Positive solutions for nonhomogeneous m-point boundary value problems

Let a is an element of C[0, 1], b is an element of C([0, 1], (-infinity, 0]). Let d is an element of R and d > 0. Let phi(1) (t) be the unique solution of the linear boundary value problem u"(t) + a(t)u'(t) + b(t)u(t) = 0, t is an element of (0, 1), u(0) = 0, u(1) = 1. We study the existence of positive solutions for the m-point boundary value problem u" + a(t)u' + b(t)u + h(t)f (u) = 0, u(0) = 0, u(1) - (i=1)Sigma(m-2) alpha(i)u(xi(i)) = d, where xi(i) is an element of (0, 1) and alpha(i) is an element of (0, infinity) (for i is an element of {1,..., m - 2}) are given constants satisfying (i=1)Sigma(m-2) alpha(i) x phi(1)(xi(i)) < 1. Under suitable conditions, we show that there exists a positive number d* such that the problem has at least one solution for 0 < d < d* and no solution for d > d*. (C) 2004 Elsevier Ltd. All rights reserved.