On the unique solvability of a family of multipoint-integral boundary value problems for a third order differential equation

A family multipoint - integral boundary value problems for a third order differential equation with variable coefficients is considered. The questions of a existence unique solution of the considered problem and ways of its construction are investigated. The family multipoint - integral boundary value problems for the differential equation of third order with variable coefficients is reduced to a family multipoint - integral boundary value problems for a system of three differential equations by introducing new functions. For solve of resulting family of multipoint - integral boundary value problems is applied a parametrization method. An algorithms of finding the approximate solution to the family multipoint - integral boundary value problems for the system of three differential equations are proposed and their convergence is proved. The conditions of the unique solvability of the family multipoint - integral boundary value problems for the system of three differential equations are obtained in the terms of initial data. The results also formulated relative to the original of the family multipoint - integral boundary value problems for the differential equation of third order with variable coefficients.