Continuity in a parameter of solutions to boundary-value problems in Sobolev spaces

TL;DR

This work analyzes the continuous dependence of solutions to the most general linear boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. It provides a constructive criterion (Condition (0) plus Limit Conditions I–II) ensuring solutions depend continuously on a parameter, and establishes a two-sided convergence estimate via a discrepancy measure. The results extend prior parameter-continuity theory to higher-order systems with very general boundary conditions and arbitrary Sobolev indices, yielding well-posedness and stability insights for parameter-dependent problems. The framework uses a reduction to first-order systems, leverages Fredholm properties of the boundary-value map, and yields explicit convergence rates for solutions.

Abstract

We consider the most general class of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of an arbitrary order whose solutions and right-hand sides belong to appropriate Sobolev spaces. For parameter-dependent problems from this class, we prove a constructive criterion for their solutions to be continuous in the Sobolev space with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of these solutions to the solution of the nonperturbed problem.

Theorems & Definitions (7)

• Definition 3.1
• Theorem 3.2
• Corollary 3.3
• Remark 3.4
• Theorem 3.5
• proof : Proof of Theorem \ref{['nep v']}
• proof : Proof of Theorem \ref{['3.6.th-bound v']}