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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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