### Abstract

We consider the Sturm-Liouville boundary value problem-(p(x)u'(x))'+q(x)u(x)=f(x, u(x))+h(x), x?(0, p),c_{00}u(0)+c_{01}u'(0)=0, c_{10}u(p)+c_{11}u'(p)=0,where p?C^{1}([0, p]), q?C^{0}([0, p]), with p(x)>0, x?[0, p], c^{2}_{i0}+c^{2}_{i1}>0, i=0, 1, h?L^{2}(0, p), and f:[0, p]×R?R is a Carathéodory function. We assume that the rate of growth of f(x, ?) is at most linear as ??8, but the asymptotic behaviour may be different as ??±8, so the non-linearity is termed "jumping." Conditions for existence of solutions of this problem are usually expressed in terms of "non-resonance" with respect to the standard Fucík spectrum. In this paper we give conditions for both existence and non-existence of solutions in terms of a slightly different idea of the spectrum. These conditions extend the usual Fucík spectrum conditions. © 2001 Academic Press.

Original language | English |
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Pages (from-to) | 215-227 |

Number of pages | 13 |

Journal | Journal of Differential Equations |

Volume | 170 |

Issue number | 1 |

DOIs | |

Publication status | Published - 10 Feb 2001 |