On the continuity of multivalued mappings and the stability of fixed points (Q1073329)

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.