About sign-constancy of Green’s function of a two-point problem for impulsive second order delay equations

摘要

We consider the following second order differential equation with delay $$ egin{cases} egin{split} (Lx)(t)&equiv{x''(t)+sum_{j=1}^p {a_{j}(t)x'(t-au_{j}(t))}+sum_{j=1}^p {b_{j}(t)x(t-heta_{j}(t))}}=f(t), & tin[0,omega] end{split} x(t_k)=gamma_{k}x(t_k-0), x'(t_k)=delta_{k}x'(t_k-0), quad k=1,2,dots,r. end{cases} $$ In this paper we find sufficient conditions of positivity of Green's functions for this impulsive equation coupled with two-point boundary conditions in the form of theorems about differential inequalities. Choosing the test function in these theorems, we obtain simple sufficient conditions.