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

Abstract

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: $$\begin{array}{lll}x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))+e(t),\ t\in(0,1),\\x^{(i)}(0) = 0, i=0,1,\ldots,n-1,\ i\neq p, \\x^{(k)}(1) = \sum\limits_{j=1}^{m-2}{\beta_j}x^{(k)}(\eta_j),\end{array}$$ where $${f:[0,1]\times \mathbb{R}^n \to \mathbb{R}=(-\infty,+\infty)}$$ is a continuous function, $${e(t)\in L^1[0,1], p, k\in\{0,1,\ldots,n-1\}}$$ 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.