On the positivity of the Green function of linear boundary value problems for fourth-order equations on a graph (Q1588999)

Let \(\{\gamma_i\}^m_1\) be a set of rectilinear intervals in the space \(\mathbb{R}^n\). Let the graph \(\Gamma\) consist of the union of all edges \(\gamma_i\) (augmented by some additional points). Consider the ordinary differential equation \((p(x)u'')''= f(x)\) (subject to certain conditions on some points in \(\Gamma\)). Then the authors prove that the Green function \(G(x,s)\) is strictly positive on \(\Gamma\times\Gamma\).