Approximation properties of multipoint boundary-value problems

We consider a wide class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the normed complex space $(C^{(n)}{)}^m$ of $n\ge r$ times continuously differentiable functions $y:[a,b]\to {\mathbb{C}}^m$. The boundary conditions for these problems are of the most general form $By=q$, where $B$ is an arbitrary continuous linear operator from $(C^{(n)}{)}^m$ to ${\mathbb{C}}^{rm}$. We prove that the solutions to the considered problems can be approximated in $(C^{(n)}{)}^m$ by solutions to some multipoint boundary-value problems. The latter problems do not depend on the right-hand sides of the considered problem and are built explicitly.