二阶时滞微分方程的线性化振动性(英文)

Linearized Oscillations for Second Order Delay Differential Equations

考虑二阶变系数非线性时滞微分方程(1):y'(t)+A(t)y'(t)+B(t)f(y(t-r))=0,t≥0.我们证明在适当条件下,方程(1)的最终位于某带形域内的每个解振动,只要相关联的常系数线性方程(2)z'(t)+αz'(t)+βz(t-r)=0其相同的结论为真.我们也证明了在适当条件下,方程(1)有一个正解,只要方程(2)有一个正解.所得结果解决了Gyǒri和Ladas在文中提出的一个公开问题.

Consider the second order delay differential equation(1) y〃(t)+A(t)y′(t)+B(t)f[y(t-r)]=0, t≥0.We prove that under appropriate hypotheses, every solution which lies eventually in a certain strip oscillates provided that the same is true for an associated linear equation with constant coefficients of the form z〃t(t)+αz′(t)+βz(t-r)=0, t≥0. A partial converse is also presented, where we show that under appropriate hypotheses, Eq. (1) has apositive solution. These results solve an open problem of Gy(o|¨)ri and Ladas~[1].