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Oscillatory properties of second order nonlinear differential equations. (English) Zbl 1205.34034

The authors derive necessary and sufficient criteria for the oscillation, nonoscilation, and strong oscillation of the second order nonlinear differential equation
\[ u''+f(t,u,u')=0 \]
on \([a,\infty)\), where \(f:[a,\infty)\times{\mathbb R}^2\to{\mathbb R}\) is continuous and \(f(t,x,y)\geq0\) on \([a,\infty)\times{\mathbb R}^2\). The results generalize several criteria known for the case of \(f(t,x,y)=h(t)\,g(x)\). The proofs involve, in particular, iterated exponential and logarithmic functions and the Kiguradze lemma. Several examples are also presented to illustrate the sharpness of the conditions.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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