scholarly journals Some Modified Extragradient Methods for Solving Split Feasibility and Fixed Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-32 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Split Feasibility Problem ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Extragradient Methods ◽  
Infinite Dimensional ◽  
Point Set ◽  
Split Feasibility

We consider and study the modified extragradient methods for finding a common element of the solution setΓof a split feasibility problem (SFP) and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of infinite-dimensional Hilbert spaces. We propose an extragradient algorithm for finding an element ofFix(S)∩ΓwhereSis strictly pseudocontractive. It is proven that the sequences generated by the proposed algorithm converge weakly to an element ofFix(S)∩Γ. We also propose another extragradient-like algorithm for finding an element ofFix(S)∩ΓwhereS:C→Cis nonexpansive. It is shown that the sequences generated by the proposed algorithm converge strongly to an element ofFix(S)∩Γ.

scholarly journals The Mann-Type Extragradient Iterative Algorithms with Regularization for Solving Variational Inequality Problems, Split Feasibility, and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-19
Author(s):  
Lu-Chuan Ceng ◽  
Himanshu Gupta ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Split Feasibility

The purpose of this paper is to introduce and analyze the Mann-type extragradient iterative algorithms with regularization for finding a common element of the solution setΞof a general system of variational inequalities, the solution setΓof a split feasibility problem, and the fixed point setFix(S)of a strictly pseudocontractive mappingSin the setting of the Hilbert spaces. These iterative algorithms are based on the regularization method, the Mann-type iteration method, and the extragradient method due to Nadezhkina and Takahashi (2006). Furthermore, we prove that the sequences generated by the proposed algorithms converge weakly to an element ofFix(S)∩Ξ∩Γunder mild conditions.

scholarly journals Iterative Methods for Finding Solutions of a Class of Split Feasibility Problems over Fixed Point Sets in Hilbert Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1012
Author(s):  
Suthep Suantai ◽  
Narin Petrot ◽  
Montira Suwannaprapa
Keyword(s):  
Fixed Point ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Split Feasibility Problem ◽  
Inverse Image ◽  
Feasibility Problem ◽  
Point Sets ◽  
Fixed Point Set ◽  
Split Feasibility ◽  
Fixed Point Sets

We consider the split feasibility problem in Hilbert spaces when the hard constraint is common solutions of zeros of the sum of monotone operators and fixed point sets of a finite family of nonexpansive mappings, while the soft constraint is the inverse image of a fixed point set of a nonexpansive mapping. We introduce iterative algorithms for the weak and strong convergence theorems of the constructed sequences. Some numerical experiments of the introduced algorithm are also discussed.

scholarly journals A General Composite Algorithms for Solving General Equilibrium Problems and Fixed Point Problems in Hilbert Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-25
Author(s):  
Rattanaporn Wangkeeree ◽  
Uthai Kamraksa ◽  
Rabian Wangkeeree
Keyword(s):  
Fixed Point ◽  
General Equilibrium ◽  
Hilbert Spaces ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Fixed Point Set ◽  
Strictly Pseudocontractive Mapping ◽  
Asymptotically Nonexpansive ◽  
Point Set ◽  
The Common

We introduce a general composite algorithm for finding a common element of the set of solutions of a general equilibrium problem and the common fixed point set of a finite family of asymptotically nonexpansive mappings in the framework of Hilbert spaces. Strong convergence of such iterative scheme is obtained which solving some variational inequalities for a strongly monotone and strictly pseudocontractive mapping. Our results extend the corresponding recent results of Yao and Liou (2010).

scholarly journals Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-25 ◽  
Author(s):  
L. C. Ceng ◽  
A. Petruşel ◽  
J. C. Yao
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Approximation Method ◽  
General System ◽  
Solution Set ◽  
Common Element ◽  
Feasibility Problem ◽  
Extragradient Methods ◽  
System Of Variational Inequalities ◽  
Split Feasibility

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.

scholarly journals Applications of Fixed-Point and Optimization Methods to the Multiple-Set Split Feasibility Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Yonghong Yao ◽  
Rudong Chen ◽  
Giuseppe Marino ◽  
Yeong Cheng Liou
Keyword(s):  
Fixed Point ◽  
Inverse Problems ◽  
Linear Operator ◽  
Linear Transformation ◽  
Convex Sets ◽  
Split Feasibility Problem ◽  
Optimization Methods ◽  
Feasibility Problem ◽  
Convex Feasibility ◽  
Split Feasibility

The multiple-set split feasibility problem requires finding a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It can be a model for many inverse problems where constraints are imposed on the solutions in the domain of a linear operator as well as in the operator’s range. It generalizes the convex feasibility problem as well as the two-set split feasibility problem. In this paper, we will review and report some recent results on iterative approaches to the multiple-set split feasibility problem.

scholarly journals Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Shenghua Wang ◽  
Shin Min Kang
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Fixed Point Set ◽  
Point Set

We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

scholarly journals Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-27 ◽  
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Strong Convergence ◽  
Infinite Family ◽  
Accretive Operator ◽  
Common Element ◽  
Point Problem ◽  
Fixed Point Set ◽  
Extragradient Methods ◽  
Differentiable Norm ◽  
Implicit And Explicit

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

scholarly journals A general convergence theorem for multiple-set split feasibility problem in Hilbert spaces

2015 ◽  
Vol 31 (3) ◽  
pp. 349-357
Author(s):  
ABDUL RAHIM KHAN ◽  
◽  
MUJAHID ABBAS ◽  
YEKINI SHEHU ◽  
◽  
...  
Keyword(s):  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Split Feasibility Problem ◽  
Contractive Mapping ◽  
Convergence Result ◽  
Feasibility Problem ◽  
Infinite Dimensional ◽  
Split Feasibility ◽  
General Convergence

We establish strong convergence result of split feasibility problem for a family of quasi-nonexpansive multi-valued mappings and a total asymptotically strict pseudo-contractive mapping in infinite dimensional Hilbert spaces.

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