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.