Another approach to K-subadditivity

Abstract

In the paper the notion of weakly K-subadditive set-valued maps is introduced in such a way that F is weakly K-superadditive if and only if \(-F\) is weakly K-subadditive. This new definition is a natural generalization of K-subadditive set-valued maps from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021), for which opposite set-valued maps need not be K-subadditive. Among others, we prove that every weakly K-subadditive set-valued map which is K–upper bounded on a “large” set has to be locally weakly K-upper bounded and weakly K-lower bounded at every point of the domain. This theorem completes an analogous result for K-subadditive set-valued maps which are weakly K-upper bounded on “large” sets from Jabłońska and Nikodem (Aequ Math 95:1221–1231, 2021).