Compatible and incompatible nonuniqueness conditions for the classical Cauchy problem (Q554944)

Summary: Sufficient conditions for nonuniqueness of the classical Cauchy problem \[ \dot x = f(t, x),\quad x(t_0) = x_0 \] are given. As the essential tool serves a method which estimates the ``distance'' between two solutions with an appropriate Lyapunov function and permits to show that under certain conditions the ``distance'' between two different solutions vanishes at the initial point. In the second part, attention is paid to conditions that are obtained by a formal inversion of uniqueness theorems of Kamke-type but cannot guarantee nonuniqueness because they are incompatible.