ON A BOUNDARY VALUE PROBLEM WITH INTEGRAL CONDITIONS FOR A SYSTEM OF DIFFERENTIAL EQUATIONS WITH MANY TRANSFORMED ARGUMENTS

A.M. Samoilenko's numerical-analytic method is 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 integral boundary conditions is considered at this paper. To investigate the existence and approximate construction of the solution of such boundary value problem it is proposed a traditional scheme of the numerical-analytic method with a determining equation, as well as a modified scheme without a determining equation. In the case of a traditional scheme it is constructed a recurrent sequence of functions that depend on parameter, each of which satisfies given boundary conditions. It is shown that under typical for numerical-analytic method assumptions, this sequence uniformly convergences to the limit function. It is established the value of the parameter at which the limit function will be an exact solution of the original boundary value problem. Approximate determining function and approximate determining equation put into consideration, and on the basis of them sufficient conditions for the solvability of this boundary value problem are obtained. The necessary conditions for the solvability of the considered boundary value problem and an estimation of the deviation of the approximate solution from the exact solution were also obtained. In the case of the modified scheme it is constructed a recurrent sequence of functions, each of which satisfies the specified boundary conditions. Under the typical for the numerical-analytic method assumptions, the uniform convergence of this sequence to the limit function, which is the exact solution of the considered boundary value problem, is proved. It is established the uniqueness of this solution and it is obtained an estimation of the deviation of the approximate solution from the exact solution. The proposed modified scheme of the numerical-analytic method is illustrated by concrete examples.