On Small Solutions of Second Order Differential Equations with Random Coefficients

Laszlo Hatvani and Laszlo Stacho

Address. Bolyai Institute, Aradi vertanuk tere 1, H-6720, Szeged, Hungary

E-mail: hatvani@math.u-szeged.hu

stacho@math.u-szeged.hu

Abstract. We consider the equation

x''+a^2(t)x=0, a(t):=a_k if t_{k-1} \le t \less t_k,

for k=1,2,\ldots, where $\{a_k\}$ is a given increasing sequence of positive numbers, and $\{t_k\}$ is chosen at random so that $\{t_k-t_{k-1}\}$ are totally independent random variables uniformly distributed on interval $[0,1]$. We determine the probability of the event that all solutions of the equation tend to zero as $t\to \infty$.

AMS classification. 34F05, 34D20, 60K40

Key words. Asymptotic stability, energy method, small solution