Oscillatory properties of second order nonlinear differential equations (Q976548)

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.