  We consider the nonlinear equation $$-u'' = f(u) + h , \quad \text{on} \quad
(-1,1),$$ 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
$k \geq 0$ 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.
