---
tag: 4D090WJ84H
kind: statement
role: theorem
name: Carathéodory's existence theorem
msc: [34A12, 34A06]
uses: [075VBY4K4X, 098YP831QY]
about: [0FXDW992DS]
sources:
  - url: https://en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem
same_as:
  wikidata: Q5037752
  wikipedia: Carathéodory's existence theorem
---

## Statement

Let $t_0 \in \R$, $x_0 \in \R^n$ and $a, b > 0$, and let $R = [t_0 - a, t_0 + a] \times \set{x \in \R^n : \norm{x - x_0} \le b}$.
Let $f : R \to \R^n$ satisfy these three conditions.

1. For each $x$, the map $t \mapsto f(t, x)$ is Lebesgue measurable.
2. For almost every $t$, the map $x \mapsto f(t, x)$ is continuous.
3. There is a Lebesgue integrable function $m : [t_0 - a, t_0 + a] \to [0, \infty)$ with $\norm{f(t, x)} \le m(t)$ for all $(t, x) \in R$.

Then there are $\beta \in (0, a]$ and an absolutely continuous function $x : [t_0 - \beta, t_0 + \beta] \to \R^n$ with $\norm{x(t) - x_0} \le b$ for all $t$, with $x(t_0) = x_0$, and with $x'(t) = f(t, x(t))$ for almost every $t$.

## Remarks

Such a function $x$ is called a solution in the sense of Carathéodory; it is the same as a continuous solution of the integral equation $x(t) = x_0 + \int_{t_0}^{t} f(s, x(s))\, ds$.
One can take any $\beta$ with $\int_{t_0 - \beta}^{t_0 + \beta} m(s)\, ds \le b$.
The proof follows that of the [[Peano existence theorem|098YP831QY]]: one builds approximate solutions with delayed arguments.
The bound by $m$ makes them equicontinuous, and the dominated convergence theorem allows passage to the limit.
Example: let $f(t, x) = 0$ for $t < 0$ and $f(t, x) = 1$ for $t \ge 0$, with $x(0) = 0$.
The function $x(t) = \max(t, 0)$ is a Carathéodory solution.
There is no solution that is differentiable at every point, because a derivative cannot have a jump discontinuity, by Darboux's theorem.
So the theorem covers equations with discontinuous time dependence, where the classical theory gives nothing.

## Explanations

### L1 human

Classical existence theorems ask the right side of $x' = f(t, x)$ to be continuous.
In applications the forcing is often switched on and off, so $f$ jumps in time.
Carathéodory's theorem still gives a solution, if one only asks the equation to hold at almost every time.
The solution is then a continuous curve with corners where the forcing jumps.

### L2 human

The theorem replaces continuity in $t$ by measurability, and boundedness of $f$ by an integrable bound.
Solutions are absolutely continuous functions, which is the natural class for the fundamental theorem of calculus of the Lebesgue integral.
This framework underlies control theory, where controls are measurable functions of time.
As for Peano's theorem, there is no uniqueness in general; a Lipschitz condition in $x$ with an integrable Lipschitz constant gives uniqueness, as in the [[Picard–Lindelöf theorem|0P6KRD5B7A]].