Compatible and Incompatible Nonuniqueness Conditions for the Classical Cauchy Problem

In the first part of this paper sufficient conditions for nonuniqueness of the classicalCauchy problem x˙=f(t,x), x(t0)=x0 are given. As the essential tool serves amethod which estimates the “distance” between two solutions with an appropriateLyapunov function and permits to show that under certain conditions the “distance” between two different solutions vanishes at the initial point. In the second part attentionis paid to conditions that are obtained by a formal inversion of uniquenesstheorems of Kamke-type but cannot guarantee nonuniqueness because they are incompatible