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].

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We study a nonlinear countable-point boundary-value problem for a differential equation unresolved with respect to the derivative. This equation and a nonlinear boundary condition are defined in the Banach space of bounded number sequences. We study the reducibility of the posed problem to a multipoint boundary-value problem in a finite-dimensional space.It is clear that the possibility of the constructive investigation of various boundary-value problems for ordinary differential equations substantially depends on the dimension of the space in which a boundary-value problem is considered. At present, boundary-value problems of various types are best studied in finite-dimensional spaces. The investigation of these problems in abstract Banach spaces was carried out in relatively few works, mainly devoted to periodic boundary-value problems. The study of boundary-value problems in the Banach space M of bounded sequences x = (x 1 , x 2 , x 3 , . . . ) of real numbers with norm x = sup i {|x i |, i = 1, 2, 3, . . . } was originated in [1][2][3][4], where a periodic boundary-value problem for first-order and second-order equations and a twopoint boundary-value problem for a nonlinear first-order equation with linear boundary condition were investigated. Countable-point boundary-value problems for ordinary nonlinear differential equations of normal form defined in the space M were studied in [5,6].The present paper is a logic continuation of [5,6]. We consider here a boundary-value problem with nonlinear boundary condition that contains an unbounded countable set of boundary moments on the positive semiaxis, whereas the equation and the boundary condition of the problem are defined in the space M and, moreover, the equation is unresolved with respect to the derivative. In the case where the set of boundary moments belongs to the finite segment [0, T ], the initial boundary-value problem is reduced to an analogous multipoint boundary-value problem in a finite-dimensional space. Note that a three-point boundary-value problem with linear boundary condition for an equation in a finite-dimensional space that is unresolved with respect to the derivative was considered in [7].

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