##
**On Kamenev-type oscillation theorems for second-order differential equations with damping.**
*(English)*
Zbl 0987.34024

This is an interesting paper dealing with oscillation criteria for nonlinear second-order ordinary differential equations with damping. The results of this paper improve a few Kamenev-type oscillation theorems. The methodology of this paper is stimulating and is different from that of many previous papers of this kind. The author provides a more unified approach for the study of Kamenev-type oscillation theorems. Three examples are given. They are new and not covered by any of the known results.

Reviewer: Jurang Yan (Taiyaun)

### MSC:

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

34K11 | Oscillation theory of functional-differential equations |

PDF
BibTeX
XML
Cite

\textit{J. S. W. Wong}, J. Math. Anal. Appl. 258, No. 1, 244--257 (2001; Zbl 0987.34024)

Full Text:
DOI

### References:

[1] | Atkinson, F.V., On second order nonlinear oscillations, Pacific J. math., 5, 643-647, (1955) · Zbl 0065.32001 |

[2] | Baker, J.W., Oscillation theorems for a second-order damped nonlinear differential equation, SIAM J. appl. math., 25, 37-40, (1973) · Zbl 0239.34015 |

[3] | Bobisud, L.E., Oscillation of nonlinear differential equations with small nonlinear damping, SIAM J. appl. math., 18, 74-76, (1970) · Zbl 0193.05704 |

[4] | Bobisud, L.E., Oscillation of solutions of damped nonlinear differential equations, SIAM J. appl. math., 18, 601-606, (1970) · Zbl 0206.38005 |

[5] | Butler, G.J., The oscillatory behavior of a second-order nonlinear differential equation with damping, J. math. anal. appl., 57, 273-289, (1977) · Zbl 0348.34022 |

[6] | Chen, B.-S., Oscillation criteria for nonlinear second-order damped differential equations, Acta math. appl. sinica, 15, 166-173, (1992) · Zbl 0759.34027 |

[7] | Erbe, L., Oscillation criteria for second-order nonlinear differential equations, Aunali di Mathematica pura appl., 94, 257-268, (1972) · Zbl 0296.34026 |

[8] | Erbe, L.H.; Kong, Q.; Ruan, S., Kamenev-type theorems for second-order matrix differential system, Proc. amer. math. soc., 117, 957-962, (1993) · Zbl 0777.34024 |

[9] | Fite, W.B., Concerning the zeros of the solutions of certain differential equations, Trans. amer. math. soc., 19, 341-352, (1918) · JFM 46.0702.02 |

[10] | Grace, S.R.; Lalli, B.S.; Yeh, C.C., Oscillation theorems for nonlinear second-order differential equations with a nonlinear damping term, SIAM J. math. anal., 15, 1082-1093, (1984) · Zbl 0563.34042 |

[11] | Hartman, P., On non-oscillatory linear differential equations of second order, Am. J. math., 74, 389-400, (1952) · Zbl 0048.06602 |

[12] | Kamenev, I.V., An integral criterion for oscillation of linear differential equation of second order, Mat. zametki, 23, 249-251, (1978) · Zbl 0386.34032 |

[13] | Kiguradze, I.T., On condition for oscillation of solutions of the equation u″+a(t)|u|nsgnu=0, Casopis pest. mat., 89, 492-495, (1962) · Zbl 0138.33504 |

[14] | Kwong, M.K.; Zettl, A., Integral inequalities and second-order linear oscillation, J. diff. eq., 45, 16-33, (1982) · Zbl 0498.34022 |

[15] | Leighton, W., The detection of oscillation of solutions of a second-order linear differential equation, Duke math. J., 17, 57-62, (1950) · Zbl 0036.06101 |

[16] | Naito, M., Oscillation criteria for a second-order differential equation with a damping term, Hiroshima math. J., 4, 347-354, (1974) |

[17] | Philos, Ch.G., On a Kamenev’s integral criterion for oscillation of linear differential equations of second order, Utilitas math., 24, 277-289, (1983) · Zbl 0528.34035 |

[18] | Philos, Ch.G., Oscillation theorems for linear differential equations of second order, Arch. math., 53, 482-492, (1989) · Zbl 0661.34030 |

[19] | Sobol, I.M., Investigation with the aid of polar coordinates of the asymptotic behavior of solutions of a linear differential equation of the second order, Math. sb., 28, 707-714, (1951) |

[20] | Wintner, A., A criterion of oscillatory stability, Q. J. appl. math., 7, 115-117, (1949) · Zbl 0032.34801 |

[21] | Wong, J.S.W., Oscillation and nonoscillation of solutions of second-order linear differential equations with integrable coefficients, Trans. amer. math. soc., 144, 197-215, (1969) · Zbl 0195.37402 |

[22] | Wong, J.S.W., Oscillation theorems for second-order nonlinear differential equations, Bull. inst. math. acad. sinica, 3, 283-309, (1975) · Zbl 0316.34035 |

[23] | Wong, J.S.W., An oscillation criterion for second-order nonlinear differential equations with iterated integral averages, Diff. int. eq., 6, 83-91, (1993) · Zbl 0771.34026 |

[24] | Wong, J.S.W., Oscillation criteria for second-order nonlinear differential equations involving general means, J. math. anal. appl., 247, 489-505, (2000) · Zbl 0964.34028 |

[25] | Wong, J.S.W.; Yeh, C.C., An oscillation criterion for second-order sublinear differential equations, J. math. anal. appl., 171, 346-351, (1992) · Zbl 0767.34020 |

[26] | Wong, R., Asymptotic approximations of integrals, (1989), Academic Press New York · Zbl 0679.41001 |

[27] | Yan, Jurang, A note on an oscillatory criterion for an equation with damped term, Proc. amer. math. soc., 90, 277-280, (1984) · Zbl 0542.34028 |

[28] | Yan, Jurang, Oscillation theorems for second-order linear differential equations with damping, Proc. amer. math. soc., 98, 276-282, (1986) · Zbl 0622.34027 |

[29] | Yeh, C.C., Oscillation theorems for nonlinear second-order differential equations with a damped term, Proc. amer. math. soc., 84, 397-402, (1982) · Zbl 0498.34023 |

[30] | Yelchin, M., Sur le condition pour q’une solution d’un systéme linéaire du second ordre posséde deux zeros, Dokl. akad. nauk SSSR, 51, 573-576, (1946) · Zbl 0063.09068 |

[31] | Zlámal, M., Oscillation criterions, Casopis pest. mat., 75, 213-217, (1950) · Zbl 0040.19502 |

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.