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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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