Some qualitative properties of solutions of higher-order lower semicontinus differential inclusions

Let $n,k\in{\bf N}$, let $T>0$, and let $F:[0,T]\times({\bf R}^n)^k\to 2^{{\bf R}^n}$ be a lower semicontinuos and bounded multifunction, with nonempty closed values. We prove that there exists a bounded and upper semicontinuous multifunction $G:{\bf R}\times({\bf R}^n)^k\to2^{{\bf R}^n}$, with nonempty compact convex values, such that every generalized solution $u:[0,T]\to{\bf R}^n$ of the differential inclusion $u^{(k)}\in G(t,u,u^\prime,\ldots,u^{(k-1)})$ is a generalized solution to the differential inclusion $u^{(k)}\in F(t,u,u^\prime,\ldots,u^{(k-1)})$. As an application, we prove an existence and qualitative result for the generalized solutions of the Cauchy problem associated to the inclusion $u^{(k)}\in F(t,u,u^\prime,\ldots,u^{(k-1)})$. In particular, we prove that if $F$ is lower semicontinuous and bounded, with nonempty closed values, then the solution multifunction admits an upper semicontinuous multivalued selection with nonempty compact connected values. Finally, by applying the latter result, we prove an analogous existence and qualitative result for the generalized solutions of the Cauchy problem associated to the differential equation $g(u^{(k)})=f(t,u,u^\prime,\ldots,u^{(k-1)})$, where $f$ is continuous. We only assume that $g$ is continuous and locally nonconstant.