Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems
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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 1
Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan 3 Department of Computer Science and Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan 2
Correspondence should be addressed to Chia-Ping Chen, [email protected] Received 21 May 2010; Accepted 22 December 2010 Academic Editor: Satit Saejung Copyright q 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. We consider the following hierarchical equilibrium problem and variational inequality problem abbreviated as HEVP: find a point x∗ ∈ EPF, B such that Ax∗ , x − x∗ ≥ 0, for all x ∈ EPF, B, where A, B are two monotone operators and EPF, B is the solution of the equilibrium problem of finding z ∈ C such that Fz, y Bz, 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 {xs,t }. We prove that the net {xs,t } hierarchically converges to the solution of HEVP; that is, for each fixed t ∈ 0, 1, the net {xs,t } converges in norm, as s → 0, to a solution xt ∈ EPF, B of the equilibrium problem, and as t → 0, the net {xt } converges in norm to the unique solution x∗ of HEVP.
1. Introduction Let H be a real Hilbert space with inner product ·, · and norm · , respectively, and let C be a nonempty closed convex subset of H. Recall that a mapping A of C into H is called monotone if Au − Av, u − v ≥ 0,
1.1
for all u, v ∈ C and A : C → H is called α-inverse strongly monotone mapping if there exists a positive real number α such that Au − Av, u − v ≥ αAu − Av2 ,
1.2
2
Fixed Point Theory and Applications
for all u, v ∈ C. It is obvious that any α-inverse strongly monotone mapping A is monotone and 1/α-Lipschitz continuous. Recently, the following problem has attracted much attention: find hierarchically a fixed point of a nonexpansive mapping T with respect to a nonexpansive mapping P , namely, Find x ∈ FixT such that x − P x, x − x ≤ 0,
∀x ∈ FixT .
1.3
Some algorithms for solving the hierarchical fixed point problem 1.3 have been introduced by many authors. For related works, please see, for instance, 1–9 and the references therein. Remark 1.1. It is not hard to check that solving 1.3 is equivalent to the fixed point problem Find x ∈ C such that x projFixT · P x,
1.4
where projFixT stands for the metric projection on the closed convex set FixT . By using the definition of the normal cone to FixT , th
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