Positive solutions of nonlinear three-point boundary-value problems

Let a ∈ C[0, 1], b ∈ C([0, 1], (−∞, 0]). Let φ1(t ) be the unique solution of the linear boundary value problem u (t )+ a(t)u (t )+b(t)u(t) = 0, t∈ (0, 1), u(0) = 0, u(1) = 1. We study the existence of positive solutions to the nonlinear boundary-value problem u (t )+ a(t)u (t )+b(t)u(t) +h(t)f (u) = 0, t∈ (0, 1), u(0) = 0, αu(η)= u(1), where 0 < η < 1 and 0 < αφ1(η) < 1 are given, h ∈ C([0, 1], [0,∞)) satisfying that there exists x0 ∈ [0, 1] such that h(x0) > 0, and f ∈ C([0,∞), [0,∞)). We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones.  2003 Elsevier Science (USA). All rights reserved.