Half-eigenvalues of periodic Sturm-Liouville problems

Paul A. Binding, Bryan P. Rynne

Research output: Contribution to journalArticle

16 Citations (Scopus)

Abstract

We consider the nonlinear Sturm-Liouville problem -(pu')'+qu=au+-bu-+?u, in (0,2p), u(0)=u(2p), (pu)'(0)=(pu)'(2p), where 1/p, q?L1(0,2p), with p>0 a.e. on (0,2p), a, b?L1(0,2p), ? is a real parameter, and u±(t)=max{±u(t),0} for t?[0,2p]. Values of ? for which (1)-(2) has a non-trivial solution u will be called half-eigenvalues while the corresponding solutions u will be called half-eigenfunctions. The set of half-eigenvalues will be denoted by SH. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties associated with SH. These properties yield results on the existence and non-existence of solutions of the problem -(pu')'+qu=f(t,u)+h, in (0,2p) (together with (2)), where h?L1(0,2p), f:[0,2p]× R?R is a Carathéodory function, and the limits a(t):=lim??8 f(t,?)/?, b(t):=lim??-8 f(t,?)/?, exist for a.e. t?[0,2p]. When the functions a and b are constant the set SH is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2004 Elsevier Inc. All rights reserved.

Original languageEnglish
Pages (from-to)280-305
Number of pages26
JournalJournal of Differential Equations
Volume206
Issue number2
DOIs
Publication statusPublished - 15 Nov 2004

Keywords

  • Fučík spectrum
  • Half-eigenvalues
  • Jumping nonlinearity
  • Perodic Sturm-Liouville problems

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