On solvability of the most general linear boundary-value problems in spaces of smooth functions

TL;DR

This paper develops a comprehensive theory for the solvability of general linear boundary-value problems for systems of first-order ODEs in smooth function spaces, covering overdetermined, underdetermined, and high-derivative boundary conditions.

Contribution

It introduces the characteristic matrix approach, linking its index and Fredholm properties to the boundary-value problem's solvability, including limit theorems for these matrices.

Findings

Characteristic matrix index equals the problem's Fredholm index.

Fredholm numbers match the dimensions of kernel and cokernel.

Limit theorems establish stability of solvability conditions.

Abstract

In the paper we develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of first-order ordinary differential equations in spaces of smooth functions on a finite interval. This problems are set with boundary conditions in generic form, that covers overdetermined and underdetermined cases. They also may contain derivatives, whose orders exceed the order of the differentiall system. Our study is based on using of the so called characteristic matrix of the problem, whose index and Fredholm numbers (i.e., the dimensions of the problem kernel and co-kernel) coincide, respectively, with the index and Fredholm numbers of the inhomogeneous boundary-value problem. We also prove a limit theorems for a sequence of characteristic matrices.