Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 697248, 11 pages
doi:10.1155/2011/697248
Research Article

An Implicit Extragradient Method for Hierarchical Variational Inequalities

Yonghong Yao1 and Yeong Cheng Liou2

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 20 September 2010; Accepted 7 November 2010

Academic Editor: Jen Chih Yao

Copyright © 2011 Yonghong Yao and Yeong Cheng Liou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

As a well-known numerical method, the extragradient method solves numerically the variational inequality V I ( 𝐶 , 𝐴 ) of finding 𝑢 ∈ 𝐶 such that ⟨ 𝐴 𝑢 , 𝑣 − 𝑢 ⟩ ≥ 0 , for all 𝑣 ∈ 𝐶 . In this paper, we devote to solve the following hierarchical variational inequality H V I ( 𝐶 , 𝐴 , 𝑓 ) Find ̃ 𝑥 ∈ V I ( 𝐶 , 𝐴 ) such that ⟨ ( 𝐼 − 𝑓 ) ̃ 𝑥 , 𝑥 − ̃ 𝑥 ⟩ ≥ 0 , for all 𝑥 ∈ V I ( 𝐶 , 𝐴 ) . We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality H V I ( 𝐶 , 𝐴 , 𝑓 ) . It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of H V I ( 𝐶 , 𝐴 , 𝑓 ) in Hilbert spaces. As a special case, we obtain the minimum norm solution of the variational inequality V I ( 𝐶 , 𝐴 ) .