scholarly journals An Extension of Subgradient Method for Variational Inequality Problems in Hilbert Space

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xueyong Wang ◽  
Shengjie Li ◽  
Xipeng Kou
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Nonexpansive Mapping ◽  
Iterative Process ◽  
Subgradient Method ◽  
Variational Inequality Problems ◽  
Mild Conditions ◽  
Weak Convergence Theorem

An extension of subgradient method for solving variational inequality problems is presented. A new iterative process, which relates to the fixed point of a nonexpansive mapping and the current iterative point, is generated. A weak convergence theorem is obtained for three sequences generated by the iterative process under some mild conditions.

WEAK CONVERGENCE OF AN IMPLICIT ITERATIVE PROCESS WITH ERRORS FOR AN ASYMPTOTICALLY QUASI I-NONEXPANSIVE MAPPING IN BANACH SPACES

2011 ◽  
Vol 04 (02) ◽  
pp. 309-319 ◽  
Author(s):  
Farrukh Mukhamedov ◽  
Mansoor Saburov
Keyword(s):  
Fixed Point ◽  
Weak Convergence ◽  
Nonexpansive Mapping ◽  
Common Fixed Point ◽  
Banach Spaces ◽  
Iterative Process

In this paper we prove the weak convergence of the implicit iterative process with errors to a common fixed point of an asymptotically quasi I-nonexpansive mapping T and an asymptotically quasi-nonexpansive mapping I in Banach spaces.

scholarly journals Strong Convergence of an Iterative Algorithm for Hierarchical Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Poom Kumam ◽  
Thanyarat Jitpeera
Keyword(s):  
Fixed Point ◽  
Variational Inequality ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Algorithm ◽  
Variational Inequality Problem ◽  
Fixed Point Set ◽  
Mild Conditions ◽  
Hierarchical Problems ◽  
Point Set

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.

scholarly journals A Mean Extragradient Method for Solving Variational Inequalities

Symmetry ◽  
2021 ◽  
Vol 13 (3) ◽  
pp. 462
Author(s):  
Apichit Buakird ◽  
Nimit Nimana ◽  
Narin Petrot
Keyword(s):  
Hilbert Space ◽  
Variational Inequality ◽  
Weak Convergence ◽  
Numerical Experiments ◽  
Variational Inequality Problem ◽  
Extragradient Method ◽  
Mean Value ◽  
Inequality Problem ◽  
The Mean ◽  
Mean Value Method

We propose a modified extragradient method for solving the variational inequality problem in a Hilbert space. The method is a combination of the well-known subgradient extragradient with the Mann’s mean value method in which the updated iterate is picked in the convex hull of all previous iterates. We show weak convergence of the mean value iterate to a solution of the variational inequality problem, provided that a condition on the corresponding averaging matrix is fulfilled. Some numerical experiments are given to show the effectiveness of the obtained theoretical result.

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