### Abstract

We consider the m-point boundary value problem consisting of the equation - u^{?} = f (u), on (0, 1), where f : R ? R is C^{1}, with f (0) = 0, together with the boundary conditions u (0) = 0, u (1) = underover(?, i = 1, m - 2) a_{i} u (?_{i}), where m = 3, ?_{i} ? (0, 1) and a_{i} > 0 for i = 1, ..., m - 2, with underover(?, i = 1, m - 2) a_{i} < 1 . We first show that the spectral properties of the linearisation of this problem are similar to the well-known properties of the standard Sturm-Liouville problem with separated boundary conditions (with a minor modification to deal with the multi-point boundary condition). These spectral properties are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the above problem. Finally, we use the global bifurcation theorem to obtain nodal solutions (that is, sign-changing solutions with a specified number of zeros) of the above problem, under various conditions on the asymptotic behaviour of f. © 2006 Elsevier Ltd. All rights reserved.

Original language | English |
---|---|

Pages (from-to) | 3318-3327 |

Number of pages | 10 |

Journal | Nonlinear Analysis: Theory, Methods and Applications |

Volume | 67 |

Issue number | 12 |

DOIs | |

Publication status | Published - 15 Dec 2007 |

### Keywords

- Eigenvalues
- Multi-point boundary value problem
- Nonlinear boundary value problem