A boundary value problem for a system of differential equations with finite quantity of transformed arguments in the case of multipoint boundary conditions is considered in this paper.

To investigate the existence and approximate construction of the solution of such boundary value problem it is proposed a traditional scheme of the A.M. Samoilenko’s numerical-analytic method with a determining equation.

A recurrent sequence of functions that depend on parameter, each of which satisfies given boundary conditions, is constructed. It is shown that under typical for numerical-analytic method assumptions, this sequence uniformly convergences to the limit function. It is established the value of the parameter at which the limit function will be an exact solution of the original boundary value problem. Approximate determining function and approximate determining equation put into consideration, and on the basis of them sufficient conditions for the solvability of this boundary value problem are obtained. The necessary conditions for the solvability of the considered boundary value problem and an estimation of the deviation of the approximate solution from the exact solution were also obtained.

The proposed scheme of the numerical-analytic method is illustrated by concrete example.

[1] Samoilenko A.M., Ronto N.I. Numerical-Analytic Methods for the Investigation of Solutions of Boundary-Value Problems. Naukova Dumka, Kiev, 1985. (in Russian)
[2] Samoilenko A.M., Ronto N.I. Numerical-Analytic Methods in the Theory of Boundary-Value Problems for Ordinary Differential Equations. Naukova Dumka, Kiev, 1992. (in Russian)
[3] Samoilenko A.M., Ronto M. Numerical-Analytic Methods in the Theory of Boundary-Value Problems. World Scientific, River Edge, NJ, 2000. DOI: https://doi.org/10.1142/3962
[4] Ronto M.I., Samoilenko A.M., Trofimchuk S.I. The theory of the numerical-analytic method: Achievements and new trends of development. III. Ukr. Math. J. 1998, 50 (7), 960–979. URL: https://umj.imath.kiev.ua/index.php/umj/article/view/4876 (in Russian)
[5] Filipchuk M.P. Two-point boundary value problem for a system with many transformed arguments. Bukovinian Math. J. 2017, 5 (1-2), 139–143. URL: https://tinyurl.com/bmj2017-1and2-17 (in Ukrainian)
[6] Filipchuk M.P. On a two-point boundary value problem for a system of differential equations with many transformed arguments. Bukovinian Math. J. 2021, 9 (1), 284–290. DOI: https://doi.org/10.31861/bmj2021.01.24 (in Ukrainian)
[7] Filipchuk M.P., Filipchuk O.I. On a boundary value problem with integral conditions for a system of differential equations with many transformed arguments. Bukovinian Math. J. 2024, 12 (1), 107–119. DOI: https://doi.org/10.31861/bmj2024.01.10 (in Ukrainian)

[8] Filipchuk M.P., Bigun Ya.I. Numerical-analytic method for the investigation of multipoint boundaryvalue problems for systems of differential equations with transformed argument. Ukr. Math. J. 1998, 50 (11), 1805–1810. DOI: https://doi.org/10.1007/BF02524491 (translation of Ukr. Math. J. 1998, 50 (11), 1581–1585. URL: https://umj.imath.kiev.ua/index.php/umj/article/view/4815 (in Ukrainian)