Solution curves of 2m-th order boundary-value problems

Abstract

We consider a boundary-value problem of the form Lu = (λƒ (u), where L is a 2m-th order disconjugate ordinary differential operator (m ≥ 2 is an integer), λ ∈ [0, ∞), and the function ƒ : ℝ → ℝ is C2 and satisfies ƒ(ξ) > 0, ξ ∈ ℝ. Under various convexity or concavity type assumptions on ƒ we show that this problem has a smooth curve, S0, of solutions (λ, u), emanating from (λ, u) = (0, 0), and we describe the shape and asymptotes of S0. All the solutions on S0 are positive and all solutions for which u is stable lie on S0.