An Implicit Extragradient Method for Hierarchical Variational Inequalities
<p/> <p>As a well-known numerical method, the extragradient method solves numerically the variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i1.gif"/></inline-formula> of finding <inline-formula> <graphic file="1687-1812...
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doaj-17c7c5780c4e4a908429bd0f547ac2112020-11-24T22:00:04ZengSpringerOpenFixed Point Theory and Applications1687-18201687-18122011-01-0120111697248An Implicit Extragradient Method for Hierarchical Variational InequalitiesLiou YeongChengYao Yonghong<p/> <p>As a well-known numerical method, the extragradient method solves numerically the variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i1.gif"/></inline-formula> of finding <inline-formula> <graphic file="1687-1812-2011-697248-i2.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2011-697248-i3.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2011-697248-i4.gif"/></inline-formula>. In this paper, we devote to solve the following hierarchical variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i5.gif"/></inline-formula> Find <inline-formula> <graphic file="1687-1812-2011-697248-i6.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2011-697248-i7.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2011-697248-i8.gif"/></inline-formula>. We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i9.gif"/></inline-formula>. It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of <inline-formula> <graphic file="1687-1812-2011-697248-i10.gif"/></inline-formula> in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i11.gif"/></inline-formula>.</p>http://www.fixedpointtheoryandapplications.com/content/2011/697248 |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Liou YeongCheng Yao Yonghong |
spellingShingle |
Liou YeongCheng Yao Yonghong An Implicit Extragradient Method for Hierarchical Variational Inequalities Fixed Point Theory and Applications |
author_facet |
Liou YeongCheng Yao Yonghong |
author_sort |
Liou YeongCheng |
title |
An Implicit Extragradient Method for Hierarchical Variational Inequalities |
title_short |
An Implicit Extragradient Method for Hierarchical Variational Inequalities |
title_full |
An Implicit Extragradient Method for Hierarchical Variational Inequalities |
title_fullStr |
An Implicit Extragradient Method for Hierarchical Variational Inequalities |
title_full_unstemmed |
An Implicit Extragradient Method for Hierarchical Variational Inequalities |
title_sort |
implicit extragradient method for hierarchical variational inequalities |
publisher |
SpringerOpen |
series |
Fixed Point Theory and Applications |
issn |
1687-1820 1687-1812 |
publishDate |
2011-01-01 |
description |
<p/> <p>As a well-known numerical method, the extragradient method solves numerically the variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i1.gif"/></inline-formula> of finding <inline-formula> <graphic file="1687-1812-2011-697248-i2.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2011-697248-i3.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2011-697248-i4.gif"/></inline-formula>. In this paper, we devote to solve the following hierarchical variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i5.gif"/></inline-formula> Find <inline-formula> <graphic file="1687-1812-2011-697248-i6.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1812-2011-697248-i7.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2011-697248-i8.gif"/></inline-formula>. We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i9.gif"/></inline-formula>. It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of <inline-formula> <graphic file="1687-1812-2011-697248-i10.gif"/></inline-formula> in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality <inline-formula> <graphic file="1687-1812-2011-697248-i11.gif"/></inline-formula>.</p> |
url |
http://www.fixedpointtheoryandapplications.com/content/2011/697248 |
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