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Oscillation theorems for a class of linear fourth order differential equations. (English) Zbl 0542.34030
Consider the equation (*) $$y^{(4)}+P(t)y''+R(t)y'+Q(t)y=0$$, where P(t), R(t), Q(t) are real-valued continuous functions on $$I=[a,\infty)$$, $$- \infty<a<\infty$$. Assume that (A): P(t)$$\leq 0$$, R(t)$$\leq 0$$, $$R^ 2(t)\leq 2P(t)Q(t)$$ for all $$t\in I$$ and Q(t) is not identical to zero in any interval of I. (B): $$\int^{\infty}_{\tau_ 0}t^{2+\alpha}Q(t)dt=-\infty, \tau_ 0<\max \{a,0\}$$, for some $$0\leq \alpha<1$$. The author proves the following theorem: Suppose that $$\int^{\infty}_{\tau_ 0}tP(t)dt>-\infty,$$ Q(t)$$\leq R(t)$$ for all $$t\geq \tau_ 0$$ and that (A) and (B) hold, or (B) holds and $$\int^{\infty}_{\tau_ 0}tP(t)dt>-\infty,$$ P(t)$$\leq R(t)\leq 0$$, 2Q(t)$$\leq R(t)$$ for all $$t\in I$$. Then equation (*) is oscillatory and there exists a fundamental system of solutions of (*) such that two solutions of this system are oscillatory, other solutions of this system are nonoscillatory and one of them tends monotonically to $$\infty$$ as $$t\to \infty$$ and the other tends to zero if $$\int^{\infty}_{\tau_ 0}s^{2+\alpha}R(s)ds>-\infty.$$
Reviewer: T.S.Liu

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
##### Keywords:
oscillatory solution
Full Text:
##### References:
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