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

Let ξi ∈ (0, 1), ai ∈ (0,∞), i = 1, . . . , m − 2, be given constants satisfying m−2 i=1 ai = 1 and 0 < ξ1 < ξ2 < ··· < ξm−2 < 1. We show the existence of solutions for the m-point boundary value problem x = f (t,x,x ), t ∈ (0, 1), x (0) = 0, x(1) = m−2 i=1 aix(ξi ), where f : [0, 1] × R2 →R is continuous and f (t,r1, 0) 0, f (t,r2, 0) 0 for some r1, r2 with r1 < r2. Our analysis is based on the nonlinear alternative of Leray–Schauder.  2004 Elsevier Inc. All rights reserved