Approximation properties of solutions to multipoint boundary-value problems

Abstract

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to {\mathbb{C}}^m$ belong to the Sobolev space $(W_1^r{)}^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)}{)}^m\to {\mathbb{C}}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^r{)}^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^r{)}^m$ and $(C^{(r-1)}{)}^m$.

13 pages, revised version, Russian

References in corpus (1)

• A criterion for continuity in a parameter of solutions to generic boundary-value problems for higher-order differential systems