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

Let T ⊂ R be a time-scale, with a = infT, b = supT. We consider the nonlinear boundary value problem − p(t)uΔ(t ) Δ + q(t)uσ (t ) = λf t,uσ (t ) , on T, (1) u(a) = u(b) = 0, (2) 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 ×R+. 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 λmax 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).  2004 Elsevier Inc. All rights reserved