On the qualitative theory of lower semicontinuous differential inclusions
The author is concerned in this paper with the qualitative properties of the set of Carathéodory solutions for a differential inclusion of the form (*) \(\dot x(t)\in F(t,x(t))\), \(t\in [0,T]\), \(x(0)=\xi \in {\mathfrak R}^ n\). For any \(M>0\), consider the cone \(\Gamma^ m\subset {\mathfrak R}\times {\mathfrak R}^ n:\) \(\Gamma^ m=\{(t,x)\); \(| x| \leq Mt\}\). A function f defined on \({\mathfrak R}^{n+1}\) is said to be \(\Gamma^ m\)- continuous at a point (t,x) if \(f(t_ m,x_ m)\to f(t,x)\) as \(m\to \infty\), whenever \(t_ m\to t\), \(x_ m\to x\), and \((t_ m-t,x_ m- x)\in \Gamma^ m\) for every \(m\geq 1\). Introducing the notion of \(\Gamma^ m\)-quasicontinuity, which is a Carathéodory-type extension of \(\Gamma^ m\)-continuity, it is shown that most of the qualitative theorems concerning O.D.E.s with continuous right-hand side can be proved in the directionally quasicontinuous case as well. In particular, the set of trajectories of (*) is compact and connected and depends upper semicontinuously on \(\xi\).
scientific article; zbMATH DE number 4105367