About Positivity of Green's Functions for Nonlocal Boundary Value Problems with Impulsive Delay Equations

The impulsive delay differential equation is considered (Lx)(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 = t0 < t1 < t2 < ⋯<t k < t k+1 = b, x(ζ) = 0, ζ ∉ [a, b], with nonlocal boundary condition lx = ∫(a)(b) φ(s)x'(s)ds + θx(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.