一类二阶脉冲线性非振动微分方程解的渐近性态

Asymptotical behaviors for a second order linear nonoscillatory ODE with impulses

研究了一类二阶线性非振动脉冲微分方程(a(t)x′)′=p(t)x+∑∞n=1anδ(t-tn)x解的有界性和趋零性,其中a(t)为正的连续可微函数,p(t)为非负连续函数,且不最终恒为零,an≥0(n∈N),δ(t)是δ-函数.充分考虑脉冲的影响,通过建立脉冲微分方程与相应的常微分方程解的比较不等式,得到了判断脉冲微分方程解有界和趋零的充要条件.

The structure and asymptotical behaviors of solutions for a second order linear ordinary differential equation with impulsesa(t)x′′=p(t)x+∑∞n=1a_nδt-t_nxare investigated, where δ(t) is the Dirac δ-function, and 0≤t_0<t_1<…<t_n<…(t_n→∞ as n→∞),a_n>0 for n∈N, a(t)>0 is a continuous function on [t_0,+∞). Some necessary and sufficient conditions are obtained in this paper.