The Gronwall inequality

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 $f,g:[a,b]\times U\to \mathbf{X}$ be continuous functions and let $y,z:[a,b]\to U$ 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 $$\Vert g(t,x_2)-g(t,x_1)\Vert \le C\Vert x_2-x_1\Vert$$ and a continuous function $\ddot{I}•:[a,b]\to [0,\infty )$ so that $$\Vert f(t,y(t))-g(t,y(t))\Vert \le \ddot{I}•(t).$$ Then for $t\in [a,b]$ $$\Vert y(t)-z(t)\Vert \le e^{C|t-a|}\Vert y_0-z_0\Vert +e^{C|t-a|}{\int }_a^t{e}^{-C|s-a|}\ddot{I}•(s)ds.$$

4 pages