Semipositone m-point boundary-value problems

Abstract

We study the m-point nonlinear boundary-value problem -[p(t)u' (t)]' = λƒ (t, u(t)), 0 < t < 1, u'(0) = 0, ∑m-2i=1 αiu(ηi) = u(1), where 0 < η1 < η2 < ··· < ηm-2 < 1, αi > 0 for 1 ≤ i ≤ m - 2 and ∑m-2i=1 αi < 1, m ≥ 3. We assume that p(t) is non-increasing continuously differentiable on (0, 1) and p(t) > 0 on [0, 1]. Using a cone-theoretic approach we provide sufficient conditions on continuous ƒ(t, u) under which the problem admits a positive solution.