Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad $\text{a.e. on }(-1,1)\text{,}$ where $h\in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm

1. = \sum^{m^\pm}{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^{\pm }\ge 1$ are integers, ${\alpha }^{\pm }=({\alpha }_1^{\pm },\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_m^{\pm })\in [0,1{)}^{m^{\pm }}$, ${\eta }^{\pm }\in (-1,1{)}^{m^{\pm }}$, and we suppose that $$\sum _{i=1}^{m^{\pm }}{\alpha }_i^{\pm }<1.$$ We also suppose that $f : $\mathbb{R}$ \to $\mathbb{R}$$iscontinuous,and$$ 0 < f{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty. $$Weallow$f_{\infty} \ne f_{-\infty}$ — such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad $\text{on }(-1,1)\text{,}$ where $\lambda ,a,b>0$, and $u^{\pm}(x) =\max{\pm u(x),0}$for$x \in [-1,1]$. The problem (2)-(3) is positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the Fucik spectrum’ of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach.