One-dimensional parameter-dependent boundary-value problems in H\"older spaces

TL;DR

This paper investigates linear boundary-value problems for systems of differential equations within complex H"older spaces, establishing criteria for solution continuity relative to parameters and providing convergence estimates.

Contribution

It introduces a constructive criterion ensuring solution continuity in H"older spaces for parameter-dependent problems and offers a two-sided convergence estimate.

Findings

Solution continuity in H"older spaces is characterized by a new criterion.

A two-sided estimate for convergence rate of solutions is established.

Results apply to the most general class of linear boundary-value problems in this setting.

Abstract

We study the most general class of linear boundary-value problems for systems of $r$-th order ordinary differential equations whose solutions range over the complex H\"older space $C^{n+r,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0<\alpha\leq1$. We prove a constructive criterion under which the solution to an arbitrary parameter-dependent problem from this class is continuous in $C^{n+r,\alpha}$ with respect to the parameter. We also prove a two-sided estimate for the degree of convergence of this solution to the solution of the corresponding nonperturbed problem.