Korol I. Investigation and Construction of Solutions of the Boundary Value Problems

The thesis is devoted to the development and justification of a new numerical-analytic method for investigating and constructing periodical solutions for systems of ordinary differential equations, and solutions satisfying linear boundary conditions. The successive approximations are build, the conditions for their converging, the error estimates and the connections between the limit function and the exact solution are found. The modifications of this method, which allow to investigate such problems both in noncritical and critical cases are developed. The method is generalized for investigating the boundary value problems for autonomous, impulsive, differential-operator systems, systems with delay and is illustrated by examples. The possible periods and coefficient conditions for existence of discontinuous cycles of two-dimensional impulsive linear autonomous systems are obtained. The theory of systems of linear differential equations with singular matrix near derivative and with impulse action which is reduceable to the central canonical form is developed: the general solution is constructed, the necessary and sufficient conditions for existence of solutions of the initial problem, periodical solutions and solutions of the Noetherian boundary value problem are found, all such solutions are constructed. The necessary and sufficient conditions for existing of solutions of linear Noetherian boundary value problem for systems of linear differential equations with singular matrix near derivative and the interface conditions are found.