Linearized oscillations for odd-order neutral delay differential equations.

*(English)*Zbl 0717.34074Summary: Consider the nth order nonlinear neutral delay differential equation
\[
(1)\quad d^ n/dt^ n[x(t)-p(t)g(x(t-\tau))]+q(t)h(x(t-\sigma))=0,
\]
where \(n\geq 1\) is an odd integer. We prove that, under appropriate hypotheses, equation (1) oscillates provided that the same is true for an associated linear equation with constant coefficients of the form
\[
d^ n/dt^ n[y(t)-p_ 0y(t-\tau)]+q_ 0y(t-\sigma)=0.
\]
A partial converse is also presented, where we show that, under appropriate hypotheses, equation (1) has a positive solution.

##### 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 |

34K40 | Neutral functional-differential equations |

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\textit{G. Ladas} and \textit{C. Qian}, J. Differ. Equations 88, No. 2, 238--247 (1990; Zbl 0717.34074)

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##### References:

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