##
**On oscillation of solutions of forced nonlinear neutral differential equations of higher order.**
*(English)*
Zbl 1080.34522

Summary: Necessary and sufficient conditions are obtained for every bounded solution of
\[
[y (t) - p (t) y (t - \tau )]^{(n)} + Q (t) G \bigl (y (t - \sigma )\bigr ) = f (t), \quad t \geq 0, \tag{\(*\)}
\]
to oscillate or tend to zero as \(t \to \infty \) for different ranges of \(p (t)\). It is shown, under some stronger conditions, that every solution of \((*)\) oscillates or tends to zero as \(t \to \infty \). Our results hold for linear, a class of superlinear and other nonlinear equations and answer a conjecture by G.Ladas and Y.G.Sficas [J.Aust.Math.Soc., Ser.B 27, 502-511 (1986; Zbl 0566.34055)], and generalize some known results.

### MSC:

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

34C15 | Nonlinear oscillations and coupled oscillators for ordinary differential equations |

34K40 | Neutral functional-differential equations |

### Citations:

Zbl 0566.34055
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XMLCite

\textit{N. Parhi} and \textit{R. N. Rath}, Czech. Math. J. 53, No. 4, 805--825 (2003; Zbl 1080.34522)

### References:

[1] | Ming-Po-Chen, Z. C. Wang, J. S. Yu and B. G. Zhang: Oscillation and asymptotic behaviour of higher order neutral differential equations. Bull. Inst. Math. Acad. Sinica 22 (1994), 203–217. · Zbl 0818.34040 |

[2] | Q. Chuanxi and G. Ladas: Oscillation of higher order neutral differential equations with variable coefficients. Math. Nachr. 150 (1991), 15–24. · Zbl 0728.34075 |

[3] | D. A. Georgiou and C. Qian: Oscillation criteria in neutral equations of nth order with variable coefficients. Internat. J. Math. Math. Sci. 14 (1991), 689–696. · Zbl 0748.34043 |

[4] | K. Gopalsamy, B. S. Lalli and B. G. Zhang: Oscillation in odd order neutral differential equations. Czechoslovak Math. J. 42 (1992), 313–323. · Zbl 0778.34050 |

[5] | K. Gopalsamy, S. R. Grace and B. S. Lalli: Oscillation of even order neutral differential equations. Indian J. Math. 35 (1993), 9–25. · Zbl 0809.34080 |

[6] | S. R. Grace: On the oscillation of certain forced functional differential equation. J. Math. Anal. Appl. 202 (1996), 555–577. · Zbl 0877.34049 |

[7] | I. Gyori and G. Ladas: Oscialltion Theory of Delay-Differential Equations with Applications. Clarendon Press, Oxford, 1991. |

[8] | T. H. Hildebrandt: Introduction to the Theory of Integration. Academic Press, New York, 1963. |

[9] | I. T. Kiguradze: On the oscillation of solutions of the equation \(\tfrac{{\operatorname{d} ^m u}}{{\operatorname{d} t^m }} + a\left( t \right)u^m\) sign u = 0. Mat. Sb. 65 (1964), 172–187. · Zbl 0135.14302 |

[10] | G. Ladas and Y. G. Sficas: Oscillations of higher order neutral equations. Austral. Math. Soc. Ser. B 27 (1986), 502–511. · Zbl 0566.34055 |

[11] | G. Ladas, C. Qian and J. Yan: Oscillations of higher order neutral differential equations. Portugal. Math. 48 (1991), 291–307. · Zbl 0736.34063 |

[12] | G. S. Ladde, V. Lakshmikantham and B. G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker INC., New York, 1987. · Zbl 0832.34071 |

[13] | X. Z. Liu, J. S. Yu and B. G. Zhang: Oscillation and nonoscillation for a class of neutral differential equations. Differential Equations Dynam. Systems 1 (1993), 197–204. · Zbl 0873.34056 |

[14] | N. Parhi and P. K. Mohanty: Oscillation of solutions of forced neutral differential equations of n-th order. Czechoslovak Math. J. 45 (1995), 413–433. · Zbl 0842.34076 |

[15] | N. Parhi and P. K. Mohanty: Maintenance of oscillation of neutral differential equations under the effect of a forcing term. Indian J. Pure Appl. Math. 26 (1995), 909–919. · Zbl 0838.34082 |

[16] | N. Parhi and P. K. Mohanty: Oscillatory behaviour of solutions of forced neutral differential equations. Ann. Polon. Math. 65 (1996), 1–10. · Zbl 0874.34063 |

[17] | N. Parhi and P. K. Mohanty: Oscillations of neutral differential equations of higher order. Bull. Inst. Math. Acad. Sinica 24 (1996), 139–150. · Zbl 0858.34059 |

[18] | N. Parhi: Oscillation of higher order differential equations of neutral type. Czechoslovak Math. J. 50 (2000), 155–173. · Zbl 1045.34043 |

[19] | N. Parhi and R. N. Rath: On oscillation criteria for a forced neutral differential equation. Bull. Inst. Math. Acad. Sinica 28 (2000), 59–70. · Zbl 0961.34059 |

[20] | N. Parhi and R. N. Rath: Oscillation criteria for forced first order neutral differential equations with variable coefficients. J. Math. Anal. Appl. 256 (2001), 525–541. · Zbl 0982.34057 |

[21] | N. Parhi and R. N. Rath: On oscillation and asymptotic behaviour of solutions of forced first order neutral differential equations. Proc. Indian. Acad. Sci. (Math. Sci.), Vol. 111. 2001, pp. 337–350. · Zbl 0995.34058 |

[22] | H. L. Royden: Real Analysis. 3rd edition, MacMillan Publ. Co., New York, 1989. |

[23] | J. H. Shen: New oscillation criteria for odd order neutral equations. J. Math. Anal. Appl. 201 (1996), 387–395. · Zbl 0860.34040 |

[24] | D. Tang: Oscillation of higher order nonlinear neutral functional differential equation. Ann. Differential Equations 12 (1996), 83–88. · Zbl 0849.34059 |

[25] | J. S. Yu, Z. C. Wang and B. G. Zhang: Oscillation of higher order neutral differential equations. Rocky Mountain J. Math. To appear. |

[26] | B. G. Zhang and K. Gopalsam: Oscillations and nonoscillations in higher order neutral equations. J. Math. Phys. Sci. 25 (1991), 152–165. |

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