Matrix boundary value problem in the case of parametric resonance
The study of nonlinear Noetherian matrix boundary value problems for ordinary di erential equations is associated with numerous applications of such problems in the theory of nonlinear oscillations in mechanics, biology, electrical engineering, theory of management, theory of motion stability, particularly in problems associated with di erent cases of the parametric resonance. Research papers of Yu.A Mitropolskii, A.M. Samoilenko, N.A. Perestyuk, A.A. Boichuk, M.I. Ronto, I.G. Malkin, P.A. Proskuryakov, V.A. Yakubovich, V.M. Starzhinsky, D.I. Martynyuk, E.A. Grebenikov, Y.A. Ryabov and other scientists are dedicated to various aspects of the theory of boundary value problems. Research papers of such foreign scientists as G.D. Birkho, G.A. Bliss, R. Conti, J. Hale, W.T. Reid, S. Schwabik, O. Veivoda, D. Wexler, and others are also dedicated to the theory of boundary value problems. These methods are used in the analysis of boundary value problems for various classes of systems: boundary value problems for systems of ordinary di erential equations, matrix, boundary value problems for systems of ordinary di erential equations, autonomous di erential systems, for operator equations in functional spaces. In recent years, considerable attention is paid to the research of boundary value problems, which linear part is not reversible operator, and, in particular, in the case where the number of boundary conditions does not coincide with the dimensionality of solution. Note that in scientific literature this class of boundary value problems has been called Noetherian. The aim of this article is to obtain solvability conditions and solution constructions of Noetherian weakly nonlinear matrix boundary value problems for systems of ordinary di erential equations in the case of parametric resonance, in this case they use original techniques for solving generalized matrix equations of Sylvester with usage of projectors and pseudo inverse (by Moore-Penrose) matrixes and the operator, which leads to a linear algebraic matrix equation of Sylvester to traditional linear algebra system with a rectangular matrix. A generalized method of Green’s operator built for traditional Noetherian boundary value problems for systems of ordinary di erential equations in the works of A.M. Samoilenko and A.A. Boichuk are also used. As opposed to researches of periodic boundary value problems in the case of parametric resonance of V.A. Yakubovich and V. M. Starzhinsky, this article is devoted to the investigation of more general Noetherian matrix boundary value problems for systems of ordinary di erential equations. Obtained solvability conditions and a scheme for constructing solutions of nonlinear Noetherian matrix boundary value problems for systems of di erential equations in the case of parametric resonance generalize similar results, which are presented in papers of A.A. Boichuk and S.A. Krivosheya for periodic matrix boundary value problems for Riccati equation in the absence thereof parametric resonance. In addition, received solvability conditions and a scheme of constructing solutions stipulate the inhomogeneity dependence of the linear part of the matrix boundary value problem, and hence the solutions of the equation for generating constants from a small parameter too. Obtained results are illustrated by the example of a matrix periodic boundary value problem for Riccati equation in the case of parametric resonance.