A necessary and sufficient condition for the oscillation of higher-order neutral equations. (English) Zbl 0684.34068

Summary: Consider the higher-order neutral delay differential equation \[ (*)\quad d^ n/dt^ n(x(t)+\sum^{L}_{i=1}p_ ix(t-\tau_ i)- \sum^{M}_{j=1}r_ jx(t-\rho_ j))+\sum^{N}_{k=1}q_ kx(t-u_ k)=0, \] where the coefficients and the delays are nonnegative constants with \(n\geq 1\) odd. Then a necessary and sufficient condition for the oscillation of (*) is that the characteristic equation \[ F(\lambda):=\lambda^ n+\lambda^ n\sum^{L}_{i=1}p_ ie^{- \lambda \tau_ i}-\lambda^ n\sum^{M}_{j=1}r_ je^{-\lambda \rho_ j}+\sum^{N}_{k=1}q_ ke^{-\lambda u_ k}=0 \] has no real roots.


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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