Parameter-dependent one-dimensional boundary-value problems in Sobolev spaces

TL;DR

This paper develops criteria for the continuous dependence of solutions on parameters in a broad class of boundary-value problems for differential systems within Sobolev spaces, including estimates of convergence.

Contribution

It introduces a constructive criterion ensuring solution continuity with respect to parameters and provides convergence estimates for parameter-dependent boundary-value problems.

Findings

Solutions depend continuously on parameters under specified conditions.

A two-sided estimate for convergence degree is established.

Results apply to a broad class of multipoint boundary-value problems.

Abstract

We consider the most general class of linear boundary-value problems for higher-order ordinary differential systems whose solutions and right-hand sides belong to the corresponding Sobolev spaces. For parameter-dependent problems from this class, we obtain a constructive criterion under which their solutions are continuous in the Sobolev space with respect to the parameter. We also obtain a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem. These results are applied to a new broad class of parameter-dependent multipoint boundary-value problems.