Opuscula Mathematica Dedicated to the Memory of Professor Zdzisław Kamont ABOUT SIGN-CONSTANCY OF GREEN’S FUNCTIONS FOR IMPULSIVE SECOND ORDER DELAY EQUATIONS

Abstract. We consider the following second order differential equation with delay{ (Lx)(t) ≡ x′′(t) +∑pj=1 bj(t)x(t − θj(t)) = f(t), t ∈ [0, ω], x(tj) = γjx(tj − 0), x′(tj) = δjx′(tj − 0), j = 1, 2,..., r. 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 ∑p i=1 bi(t