Abstract. We give -- among others -- a necessary and sufficient condition for the asymptotic stability of the solutions of the differential equation $x^{\prime\prime }+h(t)x^\prime +x=0$ under the restrictions that $h(t)=2h_n>0$ on $[t_n,t_n+\tau_n]$, and $h(t)=0$ elsewhere, $0< \tau_n\le{\pi /2}$ and $t_n$ is a multiple of $\pi,n\in{\msbm N}$.

AMS Subject Classification (1991): 34D20, 34D05

Received October 24, 1994. (Registered under 5681/2009.)