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More on oscillation of $$n$$th-order equations. (English) Zbl 0841.34034
The 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.
Reviewer: O.Došlý (Brno)
##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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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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