On approximate solution of boundary-value problems by the least squares method

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.