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

Fordyce A. Davidson (Lead / Corresponding author), Bryan P. Rynne

    Research output: Contribution to journalArticlepeer-review

    5 Citations (Scopus)

    Abstract

    Let T ⊂ ℝ be a time-scale, with a = inf T, b = sup T. We consider the nonlinear boundary value problem - [p(t)uΔ(t)]Δ + q(t) uσ(t) = λf(t, uσ(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) > fξ (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 λmax > 0 (possibly λmax = ∞), such that, if 0 ≤ λ < λmax then (1)-(2) has a unique solution u(λ), while if λ ≥ λ max then (1)-(2) has no solution. The value of λmax 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 languageEnglish
    Pages (from-to)491-504
    Number of pages14
    JournalJournal of Mathematical Analysis and Applications
    Volume300
    Issue number2
    DOIs
    Publication statusPublished - 15 Dec 2004

    Keywords

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

    Fingerprint Dive into the research topics of 'Curves of positive solutions of boundary value problems on time-scales'. Together they form a unique fingerprint.

    Cite this