The Fucik Spectrum of General Sturm–Liouville Problems

Abstract

Consider the boundary value problem$\text{−(pu′)′+qu=αu}{}^{\mathrm{+}}\text{−βu}{}^{\mathrm{-}}\text{,}\text{in}\text{(0, π),}$$\text{c}{}_{00}\text{u(0)+c}{}_{01}\text{u′(0)=0,}\text{c}{}_{10}\text{u(π)+c}{}_{11}\text{u′(π)=0,}$ where u^±=max{±u, 0}. The set of points (α, β)∈$\text{R}$² for which this problem has a non-trivial solution is called the Fucik spectrum. When p≡1, q≡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.