Solvability of linear boundary-value problems for ordinary differential systems in the space $C^{n}$

TL;DR

This paper investigates the solvability of linear boundary-value problems for first-order ODE systems with general boundary conditions, establishing Fredholm properties, indices, and limit theorems in spaces of continuously differentiable functions.

Contribution

It extends the theory of boundary-value problems by analyzing general boundary conditions, including overdetermined and underdetermined cases, and characterizes the associated operators as Fredholm with specific properties.

Findings

The boundary-value problem operator is Fredholm on suitable spaces.

The index and $d$-characteristics of the operator are explicitly determined.

Limit theorems for characteristic matrices and $d$-characteristics are established.

Abstract

We study linear boundary-value problems for systems of first-order ordinary differential equations with the most general boundary conditions in the normed spaces of continuously differentiable functions on a finite closed interval. The boundary conditions are allowed to be overdetermined or underdetermined with respect to the differential system and may contain arbitrary derivatives of the unknown functions. We prove that the problem operator is Fredholm on appropriate pairs of normed spaces, find its index and $d$-characteristics, and prove limit theorems for sequences of the characteristic matrices of the boundary-value problems under study and $d$-characteristics of the corresponding Fredholm operators.