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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 $$ (*)\quad d\sp n/dt\sp n(x(t)+\sum\sp{L}\sb{i=1}p\sb ix(t-\tau\sb i)- \sum\sp{M}\sb{j=1}r\sb jx(t-\rho\sb j))+\sum\sp{N}\sb{k=1}q\sb kx(t-u\sb k)=0, $$ where the coefficients and the delays are nonnegative constants with $n\ge 1$ odd. Then a necessary and sufficient condition for the oscillation of (*) is that the characteristic equation $$ F(\lambda):=\lambda\sp n+\lambda\sp n\sum\sp{L}\sb{i=1}p\sb ie\sp{- \lambda \tau\sb i}-\lambda\sp n\sum\sp{M}\sb{j=1}r\sb je\sp{-\lambda \rho\sb j}+\sum\sp{N}\sb{k=1}q\sb ke\sp{-\lambda u\sb k}=0 $$ has no real roots.

34K99Functional-differential equations
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
Full Text: DOI
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