Fixed Point Theory and Applications
Volume 2011 (2011), Article ID 794203, 12 pages
doi:10.1155/2011/794203
Research Article

Algorithms Construction for Variational Inequalities

Yonghong Yao,1 Yeong-Cheng Liou,2 and Shin Min Kang3

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
3Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 4 October 2010; Accepted 19 February 2011

Academic Editor: Yeol J. Cho

Copyright © 2011 Yonghong Yao et al. 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

We devote this paper to solving the variational inequality of finding 𝑥 ∗ with property 𝑥 ∗ ∈ F i x ( 𝑇 ) such that ⟨ ( 𝐴 − 𝛾 𝑓 ) 𝑥 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 for all 𝑥 ∈ F i x ( 𝑇 ) . Note that this hierarchical problem is associated with some convex programming problems. For solving the above VI, we suggest two algorithms: Implicit Algorithmml: 𝑥 𝑡 = 𝑇 𝑃 𝑐 [ 𝐼 − 𝑡 ( 𝐴 − 𝛾 𝑓 ) ] 𝑥 𝑡 for all 𝑡 ∈ ( 0 , 1 ) and Explicit Algorithm: 𝑥 𝑛 + 1 = 𝛽 𝑛 𝑥 𝑛 + ( 1 − 𝛽 𝑛 ) 𝑇 𝑃 𝑐 [ 1 − 𝛼 𝑛 ( 𝐴 − 𝛾 𝑓 ) ] 𝑥 𝑛 for all 𝑛 ≥ 0 . It is shown that these two algorithms converge strongly to the unique solution of the above VI. As special cases, we prove that the proposed algorithms strongly converge to the minimum norm fixed point of 𝑇 .