Differential systems in Sobolev spaces with generic inhomogeneous boundary conditions

TL;DR

This paper reviews linear differential systems with general inhomogeneous boundary conditions in Sobolev spaces, analyzing their solvability, Fredholm properties, and solution continuity with respect to parameters.

Contribution

It provides a comprehensive analysis of the solvability, Fredholm properties, and parameter dependence of boundary-value problems for linear differential systems in Sobolev spaces.

Findings

Fredholm properties of the systems are established

Indexes and kernel dimensions are determined

Conditions for solution continuity are derived

Abstract

The paper contains a review of results on linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general inhomogeneous boundary conditions in Sobolev spaces. The character of the solvability of such problems is investigated, their Fredholm properties are established, and their indexes and the dimensions of their kernels and co-kernels are found. In addition, necessary and sufficient conditions of continuity in the parameter of the solutions of the introduced classes of boundary-value problems in Sobolev spaces of an arbitrary order are obtained.