Oscillation criteria for a forced second-order linear differential equation. (English) Zbl 0922.34029

Wong, James S. W.

J. Math. Anal. Appl. 231, No. 1, 235-240 (1999).

The paper deals with the forced second-order linear differential equation \[ (p(t)y')'+q(t)y=f(t), \quad t\in [0,\infty), \tag{1} \] where \(p>0\), \(q, f\) are continuous functions. The author presents two oscillation criteria for equation (1) that do not assume that \(q\) and \(f\) be of definite sign.
In theorem 1, a result of M.A. El-Sayed [Proc. Am. Math. Soc. 118, 813-817 (1993; Zbl 0777.34023)] is extended. The second criterion is derived under the assumption that the unforced equation \((p(t)y')'+ q(t)y=0\) is nonoscillatory.
Two examples are given to show how the results can be applied where previous results are inconclusive.

Reviewer: J.Ohriska (Košice)

MSC:

34C10

Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

Keywords:

oscillation; forced second-order linear differential equation

Citations:

Zbl 0777.34023

References:

[1]	Cassell, J. S., The asymptotic behaviour of a class of linear oscillators, Quart. J. Math. Oxford Ser. (2), 32, 287-302 (1981) · Zbl 0485.34023
[2]	Coppel, W. A., Stability and Asymptotic Behaviour of Differential Equations (1965), Heath: Heath Boston · Zbl 0154.09301
[3]	El-Sayed, M. A., An oscillation criterion for a forced second order linear differential equation, Proc. Amer. math. Soc., 118, 813-817 (1993) · Zbl 0777.34023
[4]	Kartsatos, A. G., Maintenance of oscillations under the effect of a periodic forcing term, Proc. Amer. Math. Soc., 33, 377-383 (1972) · Zbl 0234.34040
[5]	Keener, M. S., Solutions of a certain linear nonhomogeneous second order differential equations, Appl. Anal., 1, 57-63 (1971) · Zbl 0215.43802
[6]	Leighton, W., Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc., 13, 603-610 (1962) · Zbl 0118.08202
[7]	Rainkin, S. M., Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl., 53, 550-553 (1976) · Zbl 0328.34033
[8]	Skidmore, A.; Bowers, J. J., Oscillatory behaviour of solutions of \(ypxyfx\), J. Math. Anal. Appl., 49, 317-323 (1975) · Zbl 0312.34025
[9]	Skidmore, A.; Leighton, W., On the differential equation \(ypxyfx\), J. Math. Anal. Appl., 43, 46-55 (1973) · Zbl 0287.34031
[10]	Teufel, H., Forced second order nonlinear oscillations, J. Math. Anal. Appl., 40, 148-152 (1972) · Zbl 0211.12001
[11]	Wong, J. S.W., Second order nonlinear forced oscillations, SIAM J. Math. Anal., 19, 667-675 (1988) · Zbl 0655.34023