The Fucik spectrum of general Sturm-Liouville problems
type: Item
instance of: scholarly article
title: The Fucik spectrum of general Sturm-Liouville problems
zbMATH Open document ID: 0976.34024
author: Bryan P. Rynne
published in: Journal of Differential Equations
publication date: 2001-10-24T00:00:00Z
review text: The author studies the boundary value problem consisting of the Sturm-Liouville-like equation \[ -(pu')'+ qu= \alpha u^+ -\beta u^-\text{ on }[0,\pi] \] and a separated real boundary condition such as the Dirichlet boundary condition \(u(0)=0 = u(\pi)\), with \(p \in C^1([0,\pi],\mathbb{R})\), \(p > 0\) on \([0,\pi]\), \(q\in C^0([0,\pi],\mathbb{R})\), \(\alpha,\beta\in \mathbb{R}\), and \(u^\pm(x) = \max\{\pm u(x),0\}\). The set of points \((\alpha,\beta)\in\mathbb{R}^2\) for which this problem has a nontrivial solution is called the Fucik spectrum. This set is of importance in the study of nonlinear boundary value problems with jumping nonlinearities. When \(p\equiv 1\), \(q\equiv 0,\) and either the Dirichlet or the periodic boundary condition is imposed, the Fucik spectrum is explicitly known, and consists of a countable collection of curves in \(\mathbb{R}^2\). Here the author obtains various geometric properties of the general problem above. In particular, it is shown that the spectrum is always a countable collection of curves in \(\mathbb{R}^2\), and the asymptotic behaviour of these curves is determined using certain eigenvalues for the associated usual Sturm-Liouville equation \[ -(pv')'+qv=\lambda v\text{ on }[0,\pi]. \] A generalized Fucik spectrum is considered and is shown to have similar geometric properties. Some spectral-type properties of a positively homogeneous ``half-linear'' problem are also discussed via similar methods. These results are then used to give necessary and sufficient conditions for the solvability of a nonlinear problem, with jumping nonlinearity, in terms of the location of the ``half-eigenvalues'' of the associated half-linear problem.
reviewed by: Hongyou Wu
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 1437347
zbMATH Keywords: Fucik spectrum
zbMATH Keywords: jump nonlinearities
zbMATH Keywords: solvability of nonlinear problems
MaRDI profile type: Publication
full work available at URL: jdeq.1999.3661
OpenAlex ID: W2017014157
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DOI: 10.1006/JDEQ.1999.3661
description: scientific article; zbMATH DE number 1437347