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Teor. Veroyatnost. i Primenen., 1963, Volume 8, Issue 3, Pages 309–318 (Mi tvp4678)  

Short Communications

On the Theory of Differential Equations with Random Coenfficients

G. Ya. Lyubarskii, Yu. L. Rabotnikov

Kharkov

Abstract: The equation $\ddot u(t)+a_1(t)\dot u(t)+[\alpha(t)-\alpha(t)]u(t)=0$ is considered where the coefficient $a_1(t)$ and $a_0 (t)$ are real, piecewise continuous and periodic functions with the same period $T$ and $\alpha (t)$ is a real random function. The restrictions on the $\alpha (t)$ are essentially the following. The correlation length $\alpha $ is much shorter than the period $T$, the random function $\alpha(t)$, $\infty<t<\infty$, does not exceed the value ${\gamma/{\sqrt a(\gamma={const}<1)}}$.
The necessary and sufficient conditions are found for the boundedness of mean values $Mu^2 (t),M[u(t)\dot u(t)]$ and $M\dot u^2 (t)$.

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English version:
Theory of Probability and its Applications, 1963, 8:3, 290–298

Received: 30.11.1961

Citation: G. Ya. Lyubarskii, Yu. L. Rabotnikov, “On the Theory of Differential Equations with Random Coenfficients”, Teor. Veroyatnost. i Primenen., 8:3 (1963), 309–318; Theory Probab. Appl., 8:3 (1963), 290–298

Citation in format AMSBIB
\Bibitem{LyuRab63}
\by G.~Ya.~Lyubarskii, Yu.~L.~Rabotnikov
\paper On the Theory of Differential Equations with Random Coenfficients
\jour Teor. Veroyatnost. i Primenen.
\yr 1963
\vol 8
\issue 3
\pages 309--318
\mathnet{http://mi.mathnet.ru/tvp4678}
\transl
\jour Theory Probab. Appl.
\yr 1963
\vol 8
\issue 3
\pages 290--298
\crossref{https://doi.org/10.1137/1108033}


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