Exercise 2, § 2

Let $t \mapsto A(t)$ be a regulated map from J into $\mathscr{L}(E)$ such that, for any two points s, t of J, $A(s)$ and $A(t)$ commute. Put $B(t) = \int_{t_0}^t A(s)\,ds$. Show that the resolvent $C(t, t_0)$ of the equation $d\mathbf{x}/dt = A(t).\mathbf{x}$ is equal to $\exp(B(t))$.

If $t \mapsto A_1(t), t \mapsto A_2(t)$ are two regulated functions from J into $\mathscr{L}(E)$ such that, for any two points s, t of J, $A_1(s), A_1(t), A_2(s), A_2(t)$ commute pairwise, show that