On estimates for the Green's function for a multipoint boundary problem

We obtain new estimates for the Green's function $G(t,\,s)$ for a boundary problem of the Vallée-Poussin type: under certain hypotheses we prove the existence of non-negative functions $g(t)$, $h(t)$, $u(t)$ such that $g(t)h(s)\le|G(t,s)|\le g(t)$ and $|G(t,s)|\ge u(t)\max\limits_\tau|G(\tau,s)|$, where $h(t)$ and $u(t)$ are positive on sets of positive measure. These estimates allow us to apply effectively the methods of the theory of cones to investigate non-linear boundary problems.