Parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

TL;DR

This work develops a unified, parameter-aware framework for linear $r$-th order ODE systems in Sobolev spaces with generic boundary conditions. It establishes a complete criterion for continuity of solutions with respect to a parameter: the homogeneous problem must be uniquely solvable at the base parameter and the coefficients and boundary operator must converge in precise Sobolev and operator senses. It provides a canonical representation of the boundary operator and explicit strong/uniform convergence criteria, enabling systematic approximation of arbitrary boundary-value problems by polynomial-coefficient, multipoint-boundary problems without dependence on the RHS. The results yield robust error estimates and justify practical approximations in applications requiring stable parameter sensitivity and computational tractability. Overall, the paper advances the theory of parameter-dependent boundary-value problems in Sobolev spaces with broad generic boundary conditions and offers a principled path to tractable approximations.

Abstract

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $μ$ in a general metric space $\mathcal M$. The solutions belong to the Sobolev spaces $(W^{n+r}_p)^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$, $1\leq p\leq \infty$. The boundary conditions are of a most general form $By=c$, where $B$ is an arbitrary continuous operator from $(W^{n+r}_p)^m$ to $\mathbb{C}^{rm}$. They may thus contain derivatives of the unknown vector function of integer and/or fractional orders $\geq r$. We find necessary and sufficient conditions for the continuity of solutions with respect to the parameter $μ$. We also prove that the solutions of the original problems can be approximated in the space $(W^{n+r}_p)^m$ by solutions of ODE-systems with polynomial coefficients and multipoint boundary conditions, which do not depend on the right-hand sides of the original problem.

Theorems & Definitions (19)

• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• Theorem 2.4
• proof : Proof of Theorem \ref{['nep v']}
• Theorem 2.5
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['eqival ym L']}
• Theorem 3.2