We consider possible methods for the reduction of a countable-point nonlinear boundary-value problem with nonlinear boundary condition on a segment to a finite-dimensional multipoint problem constructed on the basis of the original problem by the truncation method. The results obtained are illustrated by examples.The truncation method was proposed by Persidskii in the middle of the last century for the investigation of properties of solutions of countable systems of differential equations and the construction of stability theory for them. He devoted several papers to the investigation of these problems, which were summarized in [1]. Later, the truncation method was used for the solution of various problems in the theory of differential, difference, and differential-difference equations in Banach spaces of bounded number sequences. This method has found especially extensive use after the appearance of [2], where it was used for the investigation of oscillatory solutions of countable systems of ordinary differential equations. In [3], the truncation method was first used for the solution of two-point boundary-value problems for countable systems of nonlinear differential equations with linear boundary conditions. For the solution of countable-point boundary-value problems with linear boundary conditions, this method was used in [4,5].In the present paper, we consider possible methods for the reduction of a countable-point boundary-value problem with nonlinear boundary conditions on a segment for a nonlinear differential equation in the space of bounded number sequences to multipoint boundary-value problems for equations in finite-dimensional spaces of increasing dimension, i.e., to known problems investigated, e.g., in [6].
