Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 642584, 16 pages
doi:10.1155/2010/642584
Research Article

Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems

Yonghong Yao,1 Yeong-Cheng Liou,2 and Chia-Ping Chen3

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China
2Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan
3Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan

Received 21 May 2010; Accepted 22 December 2010

Academic Editor: Satit Saejung

Copyright © 2010 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 consider the following hierarchical equilibrium problem and variational inequality problem (abbreviated as HEVP): find a point 𝑥 ∗ ∈ E P ( 𝐹 , 𝐵 ) such that ⟨ 𝐴 𝑥 ∗ , 𝑥 − 𝑥 ∗ ⟩ ≥ 0 , for all 𝑥 ∈ E P ( 𝐹 , 𝐵 ) , where 𝐴 , 𝐵 are two monotone operators and E P ( 𝐹 , 𝐵 ) is the solution of the equilibrium problem of finding 𝑧 ∈ 𝐶 such that 𝐹 ( 𝑧 , 𝑦 ) + ⟨ 𝐵 𝑧 , 𝑦 − 𝑧 ⟩ ≥ 0 , for all 𝑦 ∈ 𝐶 . We note that the problem (HEVP) includes some problems, for example, mathematical program and hierarchical minimization problems as special cases. For solving (HEVP), we propose a double-net algorithm which generates a net { 𝑥 𝑠 , 𝑡 } . We prove that the net { 𝑥 𝑠 , 𝑡 } hierarchically converges to the solution of (HEVP); that is, for each fixed 𝑡 ∈ ( 0 , 1 ) , the net { 𝑥 𝑠 , 𝑡 } converges in norm, as 𝑠 → 0 , to a solution 𝑥 𝑡 ∈ E P ( 𝐹 , 𝐵 ) of the equilibrium problem, and as 𝑡 → 0 , the net { 𝑥 𝑡 } converges in norm to the unique solution 𝑥 ∗ of (HEVP).