2-TYPES OF MINIMAX THEOREMS FOR VECTOR-VALUED FUNCTIONS

It seems that minimax theorems for vector-valued functions found in recent papers have something in common. Taking note of this, we improve several results in the author's recent works and state two types of minimax theorems for vector-valued functions. One theorem refers to functions with some special convexity properties; the other theorem refers to separated functions of the type f(x, y) = u(x) + v(y). The proofs are based on the existence of weak cone saddle points of f and on a condition about a pointed convex cone which induces a partial ordering in the image space of f. We need the condition (C/{0}) + cl C subset-of C, which implies the Sterna-Karwat condition for a convex cone C of a Hausdorff topological vector space.