ON upper and lower SP-θ (semi-pre-θ)-Continuous Multifunctions

Abstract

In this paper we introduce and investigate a new types of multifunctions called upper(lower) sp-θ-continuous multifunctions and upper(lower) semi-pre-θ-continuous multifunctions by using the notions in [1]. The concept of upper(lower) sp-θ-continuous multifunctions is stronger than both upper(lower) sp-continuous multifunctions [1] and upper(lower) semi-pre-θ-continuous multifunctions. And the concept of  upper(lower) semi-pre-θ-continuous multifunctions is a generalization of upper(lower) sp-θ-continuous multifunctions and stronger than upper(lower) semi-pre-continuous multifunctions. Furthermore, the relationships among these notions and other of well-known types of multifunctions are also discussed.