The Fucik Spectrum of General Sturm–Liouville Problems

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Abstract:

Consider the boundary value problem-(pu′)′+qu=αu+-u-, in (0, ),c00u(0)+c01u(0)=0, c10u()+c11u′()=0,where u±=max{±u, 0}. The set of points (α)∈ℝ2 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. Copyright 2000 Academic Press.

Document Type: Research Article

Affiliations: Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, Scotland

Publication date: February 1, 2000

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