Condition for solvability of a boundary value problem and bifurcation of its solution

Abstract

In this paper, we study the solvability of a linear inhomogeneous boundary value problem with perturbations for a system of second-order ordinary differential equations whose coefficients are real, continuous, and continuously differentiable on a segment. It is known that the boundary value problem considered in this paper is not always solvable, provided that the boundary value problem generating it for a system of second-order ordinary differential equations has no solutions for arbitrary inhomogeneities. The relationship between the considered linear inhomogeneous boundary value problem with perturbation for a system of second-order ordinary differential equations and an algebraic system is established. The coefficients of an algebraic system consist of the coefficients of a linear inhomogeneous boundary value problem with perturbation for a system of second-order ordinary differential equations. Based on the relationship between the boundary value problem under consideration and the algebraic system, a condition for the solvability of a linear inhomogeneous boundary value problem with perturbation for a system of second-order ordinary differential equations is found. It turned out that if this solvability condition is met, there is at least one solution of a linear inhomogeneous boundary value problem with perturbation for a system of second-order ordinary differential equations, which has the form of a partial sum of a convergent Laurent series.

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Published
2020-12-30
How to Cite
STANZHYTSKYI, O. M.; SHOVKOPLYAS, T. V.. Condition for solvability of a boundary value problem and bifurcation of its solution. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 108, n. 4, p. 3-17, dec. 2020. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/815>. Date accessed: 17 jan. 2021. doi: https://doi.org/10.26577/JMMCS.2020.v108.i4.01.
Keywords boundary problem with perturbation, generated boundary problem, the criterion of solvability, the critical case, the bifurcation of solution, algebraical system