二阶非线性常微分方程的振动性

Oscillatory Property For Nonlinear Second-Order Ordinary Differential Equations

本文综合采用文〔1～2〕的技巧,给出了二阶非线性微分方程 x″+a(t)|x|′sgnx=0. r>0 (1)振动性的充分条件,所得结果不同于文[2～3]关于(l)的振动性给果。

In this paper we study the oscillation of the Solutions of the nonlinear equation (*) x″+a (t) |x|~r sgnx=0, where r>0, a (t)∈c[t_0, ∞), R), t_0. We obtained following main result: Assum there exist real numbers a>1, q>2 and a function φ(t) such that (i) lim t→∞ 1/t~α integral from s to t (t-τ)^(α-1)α(τ)dτ≥φ(s) for t_0≤s<+∞; (ii)lim t→+∞ 1/t~α τ^(2/q)(t-τ)~α[φ_+(τ)]~2 dτ=+∞ and (iii)lim t→∞ integral from t_0 to t α(τ)dτ≥-λ>-∞, where φ_+(t)=max{φ(t),0}, λ>0. Then equation (*) oscillate.