An Implicit Extragradient Method for Hierarchical Variational Inequalities

• Main Authors: Liou YeongCheng, Yao Yonghong
• Format: Article
• Language: English
• Published: SpringerOpen 2011-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2011/697248
• ISSN: 1687-1820; 1687-1812

<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>