Solvability of a higher-order multi-point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we get a new general existence result for the following higher-order multi-point boundary value problem at resonance $$\begin{gathered} x^{(n)} (t) = f(t,x(t),x'(t),...,x^{(n - 1)} (t)),t \in (0,1), \hfill \\ x(0) = \sum\limits_{i = 1}^m {a_i x(\xi _i ),x'(0) = ... = x^{(n - 2)} (0) = 0,x^{(n - 1)} (1) = } \sum\limits_{j = 1}^l {\beta _j x^{(n - 1)} (\eta _j )} , \hfill \\ \end{gathered} $$ where f: [0, 1] × ℝ n → ℝ is a Caratheodory function, 0 < ξ 1 < ξ 2 < … < ξ m < 1, α i ∈ ℝ, i = 1, 2, …, m, m ≥ 2 and 0 < η 1 < … < η l < 1, β j ∈ ℝ, j = 1, …, l, l ≥ 1. In this paper, two of the boundary value conditions are responsible for resonance.