On the continuity of multivalued mappings and the stability of fixed points
type: Item
instance of: scholarly article
title: On the continuity of multivalued mappings and the stability of fixed points
zbMATH Open document ID: 0588.47063
author: Nguyen Xuan Tan
published in: Acta Mathematica Vietnamica
publication date: 1984-01-01T00:00:00Z
review text: For a multivalued mapping \(F: X\to 2^ Y\), where X, Y are \(T_ 2\)- spaces, the notions of upper resp. lower semicontinuity (u.s.c. resp. l.s.c.) are well known. With the help of these notions the notions of upper resp. lower semi-equicontinuity (u.s.e.c. resp. l.s.e.c.) of a family \(\{F_ i\), \(i\in I\}\) are given, where \(F_ i: X\to 2^ Y\) with nonempty values and X, Y are topological real vector spaces. Now in this paper it is shown that a family \(\{F_ i\), \(i\in I\}\) of u.s.c. mappings is l.s.e.c. if X is a barreled space and Y is a locally convex space. Further, for \(C\subseteq Y\) closed and convex, \(F: X\times C\to 2^ C\) u.s.c. convex and compact with nonempty closed values, the mapping \(G: X\to 2^ C\) defined by \(G(x)=\{y\in C\); \(y\in F(x,y)\}\) is convex and continuous. Moreover, if (Q,\({\mathbb{Q}},\mu)\) is a complete measurable space, Y a separable Fréchet space, \(F: Q\times X\times Y\to 2^ Y\) with nonempty closed convex values, measurable in \(t\in Q\), convex u.s.c. in (x,y)\(\in X\times Y\) and \(\overline{F(t,X,Y)}\) compact for any fixed \(t\in Q\), then the mapping \(L: Q\times X\to 2^ Y\) defined by \(L(t,x)=\{y\in Y\); \(y\in F(t,x,y)\}\) is measurable in t and convex continuous in x. The paper is concluded with some properties on the continuity of solutions of some variational inequalities, which are applications of the above results.
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 46A08
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 28B20
zbMATH DE Number: 3944633
zbMATH Keywords: multivalued mapping
zbMATH Keywords: lower semicontinuity
zbMATH Keywords: lower semi-equicontinuity
zbMATH Keywords: barreled space
zbMATH Keywords: measurable space
zbMATH Keywords: separable Fréchet space
zbMATH Keywords: continuity of solutions of some variational inequalities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3944633