Апроксимативні властивості розв’язків багатоточкових крайових задач

A. A. Murach; O. B. Pelekhata; V. O. Soldatov

doi:10.37863/umzh.v73i3.6505

A. A. Murach Institute of Mathematics, NAS of Ukraine
O. B. Pelekhata National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”
V. O. Soldatov Institute of Mathematics, NAS of Ukraine https://orcid.org/0000-0001-7496-5524

DOI: https://doi.org/10.37863/umzh.v73i3.6505

Abstract

UDC 517.927

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$ known as the general boundary-value problems.
Their solutions $y\colon[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m,$ and the boundary conditions are given in the form $By=q$ where $B\colon(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator.
For such a problem, we prove that its solution can be approximated in $(W_1^{r})^m$ with arbitrary precision 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 error of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}.$

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Approximation properties of solutions to multipoint boundary-value problems

Abstract

References