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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:
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