### Abstract

Let T ? R be a time-scale, with a = inf T, b = sup T. We consider the nonlinear boundary value problem - [p(t)u^{?}(t)]^{?} + q(t) u^{s}(t) = ?f(t, u^{s}(t)), on T, u(a) = u(b) = 0, where ? ? R_{+} :=[0, 8), and f : T × R ? R satisfies the conditions f(t, ?) > 0, (t, ?) ? T × R, f(t, ?) > f_{?} (t, ?) ?, (t, ?) ? T × RM^{+}. We prove a strong maximum principle for the linear operator defined by the left-hand side of (1), and use this to show that for every solution (?, u) of (1)-(2), u is positive on T \ {a,b}. In addition, we show that there exists ?_{max} > 0 (possibly ?_{max} = 8), such that, if 0 = ? < ?_{max} then (1)-(2) has a unique solution u(?), while if ? = ? max then (1)-(2) has no solution. The value of ?_{max} is characterised as the principal eigenvalue of an associated weighted eigenvalue problem (in this regard, we prove a general existence result for such eigenvalues for problems with general, nonnegative weights). © 2004 Elsevier Inc. All rights reserved.

Original language | English |
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Pages (from-to) | 491-504 |

Number of pages | 14 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 300 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Dec 2004 |

### Keywords

- Nonlinear boundary value problem
- Positive solutions
- Strong maximum principle
- Time-scales
- Weighted eigenvalue problem

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## Cite this

*Journal of Mathematical Analysis and Applications*,

*300*(2), 491-504. https://doi.org/10.1016/j.jmaa.2004.07.012