带有阻尼项的二阶非线性微分方程的振动性定理

OSCILLATION OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH DAMPING TERMS

考虑带有阻尼项的二阶非线性微分方程x″(t) + p(t)x′(t) + q(t) f(x(t) ) =0 ,t≥t0 ,其中 p ,q∈C[t0 ,∞ )允许变号 ,f∈C(R) ,且当x≠ 0时xf(x) >0 .借助于一个一般的Riccati变换w(t) =x′(t)f(x(t) )+ p(t)2K ,其中K >0为常数 ,给出了上述方程振动的一些新的结果 .

Consider the second order nonlinear differential equation with damped terms x″(t)+p(t)x′(t)+q(t)f(x(t)) =0,where t≥t 0, p,q∈C[t 0,∞ ) are allowed to change their signs, f∈C(R ), and xf(x)> 0 for x ≠0. By virtue of a generalized Riccati transformation w(t)=x′(t)f(x(t))+p(t)2K , where K >0 is a constant, some new oscillation criteria are obtained.