Modified Extragradient Methods for a System of Variational Inequalities in Banach Spaces

2009 ◽  
Vol 110 (3) ◽  
pp. 1211-1224 ◽  
Author(s):  
Yonghong Yao ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Yeong-Cheng Liou ◽  
Huma Yaqoob
Keyword(s):  
Variational Inequalities ◽  
Banach Spaces ◽  
Extragradient Methods ◽  
System Of Variational Inequalities

scholarly journals Modified Relaxed Extragradient Method for a General System of Variational Inequalities and Nonexpansive Mappings in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yuanheng Wang ◽  
Liu Yang
Keyword(s):  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Iterative Sequence ◽  
Monotone Mappings ◽  
System Of Variational Inequalities ◽  
Common Solution ◽  
Demiclosedness Principle

The purpose of this paper is to introduce a new modified relaxed extragradient method and study for finding some common solutions for a general system of variational inequalities with inversestrongly monotone mappings and nonexpansive mappings in the framework of real Banach spaces. By using the demiclosedness principle, it is proved that the iterative sequence defined by the relaxed extragradient method converges strongly to a common solution for the system of variational inequalities and nonexpansive mappings under quite mild conditions.

scholarly journals Mann-Type Extragradient Methods for General Systems of Variational Inequalities with Multivalued Variational Inclusion Constraints in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
Lu-Chuan Ceng ◽  
Abdul Latif ◽  
Saleh A. Al-Mezel
Keyword(s):  
Banach Space ◽  
Variational Inequalities ◽  
Banach Spaces ◽  
Nonexpansive Mappings ◽  
Extragradient Method ◽  
General System ◽  
Variational Inclusion ◽  
Uniformly Convex ◽  
Extragradient Methods ◽  
Extragradient Algorithm

We introduce Mann-type extragradient methods for a general system of variational inequalities with solutions of a multivalued variational inclusion and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type extragradient methods are based on Korpelevich’s extragradient method and Mann iteration method. We first consider and analyze a Mann-type extragradient algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another Mann-type extragradient algorithm in a smooth and uniformly convex Banach space. Under suitable assumptions, we derive some weak and strong convergence theorems. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

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