On the Cauchy problem for a class of differential inclusions with applications

Our main result is the following: Let $F:R imes R^n o 2^{R^n}$ be a multifunction, and assume that there exists a neglegible subset $Usubseteq R imes R^n$, satisfying a certain geometrical condition, such that the restriction of $F$ to $(R imes R^n)setminus U$ is bounded, lower semicontinuous with nonempty closed values, and its range belongs to a certain family ${cal A}_n$ defined below. Then, there exists a bounded multifunction $G:R imes R^n o 2^{R^n}$ such that $G$ is upper semicontinuous with nonempty compact convex values, and every generalized solution of $u^prime(t)in G(t,u(t))$ is a solution of $u^prime(t)in F(t,u(t))$. Such a result improves a celebrated result by A. Bressan, valid for lower semicontinuous multifunctions. We point out that a multifunction $F$ satisfying our assumptions can fail to be lower semicontinuous even at all points $(t,x)in R imes R^n$. We derive some existence and qualitative results for the Cauchy problem associated to such a class of multifunctions. As an application, we prove existence and qualitative results for the implicit Cauchy problem $g(u^prime)=f(t,u)$, $u(0)=\xi$, with $f$ discontinuous in $u$.