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Oscillation theorems for third order nonlinear differential equations. (English) Zbl 0760.34031
The paper deals with oscillation and asymptotic behaviour of the nonlinear third order equation $$(*)\;(r_ 2(t)(r_ 1(t)y')')'+p(t)y'+q(t)f(y)=0$$, where $$p,q$$ are nonnegative functions and $$r_ 1(t)$$, $$r_ 2(t)>0$$ for $$t\in[a,\infty)$$. Conditions on the functions $$r_ 1,r_ 2,p,q$$ are given which guarantee that no nonoscillatory solution of $$(*)$$ has property $$V_ 2$$ provided certain (nonlinear) second order equation associated with $$(*)$$ is oscillatory (a solution $$y$$ of $$(*)$$ is said to possess property $$V_ 2$$ if $$y(t)L_ ky(t)>0$$, $$k=1,2$$, $$y(t)L_ 3y(t)\leq 0$$ for large $$t$$, where $$L_ 0y=y$$, $$L_ iy=r_ i(L_{i-1}y)'$$, $$i=1,2$$, $$L_ 3y=(L_ 2y)')$$. As a corollary of these results the following oscillation criterion is proved.
Theorem. Let any condition which implies that no nonoscillatory solution of $$(*)$$ has property $$V_ 2$$ be satisfied. A solution $$y$$ of $$(*)$$ which exists on an interval $$[T,\infty)$$ is oscillatory if and only if there exists $$t_ 0\in[T,\infty)$$ such that $$2yL_ 2y-{r_ 2\over r_ 1}(L_ 1y)^ 2+py^ 2\leq 0$$ for $$t=t_ 0$$.
Reviewer: O.Došlý (Brno)

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems
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