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 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 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 Hybrid Extragradient Iterative Algorithms for Variational Inequalities, Variational Inclusions, and Fixed-Point Problems

2012 ◽  
Vol 2012 ◽  
pp. 1-32
Author(s):  
Lu-Chuan Ceng ◽  
Ching-Feng Wen
Keyword(s):  
Fixed Point ◽  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Real Hilbert Space ◽  
Extragradient Method ◽  
Iterative Algorithms ◽  
General System ◽  
Point Problem ◽  
Strictly Pseudocontractive Mapping ◽  
Common Solution

We investigate the problem of finding a common solution of a general system of variational inequalities, a variational inclusion, and a fixed-point problem of a strictly pseudocontractive mapping in a real Hilbert space. Motivated by Nadezhkina and Takahashi's hybrid-extragradient method, we propose and analyze new hybrid-extragradient iterative algorithm for finding a common solution. It is proven that three sequences generated by this algorithm converge strongly to the same common solution under very mild conditions. Based on this result, we also construct an iterative algorithm for finding a common fixed point of three mappings, such that one of these mappings is nonexpansive, and the other two mappings are strictly pseudocontractive mappings.

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 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 On Variational Inclusion and Common Fixed Point Problems inq-Uniformly Smooth Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-20
Author(s):  
Yanlai Song ◽  
Huiying Hu ◽  
Luchuan Ceng
Keyword(s):  
Fixed Point ◽  
Common Fixed Point ◽  
Iterative Algorithm ◽  
Infinite Family ◽  
General System ◽  
Iterative Sequence ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Fixed Point Set ◽  
Uniformly Smooth

We introduce a general iterative algorithm for finding a common element of the common fixed-point set of an infinite family ofλi-strict pseudocontractions and the solution set of a general system of variational inclusions for two inverse strongly accretive operators in aq-uniformly smooth Banach space. Then, we prove a strong convergence theorem for the iterative sequence generated by the proposed iterative algorithm under very mild conditions. The methods in the paper are novel and different from those in the early and recent literature. Our results can be viewed as the improvement, supplementation, development, and extension of the corresponding results in some references to a great extent.

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 Iterative Algorithms with Perturbations for Solving the Systems of Generalized Equilibrium Problems and the Fixed Point Problems of Two Quasi-Nonexpansive Mappings

2012 ◽  
Vol 2012 ◽  
pp. 1-22
Author(s):  
Rabian Wangkeeree ◽  
Uraiwan Boonkong
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Iterative Algorithms ◽  
Common Fixed Points ◽  
Common Element ◽  
Fixed Point Set ◽  
Generalized Equilibrium ◽  
Point Set ◽  
The Common

We introduce new iterative algorithms with perturbations for finding a common element of the set of solutions of the system of generalized equilibrium problems and the set of common fixed points of two quasi-nonexpansive mappings in a Hilbert space. Under suitable conditions, strong convergence theorems are obtained. Furthermore, we also consider the iterative algorithms with perturbations for finding a common element of the solution set of the systems of generalized equilibrium problems and the common fixed point set of the super hybrid mappings in Hilbert spaces.

scholarly journals Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

2015 ◽  
Vol 46 (4) ◽  
pp. 453-503
Author(s):  
Lu-Chuan Ceng
Keyword(s):  
Variational Inequalities ◽  
Iterative Algorithm ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Steepest Descent Method ◽  
Feasibility Problem ◽  
Fixed Point Set ◽  
Special Cases ◽  
Mixed Equilibrium ◽  
Split Feasibility

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

scholarly journals Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Zhao-Rong Kong ◽  
Lu-Chuan Ceng ◽  
Qamrul Hasan Ansari ◽  
Chin-Tzong Pang
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Fixed Point Set ◽  
Inequality Problem ◽  
Hybrid Extragradient Method ◽  
Point Set ◽  
Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI.

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