Exact Separation Theorem For Disjoint Closed Sets In Asplund Spaces

The separation theorem is the cornerstone of functional analysis and optimiza-tion theory.The classical convex set separation theorem plays an important role in convex analysis and optimization theory.In the nonconvex case,some researchers established various"Fuzzy"separation theorems in terms of the normal cones of the concerned sets;for example,the extremal principle is such a"Fuzzy"separation theorem.Laterly,some researchers further established various"Exact"separation theorems in terms of normal cones.In this paper,in terms of Frechet normal cones,we mainly consider the exact separation theorems for finitely many disjoint closed sets in an Asplund space.Noting that the existing exact separation results require the compactness assumption,which is quite restrictive.The main purpose of this paper is to weaken the compactness assumption,and provides exact separation result for finitely many disjoint closed sets and the relatively weak assumption.