Linear and nonlinear, second-order problems with Sturm-Liouville-type, multi-point boundary conditions

We consider the nonlinear equation $$-u^{\mathrm{\prime }\mathrm{\prime }}=f(u)+h,\text{on}(-1,1),$$ where $f:\mathbb{R}\to \mathbb{R}$ and $h : [-1,1] \to {\mathbb R}$ are continuous, together with general Sturm-Liouville type, multi-point boundary conditions at $\pm 1$. We will obtain existence of solutions of this boundary value problem under certain nonresonance' conditions, and also Rabinowitz-type global bifurcation results, which yield nodal solutions of the problem. These results rely on the spectral properties of the eigenvalue problem consisting of the equation $$-u'' = \lambda u, \quad \text{on} \quad (-1,1),$$ together with the multi-point boundary conditions. In a previous paper it was shown that, under certain optimal’ conditions, the basic spectral properties of this eigenvalue problem are similar to those of the standard Sturm-Liouville problem with single-point boundary conditions. In particular, for each integer $k\ge 0$ there exists a unique, simple eigenvalue ${\lambda }_k$, whose eigenfunctions have oscillation count' equal to $k$, where the oscillation count’ was defined in terms of a complicated Pr"ufer angle construction. Unfortunately, it seems to be difficult to apply the Pr"ufer angle construction to the nonlinear problem. Accordingly, in this paper we use alternative, non-optimal, oscillation counting methods to obtain the required spectral properties of the linear problem, and these are then applied to the nonlinear problem to yield the results mentioned above.