On the K-continuity of K-hull midconvex set-valued functions

Abstract

In the present paper it is proved the K-continuity of a set-valued function which is K-hull midconvex, K-lower semicontinuous at a point and whose values are bounded K-midconvex sets; the concept of K-hull midconvexity is introduced here and it is weaker than the one of K-midconvexity. Then a generalization of the well known Theorem of Bernstein-Doech is obtained, which gives, in the particular case when the space is locally bounded, a characterization of the K-continuity of set-valued function. The theorems presented here improve some earlier results obtained by K. Nikodem (Zeszyty Nauk Politech. Lodz 559, Rospravy Mat. 114, 1989).