Differential inclusions with non-closed, non-convex right hand side

For a class of lower semicontinuous differential inclusions with nonclosed, non-convex right hand side, the set of solutions is proved to be nonempty and connected. Existence of periodic solutions is also studied. Our results apply, in particular, to the problem x˙ ∈ ext F(x) ∩ int G(x), the right hand side being the intersection of the extreme and the interior points of two continuous multifunctions with compact, convex values.