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 \( {\left({W}_1^r\right)}^m \) 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 \( {\left({W}_1^r\right)}^m \) 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 \( {\left({W}_1^r\right)}^m \) and (C(r−1))m.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, No. 3, pp. 341–353, March, 2021. Ukrainian DOI: 10.37863/umzh.v73i3.6505.
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Murach, A.A., Pelekhata, O.B. & Soldatov, V.O. Approximation Properties of Solutions to Multipoint Boundary-Value Problems. Ukr Math J 73, 399–413 (2021). https://doi.org/10.1007/s11253-021-01951-w
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DOI: https://doi.org/10.1007/s11253-021-01951-w