Generic continuity of restricted weak upper semi-continuous set-valued mappings
type: Item
instance of: scholarly article
title: Generic continuity of restricted weak upper semi-continuous set-valued mappings
zbMATH Open document ID: 0852.54018
DOI: 10.1007/BF00419372
published in: Set-Valued Analysis
publication date: 1996-12-08T00:00:00Z
review text: A multifunction \(F : X \to Y\) from a topological space into normed linear space \(Y\) is called restricted weak usc at \(x \in X\) if given a neighbourhood \(W\) of zero in \(Y\) endowed with the weak topology there exists a neighbourhood \(U\) of \(x\) such that \(F(U) \subset F(x) + W\). For weakly compact valued multifunctions such notion reduces to weak upper semicontinuity. The paper contains characterizations of restricted weak usc multifunctions, applications to subdifferentials, relations between restricted weak and strong (norm) usc, an interesting selection theorem for \(F\) defined on a \((\sigma - \beta)\)-unfavorable space, the generic continuity theorem exhibiting conditions under which a weakly usc multifunction is single-valued and strongly usc at the points of a residual subset of its domain. The results are applied to Fréchet differentiability of continuous convex functions and certain locally Lipschitzian functions.
Mathematics Subject Classification ID: 54C60
Mathematics Subject Classification ID: 54C65
Mathematics Subject Classification ID: 91A05
zbMATH DE Number: 898038
zbMATH Keywords: Namioka space
zbMATH Keywords: Banach space
zbMATH Keywords: weak topology
zbMATH Keywords: weak upper semicontinuity
zbMATH Keywords: restricted weak usc multifunctions
zbMATH Keywords: selection theorem
author: John Giles
author: Warren B. Moors
reviewed by: Włodzimierz Ślęzak
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 898038