## Oscillation criteria in neutral equations of $$n$$-order with variable coefficients.(English)Zbl 0748.34043

The oscillatory behaviour of the $$n$$-th order neutral delay differential equation $(d^ n/dt^ n)[y(t)+P(t)y(t-\tau)]+M(t)y(t-\sigma)=0 (1)$ $$P$$, $$M$$ continuous, $$M>0$$, is considered. For $$n=1$$, $$-1\leq P\leq 0$$, $$R'\leq 0$$, $$1-R(t)\int^{t+\sigma}_{t-\tau}M(s)ds\leq 0$$, where $$R(t)=P(t-\sigma)M(t)/M(t-\tau)$$, every solution of (1) oscillates. Also some theorems connected with (1) for $$n$$ odd and positive $$M$$ are demonstrated. The assumptions of the theorems are new and extend known results.

### MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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