Constructive theory of boundary value problems for linear integral-differential equations

Abstract

The necessary and sufficient conditions for solvability of boundary value problems of the linear integral-differential equations at phase and integral constraints are obtained. A method for constructing the solution of the boundary value problem with constraints by constructing minimizing sequences is developed. The basis of the proposed method for solving the boundary value problem is the principle of immersion. The principle of immersion is created by building the solution of a class of Fredholm integral equations of the first kind. The principal difference of the proposed method is that the origin value problem at the beginning immersed to the controllability problem with fictitious controls of functional spaces, followed by reduction to the initial problem of optimal control. Solvability and construction of a solution of the boundary value problems are solved together by solving an optimization problem. Creating a general theory of boundary value problems for linear integral-differential equations with complex boundary conditions in the presence of the phase and integral constraints is a topical problem with applications in the natural sciences, economics and ecology.