In this paper we introduce the
notions of multifunctions with strongly-closed graphs and 𝜃-closed graphs. We then
extend these notions as well as the notion of multifunctions with closed graphs
and use these extensions to generalize and extend various known results on
multifunctions. These generalizations and extensions include a number of sufficient
conditions for multifunctions to be upper-semi-continuous, several generalizations
and extensions of the well-known Uniform Boundedness Principle from analysis, and
several “fixed set” theorems.
