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The purpose of this paper is to characterize in terms of scalar preinvexity (resp. prequasiinvexity) the set-valued maps which are K-preinvex (resp. K-prequasiinvex) with respect to a closed convex cone K. Moreover, as applications of our results some conditions under which a local solution of set-valued scalar optimization for (VP) is a global supper efficient solution for (VP) are given.

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Abstract. The solvability of the generalized weak vector implicit variational inequality problem, generalized strong vector implicit variational inequality problem and generalized vector variational inequality problem are proved by using a generalized Fan's KKM theorem. Our results extend and unify corresponding results of other authors.