---
tag: 00WAVB68F9
kind: definition
name: Picard operator
msc: [34A12, 47H10]
defines: 07PB7HH1T9
uses: [0J7HF2PF1H, 075VBY4K4X]
sources:
  - url: https://en.wikipedia.org/wiki/Picard%E2%80%93Lindel%C3%B6f_theorem
    locator: Proof sketch
---

## Definition

Let $J = [t_{\min}, t_{\max}]$, let $t_0 \in J$ and let $x_0 \in \mathbb{R}^n$. Let $f : J \times \mathbb{R}^n \to \mathbb{R}^n$ be a function such that $s \mapsto f(s, \gamma(s))$ is integrable on $J$ for each continuous $\gamma : J \to \mathbb{R}^n$.
The Picard operator of $f$ at $(t_0, x_0)$ maps a continuous curve $\gamma : J \to \mathbb{R}^n$ to the curve $P\gamma : J \to \mathbb{R}^n$ given by
$$(P \gamma)(t) = x_0 + \int_{t_0}^{t} f\left(s, \gamma(s)\right) ds.$$

## Remarks

Assume the [[Picard–Lindelöf hypotheses|0J7HF2PF1H]] with constants $a, L, K$. Then $P$ maps the continuous curves with values in the closed ball $\|x - x_0\| \le a$ to curves in this ball.
A fixed point of $P$ is a solution of the [[initial value problem|075VBY4K4X]]. See [[the integral equation form|015KK993C5]].
[[Picard iteration|0RJ5NHWYTP]] applies $P$ again and again, starting from the constant curve $x_0$.

## Explanations

### L1 human

The operator takes a guess for the solution and makes a new guess.
It integrates the field along the old guess, starting from $x_0$.
A curve that the operator does not change is a solution.