Continuity of the solutions of one-dimensional boundary-value problems in Hölder spaces with respect to the parameter

Abstract

We introduce the most general class of linear boundary-value problems for systems of ordinary differential equations of order $r \geq 2$ whose solutions belong to the complex Hölder space $C^{n+r,\alpha} ([a, b])$, where $n \in Z_{+},\; 0 < \alpha \leq 1$ и $[a, b] \subset R$, and $[a, b] \subset R$. We establish sufficient conditions under which the solutions of these problems continuously depend on the parameter in the H¨older space $C^{n+r,\alpha} ([a, b])$.