More on oscillation of \(n\)th-order equations.

*(English)*Zbl 0841.34034The author continues his investigation of oscillation and asymptotic properties of the forced higher-order equation of the form \((*)\) \(x^{(n)}+ p(t) x^{(n- 1)}+ H(t, x)= 0\). A typical statement is given in the following theorem.

Theorem. Suppose that \(n\) is odd, the function \(p\) is continuous and eventually nonpositive and \[ \int^\infty_{t^*} t^i\Biggl(\exp \int^t_{t^*} p(s) ds\Biggr) H(t, k) dt= -\infty \] for any \(t^*\geq 0\), every positive real constant \(k\), and some integer \(i\), where \(1\leq i\leq n- 1\). (a) Then every solution of \((*)\) with bounded \((n- i- 1)\)st derivative is oscillatory. In particular, every bounded solution of \((*)\) is oscillatory. (b) If \(i= 1\) and \(x(t)\) is an unbounded solution of \((*)\), then \(x(t) x^{(j)}(t)> 0\) for all \(j= 0, 1,\dots, n\), eventually.

Related author’s results on asymptotic behaviour of forced \(n\)th-order equations may be found in [‘Differential equations and applications’, Proc. Int. Conf., Columbus/OH (USA) 1988, Vol. II, 29-34 (1989; Zbl 0721.34029)] and in two recent papers submitted for publication into Anal. Pol. Math. and Hiroshima Math. J.

Theorem. Suppose that \(n\) is odd, the function \(p\) is continuous and eventually nonpositive and \[ \int^\infty_{t^*} t^i\Biggl(\exp \int^t_{t^*} p(s) ds\Biggr) H(t, k) dt= -\infty \] for any \(t^*\geq 0\), every positive real constant \(k\), and some integer \(i\), where \(1\leq i\leq n- 1\). (a) Then every solution of \((*)\) with bounded \((n- i- 1)\)st derivative is oscillatory. In particular, every bounded solution of \((*)\) is oscillatory. (b) If \(i= 1\) and \(x(t)\) is an unbounded solution of \((*)\), then \(x(t) x^{(j)}(t)> 0\) for all \(j= 0, 1,\dots, n\), eventually.

Related author’s results on asymptotic behaviour of forced \(n\)th-order equations may be found in [‘Differential equations and applications’, Proc. Int. Conf., Columbus/OH (USA) 1988, Vol. II, 29-34 (1989; Zbl 0721.34029)] and in two recent papers submitted for publication into Anal. Pol. Math. and Hiroshima Math. J.

Reviewer: O.Došlý (Brno)

##### MSC:

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

##### Keywords:

bounded solutions; positive solutions; oscillation; asymptotic properties; forced higher-order equation
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\textit{W. A. J. Kosmala}, Georgian Math. J. 2, No. 6, 593--602 (1995; Zbl 0841.34034)

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##### References:

[1] | L. Erbe, Oscillation, nonoscillation, and asymptotic behaviour for third-order nonlinear differential equations.Ann. Mat. Pura Appl. 110(1976), 373–391. · Zbl 0345.34023 |

[2] | J. W. Heidel, Qualitative behaviour of solutions of a third-order nonlinear differential equation.Pacific J. Math. 27(1968), 507–526. · Zbl 0172.11703 |

[3] | A. G. Kartsatos, The oscillation of a forced equation implies the oscillation of the unforced equation–small forcing.J. Math. Anal. Appl. 76(1980), 98–106. · Zbl 0443.34032 |

[4] | A. G. Kartsatos and W. A. Kosmala, The behaviour of annth-order equation with two middle terms.J. Math. Anal. Appl. 88(1982), 642–664. · Zbl 0513.34063 |

[5] | W. A. Kosmala, Properties of solutions of higher-order differential equations.Diff. Eq. Appl. 2 (1989), 29–34. |

[6] | W. A. Kosmala, Properties of solutions ofnth-order equations.Ordinary and delay differential equations, 101–105,Pitman, 1992. |

[7] | W. A. Kosmala, Oscillation of a forced higher-order equation.Ann. Polonici Math. (to appear). · Zbl 0817.34021 |

[8] | W. A. Kosmala, Behavior of bounded positive solutions of higher-order differential equations.Hiroshima Math. J. (to appear). · Zbl 0835.34037 |

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