---
tag: 7XTBR347P5
kind: statement
role: theorem
name: Comparison theorem
msc: [34A40]
uses: [04SW7WECS8, 055VVSN00V, 07EJHQYR9A]
about: [0FXDW992DS]
sources:
  - url: https://en.wikipedia.org/wiki/Comparison_theorem
---

## Statement

Let $D \subseteq \R^2$ be open and let $f : D \to \R$ be continuous and locally Lipschitz in its second variable.
Let $a < b$, let $u : [a, b] \to \R$ be a [[solution|04SW7WECS8]] of $u' = f(t, u)$, and let $v : [a, b] \to \R$ be differentiable, with the graphs of $u$ and $v$ in $D$.
Assume that $v'(t) \le f(t, v(t))$ for each $t \in [a, b]$ and that $v(a) \le u(a)$.
Then $v(t) \le u(t)$ for each $t \in [a, b]$.

## Remarks

Proof idea: suppose $v(t_1) > u(t_1)$ and let $s$ be the last time before $t_1$ with $v(s) \le u(s)$, so $v(s) = u(s)$.
Just after $s$ the difference $w = v - u$ is positive and satisfies $w' \le f(t, v) - f(t, u) \le K w$ by the [[Lipschitz condition|055VVSN00V]].
Then $w(t) e^{-K(t - s)}$ is non-increasing and $w(s) = 0$, so $w \le 0$ there, a contradiction; this is the differential form of [[Grönwall's inequality|07EJHQYR9A]].
Example: $v(t) = 0$ satisfies $v' \le 1 + v^2$, so the solution $u = \tan t$ of $u' = 1 + u^2$, $u(0) = 0$, is $\ge 0$ on $[0, \pi/2)$; with $v(t) = t$, which satisfies $v' = 1 \le 1 + v^2$, it gives $\tan t \ge t$ there.
The Lipschitz hypothesis is needed: for $f(t, x) = 2\sqrt{\abs{x}}$, the solution $u = 0$ and the function $v = t^2$ satisfy $v' = 2\sqrt{v}$ and $v(0) = u(0)$, but $v > u$ for $t > 0$.
Without Lipschitz conditions the conclusion holds when $u$ is replaced by the maximal solution of the initial value problem.

## Explanations

### L1 human

If a quantity always grows at most as fast as the rule $u' = f(t, u)$ allows, and starts no higher, then it stays below the solution of that rule.
One can bound solutions of a hard equation by solutions of an easier one.

### L2 human

Comparison arguments give a priori bounds, blow-up times and global existence results for scalar equations.
For systems, one applies the theorem to a scalar quantity such as $\norm{x(t)}^2$ or a [[Lyapunov function|04P7B5XC22]].
The [[Bihari–LaSalle inequality|9KMJSKNZ72]] and Grönwall's inequality are comparison results in integral form.
The [[Sturm comparison theorem|W7VXNYZQN2]] is a different result, about zeros of solutions of second-order linear equations.