A Simple Proof of Gronwall’s Inequality

Edited by Leun Kim

[latexpage]
This post introduces a simple proof of Gronwall’s inequality (Thomas Hakon Gronwall, 1877~1932) of the integral form. Let $\alpha, \beta, u \in C^0([a,b])$ with $a<b$. Assume that $\beta \geqslant 0$ on $[a,b]$. Suppose that $$u(t) \leqslant \alpha (t) + \int_{a}^{t} \beta(s) u(s) ds \;\;\; for\;\;\; t \in [a,b].$$

Then the following holds.

(1) $u(t) \leqslant \alpha (t) + \int_{a}^t \alpha(s) \beta(s) e^{\int_{s}^t \beta(r) dr} ds$ for $t \in [a,b]$.
(2) If $\alpha$ is non-decreasing, $u(t) \leqslant \alpha(t) e^{\int_{a}^t \beta(r) dr}$ for $t \in [a,b]$.

Proof.
(1) Define $$v(s) = e^{- \int_{a}^s \beta (r) dr} \int_{a}^s \beta(r) u(r) dr \;\;\; for \;\;\; s \in [a,t].$$ Differentiating both sides, we have

\begin{eqnarray} v\;'(s) &=& \beta(s) e^{-\int_{a}^s \beta (r) dr} \left( u(s) – \int_{a}^s \beta (r) u(r) dr \right) \\ & \leqslant & \beta (s) e^{-\int_{a}^s \beta (r) dr} \alpha(s).  \end{eqnarray}

Integrating both sides from $a$ to $t$ respect to $s$, we get

$$v(t) = \int_{a}^t v\;'(s) ds \leqslant \int_{a}^t \beta (s) e^{-\int_{a}^s \beta (r) dr} \alpha(s) ds.$$

But from the definition of $v$, we have

\begin{eqnarray} \int_{a}^t \beta(s) u(s) ds &=& e^{\int_{a}^t \beta(r) dr} v(t) \\ & \leqslant & e^{\int_{a}^t \beta (r) dr} \int_{a}^t \alpha (s) \beta (s) e^{- \int_{a}^s \beta (r) dr} ds \\ &=& \int_{a}^t \alpha (s) \beta (s) e^{\int_{s}^t \beta(r) dr} ds \end{eqnarray} which proves (1).

(2) Since $\alpha$ is non-decreasing, $\alpha(s) \leqslant \alpha(t)$ for $s \leqslant t$. Thus
\begin{eqnarray} u(t) & \leqslant & \alpha(t) + \int_{a}^t \alpha(s) \beta(s) e^{\int_{s}^t \beta(r) dr} ds \\ & \leqslant & \alpha (t) + \alpha(t) \int_{a}^t \beta (s) e^{\int_{s}^t \beta (r) dr} ds \\ &=& \alpha(t) – \alpha(t) \int_{a}^t \frac{d}{ds} e^{\int_{s}^t \beta(r) dr} ds \\ &=& \alpha(t) – \alpha(t) ( 1 – e^{\int_{a}^t \beta (r) dr} ) \\ &=& \alpha(t) e^{\int_{a}^t \beta (r) dr}. \end{eqnarray}

Leun Kim

Ph.D Candidate at The University of Tokyo
I was born and raised in Daegu, S. Korea. I majored in electronics and math in Seoul from 2007 to 2012. I've had a great interest in math since freshman year, and I studied PDE in Osaka, Japan from 2012-2014. I worked at a science museum and HUFS from 2014 in Seoul. Now I'm studying PDE in Tokyo, Japan. I also developed an interest in music, as I met a great piano teacher Oh in 2001, and joined an indie metal band in 2008. In my spare time, I enjoy various things, such as listening music, blogging, traveling, taking photos, and playing Go and Holdem. Please do not hesitate to contact me with comments, email, guestbook, and social medias.