scholarly journals Strong convergence of a hybrid method for pseudomonotone variational inequalities and fixed point problems

2012 ◽  
Vol 20 (1) ◽  
pp. 489-504
Author(s):  
Xin Yu ◽  
Yonghong Yao ◽  
Yeong-Cheng Liou
Keyword(s):  
Strong Convergence ◽  
Hybrid Method ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Infinite Family ◽  
Common Fixed Points ◽  
Common Element ◽  
Lipschitz Continuous ◽  
Cq Method ◽  
Necessary And Sufficient

AbstractIn this paper, we suggest a hybrid method for finding a common element of the set of solution of a pseudomonotone, Lipschitz-continuous variational inequality problem and the set of common fixed points of an infinite family of nonexpansive mappings. The proposed iterative method combines two well-known methods: extragradient method and CQ method. We derive a necessary and sufficient condition for the strong convergence of the sequences generated by the proposed method.

scholarly journals A New Iterative Method for Finding Common Solutions of a System of Equilibrium Problems, Fixed-Point Problems, and Variational Inequalities

2010 ◽  
Vol 2010 ◽  
pp. 1-27 ◽  
Author(s):  
Jian-Wen Peng ◽  
Soon-Yi Wu ◽  
Jen-Chih Yao
Keyword(s):  
Fixed Point ◽  
Equilibrium Problems ◽  
Iterative Scheme ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
Infinite Family ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
System Of Equilibrium Problems ◽  
New Iterative Method

We introduce a new iterative scheme based on extragradient method and viscosity approximation method for finding a common element of the solutions set of a system of equilibrium problems, fixed point sets of an infinite family of nonexpansive mappings, and the solution set of a variational inequality for a relaxed cocoercive mapping in a Hilbert space. We prove strong convergence theorem. The results in this paper unify and generalize some well-known results in the literature.

scholarly journals Strong Convergence of a General Iterative Method for a Countable Family of Nonexpansive Mappings in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Chin-Tzong Pang ◽  
Eskandar Naraghirad
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Accretive Operator ◽  
Common Fixed Points ◽  
Countable Family ◽  
Convergence Theorems ◽  
General Iterative Method

We introduce a general algorithm to approximate common fixed points for a countable family of nonexpansive mappings in a real Banach space. We prove strong convergence theorems for the sequences produced by the methods and approximate a common fixed point of a countable family of nonexpansive mappings which solves uniquely the corresponding variational inequality. Furthermore, we apply our results for finding a zero of an accretive operator. It is important to state clearly that the contribution of this paper in relation with the previous works (Marino and Xu, 2006) is a technical method to prove strong convergence theorems of a general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces. Our results improve and generalize many known results in the current literature.

scholarly journals Strong Convergence of a Hybrid Iteration Scheme for Equilibrium Problems, Variational Inequality Problems and Common Fixed Point Problems, of Quasi-ϕ-Asymptotically Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Jing Zhao
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Monotone Mapping ◽  
Common Fixed Points ◽  
Iteration Scheme ◽  
Common Element ◽  
Asymptotically Nonexpansive Mappings ◽  
Asymptotically Nonexpansive

We introduce an iterative algorithm for finding a common element of the set of common fixed points of a finite family of closed quasi-ϕ-asymptotically nonexpansive mappings, the set of solutions of an equilibrium problem, and the set of solutions of the variational inequality problem for aγ-inverse strongly monotone mapping in Banach spaces. Then we study the strong convergence of the algorithm. Our results improve and extend the corresponding results announced by many others.

scholarly journals Strong Convergence of A Hybrid Method for Infinite Family of Nonexpansive Mapping and Variational Inequality

2021 ◽  
Vol 27 (1) ◽  
pp. 90-102
Author(s):  
Savita Rathee ◽  
Monika Swami
Keyword(s):  
Variational Inequality ◽  
Strong Convergence ◽  
Hybrid Method ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Infinite Family ◽  
The Other ◽  
Iterative Technique ◽  
Expansive Mapping ◽  
Typical Component

The motivation behind this paper is to use hybrid method for searching a typical component of the set of fixed point of an infinite family of non expansive mapping and the set of monotone, Lipschtiz continuous variational inequality problem. The contemplated method is combination of two method one is extragradient method and the other one is DQ method. Also, we demonstrate the strong convergence of the designed iterative technique, under some warm conditions.

scholarly journals Strong Convergence of a Hybrid Projection Algorithm for Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in a Banach Space

2009 ◽  
Vol 2009 ◽  
pp. 1-22 ◽  
Author(s):  
Wariam Chuayjan ◽  
Sornsak Thianwan
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Nonexpansive Mappings ◽  
Common Fixed Points ◽  
Projection Algorithm ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Inverse Strongly Monotone Operator ◽  
Hybrid Projection Algorithm

We introduce and study a new hybrid projection algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly-monotone operator in a Banach space. Under suitable assumptions, we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space. The results obtained in this paper extend and improve the several recent results in this area.

scholarly journals A New Iterative Algorithm for the Set of Fixed-Point Problems of Nonexpansive Mappings and the Set of Equilibrium Problem and Variational Inequality Problem

2011 ◽  
Vol 2011 ◽  
pp. 1-24 ◽  
Author(s):  
Atid Kangtunyakarn
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Equilibrium Problem ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Fixed Point Problems ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Pseudocontractive Mappings

We introduce a new iterative scheme and a new mapping generated by infinite family of nonexpansive mappings and infinite real number. By using both of these ideas, we obtain strong convergence theorem for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of fixed-point problems of infinite family of nonexpansive mappings. Moreover, we apply our main result to obtain strong convergence theorems for finding a common element of the set of solution of equilibrium problem and the set of variational inequality and the set of common fixed point of pseudocontractive mappings.

scholarly journals A General Approximation Scheme for Solutions of Various Problems in Fixed Point Theory

2013 ◽  
Vol 2013 ◽  
pp. 1-18 ◽  
Author(s):  
Eric U. Ofoedu
Keyword(s):  
Fixed Points ◽  
Fixed Point Theory ◽  
Null Space ◽  
Nonexpansive Mappings ◽  
Infinite Family ◽  
Point Theory ◽  
Common Fixed Points ◽  
Iterative Sequence ◽  
Generalized Mixed Equilibrium Problem ◽  
Common Element

It is our aim to prove strong convergence of a new iterative sequence to a common element of the solution set of a generalized mixed equilibrium problem; the null space of an inverse strongly monotone operator; the set of common fixed points of a countable infinite family of nonexpansive mappings; and the set of fixed points of a continuous pseudocontractive mapping. Moreover, the common element is also a unique solution of a variational inequality problem and optimality condition for a certain minimization problem. Our theorems generalize, improve, and unify several recently announced results.

scholarly journals Strong Convergence Theorem for Bregman Strongly Nonexpansive Mappings and Equilibrium Problems in Reflexive Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinhua Zhu ◽  
Shih-sen Chang ◽  
Min Liu
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Hybrid Method ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Reflexive Banach Spaces

By using a new hybrid method, a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of Bregman strongly nonexpansive mappings in a reflexive Banach space is proved.

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