Research Article Optimality Conditions of Globally Efficient Solution for Vector Equilibrium Problems with Generalized Convexity | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 898213 13 pages doi 2009 898213 Research Article Optimality Conditions of Globally Efficient Solution for Vector Equilibrium Problems with Generalized Convexity Qiusheng Qiu Department of Mathematics Zhejiang Normal University Jinhua Zhejiang 321004 China Correspondence should be addressed to Qiusheng Qiu qsqiu@ Received 19 March 2009 Accepted 21 September 2009 Recommended by Yeol Je Cho We study optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity. The necessary and sufficient conditions of globally efficient solution for the vector equilibrium problems are obtained. The Kuhn-Tucker condition of globally efficient solution for vector equilibrium problems is derived. Meanwhile we obtain the optimality conditions for vector optimization problems and vector variational inequality problems with constraints. Copyright 2009 Qiusheng Qiu. 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. 1. Introduction Throughout the paper let X Y and Z be real Hausdorff topological vector spaces D c X a nonempty subset and 0Y denotes the zero element of Y. Let C c Y and K c Z be two pointed convex cones see 1 such that int C 0 int K 0 where int C denotes the interior of C. Let g D Z be a mapping and let F D X D Y be a mapping such that F x x 0 for all x e D. For each x e D we denote F x D UyeDF x y and define the constraint set A x e D g x e -K which is assumed to be nonempty. Consider the vector equilibrium problems with constraints for short VEPC finding x e A such that F x y e - P Vy e A VEPC where P u 0Y is a convex cone in Y. 2 Journal of Inequalities and Applications Vector equilibrium problems which contain vector optimization problems vector .
