Continuity in a parameter of solutions to generic boundary-value problems

TL;DR

This paper defines a broad class of linear boundary-value problems for differential systems with solutions in complex H"older spaces and establishes conditions for the solutions' continuous dependence on parameters.

Contribution

It introduces the most general class of boundary-value problems with derivatives in boundary conditions and provides a constructive criterion for the solutions' continuity in the H"older space.

Findings

Established a constructive criterion for solution continuity in parameter-dependent problems.

Extended the class of boundary-value problems to include derivatives in boundary conditions.

Proved solutions depend continuously on parameters in the specified function space.

Abstract

We introduce the most general class of linear boundary-value problems for systems of first-order ordinary differential equations whose solutions belong to the complex H\"older space $C^{n+1,\alpha}$, with $0\leq n\in\mathbb{Z}$ and $0\leq\alpha\leq1$. The boundary conditions can contain derivatives $y^{(r)}$, with $1\leq r\leq n+1$, of the solution $y$ to the system. For parameter-dependent problems from this class, we obtain constructive criterion under which their solutions are continuous in the normed space $C^{n+1,\alpha}$ with respect to the parameter.