On the approximate solution of autonomous boundary-value problems by the Newton method.

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]