Picard–Lindelöf theorem in integral form

Statement

Let $J = [t_{\min}, t_{\max}]$ and $t_0 \in J$, let $x_0 \in \mathbb{R}^n$, and let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ satisfy the Picard–Lindelöf hypotheses at $(t_0, x_0)$ with constants $a, L, K$. Let $B$ be the closed ball of radius $a$ around $x_0$. Then there is exactly one continuous curve $\alpha : J \to B$ such that $$\alpha(t) = x_0 + \int_{t_0}^{t} f\left(s, \alpha(s)\right) ds \quad \text{for each } t \in J.$$

Remarks

The curve $\alpha$ is the unique fixed point of the Picard operator on the curves with values in $B$. By the integral equation form, this is the same curve as the solution of the Picard–Lindelöf theorem.

Explanations

L1 human

This is the Picard–Lindelöf theorem written without derivatives. It says that the integral equation has exactly one continuous solution that stays in the ball.