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

Our main result is the following: let F : R x R-n -> 2(Rn) be a multifunction, and assume that there exists a neglegible subset U subset of R x R-n, satisfying a certain geometrical condition, such that the restriction of F to (R x R-n) \ U is bounded, lower semicontinuous with non-empty closed values, and its range belongs to a certain family A(n) defined below. Then, there exists a bounded multifunction G : R x R-n -> 2(Rn) such that G is upper semicontinuous with non-empty compact convex values, and every generalized solution of u '(t) is an element of G(t,u(t)) is a solution of u '(t) is an element of 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) is an element of R x 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 ') = f(t,u),u(0) = xi, with f discontinuous in u.