About sign-constancy of Green's functions for impulsive second order delay equations

摘要

We consider the following second order differential equation with delay [egin{cases} (Lx)(t)equiv{x''(t)+sum_{j=1}^p {b_{j}(t)x(t-heta_{j}(t))}}=f(t), quad tin[0,omega], x(t_j)=gamma_{j}x(t_j-0), x'(t_j)=delta_{j}x'(t_j-0), quad j=1,2,ldots,r. end{cases}] In this paper we find necessary and sufficient conditions of positivity of Green's functions for this impulsive equation coupled with one or two-point boundary conditions in the form of theorems about differential inequalities. By choosing the test function in these theorems, we obtain simple sufficient conditions. For example, the inequality (sum_{i=1}^p{b_i(t)left(rac{1}{4}+right)}lt rac{2}{omega^2}) is a basic one, implying negativity of Green's function of two-point problem for this impulsive equation in the case (0lt gamma_ileq{1}), (0lt delta_ileq{1}) for (i=1,ldots ,p).