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We study the problem of determination of conditions for the existence of solutions to weakly nonlinear periodic boundary-value problems for systems of ordinary differential equations with switchings and construct these solutions. The critical case where the equation for generating amplitudes of weakly nonlinear periodic boundary-value problem with switchings does not turn into the identity is analyzed. We improve the classification of critical and noncritical cases and construct an iterative algorithm for finding solutions of the weakly nonlinear periodic boundary-value problem with switchings in the critical case on the basis of the generalized Newton–Kantorovich theorem. We investigate the case of a nonlinear equation whose dimension does not coincide with the dimension of the unknown in the case where the rank of Jacobian of the nonlinear equation is complete. As an example of application of the constructed iterative scheme based on the generalized Newton–Kantorovich theorem, we obtain approximations to the solutions of periodic boundary-value problem for the mathematical model of nonisothermal chemical reaction. To check the accuracy of the obtained approximations, we evaluate the discrepancies in the original equation. We establish an estimate for the length of the range of a small parameter, where the iterative procedure aimed at the construction of solutions to the weakly nonlinear periodic boundary-value problem with switchings in the mathematical model of nonisothermal chemical reaction is convergent. To this end, we use the conditions of convergence, in particular, the requirement of contraction for the operator used to obtain the required solution of the original problem under the assumption of applicability of the generalized Newton–Kantorovich theorem. [ABSTRACT FROM AUTHOR]

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]

For the nonlinear periodic boundary-value problem posed for an ordinary differential equation in the critical and noncritical cases, we obtain constructive conditions of its solvability and propose a scheme for finding its solutions by using the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

We establish constructive conditions of solvability of linear differential-algebraic boundary-value problems with impulsive action and propose a scheme for the construction of solutions of these problems. [ABSTRACT FROM AUTHOR]

We establish constructive necessary and sufficient conditions of solvability and propose a scheme for the construction of solutions to a nonlinear boundary-value problem unsolved with respect to the derivative. We also suggest convergent iterative schemes for finding approximate solutions of this problem. As an example of application of the proposed iterative scheme, we find approximations to the solutions of periodic boundary-value problems for a Rayleigh-type equation unsolved with respect to the derivative, in particular, in the case of a periodic problem for the equation used to describe the motion of satellites on elliptic orbits. [ABSTRACT FROM AUTHOR]

The investigation of differential-algebraic boundary-value problems was originated in the works by Weierstrass, Luzin, and Gantmakher. The works by Campbell, Boyarintsev, Chistyakov, Samoilenko, Perestyuk, Yakovets, Boichuk, Ilchmann, and Reis were devoted to the systematic study of differential-algebraic boundary-value problems. At the same time, the investigation of differential-algebraic boundary-value problems is closely connected with the analysis of linear boundary-value problems for ordinary differential equations originated in the works by Poincaré, Lyapunov, Krylov, Bogolyubov, Malkin, Myshkis, Grebenikov, Ryabov, Mitropolskii, Kiguradze, Samoilenko, Perestyuk, and Boichuk. The investigation of linear differential-algebraic boundary-value problems is connected with numerous applications of the corresponding mathematical models to the theory of nonlinear oscillations, mechanics, biology, radio-engineering, and the theory of stability of motion. Thus, the problem of generalization of the results obtained by Campbell, Samoilenko, and Boichuk to the case of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative is quite actual. In particular, this is true for the problem of finding necessary and sufficient conditions for the existence of solutions of nonlinear integrodifferential boundary-value problems unsolved with respect to the derivative. In the present paper, we establish conditions for the existence of solutions of the nonlinear integrodifferential boundary-value problem unsolved with respect to the derivative and present a constructive scheme for their finding. [ABSTRACT FROM AUTHOR]

The investigations of linear differential-algebraic boundary-value problems are closely connected with extensive applications of the corresponding mathematical models in the theory of nonlinear oscillations, mechanics, biology, radio-engineering, and the theory of stability of motion. Thus, the problem of generalization of the results obtained by S. Campbell, A. M. Samoilenko, and O. A. Boichuk to the case of nonlinear boundary-value problems unsolved with respect to the derivative seems to be quite urgent. In particular, this is true for finding necessary and sufficient conditions for the existence of the required solutions of nonlinear integrodifferential boundary-value problems with deviating argument unsolved with respect to the derivative. We establish conditions for the existence of solutions of a nonlinear integrodifferential boundary-value problem with deviating argument unsolved with respect to the derivative and propose a constructive scheme for finding these solutions. [ABSTRACT FROM AUTHOR]

