变系数高阶中立型微分方程的振动性(英文)

Oscillations of High Order Neutral Differential Equations with Variable Coefficients

考虑变系数高阶中立型微分方程(NDDE)d^n/(dt^n)[y(t)+p(t)y(t-τ)]+sum from n=1 to ∞q^i(t)y(t-σ_i)=0 (1)其中p(t)、g_i(t)都是区间[T,∞)上连续的实值函数.p(t)有界,q_i(t)≥0(i=1,2,···,m)且至少有一个q_i(t)最终大于某一任意小的正数.τ≥0,σ_i≥0.m≥1,n≥1均为正整数. 本文研究了方程(1)在p(t)≥一1及p(t)≤-1等情况下解的渐近性和振动性,获得了一系列使解振动的充分条件.特别,p(t)有时可以是变号函数.

Consider the high order neutral differential equation where p(t), q_i(t)∈C[T, ∞), p(t) is bounded, q_i(t)≥0 (i=1,2,…,m) and there is at least one q_i(t) which is greater than a positive number eventually, τ≥0, σ≥0.m≥1 and n≥1 are positive integers. In this paper, author studies the asymptotic and oscillatory behavior of solutions of (1) with p(t)≥-1 or p(t)≤-1. A series of sufficient conditions for the oscillation of solutions have been obtained. Particularly, p(t) can be a function of variable sign in some case.