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...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2011-01-01
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| Series: | Fixed Point Theory and Applications |
| Online Access: | http://www.fixedpointtheoryandapplications.com/content/2011/697248 |
| _version_ | 1819032908660211712 |
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| author | Liou YeongCheng Yao Yonghong |
| author_facet | Liou YeongCheng Yao Yonghong |
| author_sort | Liou YeongCheng |
| collection | DOAJ |
| 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> |
| first_indexed | 2024-12-21T07:09:25Z |
| format | Article |
| id | doaj.art-17c7c5780c4e4a908429bd0f547ac211 |
| institution | Directory Open Access Journal |
| issn | 1687-1820 1687-1812 |
| language | English |
| last_indexed | 2024-12-21T07:09:25Z |
| publishDate | 2011-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Fixed Point Theory and Applications |
| spelling | doaj.art-17c7c5780c4e4a908429bd0f547ac2112022-12-21T19:12:00ZengSpringerOpenFixed 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 |
| spellingShingle | Liou YeongCheng Yao Yonghong An Implicit Extragradient Method for Hierarchical Variational Inequalities Fixed Point Theory and Applications |
| title | 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_short | An Implicit Extragradient Method for Hierarchical Variational Inequalities |
| title_sort | implicit extragradient method for hierarchical variational inequalities |
| url | http://www.fixedpointtheoryandapplications.com/content/2011/697248 |
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