Existence results of a m-point boundary value problem at resonance

Let xi(i) (0, 1), a(i) is an element of (0, infinity), i = 1,..., m - 2, be given constants satisfying Sigma(i=1)(m-2) a(i) = 1 and 0 < xi(1) < xi(2) < (...) < xi(m-2) < 1. We show the existence of solutions for the in-point boundary value problem x" = f(t, x, x'), t is an element of (0, 1), x'(0) = 0, x(1) = Sigma(i=1)(m-2) a(i)x(xi(1)), where f: [0, 1] x R-2 --> R is continuous and f (t, r(1), 0) less than or equal to 0, f(t, r(2), 0) greater than or equal to 0 for some r(1), r(2) with r(1) < r(2). Our analysis is based on the nonlinear alternative of Leray-Schauder. (C) 2004 Elsevier Inc. All rights reserved.