About positivity of Green's functions for nonlocal boundary value problems with impulsive delay equations

The impulsive delay differential equation is considered (L x) (t) = x ′ (t) + ∑ i = 1 m p i (t) x (t - τ i (t)) = f (t), t ∈ [ a, b ], x (t j) = β j x (t j - 0), j = 1,., k, a = t 0 < t 1 < t 2 < < t k < t k + 1 = b, x (ζ) = 0, ζ ∉ [ a, b ], with nonlocal boundary condition l x = ∫ a b φ s x ′ s d s + θ x a = c, φ ∈ L ∞ a, b; θ, c ∈ R. Various results on existence and uniqueness of solutions and on positivity/negativity of the Green's functions for this equation are obtained.