Parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

TL;DR

This paper investigates parameter-dependent boundary-value problems for high-order ODE systems in Sobolev spaces, establishing conditions for solution continuity and approximation by simpler polynomial-coefficient problems.

Contribution

It provides necessary and sufficient conditions for the continuity of solutions with respect to parameters and introduces approximation methods using polynomial-coefficient systems.

Findings

Conditions for solution continuity in parameter-dependent problems

Solutions can be approximated by polynomial-coefficient systems

General boundary conditions including fractional derivatives

Abstract

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $\mu$ 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 $\mu$. 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.