On the boundary problem of ordinary differential equations

Abstract

Building the solution of the boundary problem of ordinary differential equations with local and non-local relations, as well as with state constraints is little-investigated problem of the qualitative theory of differential equations. The paper presents a method of solving the boundary problem of ordinary differential equations with boundary conditions if there are phase and integral constraints. The method is based on immersion principle, which is based on the general solution of the Fredholm integral equation of the first kind, which allows to reduce the initial boundary problem into the special problem of the optimal equation. Necessary and sufficient conditions for an existence of solving the boundary problem, as well as the building of its solution are obtained. The essence of the method is that in the first phase of the study by transformation of and bringing in a fictitious control the initial problem is immersed in the task of manageability. Then, the existence of solutions of the initial problem and the construction of its solution is carried out by solving the problem of optimal control of a special kind. In this approach, the necessary and sufficient conditions for existence of solving the boundary problem can be obtained from the condition for achieving the lower limit of the functional on a given set, and solutions of the original boundary problem are limit points of minimizing sequences.