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Keywords:
Asymptotic stability; energy method; small solution
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 $.
References:
[1] V. I. Arnold: Mathematical Methods of Classical Mechanics. Springer-Verlag, New York, 1989. MR 0997295
[2] M. Biernacki: Sur l’équation différentielle $x^{\prime \prime }+A(t)x=0$. Prace Mat.-Fiz., 40 (1933), 163–171. Zbl 0006.20001
[3] T. A. Burton, J. W. Hooker: On solutions of differential equations tending to zero. J. Reine Angew. Math., 267 (1974), 154–165. MR 0348204 | Zbl 0298.34042
[4] Kai Lai Chung: A Course in Probability Theory. Academic Press, New York, 1974. MR 0346858
[5] Á. Elbert: Stability of some difference equations. In Advances in Difference Equations (Proceedings of the Second International Conference on Difference Equations, Veszprém, Hungary, August 7–11, 1995), Gordon and Breach Science Publisher, London, 1997; 165–187. MR 1636322
[6] Á. Elbert: On asymptotic stability of some Sturm-Liouville differential equations. General Seminar of Mathematics, University of Patras, 22-23 (1996/97), (to appear).
[7] J. R. Graef, J. Karsai: On irregular growth and impulses in oscillator equations. In Advances in Difference Equations (Proceedings of the Second International Conference on Difference Equations, Veszprém, Hungary, August 7–11, 1995), Gordon and Breach Science Publisher, London, 1997; 253–262. MR 1636328
[8] P. Hartman: On a theorem of Milloux. Amer. J. Math., 70 (1948), 395–399. MR 0026194 | Zbl 0035.18204
[9] P. Hartman: The existence of large or small solutions of linear differential equation. Duke Math. J., 28 (1961), 421–430. MR 0130432
[10] P. Hartman: Ordinary Differential Equations. Wiley, New York, 1964. MR 0171038 | Zbl 0125.32102
[11] L. Hatvani: On the asymptotic stability by nondecrescent Lyapunov function. Nonlinear Anal., 8 (1984), 67–77. MR 0732416 | Zbl 0534.34054
[12] L. Hatvani: On the existence of a small solution to linear second order differential equations with step function coefficients. Differential and Integral Equations, (to appear). MR 1639105 | Zbl 0916.34013
[13] I. T. Kiguradze, T. A. Chanturia: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Acad. Publ., Dordrecht, 1993. MR 1220223 | Zbl 0782.34002
[14] J. W. Macki: Regular growth and zero-tending solutions. In Ordinary Differential Equations and Operators, Lecture Notes of Math. 1032, Springer-Verlag, Berlin, 1982; 358–374. MR 0742649
[15] H. Milloux: Sur l’équation différentielle $x^{\prime \prime }+A(t)x=0$. Prace Mat.-Fiz., 41 (1934), 39–54. Zbl 0009.16402
[16] P. Pucci, and J. Serrin: Asymptotic stability for ordinary differential systems with time dependent restoring potentials. Archive Rat. Mech. Anal., 132 (1995), 207–232. MR 1365829
[17] J. Terjéki: On the conversion of a theorem of Milloux, Prodi and Trevisan. In Differential Equations, Lecture Notes in Pure and Appl. Math. 118, Springer-Verlag, Berlin, 1987; 661–665. MR 1021771
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