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We establish necessary and sufficient conditions for the existence of solutions of a nonlinear matrix boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. We construct a convergent iterative scheme for finding approximate solutions of the problem. As an example of application of the proposed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Riccati-type equation with parametric perturbation. To check the accuracy of the obtained approximations, we introduce residuals in the original equation. [ABSTRACT FROM AUTHOR]

We set forth solvability conditions and construction of the generalized Green operator for Noetherian linear boundary value problem for the matrix differential equations and solvability conditions and the constructive scheme for constructing solutions of nonlinear Noetherian boundary value problem for matrix differential equation. We show that the principal results in the theory of weakly nonlinear periodic oscillations remain valid for nonlinear Noetherian boundary value problem for matrix differential equation. The study is illustrated by the periodic problem for a Duffing-type matrix differential equation. [ABSTRACT FROM AUTHOR]

We set forth solvability conditions and construction of the generalized Green's operator for a Noetherian matrix linear differential-algebraic boundary value problem. Sufficient conditions of reducibility of a generalized matrix differential-algebraic operator to a conventional differential-algebraic equation with an unknown column vector are established. To solve a matrix differential-algebraic boundary value problem, we employ a special solvability conditions and the construction of a general solution to a matrix Sylvester-type equation. [ABSTRACT FROM AUTHOR]

Necessary and sufficient conditions are established for the existence of solutions of the nonlinear Noetherian boundary-value problem for a system of differential equations in the case of parametric resonance. We propose a convergent iterative algorithm for finding approximate solutions of the nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the case of parametric resonance. As an example, we apply the constructed iterative algorithm to find approximate solutions of the periodic boundary-value problem for the Duffing-type equation with parametric perturbation. The accuracy of the obtained approximate solutions of the periodic boundary-value problem for the Duffing-type equation is controlled by using deviations from the original equation. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of a weakly nonlinear Noetherian boundary-value problem for a system of ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]