Nonexistence of solutions for differential inclusions with upper semicontinuous nonconvex right-hand side (Q1113343)

In this note the author considers the differential inclusion \((1)\quad \dot x(t)=f(x(t),u(t)),\) \(x(0)=0\in R^ n\), \((2)\quad u(t)\in R(x(t)),\) where \(f: R^ n\times R^ m\to R^ n\) is continuous and \(R: R^ n\to R^ m\) is an upper semicontinuous multifunction with compact convex values. The author gives two counterexamples showing that these conditions are not sufficient to ensure the existence property for (1), (2).