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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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