On the continuity of multivalued mappings and the stability of fixed points
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.
continuity of solutions of some variational inequalities
scientific article; zbMATH DE number 3944633