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On Small Solutions of Second Order Differential Equations
with Random Coefficients

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*
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