Linearized oscillation of odd order nonlinear neutral delay differential equations (I)

In this paper, we proved that the odd order nonlinear neutral delay differential equation[x(t)−p(t)g(x(t−τ))](n)+q(t)h(x(t−σ))=0[x(t)−p(t)g(x(t−τ))](n)+q(t)h(x(t−σ))=0 has the same oscillatory character as its linearized equation[x(t)−p0x(t−τ)](n)+q0x(t−σ)=0[x(t)−p0x(t−τ)](n)+q0x(t−σ)=0 under some rather relaxed conditions on g(u)g(u) and h(u)h(u), where p0=limt→∞p(t)p0=limt→∞p(t), q0=limt→∞q(t)q0=limt→∞q(t).