一类二阶非线性微分方程的非振动准则

Nonoscillation Criteria for A Class of Second Order Nonlinear Differential Equations

本文用能量函数方法得到了一类二阶非线性微分方程的几个非振动准则,推广了Lynn H Erbe的结果。一九八五年Lynn H·Erbe在文[1]中,用能量函数的方法讨论了二阶非线性微分方程y’’+q(x)y~γ=0 γ>0是既约分数,q(x)是正的局部有界变差函数,对前人的若干非振动准则作了改进。我们把这种方法进一步推广用于另一类二阶非线性微分方程(r(x)y’)’+q(x)y~γ=0,也能得到新的一些非振动准则。现陈述于后。

In this paper, a few nonoscillation criteria are presented for a clas sof second order nonlinear differential equations, which are supposed to be the developmentof the results by lynn H. Erbe. In equation (r(x)y')'+ q(x)y' = 0 (1) Let γ>0,is the quotiont of odd positive integers and r(x)q(x)∈CBV loc[a,∞], r(x) q (x)>0 has the Jordan representation.(i) Ifγ>1, the(1)is nonoscillatory in case any of following hold:(a) integral from n=a to ∞(q(t))^(1/γ+1) (γ(t))^(-γ/γ+1) (Q +(t))^(γ/γ+1) dt<∞ and(?)Q_~1/2 (x) integral from n=x to∞(q(t))^(1/γ+1)(r(t))^(-γ/γ+1) (Q_(t))^(γ/γ+1) dt = 0(b) (?) Q_^(1/2) (x) integral from n=x to ∞ q(t)Q_^(γ/γ+1) (t)dt = 0(ii)If γ>1 and r' (x)>0, (iii)If 0<γ<1 equation(1)has similar results.