On the solvability of Fredholm boundary-value problems in fractional Sobolev spaces

TL;DR

This paper investigates the solvability of Fredholm boundary-value problems for linear ODE systems with general boundary conditions in fractional Sobolev spaces, establishing their Fredholm properties and calculating indices and kernel dimensions.

Contribution

It proves the Fredholm property for these boundary-value problems in fractional Sobolev spaces and provides explicit formulas for indices and kernel dimensions, with constructive examples.

Findings

Fredholm property established for the problems.

Explicit formulas for indices and kernel dimensions.

Constructive examples illustrating the results.

Abstract

Systems of linear ordinary differential equations with the most general inhomogeneous boundary conditions in fractional Sobolev spaces on a finite interval are studied. The Fredholm property of such problems in corresponding pairs of Banach spaces is proved, and their indices and dimensions of kernels and cokernels are found. Examples are given that show the constructive character of the obtained results.