The extension of continuity and generalized results for set-valued maps in convex analysis

The paper is organized into two parts. In the first part, we establish some new vectorial set-valued variant of Hahn–Banach theorems with a K-concave continuous (resp. lower semicontinuous, upper semicontinuous) set-valued map dominated by a K-convex set valued map. The existence of the extension of continuity for dominated set-valued maps in order-complete topological vector spaces is also given. The main purpose of the second part is to present some applications to the existence theorems of K-quasigradients of K-convex set-valued maps, a generalized Lagrange multiplier theorem and a generalized sandwich theorem.