---
tag: 07EJHQYR9A
kind: statement
role: lemma
name: Grönwall's inequality
aliases: [gronwall-lemma, gronwall-bellman-inequality]
msc: [26D10, 34A40]
uses: [0Q1D85B8G6]
sources:
  - url: https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality
    locator: Differential form
  - url: https://en.wikipedia.org/wiki/Gr%C3%B6nwall%27s_inequality
    locator: Integral form for continuous functions
same_as:
  wikidata: Q510733
  wikipedia: Grönwall's inequality
---

## Statement

Let $a < b$ and let $I$ be an [[interval|0Q1D85B8G6]] of the form $[a, b]$, $[a, b)$ or $[a, \infty)$. Let $I^\circ$ be the interior of $I$.
Differential form: let $\beta, u : I \to \mathbb{R}$ be continuous, let $u$ be differentiable on $I^\circ$, and assume $u'(t) \le \beta(t)\, u(t)$ for each $t \in I^\circ$.
Then
$$u(t) \le u(a) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$
Integral form: let $\beta, u : I \to \mathbb{R}$ be continuous with $\beta \ge 0$, let $\alpha : I \to \mathbb{R}$ be non-decreasing, and assume
$$u(t) \le \alpha(t) + \int_a^t \beta(s)\, u(s)\, ds \quad \text{for each } t \in I.$$
Then
$$u(t) \le \alpha(t) \exp\left( \int_a^t \beta(s)\, ds \right) \quad \text{for each } t \in I.$$

## Remarks

The differential form needs no assumption on the signs of $\beta$ and $u$. Its proof shows that $u(t) \exp\left(-\int_a^t \beta\right)$ is non-increasing.
The integral form does not need $u$ to be differentiable.
For constant $\alpha = 0$, the integral form gives $u \le 0$. This is the step that proves [[uniqueness of solutions|025PKW8AMR]].
A variant with an additional constant term gives the [[Grönwall bound|0FRX8JD7QH]], and there is a [[discrete version|0NTM3J5MFE]].

## History

The inequality is named after Thomas Hakon Grönwall. He proved it in 1919, in the integral form with constant $\alpha$ and $\beta$.
Richard Bellman proved a slightly more general integral form in 1943.

## Explanations

### L1 human

If a quantity grows at most at a rate proportional to itself, it grows at most exponentially.
The inequality compares the quantity with the solution of the equation $v' = \beta v$.
This is the main tool to compare two solutions of a differential equation.