SOLVABILITY OF A BOUNDARY VALUE PROBLEM OF THE SAMARSKY-IONKIN TYPE FOR DIFFERENTIAL EQUATIONS WITH INVOLUTION

In this paper, we investigate a non-local boundary value problem for a second-order differential equation with an involutive transformation. The aim of this work is to apply the parametrization method developed by Professor D. Dzhumabayev to study the solvability of non-local boundary value problems in the context of differential equations with involutive transformations. It is known that the Cauchy problem for such equations may not have a unique solution. Therefore, parameters μ₁= y(½), μ₂= y′(½) are introduced, and variables y(x)=u(x)+μ₁+μ₂(x-1/2) are replaced. The parameter values are determined at the midpoint of the interval, which guarantees the existence of a unique solution for the Cauchy problem of the original equation. The performed variable substitution formally divides the problem into two components: the Cauchy problem for the initial equation and a system of linear equations with respect to the introduced parameters. By solving the Cauchy problem and substituting its solution into the boundary conditions, one can obtain a system of linear equations with respect to the parameters. If the matrix of this system is invertible, then the problem has a unique solution. In the case when the matrix is non-invertible, two scenarios are possible: either the boundary value problem is unsolvable, or it has multiple solutions. For the second case, the paper defines the eigenvalues and solvability conditions of the boundary value problem.

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