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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[a,). 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 k y(t)>0, k=1,2, y(t)L 3 y(t)0 for large t, where L 0 y=y, L i y=r i (L i-1 y) ' , i=1,2, L 3 y=(L 2 y) ' ). 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,) is oscillatory if and only if there exists t 0 [T,) such that 2yL 2 y-r 2 r 1 (L 1 y) 2 +py 2 0 for t=t 0 .

Reviewer: O.Došlý (Brno)
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A34Nonlinear ODE and systems, general