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Keywords:
Asymptotic stability; energy method; small solution
Summary:
We consider the equation $x^{\prime \prime }+a^2(t)x=0,\qquad a(t):=a_k\ \hbox{ if }t_{k-1}\le t<t_k,\ \hbox{ for }k=1,2,\ldots ,$ where $\lbrace a_k\rbrace$ is a given increasing sequence of positive numbers, and $\lbrace t_k\rbrace$ is chosen at random so that $\lbrace t_k-t_{k-1}\rbrace$ 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 $t\rightarrow \infty$.
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