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 ∈ [0, 1], subject to the boundary conditions x(0) = 0, x (0) = 0, x (1) = m−2 ∑ i=1 αi x (ηi ), where f : [0, 1] × R6 −→ R, F,G, H are three operators, αi (i = 1, . . . ,m − 2) ≥ 0, 0 < η1 < η2 < · · · < ηm−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.