Solvability of functional differential equations with multi-point boundary value problems at resonance

In this paper, we discuss the following third order functional differential equations x'''(t) = f(t, x(t), (Fx)(t), x'(t), (Gx')(t), x ''(t), (Hx '')(t)), t epsilon [0, 1], subject to the boundary conditions x=(0) = 0, x ''(0) = 0, x'(1) = Sigma(m-2)(i=1)alpha ix'(eta(i)), where f : [0, 1] x R-6 -> R, F, G, H are three operators, alpha(i) (i = 1, ..., m - 2) >= 0, 0 < eta(1) < eta(2) < ... < eta(m-2) < 1. Under some appropriate conditions, some existence and multiplicity results are given for the problem at resonance by using a priori estimates and the topological degree theory of Mawhin. (C) 2007 Elsevier Ltd. All rights reserved.