We consider the nonlinear boundary value problem consisting of the equation \tag{ 1} -u" = f(u) + h, \quad \text{ a.e. on $(-1,1)$,} 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, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{ m^\pm}$, $\eta^\pm \in (-1,1)^{ m^\pm}$, and we suppose that $$ \sum_{ i=1}^{ m^\pm} \alpha_i^\pm < 1 . $$ We also suppose that $f : \mathbb{ R} \to \mathbb{ R}$ is continuous, and $$ 0 < f_{ \pm\infty}:=\lim_{ s \to \pm\infty} \frac{ f(s)}{ s} < \infty. $$ We allow $f_{ \infty} \ne f_{ -\infty}$ --- such a nonlinearity $f$ is { \em jumping}. Related to (1) is the equation \tag{ 3} -u" = \lambda(a u^+ - b u^-), \quad \text{ on $(-1,1)$,} 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.
Categories: math.CA