn},{cn},{dn}be four real sequences in[0,1]and satisfy the conditions: (i)an+cn≤1,bn+dn≤1; (ii)an,bn,dn→0asn→∞andcn=o(an); (iii)Σn=0∞an=∞. For somex0,z0∈D, let{un},{vn},{wn}be any bounded sequences inD, and let{xn},{zn}be the modified Ishikawa and Mann iterative sequences with errors, respectively. Then the convergence of{xn}is equivalent to that of{zn}.

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.

Counterexamples Concerning Support Theorems for Convex Sets in Hilbert Space

1988 ◽  
Vol 31 (1) ◽  
pp. 121-128 ◽  
Author(s):  
R. R. Phelps
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Convex Function ◽  
Convex Subset ◽  
Convex Sets ◽  
Lower Semicontinuous ◽  
Nonempty Closed Convex Subset ◽  
Common Point ◽  
Support Points ◽  
Support Theorems

AbstractThe Bishop-Phelps theorem guarantees the existence of support points and support functionals for a nonempty closed convex subset of a Banach space; equivalently, it guarantees the existence of subdifferentials and points of subdifferentiability of a proper lower semicontinuous convex function on a Banach space. In this note we show that most of these results cannot be extended to pairs of convex sets or functions, even in Hilbert space. For instance, two proper lower semicontinuous convex functions need not have a common point of subdifferentiability nor need they have a subdifferential in common. Negative answers are also obtained to certain questions concerning density of support points for the closed sum of two convex subsets of Hilbert space.

scholarly journals Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Kyung Soo Kim
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Nonempty Closed Convex Subset ◽  
Common Element ◽  
Mild Conditions ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive ◽  
Reversible Semigroup ◽  
Invariant Means

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space with respect to a sequence of asymptotically left invariant means{μn}defined on an appropriate invariant subspace ofl∞(S), whereSis a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.

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