On \(\varvec{St^{\alpha \beta }_{\gamma }}\)-Graph and \(\varvec{St^{\alpha \beta }_{\gamma }}\)-Pointwise Convergence of Sequences of Set-Valued Functions Defined on Probabilistic Normed Spaces

Abstract

The notions of graphical and pointwise limits of a sequence of set-valued functions defined from one metric space into another were studied by Aubin and Frankowska. In this study, by using the concept of \(\alpha \beta\)-density of subsets of natural numbers, we introduce the notions of \(St^{\alpha \beta }_{\gamma }\)-graphical and \(St^{\alpha \beta }_{\gamma }\)-pointwise limits of a sequence of set-valued functions defined from one probabilistic normed space into another. Subsequently, the article introduces the notions of \(St^{\alpha \beta }_{\gamma }\)-graph and \(St^{\alpha \beta }_{\gamma }\)-pointwise convergence of these sequences. Moreover, we look at the correspondence between these convergences and establish some associated theorems.