Heriot-Watt Mathematics Report Series
HWM98-34, 22 Oct 1998
The Fucik spectrum of general Sturm-Liouville problems
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.
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