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We investigate a weakly nonlinear boundary-value problem for a system of nonlinear ordinary differential equations with discontinuous right-hand side and periodic boundary conditions. A modified iterative scheme is proposed for the construction of approximate solutions of this problem. Unlike our previous results, this scheme is based on the refined expansion of the right-hand side of the analyzed differential equation and a new choice of the initial approximation. The proposed iterative procedure is applied to a mathematical model of nonisothermal chemical reaction with periodic switchings of the control functions. In the considered example, the convergence of our iterative scheme to the required solution is illustrated by the analysis of discrepancies in the boundary-value problem. We establish estimates for the range of a small parameter within which the operator used for the construction of the required solutions is contractive. The results of numerical simulations confirm the fact that the proposed computational scheme guarantees better accuracy than the method of simple iterations. [ABSTRACT FROM AUTHOR]

We use the scheme of the classical least-squares method for the construction of approximate pseudosolutions of a linear matrix boundary-value problem for a system of differential-algebraic equations. [ABSTRACT FROM AUTHOR]

We establish constructive conditions of solvability and an algorithm for the construction of solutions of an autonomous periodic boundary-value problem for a matrix differential equation. [ABSTRACT FROM AUTHOR]

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 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]

We establish constructive conditions for the existence of solutions of an autonomous Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case and develop a modified iterative procedure for finding its solutions. [ABSTRACT FROM AUTHOR]

We investigate the problem of the determination of conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems for systems of ordinary differential equations and the construction of these solutions. We consider the special critical case where the equation for finding the generating solution of a weakly nonlinear Noetherian boundary-value problem turns into an identity. We improve the classification of critical cases and construct an iterative algorithm for finding solutions of weakly nonlinear Noetherian boundary-value problems in the special critical case. [ABSTRACT FROM AUTHOR]

Using the least squares method, we construct a new iterative procedure for finding solutions of a weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case in the form of an expansion of a solution in a generalized Fourier polynomial in the neighborhood of the generating solution. We obtain an estimate for the range of values of the small parameter for which this iterative procedure converges to the required solution. [ABSTRACT FROM AUTHOR]

We establish constructive conditions for the bifurcation of solutions and construct an iteration procedure for finding solutions of a linear Noether boundary-value problem for a system of ordinary differential equations with pulse action in the critical case. We obtain an estimate for the range of values of a small parameter for which the iteration procedure converges. [ABSTRACT FROM AUTHOR]

We establish constructive conditions for the appearance of solutions of a linear Fredholm boundary-value problem for a system of ordinary differential equations in the critical case and propose an iterative procedure for finding these solutions. The range of values of a small parameter for which the indicated iterative procedure is convergent is estimated. [ABSTRACT FROM AUTHOR]

We find an estimate for the range of values of a small parameter for which the convergence of an iterative procedure for the construction of solutions of an autonomous weakly nonlinear Noether boundary-value problem for a system of ordinary differential equations in the critical case is preserved. [ABSTRACT FROM AUTHOR]

We study the problems of finding the conditions for the existence of solutions of a linear Noetherian boundary-value problem for a hybrid system of ordinary differential equations and difference equations and the construction of these solutions. We also propose an algorithm for the construction of solutions of linear Noetherian boundary-value problems for hybrid systems of difference-differential equations in the critical and noncritical cases. [ABSTRACT FROM AUTHOR]