Linearized oscillations for odd-order neutral delay differential equations

AbstractConsider the nth order nonlinear neutral delay differential equation dndtn [x(t) − p(t) g(x(t − τ))] + q(t) h(x(t − σ)) = 0, (1) where n ⩾ 1 is an odd integer. We prove that, under appropriate hypotheses, Eq. (1) oscillates provided that the same is true for an associated linear equation with constant coefficients of the form dndtn [y(t) − p0y(t − τ)] + q0y(t − σ) = 0. A partial converse is also presented, where we show that, under appropriate hypotheses, Eq. (1) has a positive solution