An oscillation criterion for a class of ordinary differential equations. (Russian) Zbl 0768.34018

The paper deals with oscillatory and asymptotic behaviour of the nonlinear differential equation (*) \(u^{(n)} + u^{(n-2)} = f(t,u,\dots,u^{(n-1)})\), \(n\geq 3\), where the nonlinearity \(f: [0,\infty) \times \mathbb{R}^ n \times \mathbb{R}^ n \to \mathbb{R}\) satisfies the sign condition \(f(t,x_ 1,\dots,x_ n)x_ 1 \leq 0\). As a corollary of the general results for (*), the following oscillation criterion concerning the Emdem-Fowler type equation (**) \(u^{(n)} + u^{(n-2)} = p(t) | u|^ \lambda\text{sgn }u\), \(\lambda\neq 1\), \(p(t) < 0\), is obtained.
Theorem. Let \(n\) be even. Then the condition \(\int^ \infty t^{\sigma_ n(\lambda)}p(t)dt = -\infty\), where \(\sigma_ n(\lambda) = (n-3)\lambda\) if \(0 < \lambda < 1\) and \(\sigma_ n(\lambda) = (n-3)\), if \(\lambda > 1\), is necessary and sufficient for all eventually nonvanishing solutions of (**) to be oscillatory.
This theorem and some other statements of the paper reveal the surprising fact that concerning oscillation properties, equation (*) behaves like the equation \(u^{(n-2)} = p(t)u\) rather than like \(u^{(n)} = p(t)u\).
Reviewer: O.Došlý (Brno)


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