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**Oscillation theorems for second-order nonlinear differential equations with damped term.**
*(English)*
Zbl 1061.34020

Summary: Several oscillation criteria are given for the damped nonlinear second-order differential equation \((a(t)[y'(t)]^\sigma)'+p (t)[y'(t)]^\sigma + q(t)(y)t))=0\), where \(\sigma>0\) is any quotient of odd integers, \(a\in C(\mathbb{R}, (0,\infty))\), \(p(t)\) and \(q(t)\) are allowed to change sign on \([t_0, \infty)\), and \(f\in C^1(\mathbb{R},\mathbb{R})\) such that \(xf(x)>0\) for \(x\neq 0\). Our results improve and extend some known oscillation criteria. Examples are inserted to illustrate our results.

### MSC:

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

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\textit{W.-T. Li} and \textit{P. Zhao}, Math. Comput. Modelling 39, No. 4--5, 457--471 (2004; Zbl 1061.34020)

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### References:

[1] | Li, H.J, Oscillation criteria for second order linear differential equations, J. math. anal. appl., 194, 217-234, (1995) · Zbl 0836.34033 |

[2] | Li, W.T, Oscillation of certain second order nonlinear differential equations, J. math. appl. appl., 217, 1-14, (1998) · Zbl 0893.34023 |

[3] | Wong, P.J.Y; Agarwal, R.P, Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. math. anal. appl., 198, 337-354, (1996) · Zbl 0855.34039 |

[4] | Wong, P.J.Y; Agarwal, R.P, The oscillation and asymptotically monotone solutions of second order quasilinear differential equations, Appl. math. comput., 79, 207-237, (1997) |

[5] | Li, W.T; Yan, J.R, An oscillation criterion for second order superlinear differential equations, Indian J. pure and appl. math., 28, 6, 735-740, (1997) · Zbl 0880.34033 |

[6] | Graef, J.R; Spikes, P.W, On the oscillatory behavior of solutions of second order non-linear differential equations, Czechoslovak math. J., 36, 275-284, (1986) · Zbl 0627.34034 |

[7] | Kwong, M.K; Wong, J.S.W, An application of integral inequality to second order non-linear oscillation, J. differential equations, 46, 63-77, (1982) · Zbl 0503.34021 |

[8] | Yan, J.R, Oscillation theorems for second order linear differential equations with damping, (), 276-282 · Zbl 0622.34027 |

[9] | Yan, J.R, A note on an oscillation criterion for equations with damped term, (), 277-280 · Zbl 0542.34028 |

[10] | Yeh, C.C, Oscillation theorems for nonlinear second order differential equations with damped term, (), 397-402 · Zbl 0498.34023 |

[11] | Grace, J.R; Lalli, B.S, Asymptotic and oscillation behavior of solutions of a class of second order differential equations with deviating arguments, J. math anal. appl., 145, 112-136, (1990) · Zbl 0684.34069 |

[12] | Grace, J.R; Lalli, B.S, Oscillation and convergence to zero of solutions of damped second order differential equations, J. math anal. appl., 102, 539-548, (1984) · Zbl 0575.34023 |

[13] | Kartsatos, A.G, Recent results on oscillation of solutions of forced and perturbed nonlinear differential equations of even order, (), 17-72 |

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

[15] | Li, W.T, Positive solutions of second order nonlinear differential equations, J. math. anal. appl., 221, 326-337, (1998) · Zbl 0966.34030 |

[16] | 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 |

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

[18] | Philos, Ch.P; Purnaras, I.K, Oscillation in superlinear differential equations of second order, J. math. anal. appl., 165, 1-11, (1992) · Zbl 0756.34036 |

[19] | Rogovchenko, Y.R, Note on “oscillation criteria for second order linear differential equations”, J. math. anal. appl., 203, 560-563, (1996) · Zbl 0862.34024 |

[20] | Thandapani, E; Gyori, I; Lalli, B.S, An application of discrete inequality to second order nonlinear oscillation, J. math. anal. appl., 186, 200-208, (1994) · Zbl 0823.39004 |

[21] | Thandapani, E; Pandian, S, On the oscillatory behavior of solutions of second order nonlinear difference equations, Z. anal. anwendungent, 13, 347-358, (1994) · Zbl 0803.39004 |

[22] | Wong, J.S, On the second order nonlinear oscillations, Funkcial. ekvac., 11, 207-234, (1968) |

[23] | Wong, J.S, Oscillation criteria for second order nonlinear differential equations involving integral averages, Canad. J. math., 45, 1094-1103, (1993) · Zbl 0797.34037 |

[24] | Wong, J.S, An oscillation criterion for second order nonlinear differential equations, (), 109-112 · Zbl 0603.34025 |

[25] | Wong, J.S, Oscillation theorems for second order nonlinear differential equations, (), 1069-1077 · Zbl 0694.34027 |

[26] | Wong, J.S, Oscillation criteria for second order nonlinear differential equations with integrable coefficients, (), 389-395 · Zbl 0760.34032 |

[27] | Yu, Y.H, Leighton type oscillation criterion and Sturm type comparison theorem, Math. nachr., 153, 137-143, (1991) · Zbl 0795.34025 |

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