tailieunhanh - Báo cáo hóa học: " Research Article Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Hierarchical Convergence of a Double-Net Algorithm for Equilibrium Problems and Variational Inequality Problems | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010 Article ID 642584 16 pages doi 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 1 Department of Mathematics Tianjin Polytechnic University Tianjin 300160 China 2 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 Correspondence should be addressed to Chia-Ping Chen cpchen@ 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. We consider the following hierarchical equilibrium problem and variational inequality problem abbreviated as HEVP find a point x e EP F B such that Ax x - x 0 for all x e EP F B where A B are two monotone operators and EP F B is the solution of the equilibrium problem of finding z e C such that F z y Bz y - z 0 for all y e 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 e 0 1 the net xs t converges in norm as s 0 to a solution xt e EP F 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 II II respectively and let C be a nonempty closed convex subset of H. Recall that a mapping A of C into H is called .