×

Oscillation criteria for nonlinear differential equations with damping. (English) Zbl 1034.34041

The author presents oscillation criteria for the damped nonlinear differential equation \[ (r(t)x')'+ p(t)x' + q(t)f(x)=0, \tag{*} \] where \(r,f\) are continuously differentiable functions, \(r(t)>0\), \(p,q\) are continuous, \(xf(x)>0\) for \(x\neq 0\), and it is supposed that there exists \(\lambda>0\) such that \(f'(x)>\lambda\) for \(x\in \mathbb{R}\). Under the last assumption on the derivative \(f'\), equation (*) is, in a certain sense, the Sturmian majorant of the linear equation \[ (r(t)y')'+p(t)y+ \lambda q(t)y=0. \tag{**} \] Using a combination of the generalized Riccati substitution and the \(H\)-function averaging technique, conditions on the functions \(r,p,q\) are given which guarantee that (**) is oscillatory and this, in turn, implies that (*) possesses no nonoscillatory solution which is extensible up to \(\infty\). The obtained oscillation criteria are illustrated by a number of examples and corollaries.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Wintner, A., A criterion of oscillatory stability, Quart. appl. math., 7, 115-117, (1949) · Zbl 0032.34801
[2] Hartman, P., On nonoscillatory linear differential equations of second order, Amer. J. math., 74, 389-400, (1952) · Zbl 0048.06602
[3] Kamenev, I., Integral criteria of linear differential equations of second order, Math. zametki, 23, 249-251, (1978) · Zbl 0386.34032
[4] Kong, Q., Interval criteria for oscillation of second order linear ordinary differential equations, J. math. anal. appl., 229, 258-270, (1999) · Zbl 0924.34026
[5] Li, H., Oscillation critria for second order linear differential equations, J. math. anal. appl., 194, 217-234, (1995)
[6] Philos, Ch., Oscillation theorems for linear differential equations of second order, Arch. math. (basel), 53, 482-492, (1995) · Zbl 0661.34030
[7] Li, W.; Agarwal, R., Interval oscillation criterion for second order nonlinear differential equations with damping, Comp. math. appl., 40, 217-230, (2000) · Zbl 0959.34026
[8] Hartman, P., Ordinary differential equations, (1964), Wiley New York · Zbl 0125.32102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.