We determine necessary and sufficient conditions for the existence of solutions of a nonlinear boundaryvalue problem for a system of difference-algebraic equations in the case of parametric resonance. We construct a convergent iterative algorithm for finding approximations to the solutions of nonlinear boundaryvalue problem for the system of difference-algebraic equations in the case of parametric resonance. As an example of application of the constructed iterative scheme, we obtain approximations to the solutions of a two-point boundary-value problem for a system of Mathieu-type difference-algebraic equations with parametric perturbation. To check the accuracy of the obtained approximations to the solutions of the two-point boundary-value problem for the system of Mathieu-type difference-algebraic equations with parametric perturbation, we use the discrepancies in the original equation. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions to the linear boundary-value problem for a Fredholm-type matrix integrodifferential system with degenerate kernel in the critical case. We also determine the structure of generalized Green operator of this problem. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions of solvability of the nonlinear boundary-value problem in the critical case. By using the Newton–Kantorovich method, we propose a new iterative scheme for the construction of solutions of the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the solvability of the nonlinear boundary-value problem in the critical case and develop a scheme for the construction of solutions of this problem. By using the Newton–Kantorovich method, we propose a new iterative scheme for the determination of solutions to the weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case. As examples of application of the constructed iterative scheme, we obtain approximations to the solutions of a periodic boundary-value problem for the Duffing and Liénard equations. To check the accuracy of the established approximations to the solutions of the periodic boundary-value problem for the Duffing and Liénard equations, we use discrepancies in the original equations. [ABSTRACT FROM AUTHOR]

We establish constructive conditions for the solvability of a linear Noetherian boundary-value problem for a matrix difference equation and propose a procedure of construction of its solutions. We suggest an original scheme of regularization of a linear Noetherian boundary-value problem for a linear degenerate system of difference equations. [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 found solvability conditions and a construction of the generalized Green operator for a linear matrix boundary-value problem; we present an operator that reduces a linear matrix equation to the conventional linear Noether boundary-value problem. To solve a linear matrix system, we use the operator that reduces a linear matrix equation to a linear algebraic equation with a rectangular matrix. [ABSTRACT FROM AUTHOR]

In monographs of N. V. Azbelev, A. M. Samoilenko, and A. A. Boichuk, constructive methods of study of Noether boundary-value problems have been developed. These methods continue the investigation of periodic problems stated by H. Poincaré, A. M. Lyapunov, N. M. Krylov, N. N. Bogolyubov, I. G. Malkin, and O. Veivoda by the methods of small parameter. We propose an improved scheme of study of autonomous Noether boundary-value problems for nonlinear systems in critical cases. In the case of multiple roots of the equation for generating constants, we obtain sufficient conditions of existence of solutions to an autonomous boundary-value problem not solved with respect to the derivative. The effectiveness of the scheme proposed is illustrated by an example of the periodic problem for the Liénard equation. [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 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 and propose an iterative scheme for the determination of solutions of the Fredholm weakly nonlinear boundary-value problem for a system of ordinary differential equations in the case of multiple roots of the equation for generating constants. [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 second-order ordinary differential equations in the critical case. [ABSTRACT FROM AUTHOR]

We establish the conditions of regularization of the linear boundary-value problem for a system of impulsive ordinary differential equations. We also construct a generalized Green's operator and determine the form of linear impulsive perturbation of the regularized linear boundary-value problem. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the existence of solutions of a nonlinear autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in a special critical case. The specific feature of the considered problem is the inapplicability of the traditional scheme developed in the works of Malkin, Samoilenko, Grebenikov, Ryabov, and Boichuk to the investigation of the critical boundary-value problem and its solution. For the construction of solutions of a nonlinear Noetherian boundary-value problem in a special critical case, we propose a scheme that combines the Newton method and the least-squares method. The efficiency of the proposed approach is illustrated by an example of a periodic problem for a Hill-type 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]

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 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]