The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solution

Abstract

The problem with non-separated multipoint-integral conditions for high-order differential equations is considered. An interval is divided into m parts, the  values of a solution at the beginning points of the subintervals are considered as additional parameters, and the high-order differential equations are  reduced to the Cauchy problems on the subintervals for system of differential equations with parameters. Using the solutions to these problems, new  general solutions to high-order differential equations are introduced and their properties are established. Based on the general solution, non-separated  multipoint-integral conditions, and continuity conditions of a solution at the interior points of the partition, the linear system of algebraic equations with  respect to parameters is composed. Algorithms of the parametrization method are constructed and their convergence is proved. Sufficient conditions for  the unique solvability of considered problem are set. It is shown that the solvability of boundary value problems is equivalent to the solvability of systems  composed. Methods for solving boundary value problems are proposed, which are based on the construction and solving these systems.