关键词: coincidence degree, multi-point boundary value problem, resonance

Abstract:

This article deals with the following second-order multi-point boundary value problem
$$x''(t)=f(t, x(t), x'(t))+e(t), \ \ \ t\in (0,1), $$
$$x'(0)=\sum\limits_{i=1}^{m}\alpha_{i}x'(\xi_{i}), \ \ \ x(1)=\sum\limits_{j=1}^{n}\beta_{j}x(\eta_{j}). $$
Under the resonance conditions $\sum\limits_{i=1}^{m}\alpha_{i}=1, sum\limits_{j=1}^{n}\beta_{j}=1, \sum\limits_{j=1}^{n}\beta_{j}\eta_{j}=1$ , by applying the coincidence degree theory, some existence results of the problem are established. The emphasis here is that the dimension
of the linear operator is two. In this paper, we extend and improve some previously known results like the ones in the references.

Key words: coincidence degree, multi-point boundary value problem, resonance