Summary: We provide sufficient conditions for the existence of solutions to multipoint boundary value problems for nonlinear ordinary differential equations. We consider the case where the solution space of the associated linear homogeneous boundary value problem is less than 2. When this solution space is trivial, we provide criteria for the existence of solutions to boundary value problems of the form
\[ y^{(n)}(t)+a_{n-1}(t)y^{(n-1)}(t)+\cdots+a_0(t)y(t)=g(y(t)); \quad 0\leq t\leq 1 \]
subject to
\[ \sum^n_{j=1}b_{ij}(0)y^{(j-1)}(0)+\sum^n_{j=1}b_{ij}(1)y^{(j-1)}(t_1)+\cdots+\sum^n_{j=1}b_{ij}(N)y^{(j-1)}(t_N)=0 \]
for \(i=1,2,\dots, n\), and establish existence results via Schauder’s fixed point theorem. In the resonance case, we use a projection scheme to provide criteria for the solvability of our nonlinear boundary value problem. We accomplish this by analyzing a link between the behavior of the nonlinearity and the solution set of the associated linear homogeneous boundary value problem.