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An oscillation criterion for nonlinear third-order differential equations. (English) Zbl 0878.34025
Oscillation theorems are proved for third-order nonlinear DE of type $y^{(3)}+ q(t)y'= f(t,y,y',y'')\quad\text{in }I=(a,\infty)\tag{1}$ for $$a>0$$, $$q\in C^1(I,\mathbb{R})$$, $$f\in C (I\times\mathbb{R}^3,\mathbb{R})$$, $$q(t)\geq q_0>0$$, $$q'(t)\leq 0$$ in $$I$$, and various conditions on $$f$$. The main theorem gives sufficient conditions for
(i) every solution of (1) with a zero to be oscillatory at $$\infty$$; and
(ii) every solution of (1) without zeros to satisfy $$y^{(j)}(t)\to 0$$ as $$t\to\infty$$ for $$j=0,1,2$$.
This generalizes a theorem of I. T. Kiguradze [Differ. Equations 28, No. 2, 180-190 (1992); translation from Differ. Uravn. 28, No. 2, 207-219 (1992; Zbl 0788.34027)].

##### MSC:
 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34C11 Growth and boundedness of solutions to ordinary differential equations
##### Keywords:
oscillation theorems; third-order nonlinear DE
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