Well-posedness criteria for one family of boundary value problems

Шеттiк есептiң бiр үйiрiнiң қисынды шешiмдiлiк критерийлерi туралы
О критериях корректной разрешимости одного семейства краевых задач
Abdimanapova P.B. Temesheva S.M.
2023 E.A. Buketov Karaganda University Publish house

Bulletin of the Karaganda University. Mathematics Series
2023 #112 Issue 4 5 - 20 pp.

This paper considers a family of linear two-point boundary value problems for systems of ordinary differential equations. The questions of existence of its solutions are investigated and methods of finding approximate solutions are proposed. Sufficient conditions for the existence of a family of linear two-point boundary value problems for systems of ordinary differential equations are established. The uniqueness of the solution of the problem under consideration is proved. Algorithms for finding an approximate solution based on modified of the algorithms of the D.S. Dzhumabaev parameterization method are proposed and their convergence is proved. According to the scheme of the parameterization method, the problem is transformed into an equivalent family of multipoint boundary value problems for systems of differential equations. By introducing new unknown functions we reduce the problem under study to an equivalent problem, a Volterra integral equation of the second kind. Sufficient conditions of feasibility and convergence of the proposed algorithm are established, which also ensure the existence of a unique solution of the family of boundary value problems with parameters. Necessary and sufficient conditions for the well-posedness of the family of linear boundary value problems for the system of ordinary differential equations are obtained.

existence of solution , Family of linear boundary value problems , multipoint boundary value problem , necessary and sufficient condition , singular solution , well-posedness

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Almaty Technological University, Almaty, Kazakhstan
Al-Farabi Kazakh National University, Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

Almaty Technological University
Al-Farabi Kazakh National University

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