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}.

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 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 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

Strong convergence results for nonlinear mappings in real Banach spaces

2016 ◽  
Vol 25 (1) ◽  
pp. 85-92
Author(s):  
ADESANMI ALAO MOGBADEMU ◽  
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Real Banach Space ◽  
Nonempty Closed Convex Subset ◽  
Bounded Sequence ◽  
Real Sequence ◽  
Lipschitzian Mapping ◽  
Convergence Results ◽  
Increasing Function ◽  
Nonlinear Mappings

Let X be a real Banach space, K be a nonempty closed convex subset of X, T : K → K be a nearly uniformly L-Lipschitzian mapping with sequence {an}. Let kn ⊂ [1, ∞) and En be sequences with limn→∞ kn = 1, limn→∞ En = 0 and F(T) = {ρ ∈ K : T ρ = ρ} 6= ∅. Let {αn}n≥0 be real sequence in [0, 1] satisfying the following conditions: (i)P n≥0 αn = ∞ (ii) limn→∞ αn = 0. For arbitrary x0 ∈ K, let {xn}n≥0 be iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n ≥ 0. If there exists a strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that < T nx − T nρ, j(x − ρ) >≤ knkx − ρk 2 − Φ(kx − ρk) + En for all x ∈ K, then, {xn}n≥0 converges strongly to ρ ∈ F(T). It is also proved that the sequence of iteration {xn} defined by xn+1 = (1 − bn − dn)xn + bnT nxn + dnwn, n ≥ 0, where {wn}n≥0 is a bounded sequence in K and {bn}n≥0, {dn}n≥0 are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.

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 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 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.

Sign in / Sign up

Export Citation Format

Share Document