On Linear Boundary-Value Problems for Differential Systems in Sobolev spaces

TL;DR

This paper investigates Fredholm boundary-value problems for differential systems in Sobolev spaces, providing criteria for well-posedness, continuous dependence on parameters, and convergence of solutions with constructive conditions.

Contribution

It introduces new criteria for well-posedness and solution continuity of boundary-value problems, along with convergence conditions for characteristic matrices.

Findings

Criteria for correct well-posedness established

Conditions for continuous dependence on parameters derived

Convergence conditions for characteristic matrices identified

Abstract

We consider the Fredholm one-dimensional boundary-value problems in Sobolev spaces.We have obtained several important results about the indixes of functional operators, the criterion of their correct well-posedness, the criterion of the continuous dependence of the solutions of these problems on the parameter, the degree of convergence of these solutions, and sufficient constructive conditions under which the solutions of the most general class of multipoint boundary-value problems are continuous with respect to the parameter. Eeach of these boundary-value problems corresponds to a certain rectangular numerical characteristic matrix with kernel and cokernel of the same dimension as the kernel and cokernel of the boundary-value problem. The conditions for the sequence of characteristic matrices to converge are found.