Fredholm one-dimensional boundary-value problems with parameter in Sobolev spaces

TL;DR

This paper studies how solutions to linear boundary-value problems in Sobolev spaces depend on a parameter, providing criteria for continuous dependence and convergence analysis.

Contribution

It introduces a constructive criterion for the continuous dependence of solutions on a parameter in Sobolev spaces for boundary-value problems.

Findings

Established a criterion for continuous dependence on the parameter.

Analyzed the convergence degree of solutions as the parameter varies.

Focused on systems of linear differential equations in Sobolev spaces.

Abstract

For systems of linear differential equations on a compact interval, we investigate the~dependence on a parameter $\varepsilon$ of the solutions to boundary-value problems in the Sobolev spaces $W^{n}_{\infty}$. We obtain a constructive criterion of the continuous dependence of the solutions of these problems on the~parameter $\varepsilon$ for $\varepsilon=0$. The degree of convergence of these solutions is established.