Approximation Properties of Solutions to Multipoint Boundary-Value Problems

We consider a broad 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] → ℂm belong to the Sobolev space W1rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left({W}_1^r\right)}^m $$\end{document} and the boundary conditions are given in the form By = q, where B: (C(r−1))m → ℂrm is an arbitrary continuous linear operator. For this problem, we prove that its solution can be approximated in W1rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left({W}_1^r\right)}^m $$\end{document} with arbitrary accuracy by the solutions of multipoint boundaryvalue problems with the same right-hand sides. These multipoint problems are constructed explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates for the errors of solutions in the normed spaces W1rm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left({W}_1^r\right)}^m $$\end{document} and (C(r−1))m.