On parameter-dependent inhomogeneous boundary-value problems in Sobolev spaces

TL;DR

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

Contribution

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

Findings

Conditions for solution continuity with respect to parameters are established.

Solutions can be approximated by systems with polynomial coefficients.

The boundary conditions include derivatives of various orders, generalizing previous models.

Abstract

We study a wide class of linear inhomogeneous boundary-value problems for $r$th order ODE-systems depending on a parameter $\mu$ belonging to 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 $\mathbb{C}^{rm}$. Thus, they may 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, right-hand sides of the equation, and multipoint boundary conditions, which are independent of the original problem's right-hand sides.