---
tag: 00BT70ND50
kind: statement
role: theorem
name: Picard–Lindelöf theorem in integral form
msc: [34A12]
uses: [0P6KRD5B7A, 015KK993C5, 00WAVB68F9]
sources:
  - url: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
    locator: Proof sketch
---

## 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|0J7HF2PF1H]] 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|00WAVB68F9]] on the curves with values in $B$.
By the [[integral equation form|015KK993C5]], this is the same curve as the solution of the [[Picard–Lindelöf theorem|0P6KRD5B7A]].

## 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.