We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton-Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Liénard-type equation.

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