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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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