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.