| Title:
|
A note on linear perturbations of oscillatory second order differential equations (English) |
| Author:
|
Manfrin, Renato |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
46 |
| Issue:
|
2 |
| Year:
|
2010 |
| Pages:
|
105-118 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Under suitable hypotheses on $\gamma (t)$, $\lambda (t)$, $q(t)$ we prove some stability results which relate the asymptotic behavior of the solutions of $u^{\prime \prime }+ \gamma (t)u^{\prime }+\big (q(t)+ \lambda (t)\big )u=0$ to the asymptotic behavior of the solutions of $u^{\prime \prime }+ q(t)u=0$. (English) |
| Keyword:
|
second order ODE |
| Keyword:
|
boundedness of solutions |
| Keyword:
|
linear perturbations |
| MSC:
|
34C11 |
| MSC:
|
34D10 |
| idZBL:
|
Zbl 1240.34186 |
| idMR:
|
MR2684253 |
| . |
| Date available:
|
2010-06-22T22:11:31Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140307 |
| . |
| Reference:
|
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| Reference:
|
[2] Bellman, R.: Stability Theory of Differential Equations.McGraw-Hill Book Company, New York, 1953. Zbl 0053.24705, MR 0061235 |
| Reference:
|
[3] Cesari, L.: Asymptotic Behavior and Stability Problems in Ordinary Differential Equations.2nd ed., Springer-Verlag, Berlin, 1963. Zbl 0111.08701 |
| Reference:
|
[4] Galbraith, A., McShane, E. J., Parrish, G.: On the solutions of the linear second-order differential equations.Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 247–249. MR 0174817, 10.1073/pnas.53.2.247 |
| Reference:
|
[5] Knowles, I.: On stability conditions for second order linear differential equations.J. Differential Equations 34 (1979), 179–203. Zbl 0388.34029, MR 0550039, 10.1016/0022-0396(79)90003-2 |
| Reference:
|
[6] Manfrin, R.: Quadratic forms for the Liouville equation $\,w_{tt}+ \lambda ^2a(t) w=0$ with applications to Kirchhoff equation.Portugal. Math. 65 (2008), 447–484. Zbl 1160.35477, MR 2483347, 10.4171/PM/1821 |
| Reference:
|
[7] Manfrin, R.: $ L^p$ solutions of second order differential equations.Funkcial. Ekvac. 52 (2009), 255–279. Zbl 1175.34044, MR 2547105, 10.1619/fesi.52.255 |
| Reference:
|
[8] Manfrin, R.: On the boundedness of solutions of the equation $\, u^{\prime \prime }+ (1+f(t)) u=0$.Discrete Contin. Dynam. Systems 23 (2009), 991–1008. Zbl 1190.34037, MR 2461836 |
| Reference:
|
[9] Opial, Z.: Nouvelles remarques sur l’équation différentielle $u^{\prime \prime }+ a(t) u=0$.Ann. Polon. Math. 6 (1959), 75–81. Zbl 0085.07003, MR 0104864 |
| Reference:
|
[10] Trench, W. F.: On the asymptotic behavior of solutions of second order linear differential equations.Proc. Amer. Math. Soc. 14 (1963), 12–14. MR 0142844, 10.1090/S0002-9939-1963-0142844-7 |
| . |
