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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 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 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 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 necessary and sufficient conditions of solvability and a scheme of construction of the solutions for 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 periodic problem for the equation used to describe the motion of satellites on elliptic orbits. [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]