Solvability for Some Higher Order Multi-Point Boundary Value Problems at Resonance

In this paper, by using the Mawhin’s continuation theorem, we obtain an existence theorem for some higher order multi-point boundary value problems at resonance in the following form: $$begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),ldots,x^{(n-1)}(t))+e(t), tin(0,1),x^{(i)}(0) = 0, i=0,1,ldots,n-1, ineq p, x^{(k)}(1) = sumlimits_{j=1}^{m-2}{beta_j}x^{(k)}(eta_j),end{array}$$ where ${f:[0,1]times mathbb{R}^n to mathbb{R}=(-infty,+infty)}$ is a continuous function, ${e(t)in L^1[0,1], p, kin{0,1,ldots,n-1}}$ are fixed, m ≥ 3 for p ≤ k (m ≥ 4 for p > k), ${beta_j in mathbb{R}, j=1,2,ldots,m-2, 0 < eta_1 < eta_2 < cdots < eta_{m-2} <1 }$ . We give an example to demonstrate our results.