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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011, Article ID 904320, 19 pages doi:10.1155/2011/904320 Research Article The Existence of Maximum and Minimum Solutions to General Variational Inequalities in the Hilbert Lattices Jinlu Li1 and Jen-Chih Yao2 1 2 Department of Mathematics, Shawnee State University, Portsmouth, OH 45662, USA Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 804-24, Taiwan Correspondence should be addressed to Jen-Chih Yao, yaojc@math.nsysu.edu.tw Received 24 November 2010; Accepted 8 December 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2011 J. Li and J.-C. Yao. This is an open access article distributed under the Creative Commons Attribution License, which permits. | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 904320 19 pages doi 10.1155 2011 904320 Research Article The Existence of Maximum and Minimum Solutions to General Variational Inequalities in the Hilbert Lattices Jinlu Li1 and Jen-Chih Yao2 1 Department of Mathematics Shawnee State University Portsmouth OH 45662 USA 2 Department of Applied Mathematics National Sun Yat-Sen University Kaohsiung 804-24 Taiwan Correspondence should be addressed to Jen-Chih Yao yaojc@math.nsysu.edu.tw Received 24 November 2010 Accepted 8 December 2010 Academic Editor Qamrul Hasan Ansari Copyright 2011 J. Li and J.-C. Yao. 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 apply the variational characterization of the metric projection to prove some results about the solvability of general variational inequalities and the existence of maximum and minimum solutions to some general variational inequalities in the Hilbert lattices. 1. Introduction The variational inequality theory and the complementarity theory have been studied by many authors and have been applied in many fields such as optimization theory game theory economics and engineering 1-12 . The existence of solutions to a general variational inequality is the most important issue in the variational inequality theory. Many authors investigate the solvability of a general variational inequality by using the techniques of fixed point theory and the variational characterization of the metric projection in some linear normal spaces. Meanwhile a certain topological continuity of the mapping involved in the considered variational inequality must be required such as continuity and semicontinuity. A number of authors have studied the solvability of general variational inequalities without the topological continuity of the mapping. One .
