Sysoiev D. The conditions for the existence of solutions of matrix boundary value problems for systems differential equations

The thesis research is devoted to the study of the problem of finding constructive conditions for the existence and construction of solutions of matrix boundary value problems in the case of parametric resonance, depending on the eigenfunctions of the boundary value problem. The traditional study of periodic and Noetherian boundary-value problems in critical cases was associated with the assumption that the differential equation, as well as the boundary condition, are known and xed. In addition, the study of periodic problems in the case of parametric resonance was associated with the study of stability issues in particular. At the same time, the study of periodic boundary value problems in the case of parametric resonance associated with numerous applications in electronics, plasma theory, nonlinear optics, mechanics, the theory of motion stability, biology and radio engineering, the theory of nonlinear oscillations, the theory of stochastic differential equations, physics, search for minerals and machine tools involves both the construction of solutions to periodic boundary value problems and the calculation of the eigenfunction of the corresponding differential equation. Thus, the main peculiarity of this work is finding constructive conditions of existence and construction of solutions of Noetherian boundary-value problems in the case of parametric resonance, depending on the eigenfunction of the boundary value problem. Finding constructive conditions of existence and construction of solutions of boundary value problems in the case of resonance, depending on a certain function, which ensures the solvability of the boundary value problem, are known to be the characteristic peculiarity of autonomous boundary value problems. A signicant difference of this work is also the matrix record of the unknown, which generalizes the form of both matrix differential equation and the boundary condition. The solvability conditions and the scheme of constructing solutions of a nonlinear Noetherian boundary value problem for a matrix differential equation are found in the thesis research.