×

Fite-Wintner-Leighton-type oscillation criteria for second-order differential equations with nonlinear damping. (English) Zbl 1308.34043

Summary: Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zhao, A.; Wang, Y.; Yan, J., Oscillation criteria for second-order nonlinear differential equations with nonlinear damping, Computers & Mathematics with Applications, 56, 2, 542-555 (2008) · Zbl 1155.34320 · doi:10.1016/j.camwa.2008.01.014
[2] Rogovchenko, S. P.; Rogovchenko, Y. V., Oscillation theorems for differential equations with a nonlinear damping term, Journal of Mathematical Analysis and Applications, 279, 1, 121-134 (2003) · Zbl 1027.34040 · doi:10.1016/S0022-247X(02)00623-6
[3] Tiryaki, A.; Zafer, A., Oscillation of second-order nonlinear differential equations with nonlinear damping, Mathematical and Computer Modelling, 39, 2-3, 197-208 (2004) · Zbl 1049.34040 · doi:10.1016/S0895-7177(04)90007-6
[4] Agarwal, R. P.; Wang, Q. R., Oscillation and asymptotic behavior for second-order nonlinear perturbed differential equations, Mathematical and Computer Modelling, 39, 13, 1477-1490 (2004) · Zbl 1079.34020 · doi:10.1016/j.mcm.2004.07.007
[5] Baker, J. W., Oscillation theorems for a second order damped nonlinear differential equation, SIAM Journal on Applied Mathematics, 25, 1, 37-40 (1973) · Zbl 0239.34015 · doi:10.1137/0125007
[6] Bobisud, L. E., Oscillation of solutions of damped nonlinear equations, SIAM Journal on Applied Mathematics, 19, 3, 601-606 (1970) · Zbl 0206.38005 · doi:10.1137/0119059
[7] Grace, S. R., Oscillation theorems for nonlinear differential equations of second order, Journal of Mathematical Analysis and Applications, 171, 1, 220-241 (1992) · Zbl 0767.34017 · doi:10.1016/0022-247X(92)90386-R
[8] Huang, Y.; Meng, F., Oscillation criteria for forced second-order nonlinear differential equations with damping, Journal of Computational and Applied Mathematics, 224, 1, 339-345 (2009) · Zbl 1167.34325 · doi:10.1016/j.cam.2008.05.002
[9] Long, Q.; Wang, Q. R., New oscillation criteria of second-order nonlinear differential equations, Applied Mathematics and Computation, 212, 2, 357-365 (2009) · Zbl 1172.34322 · doi:10.1016/j.amc.2009.02.040
[10] Wang, Q. R.; Wu, X. M.; Zhu, S. M., Oscillation criteria for second-order nonlinear damped differential equations, Computers & Mathematics with Applications, 46, 8-9, 1253-1262 (2003) · Zbl 1059.34039 · doi:10.1016/S0898-1221(03)90216-1
[11] Yamaoka, N., Oscillation criteria for second-order damped nonlinear differential equations with \(p\)-Laplacian, Journal of Mathematical Analysis and Applications, 325, 2, 932-948 (2007) · Zbl 1108.34027 · doi:10.1016/j.jmaa.2006.02.021
[12] Zhao, X.; Meng, F., Oscillation of second-order nonlinear ODE with damping, Applied Mathematics and Computation, 182, 2, 1861-1871 (2006) · Zbl 1122.34027 · doi:10.1016/j.amc.2006.06.022
[13] Shang, N.; Qin, H., Comments on the paper: “Oscillation of second-order nonlinear ODE with damping” [Applied Mathematics and Computation 199 (2008) 644-652], Applied Mathematics and Computation, 218, 6, 2979-2980 (2011) · Zbl 1367.34036 · doi:10.1016/j.amc.2011.07.064
[14] Philos, Ch. G., Oscillation theorems for linear differential equations of second order, Archiv der Mathematik, 53, 5, 482-492 (1989) · Zbl 0661.34030 · doi:10.1007/BF01324723
[15] Wong, J. S. W., On Kamenev-type oscillation theorems for second-order differential equations with damping, Journal of Mathematical Analysis and Applications, 258, 1, 244-257 (2001) · Zbl 0987.34024 · doi:10.1006/jmaa.2000.7376
[16] Xu, Z.; Xia, Y., Kamenev-type oscillation criteria for second-order quasilinear differential equations, Electronic Journal of Differential Equations, 2005, 1-9 (2005) · Zbl 1075.34032
[17] Yan, J. R., Oscillation theorems for second order linear differential equations with damping, Proceedings of the American Mathematical Society, 98, 2, 276-282 (1986) · Zbl 0622.34027 · doi:10.2307/2045698
[18] Yeh, C. C., Oscillation theorems for nonlinear second order differential equations with damped term, Proceedings of the American Mathematical Society, 84, 3, 397-402 (1982) · Zbl 0498.34023 · doi:10.2307/2043569
[19] Pašić, M., Isoperimetric inequalities in quasilinear elliptic equations of Leray-Lions type, Journal de Mathématiques Pures et Appliquées, 75, 4, 343-366 (1996) · Zbl 0859.35028
[20] Tian, Y.; Li, F., Comparison results for nonlinear degenerate Dirichlet and Neumann problems with general growth in the gradient, Journal of Mathematical Analysis and Applications, 378, 2, 749-763 (2011) · Zbl 1211.35125 · doi:10.1016/j.jmaa.2011.01.072
[21] Walter, W., A note on Sturm-type comparison theorems for nonlinear operators, Journal of Differential Equations, 135, 2, 358-365 (1997) · Zbl 0877.34031 · doi:10.1006/jdeq.1996.3233
[22] Walter, W., Ordinary Differential Equations. Ordinary Differential Equations, Graduate Texts in Mathematics. Readings in Mathematics, 182 (1998), New York, NY, USA: Springer, New York, NY, USA · Zbl 0991.34001
[23] Carl, S., Existence and comparison results for noncoercive and nonmonotone multivalued elliptic problems, Nonlinear Analysis. Theory, Methods & Applications A, 65, 8, 1532-1546 (2006) · Zbl 1233.35090 · doi:10.1016/j.na.2005.10.028
[24] Fite, W. B., Concerning the zeros of the solutions of certain differential equations, Transactions of the American Mathematical Society, 19, 4, 341-352 (1918) · JFM 46.0702.02 · doi:10.2307/1988973
[25] Wintner, A., A criterion of oscillatory stability, Quarterly of Applied Mathematics, 7, 115-117 (1949) · Zbl 0032.34801
[26] Leighton, W., The detection of the oscillation of solutions of a second order linear differential equation, Duke Mathematical Journal, 17, 57-61 (1950) · Zbl 0036.06101 · doi:10.1215/S0012-7094-50-01707-8
[27] Wong, J. S. W., Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Transactions of the American Mathematical Society, 144, 197-215 (1969) · Zbl 0195.37402 · doi:10.1090/S0002-9947-1969-0251305-6
[28] Travis, B., \(N\) th order extension of the Wintner-Leighton theorem, Applied Mathematics and Computation, 110, 2-3, 115-119 (2000) · Zbl 1030.34030 · doi:10.1016/S0096-3003(98)10092-9
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.