THE CONTINUITY OF SET-VALUED MAPPING IN INFINITE-DIMENSIONAL SPACES

Abstract

The purpose of this paper is to investigate the upper C-continuity and the lower C-continuity of set-valued mapping (or multivalued mapping) in Hausdorff locally convex topological linear spaces by means of a convex cone (or a closed convex cone) with its nonempty interior. Specifically, in Proposition 3.1 we provide a necessary condition for the epigraph of the set-valued mapping with its nonempty interior. In Proposition 3.2, we research the C-bounded set-valued mapping in a certain given neighbourhood. In theorems 3.3, 3.5, 3.6 and corollaries 3.7, 3.8, we introduce necessary conditions for the set-valued mapping to become either upper C-semicontinuous or lower C-semicontinuous (C-u.s.c or C-l.s.c in abbreviation) In theorem 3.4, we provide a necessary and sufficient condition about the C-bounded set-valued mapping at a given point.