ON A TWO-POINT BOUNDARY VALUE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS WITH MANY TRANSFORMED ARGUMENTS

A.M. Samoilenko’s numerical-analytic method is a well-known and effective research method of solvability and approximate construction of the solutions of various boundary value problems for systems of differential equations. The investigation of boundary value problems for new classes of systems of functional- differential equations by this method is still an actual problem. A boundary value problem for a system of differential equations with finite quantity of transformed arguments in the case of linear two-point boundary conditions is considered at this paper. In order to study the questions of the existence and approximate construction of a solution of this problem, we used a modification of A.M. Samoilenko’s numerical-analytic method without determining equation, i.e. the method has an analytical component only. Sufficient conditions for the existence of a unique solution of the considered boundary value problem and an error estimation of the constructed successive approximations are obtained. The use of the developed modification of the method is illustrated by concrete examples.

AVERAGING IN MULTIFREQUENCY SYSTEMS WITH DELAY AND LOCAL INTEGRAL CONDITIONS

Multifrequency systems of dierential equations were studied with the help of averaging method in the works by R.I. Arnold, Ye.O. Grebenikov, Yu.O. Mitropolsky, A.M. Samoilenko and many other scientists. The complexity of the study of such systems is their inherent resonant phenomena, which consist in the rational complete or almost complete commensurability of frequencies. As a result, the solution of the system of equations averaged over fast variables in the general case may deviate from the solution of the exact problem by the quantity O (1). The approach to the study of such systems, which was based on the estimation of the corresponding oscillating integrals, was proposed by A.M. Samoilenko, which allowed to obtain in the works by A.M. Samoilenko and R.I. Petryshyn a number of important results for multifrequency systems with initial , boundary and integral conditions. For multifrequency systems with an argument delay, the averaging method is substantiated in the works by Ya.Y. Bihun, R.I. Petryshyn, I.V. Krasnokutska and other authors. In this paper, the averaging method is used to study the solvability of a multifrequency system with an arbitrary nite number of linearly transformed arguments in slow and fast variables and integral conditions for slow and fast variables on parts of the interval [0, L] of the system of equations. An unimproved estimate of the error of the averaging method under the superimposed conditions is obtained, which clearly depends on the small parameter and the number of linearly transformed arguments in fast variables.