### 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?L_{1}(0,2p), with p>0 a.e. on (0,2p), a, b?L_{1}(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 S_{H}. 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 S_{H}. 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?L_{1}(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 S_{H} 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 language | English |
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Pages (from-to) | 280-305 |

Number of pages | 26 |

Journal | Journal of Differential Equations |

Volume | 206 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Nov 2004 |

### Keywords

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

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## Cite this

*Journal of Differential Equations*,

*206*(2), 280-305. https://doi.org/10.1016/j.jde.2004.05.014