scholarly journals A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Songnian He ◽  
Wenlong Zhu
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Boundary Point ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Method ◽  
Minimum Norm ◽  
Mann Iteration ◽  
Boundary Point Method

LetHbe a real Hilbert space andC⊂H a closed convex subset. LetT:C→Cbe a nonexpansive mapping with the nonempty set of fixed pointsFix(T). Kim and Xu (2005) introduced a modified Mann iterationx0=x∈C,yn=αnxn+(1−αn)Txn,xn+1=βnu+(1−βn)yn, whereu∈Cis an arbitrary (but fixed) element, and{αn}and{βn}are two sequences in(0,1). In the case where0∈C, the minimum-norm fixed point ofTcan be obtained by takingu=0. But in the case where0∉C, this iteration process becomes invalid becausexnmay not belong toC. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of Tand prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projectionPC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.

scholarly journals Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Luo Yi Shi ◽  
Ru Dong Chen
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Convex Subset ◽  
Unique Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Approximation Methods ◽  
Viscosity Approximation ◽  
Viscosity Approximation Methods

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mappingTof a closed convex subsetCof a CAT(0) spaceX. Suppose that the set Fix(T)of fixed points ofTis nonempty. For a contractionfonCandt∈(0,1), letxt∈Cbe the unique fixed point of the contractionx↦tf(x)⊕(1-t)Tx. We will show that ifXis a CAT(0) space satisfying some property, then{xt}converge strongly to a fixed point ofTwhich solves some variational inequality. Consider also the iteration process{xn}, wherex0∈Cis arbitrary andxn+1=αnf(xn)⊕(1-αn)Txnforn≥1, where{αn}⊂(0,1). It is shown that under certain appropriate conditions onαn,{xn}converge strongly to a fixed point ofTwhich solves some variational inequality.

scholarly journals Parallel and Cyclic Algorithms for Quasi-Nonexpansives in Hilbert Space

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Bin-Chao Deng ◽  
Tong Chen ◽  
Baogui Xin
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Common Fixed Point ◽  
Convex Subset ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings

Let{T}i=1NbeNquasi-nonexpansive mappings defined on a closed convex subsetCof a real Hilbert spaceH. Consider the problem of finding a common fixed point of these mappings and introduce the parallel and cyclic algorithms for solving this problem. We will prove the strong convergence of these algorithms.

scholarly journals Mann iteration process for monotone nonexpansive mappings with a graph

2019 ◽  
Vol 26 (4) ◽  
pp. 629-636
Author(s):  
Monther Rashed Alfuraidan
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Convex Subset ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Mann Iteration ◽  
Iteration Sequence ◽  
Mann Iteration Process ◽  
Monotone Nonexpansive Mapping

Abstract Let {(X,\lVert\,\cdot\,\rVert)} be a Banach space. Let C be a nonempty, bounded, closed and convex subset of X and let {T:C\rightarrow C} be a G-monotone nonexpansive mapping. In this work, it is shown that the Mann iteration sequence defined by x_{n+1}=t_{n}T(x_{n})+(1-t_{n})x_{n},\quad n=1,2,\dots, proves the existence of a fixed point of G-monotone nonexpansive mappings.

scholarly journals Strong convergence for monotone bilevel equilibria with constraints of variational inequalities and fixed points using subgradient extragradient implicit rule

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Long He ◽  
Yun-Ling Cui ◽  
Lu-Chuan Ceng ◽  
Tu-Yan Zhao ◽  
Dan-Qiong Wang ◽  
...  
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Fixed Point Problem ◽  
Point Problem ◽  
Common Solution ◽  
Implicit Rule

AbstractIn a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

scholarly journals Hybrid subgradient algorithm for equilibrium and fixed point problems by approximation of nonexpansive mapping

Filomat ◽  
2020 ◽  
Vol 34 (5) ◽  
pp. 1721-1729
Author(s):  
Seyed Aleomraninejad ◽  
Kanokwan Sitthithakerngkiet ◽  
Poom Kumam
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Convergence Theorem ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Expansive Mapping ◽  
Subgradient Algorithm

In this paper anew algorithm considered on a real Hilbert space for finding acommonpoint in the solution set of a class of pseudomonotone equilibrium problem and the set of fixed points of nonexpansive mappings. We produce this algorithm by mappings Tk that are approximations of non-expansive mapping T. The strong convergence theorem of the proposed algorithms is investigated. Our results generalize some recent results in the literature.

scholarly journals Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 187-196 ◽  
Author(s):  
Kifayat Ullah ◽  
Muhammad Arshad
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Point Theory ◽  
Uniformly Convex ◽  
Weak And Strong Convergence ◽  
Uniformly Convex Banach Spaces

In this paper we propose a new three-step iteration process, called M iteration process, for approximation of fixed points. Some weak and strong convergence theorems are proved for Suzuki generalized nonexpansive mappings in the setting of uniformly convex Banach spaces. Numerical example is given to show the efficiency of new iteration process. Our results are the extension, improvement and generalization of many known results in the literature of iterations in fixed point theory.

scholarly journals On Convergence Theorems for Generalized Alpha Nonexpansive Mappings in Banach Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Buthinah A. Bin Dehaish ◽  
Rawan K. Alharbi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Mann Iteration ◽  
Approximation Theorems ◽  
Ishikawa Iteration Process ◽  
Mann Iteration Process

The present paper seeks to illustrate approximation theorems to the fixed point for generalized α -nonexpansive mapping with the Mann iteration process. Furthermore, the same results are established with the Ishikawa iteration process in the uniformly convex Banach space setting. The presented results expand and refine many of the recently reported results in the literature.

scholarly journals A New Iterative Scheme for Solving the Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Rabian Wangkeeree ◽  
Pakkapon Preechasilp
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Real Hilbert Space ◽  
Nonexpansive Mappings ◽  
Common Solution

We introduce the new iterative methods for finding a common solution set of monotone, Lipschitz-type continuous equilibrium problems and the set of fixed point of nonexpansive mappings which is a unique solution of some variational inequality. We prove the strong convergence theorems of such iterative scheme in a real Hilbert space. The main result extends various results existing in the current literature.

scholarly journals Strong Convergence Theorem of Implicit Iteration Process for Generalized Asymptotically Nonexpansive Mappings in Hilbert Space

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
Lili He ◽  
Lei Deng ◽  
Jianjun Liu
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Iteration Process ◽  
Strong Convergence Theorem ◽  
Implicit Iteration Process ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive ◽  
Implicit Iteration

LetCbe a nonempty closed and convex subset of a Hilbert spaceH, letTandS:C→Cbe two commutative generalized asymptotically nonexpansive mappings. We introduce an implicit iteration process ofSandTdefined byxn=αnx0+(1−αn)(2/((n+1)(n+2)))∑k=0n∑i+j=kSiTjxn, and then prove that{xn}converges strongly to a common fixed point ofSandT. The results generalize and unify the corresponding results.

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