Positivity of green’s matrix of nonlocal boundary value problems

We propose an approach for studying positivity of Green’s operators of a nonlocal boundary value problem for the system of n linear functional differential equations with the boundary conditions n_ix_i − Σⁿ_j=1 m_ijx_j = β_i, i = 1,...,n, where n_i and m_ij are linear bounded “local” and “nonlocal” functionals, respectively, from the space of absolutely continuous functions. For instance, (Formula presented.) can be considered. It is demonstrated that the positivity of Green’s operator of nonlocal problem follows from the positivity of Green’s operator for auxiliary “local” problem which consists of a “close” equation and the local conditions n_ix_i = α_i, i = 1,...,n.