Theorem 1

Let $t \mapsto A(t)$ be a regulated map of $J$ into $\mathscr{L}(E)$, and $t \mapsto \mathbf{b}(t)$ be a regulated map of $J$ into $E$. For every point $(t_0, \mathbf{x}_0)$ of $J \times E$ the linear equation (2) admits one and only one solution defined on all of $J$ and equal to $\mathbf{x}_0$ at the point $t_0$.