On the continuation of solutions of differential equations by vector Ljapunov functions

Abstract:

In this note we prove a continuation theorem applicable also when the estimate of the derivative of the vector Ljapunov function contains the phase coordinates explicitly. Our theorem combines and strengthens several earlier continuation results including a recent theorem of T. A. Burton, who conjectured that the monotonicity assumption on a function in his theorem may be dropped. By an example we show, however, that Burton’s theorem would be false without this assumption. Furthermore, applying our theorem we can replace this assumption by a much weaker one.