The Gronwall inequality
Ralph Howard
TL;DR
The note provides a generalized Gronwall inequality for ODEs in Banach spaces by comparing two systems $y'(t)=f(t,y(t))$ and $z'(t)=g(t,z(t))$ under a Lipschitz bound with constant $C$ on $g$ and a perturbation bound $\phi(t)$ on $f-g$. It derives an explicit stability bound using an integrating-factor argument: $\|y(t)-z(t)\| \le e^{C|t-a|}\|y_0-z_0\| + e^{C|t-a|}\int_a^t e^{-C|s-a|}\phi(s)\,ds$. The framework also covers a stronger global condition $\|f(t,x)-g(t,x)\|\le \phi(t)$ and reduces to the standard Gronwall inequality when $f=g$. Additionally, it discusses nonuniqueness of solutions when Lipschitz conditions fail and how a nearby Lipschitz system can yield useful bounds.
Abstract
We prove the following version generalization of the Gronwall inequality: Let $\mathbf X$ be a Banach space and $U\subset \mathbf X$ an open convex set in $\mathbf X$. Let $f,g\colon [a,b]\times U\to \mathbf X$ be continuous functions and let $y,z\colon [a,b]\to U$ satisfy the initial value problems \begin{align*} y'(t)&=f(t,y(t)),\quad y(a)=y_0,\\ z'(t)&=g(t,z(t)),\quad z(a)=z_0. \end{align*} Also assume there is a constant $C\ge 0$ so that $$ \|g(t,x_2)-g(t,x_1)\|\le C\|x_2-x_1\| $$ and a continuous function $φ\colon [a,b]\to [0,\infty)$ so that $$ \|f(t,y(t))-g(t,y(t))\|\le φ(t). $$ Then for $t\in [a,b]$ $$ \|y(t)-z(t)\| \le e^{C|t-a|}\|y_0-z_0\|+e^{C|t-a|}\int_a^te^{-C|s-a|}φ(s)\,ds. $$