临界状态下一阶中立型时滞微分方程的线性化振动性

Linearized oscillations for first order neutral delay differential equations in a critical state

在临界状态下建立了一阶非线性中立型时滞微分方程(x(t)-cx(t-τ))′+∑mi=1pix(t-τi)+f(t,x(t-σ1(t)),…,x(t-σn(t))=0与一个相关的二阶常微分方程振动性等价定理,进而给出了一阶非线性中立型微分方程(x(t)-cx(t-τ))′+∑mi=1pif(x(t-τi))=0与相应的线性方程振动性等价的充分条件,从而推广了文[1]的相应结果.

We established an equivalent theorem of oscillation of the following first order nonlinear delay differential equation (x(t)-cx(t-τ))′+∑mi=1p_ix(t-τ_i)+f(t,x(t-σ_1(t)),...,x(t-σ_n(t))=0and a related second order ordinary differential equation in a critical state.And then we obtain some sufficient conditions which guarantee the first order nonlinear nonautonomous differential equation(x(t)-cx(t-τ))′+∑mi=1p_if(x(t-τ_i))=0and the corresponding linear equation have the same oscillatory behavior.Some of the results in the literatureare improved.