### 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 |
---|---|

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 |

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### Keywords

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

### Cite this

*Journal of Mathematical Analysis and Applications*,

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

}

*Journal of Mathematical Analysis and Applications*, vol. 300, no. 2, pp. 491-504. https://doi.org/10.1016/j.jmaa.2004.07.012

**Curves of positive solutions of boundary value problems on time-scales.** / Davidson, Fordyce A.; Rynne, Bryan P.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Curves of positive solutions of boundary value problems on time-scales

AU - Davidson, Fordyce A.

AU - Rynne, Bryan P.

PY - 2004/12/15

Y1 - 2004/12/15

N2 - 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) us(t) = ?f(t, us(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.

AB - 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) us(t) = ?f(t, us(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.

KW - Nonlinear boundary value problem

KW - Positive solutions

KW - Strong maximum principle

KW - Time-scales

KW - Weighted eigenvalue problem

UR - http://www.scopus.com/inward/record.url?scp=8644248331&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2004.07.012

DO - 10.1016/j.jmaa.2004.07.012

M3 - Article

VL - 300

SP - 491

EP - 504

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -