Example

When $E = \mathbf{R}$ and $H$ is an interval in $\mathbf{R}$, if the function $f(t, x)$ admits a partial derivative $f'_x$ (II, p. 74) at every point $(t, x)$ of $I \times H$, such that $|f'_x(t, x)| \leq k(t)$ on $I \times H$, then condition (8) is satisfied, by the mean value theorem; we shall see later how this example generalizes to the case where $E$ is an arbitrary normed space.

If $f$ is Lipschitz on $I \times H$ then $f$ is bounded on $J \times S$ for every compact subinterval $J \subset I$ and every open ball $S \subset H$. Thus prop. 3 (IV, p. 166) can be applied, and demonstrates the existence of approximate solutions of equation (1). But we also have the following proposition, which allows us to compare two approximate solutions: