A \(C^\infty \) density theorem for differential inclusions with Lipschitz continuous right-hand sides (Q965039)

There are studied autonomous differential inclusions of the form \[ x'(t)\in F(x(t))\quad \text{for a.e. }t\in (0,T),\quad x(0)=x_0,\tag{1} \] where \(F:\mathbb R^n\to {\mathcal P}(\mathbb R^n)\) is a given set-valued map with nonempty compact convex values and \(x_0\in \mathbb R^n\). When the set-valued map \(F\) is a small blowup of a Lipschitz continuous convex compact set-valued map it is proved that \(C^{\infty }\)-solutions of problem (1) are dense in the set of all (absolutely continuous) solutions of problem (1) with respect to the supremum norm.