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

Fordyce A. Davidson, Bryan P. Rynne

Research output: Contribution to journalArticle

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) 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.

Original languageEnglish
Pages (from-to)491-504
Number of pages14
JournalJournal of Mathematical Analysis and Applications
Volume300
Issue number2
DOIs
Publication statusPublished - 15 Dec 2004

Fingerprint

Strong Maximum Principle
Principal Eigenvalue
Nonlinear Boundary Value Problems
Unique Solution
Linear Operator
Eigenvalue Problem
Existence Results
Positive Solution
Time Scales
Non-negative
Boundary Value Problem
Eigenvalue
Curve

Keywords

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

Cite this

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title = "Curves of positive solutions of boundary value problems on time-scales",
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) 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). {\circledC} 2004 Elsevier Inc. All rights reserved.",
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Curves of positive solutions of boundary value problems on time-scales. / Davidson, Fordyce A.; Rynne, Bryan P.

In: Journal of Mathematical Analysis and Applications, Vol. 300, No. 2, 15.12.2004, p. 491-504.

Research output: Contribution to journalArticle

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