---
tag: 0P6KRD5B7A
kind: statement
role: theorem
name: Picard–Lindelöf theorem
aliases: [cauchy-lipschitz-theorem, existence-and-uniqueness-theorem]
msc: [34A12]
uses: [0J7HF2PF1H, 075VBY4K4X, 08XDXP86XE, 015KK993C5]
sources:
  - url: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
    locator: Theorem
same_as:
  wikidata: Q530152
  wikipedia: Picard–Lindelöf theorem
---

## 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 $\alpha : J \to B$ with $\alpha(t_0) = x_0$ and with this property. For each $t \in J$, $\alpha$ has the derivative $f(t, \alpha(t))$ at $t$ within $J$.

## Remarks

The proof uses the [[integral equation form|015KK993C5]] and the fact that [[an iterate of the Picard operator is a contraction|08XDXP86XE]]. The limit of [[Picard iteration|0RJ5NHWYTP]] is the solution.
A common form: let $f$ be continuous on a neighbourhood of $(t_0, x_0)$ and locally Lipschitz in $x$. Then the [[initial value problem|075VBY4K4X]] has a unique solution on $[t_0 - \varepsilon, t_0 + \varepsilon]$ for some $\varepsilon > 0$.
The theorem is local. For $x' = 1 + x^2$, $x(0) = 0$, the solution $\tan t$ exists only for $|t| < \pi/2$.
Without the Lipschitz condition, solutions exist by the [[Peano existence theorem|098YP831QY]], but they need not be unique. See the [[non-uniqueness example|00K7ASZ1PR]].

## History

The theorem is named after Émile Picard and Ernst Lindelöf. It is also called the Cauchy–Lipschitz theorem, after Augustin-Louis Cauchy and Rudolf Lipschitz.

## Explanations

### L1 human

If the vector field is regular enough, the present state determines the whole motion near the start time.
Regular enough means continuous in time and Lipschitz in the state.
Then there is one and only one solution through the initial point, at least for a short time.