tailieunhanh - báo cáo hóa học:" Research Article Solving the Set Equilibrium Problems Yen-Cherng Lin and Hsin-Jung Chen"

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 Solving the Set Equilibrium Problems Yen-Cherng Lin and Hsin-Jung Chen | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2011 Article ID 945413 13 pages doi 2011 945413 Research Article Solving the Set Equilibrium Problems Yen-Cherng Lin and Hsin-Jung Chen Department of Occupational Safety and Health China Medical University Taichung 40421 Taiwan Correspondence should be addressed to Yen-Cherng Lin yclin@ Received 17 September 2010 Accepted 21 November 2010 Academic Editor Qamrul Hasan Ansari Copyright 2011 . Lin and . Chen. 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 study the weak solutions and strong solutions of set equilibrium problems in real Hausdorff topological vector space settings. Several new results of existence for the weak solutions and strong solutions of set equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems. 1. Introduction and Preliminaries Let X Y Z be arbitrary real Hausdorff topological vector spaces let K be a nonempty closed convex set of X and let C c Y be a proper closed convex and pointed cone with apex at the origin and int C Ỷ 0 that is C is proper closed with int C Ỷ 0 and satisfies the following conditions 1 AC Q C for all A 0 2 C C Q C 3 C n -C 0 . Letting A B be two sets of Y we can define relations C and C as follows 1 A cB o B - A c C 2 A CB o B - AC C. Similarly we can define the relations intC and int C if we replace the set C by int C. The trimapping f Z X K X K 2Y and mapping T K 2Z are given. The set equilibrium problem SEP i is to find an x e K such that f s x y int C 0 2 Fixed Point Theory and Applications for all y e K and for some s e T x . Such solution is called a weak solution for SEP i. We note that is equivalent to the following one f s x y - int C for all y e K and for some s e T X . For .