Geodesic
On small solutions of second order differential equations with random coefficients
Archivum mathematicum, Tome 34 (1998) no. 1, pp. 119-126

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We consider the equation \[x^{\prime \prime }+a^2(t)x=0,\qquad a(t):=a_k\ \hbox{ if }t_{k-1}\le t,\ \hbox{ for }k=1,2,\ldots ,\] 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 $.
Classification : 34D20, 34F05, 60H10, 60K40
Keywords: Asymptotic stability; energy method; small solution 
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     author = {Hatvani, L\'aszl\'o and Stach\'o, L\'aszl\'o},
     title = {On small solutions of second order differential equations with random coefficients},
     journal = {Archivum mathematicum},
     pages = {119--126},
     publisher = {mathdoc},
     volume = {34},
     number = {1},
     year = {1998},
     mrnumber = {1629680},
     zbl = {0915.34051},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_1998__34_1_a11/}
}
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Hatvani, László; Stachó, László. On small solutions of second order differential equations with random coefficients. Archivum mathematicum, Tome 34 (1998) no. 1, pp. 119-126. http://geodesic.mathdoc.fr/item/ARM_1998__34_1_a11/