## 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^{Δ}(t)]^{Δ} + q(t) u^{σ}(t) = λf(t, u^{σ}(t)), on T, u(a) = u(b) = 0, where λ ∈ ℝ_{+} :=[0, ∞), and f : T × ℝ → ℝ satisfies the conditions f(t, ξ) > 0, (t, ξ) ∈ T × ℝ, f(t, ξ) > f_{ξ} (t, ξ) ξ, (t, ξ) ∈ 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 (λ, u) of (1)-(2), u is positive on T \ {a,b}. In addition, we show that there exists λ_{max} > 0 (possibly λ_{max} = ∞), such that, if 0 ≤ λ < λ_{max} then (1)-(2) has a unique solution u(λ), while if λ ≥ λ max then (1)-(2) has no solution. The value of λ_{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).

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