Positive solutions for second-order three-point boundary value problems

We establish the existence of positive solutions for the three-point bounda ry value problem u" + a(t)f(u) = 0, u(0) = 0, u(1) - alphau(eta) = b, where b, alpha > 0, eta is an element of (0, 1), alpha eta < 1, are given. Under suitable conditions, we show that there exists a positive number b* s uch that the problem has at least one solution for 0 < b < b* and no soluti on for b > b*. (C) 2000 Elsevier Science Ltd. All rights reserved.