Definition 1

Given a positive real function $t \mapsto k(t)$ defined on $I$, one says that a map $f$ from $I \times H$ into $E$ is Lipschitz with respect to the function $k(t)$ if, for every $x \in H$, the function $t \mapsto f(t, x)$ is regulated on $I$, and if, for every $t \in I$ and every pair of points $x_1, x_2$ of $H$, one has (the "Lipschitz condition")

(8)

We shall say that $f$ is Lipschitz (without being more specific) on $I \times H$ if it is Lipschitz on this set for some constant $k \geq 0$. It is immediate that a Lipschitz function on $I \times H$ satisfies the hypotheses of lemma 1 of IV, p. 165 (the converse being false); when $f$ is Lipschitz (on $I \times H$) one says that the differential equation

(1)

is Lipschitz (on $I \times H$).