The existence of positive solution and bounded oscillation for even-order neutral differential equations with unstable type

In this paper, we consider the n-order neutral delay differential equation with unstable type((x(t)-p(t)x(t-τ))(n)=q(t)|x(t-σ)|α-1x(t-σ),t⩾t0,where n ⩾ 2 is an even integer, α ⩾ 1, τ > 0, σ > 0 and p, q ∈ C([t0, +∞), R+). We first prove that this equation always has an unbounded positive solution, then some new bounded oscillation and nonoscillation criteria are obtained.