Parameter-dependent inhomogeneous boundary-value problems in spaces of continuously differentiable functions

Abstract

We investigate a broad class of linear inhomogeneous boundary-value problems of order $r$ for systems of ordinary differential equations depending on a parameter $μ$ belonging to an abstract metric space $\mathcal M$. The solutions belong to the spaces of continuously differentiable vector-valued functions $(C^{(n+r)})^m$, $n\in\mathbb{N}\cup\{0\}$, $m, r \in \mathbb{N}$. The boundary conditions have the most general form $By=c$, where $B$ is an arbitrary continuous linear operator from $(C^{(n+r)})^m$ to $\mathbb{C}^{rm}$. Thus, they may contain derivatives of the unknown vector-valued function of integer and/or fractional orders up to $n+r$ inclusive. We establish necessary and sufficient conditions for the continuous dependence of solutions on the parameter $μ$. Furthermore, we prove that the solutions of the original problems can be approximated in the space $(C^{(n+r)})^m$ by solutions of systems of ordinary differential equations with polynomial coefficients, polynomial right-hand sides, and multipoint boundary conditions, where the approximating problems are constructed independently of the right-hand sides of the original problem.