Applications of Arzela-Ascoli

Example 1: For $f \in \mathcal{C}[0,1]$ , let

$$(Tf)(x) = \int_0^x f(t) dt$$

Then, $Tf \in \mathcal{C}[0,1]$ , so $T$ is a linear map from $\mathcal{C}[0,1]$ to itself. Let

$$\mathcal{F}:= \{ Tf : f \in \mathcal{C}[0,1], \Vert f \Vert _\infty \leq 1 \}.$$

We would like to see whether $\mathcal{F}$ is equicontinuous -

$$\vert (Tf)(x) - (Tf)(y) \vert = \vert \int_x^y f(t) dt \vert \leq \vert x - y \vert.$$

Hence, $\mathcal{F}$ is an equicontinuous family. Also,

$$\vert(Tf)(x)\vert \leq x \leq 1.$$

Hence, $\bar \mathcal{F}$ is compact.

Example 2: (Sketch) -

Theorem 22 (Peano)   Suppose $D \in \mathbb{R}^2$ is an open set, $f:D \rightarrow \mathbb{R}$ is continuous, and $(x_0, y_0) \in D$ . Then the differential equation $y' = f(x,y)$ has a local solution passing through $(x_0, y_0)$ .

Proof idea - construct a set of approximations that are equicontinuous, then use the Ascoli-Arzela to show that there is a convergent subsequence. The point where that subsequence converges to is the solution function.