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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)

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