Stability properties of solutions of linear second order differential equations with random coefficients

Journal of Differential Equations

Volumn 248, Issue 1, 2010, Pages 21-49

  Csörgo, Sándor ^a   Hatvani, László ^a  

^a UNIVERSITY OF SZEGED   (Hungary)

Author keywords

Impulses; Meissner's equation; Parametric resonance; Problem of swinging; Random oscillations; Small solution; Stochastic parametric resonance

Indexed keywords

EID: 70349780384     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2009.08.001     Document Type: Article

References (34)

1. Almira J., and Torres P.J. Invariance of the stability of Meissner's equation under perturbation of its intervals. Ann. Mat. Pura Appl. 180 (2001) 245-253
2. Arnold V.I. Mathematical Methods of Classical Mechanics (1978), Springer-Verlag, New York
3. over(n, *) = 0. Extension of a theorem by Armellini-Tonelli-Sansone. Studia Sci. Math. Hungar. 19 (1984) 151-157
4. Bobryk R.V., and Chrzeszczyk A. Parametric resonance in coupled oscillators driven by colored noise. Europhys. Lett. 68 (2004) 344-349
5. Chung K.L. A Course in Probability Theory. third edition (2001), Academic Press Inc., San Diego, CA
6. Cooper J. Parametric resonance in wave equations with a time-periodic potential. SIAM J. Math. Anal. 31 (2000) 821-835
7. Elbert A. Stability of some differential equations. Advances in Difference Equations. Veszprém, 1995 (1997), Gordon and Breach, Amsterdam 165-187
8. Elbert A. On damping of linear oscillators. Studia Sci. Math. Hungar. 38 (2001) 191-208
9. Fečkan M. Existence of almost periodic solutions for jumping discontinuous systems. Acta Math. Hungar. 86 (2000) 291-303
10. Galbraith A.S., McShane E.J., and Parrish G.B. On the solutions of linear second-order differential equations. Proc. Natl. Acad. Sci. USA 53 (1965) 247-249
11. Gnedenko B.V., and Kolmogorov A.N. Limit Distributions for Sums of Independent Random Variables (1968), Addison-Wesley Publishing Co., Reading, MA-London-Don Mills., Ont. (translated from Russian)
12. Graef J.R., and Karsai J. On the oscillation of impulsively damped halflinear oscillators. Proc. 6th Colloq. Qual. Theory Differ. Equ. Szeged, 1999. Electron. J. Qual. Theory Differ. Equ. 14 (2000) 12 pp. (electronic)
13. Graef J.R., and Karsai J. Behavior of solutions of impulsively perturbed non-halflinear oscillator equations. J. Math. Anal. Appl. 244 (2000) 77-96
14. Hartman P. On a theorem of Milloux. Amer. J. Math. 70 (1948) 395-399
15. Hartman P. The existence of large or small solutions of linear differential equations. Duke Math. J. 28 (1961) 421-430
16. Hartman P. Ordinary Differential Equations (1982), Birkhäuser, Boston
17. Hatvani L. On the existence of a small solution to linear second order differential equations with step function coefficients. Dyn. Contin. Discrete Impuls. Syst. 4 (1998) 321-330
18. Hatvani L. On small solutions of second order linear differential equations with non-monotonous random coefficients. Acta Sci. Math. (Szeged) 68 (2002) 705-725
19. Hatvani L. The growth condition guaranteeing small solutions for a linear oscillator with an increasing elasticity coefficient. Georgian Math. J. 14 (2007) 269-278
20. Hatvani L., and Stachó L. On small solutions of second order differential equations with random coefficients. Arch. Math. (Brno) 34 1 (1998) 119-126
21. Hochstadt H. A special Hill's equation with discontinuous coefficients. Amer. Math. Monthly 70 (1963) 18-26
22. Kiguradze I.T., and Chanturia T.A. Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Math. Appl. (Soviet Ser.) vol. 89 (1993), Kluwer Academic Publishers Group, Dordrecht (translated from the 1985 Russian original)
23. de L. Kronig R., and Penney W.G. Quantum mechanics in crystal lattices. Proc. R. Soc. Lond. 130 (1931) 499-513
24. Lakshmikantham V., Papageorgiou N.S., and Vasundhara J. The method of upper and lower solutions and monotone technique for impulsive differential equations with variable moments. Appl. Anal. 51 (1993) 41-58
25. Macki J.W. Regular growth and zero-tending solutions. Ordinary Differential Equations and Operators. Dundee, 1982. Lecture Notes in Math. vol. 1032 (1983), Springer, Berlin 358-374
26. 2 y = 0. Proc. Amer. Math. Soc. 17 (1966) 55-61
27. Meissner E. Über Schüttelschwingungen in Systemen mit periodisch veränderlicher Elastizität. Schweizer Bauzeitung 72 10 (1918) 95-98
28. ″ + A (t) x = 0. Prace Mat.-Fiz. 41 (1934) 39-54
29. Ruijgrok M., and Verhulst F. Parametric and autoparametric resonance. NoDEA Nonlinear Differential Equations Appl. 19 (1996) 279-298
30. Pucci P., and Serrin J. Precise damping conditions for global asymptotic stability for nonlinear second order systems. Acta Math. 170 (1993) 275-307
31. Pucci P., and Serrin J. Asymptotic stability for ordinary differential systems with time-dependent restoring potentials. Arch. Ration. Mech. Anal. 132 (1995) 207-232
32. Samoilenko A.M., and Perestyuk N.A. Impulsive Differential Equations. World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises vol. 14 (1995), World Scientific Publishing Co., Inc., River Edge, NJ (translated from Russian by Y. Chapovsky)
33. Seǐranyan A.A., and Seǐranyan A.P. Decay of resonance zones for the Meissner equation with the introduction of small dissipation. Vestnik Moskov. Univ. Ser. I Mat. Mekh. 5 (2003) 53-59 73
34. Tondl A., Ruijgrok T., Verhulst F., and Nabergoj R. Autoparametric Resonance in Mechanical Systems (2000), Cambridge University Press, Cambridge