### B P Rynne

#### Abstract

Consider the boundary value problem \begin{gather*} -(p u')' +q u = \alpha u^+ - \beta u^-, \quad \mbox{in } (0,\pi),\\ a_0 u(0) + b_0 u'(0) = 0, \quad a_1 u(\pi) + b_1 u'(\pi) = 0, \end{gather*} where $u^{\pm} = \max\{\pm u,0\}$. The set of points $(\alpha,\beta) \in \R^2$ for which this problem has a non-trivial solution is called the Fucik spectrum. When $p \equiv 1$, $q \equiv 0$, and either Dirichlet or periodic boundary conditions are imposed, the Fucik spectrum is known explicitly, and consists of a countable collection of curves, with certain geometric properties. In this paper we show that similar properties hold for the general problem above, and also for a further generalization of the Fucik spectrum. We also discuss some spectral type properties of a positively homogeneous, `half-linear' problem, and use these results to consider the solvability of a nonlinear problem with jumping nonlinearities.