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Linearized oscillations for odd-order neutral delay differential equations. (English) Zbl 0717.34074
Summary: 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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