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 An Implicit Extragradient Method for Hierarchical Variational Inequalities | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 697248 11 pages doi 2011 697248 Research Article An Implicit Extragradient Method for Hierarchical Variational Inequalities Yonghong Yao1 and Yeong Cheng Liou2 1 Department of Mathematics Tianjin Polytechnic University Tianjin 300160 China 2 Department of Information Management Cheng Shiu University Kaohsiung 833 Taiwan Correspondence should be addressed to Yonghong Yao yaoyonghong@ Received 20 September 2010 Accepted 7 November 2010 Academic Editor Jen Chih Yao Copyright 2011 Y. Yao and Y. C. 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. As a well-known numerical method the extragradient method solves numerically the variational inequality VI C A of finding u e C such that Au v - Ù 0 for all v e C. In this paper we devote to solve the following hierarchical variational inequality HVI C A f Find x e VI C A such that I - f x x - x 0 for all x e VI C A . We first suggest and analyze an implicit extragradient method for solving the hierarchical variational inequality HVI C A f . It is shown that the net defined by the suggested implicit extragradient method converges strongly to the unique solution of HVI C A f in Hilbert spaces. As a special case we obtain the minimum norm solution of the variational inequality VI C A . 1. Introduction The variational inequality problem is to find u e C such that Au v - u 0 Vv e C. The set of solutions of the variational inequality problem is denoted by VI C A . It is well known that the variational inequality theory has emerged as an important tool in studying a wide class of obstacle unilateral and equilibrium problems which arise in several branches of pure and applied sciences in a unified and general framework. Several numerical methods have been .
