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

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)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).

AB - 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).

KW - Nonlinear boundary value problem

KW - Positive solutions

KW - Strong maximum principle

KW - Time-scales

KW - Weighted eigenvalue problem

UR - https://www.scopus.com/pages/publications/8644248331

U2 - 10.1016/j.jmaa.2004.07.012

DO - 10.1016/j.jmaa.2004.07.012

M3 - Article

AN - SCOPUS:8644248331

SN - 0022-247X

VL - 300

SP - 491

EP - 504

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -