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].

No abstract

No abstract

UDC 517.9In the space of bounded number sequences, we establish sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of differential-difference equations defined on infinitedimensional tori and containing an infinite set of constant deviations of a scalar argument. Statement of the ProblemIt is known that the investigation of invariant sets (in particular, invariant tori) occupies an important place both in the theory of continuous dynamical systems (flows) and in the theory of discrete dynamical systems (cascades) defined in various normed spaces. In the last four decades, numerous fundamental results were obtained in this field of mathematics with the use of the method of the Green function of the problem of an invariant torus of a linear expansion of a dynamical system proposed by Samoilenko in 1970 (see [1,2]). In [3-6], this method was used for the investigation of invariant tori of countable systems of ordinary differential equations defined on tori. In the last ten years, several works were published (see [7][8][9][10][11][12][13][14][15]) in which this method was used for the investigation of invariant tori of countable systems of difference-differential and difference equations. In the present paper, in the space of bounded number sequences, we pose and solve the problem of finding sufficient conditions for the existence of invariant tori for linear and quasilinear countable systems of difference-differential equations defined on infinite-dimensional tori and containing an infinite set of constant different-sign deviations of a scalar argument. This problem has never been investigated before in the mathematical literature.

No abstract