Second order, multi-point problems with variable coefficients

Abstract

In this paper we consider the eigenvalue problem consisting of the equation -u" = \la r u, \quad \text{on $(-1,1)$}, where $r\in C^1[-1,1],r>0$ and $\la \in \R$, together with the multi-point boundary conditions u(\pm 1) = \sum^{m^\pm}_{i=1} \al^\pm_i u(η^\pm_i), where $m^{\pm }\ge 1$ are integers, and, for $i=1,...,m^{\pm }$, $\al_i^\pm \in \R$, ${\eta }_i^{\pm }\in [-1,1]$, with ${\eta }_i^{+}\ne 1$, ${\eta }_i^{-}\ne -1$. We show that if the coefficients $\al_i^\pm \in \R$ are sufficiently small (depending on $r$) then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients $\al_i^\pm$ are not sufficiently small then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ($r\equiv 1$), but the variable coefficient case has not been considered previously (apart from the existence of `principal' eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for existence of general solutions and also of nodal solutions --- these results rely on the spectral properties of the linear problem.