Compatible and Incompatible Nonuniqueness Conditions for the Classical CauchyProblem

Abstract

In the first part of this paper sufficient conditions for nonuniqueness of the classical Cauchy problem $\dot{x}=f(t,x)$, $x\left(t_0\right)=x_0$ 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.