Example 2

Let $G$ be a complete normed algebra over $\mathbf{R}$, and $a(t)$, $b(t)$ and $c(t)$ three maps from $J$ into $G$; the equation

is a linear differential equation; here $A(t)$ is the linear map $x \mapsto a(t)x + x b(t)$ of $G$ to itself.

For every $t \in J$, $A(t)$ is an element of the set $\mathscr{L}(E)$ of continuous linear maps from $E$ to itself (continuous endomorphisms of $E$); one knows (Gen. Top., X, p. 298) that $\mathscr{L}(E)$, endowed with the norm $\|U\| = \sup_{\|x\| \leq 1} \|Ux\|$ is a complete normed algebra over the field $\mathbf{R}$, and that $\|UV\| \leq \|U\|\ \|V\|$.

Throughout this section we shall assume that the following conditions are satisfied:

a) The map $t \mapsto A(t)$ of $J$ into $\mathscr{L}(E)$ is regulated.

b) The map $t \mapsto b(t)$ of $J$ into $E$ is regulated.

When $E$ has dimension $n$, $\mathscr{L}(E)$ is isomorphic to $\mathbf{R}^{n^2}$ (as a topological vector space) and condition a) means that each of the elements $a_{ij}(t)$ of the matrix $A(t)$ is a regulated function on $J$.

Since $\|A(t')\mathbf{x} - A(t)\mathbf{x}\| \leq \|A(t') - A(t)\|\|\mathbf{x}\|$, the map

is regulated for every $\mathbf{x} \in E$; further, for any $t \in J$ and $\mathbf{x}_1, \mathbf{x}_2$ in $E$; in other words, the right-hand side of (2) satisfies the conditions of lemma 1 of IV, p. 165 and is Lipschitz with respect to the regulated function $\|A(t)\|$ on $J \times E$. In consequence (IV, p. 173, cor. 2):