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 = t (0) < t (1) < t (2) < ⋯<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.