Solutions to a second-order multi-point boundary value problem at resonance

This article deals with the following second-order multi-point boundary value problem x''(t)=f(t,x(t),x'(t))+e(t),   t∈(0,1),x''(t)=f(t,x(t),x'(t))+e(t),   t∈(0,1),x'(0)=∑i=1mαix'(ξ1),   x(1)=∑j=1nβjx(ηj). Under the resonance conditions ∑t=1mαi=1,∑j=1nβj=1,∑j=1nβjηj=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.