ABSTRACT
In the sequel 1Z and N stand for the set of reals and the set of positive integers, respec tively. (X, IMIx) denotes areal Banach space and X* denotes its dual. B(X) (S(X)) is the closed unit ball (the unit sphere) of X, respectively. Let F : X -* 2Y be a set-valued map, where Y is also a Banach space. F is called upper semi-continuous if for any x e X and open set V containing F(x), there exists a neighborhood U of x such that for all z € (7, we have F(y) c V. We say that F is lower semi-continuous if for any x e X, xn -► x and z € F(x) there exist zn € F(xn) such that zn z. Basing on these defini tions, there is natural to introduce notions of points of lower semi-continuity and points of upper semi-continuity. We define them as follows.
