# For what functions, “$=$” instead of “$\leq$”

Theorem: Suppose $f,g\in C(U,\mathbb R^n)$ and let $f$ be locally Lipschitz-continuous in the second argument, uniformly with repsect to the first. If $x(t)$ and $y(t)$ are respective solutions of the Iinitial value problems; $x'=f(t,x), x(t_0)=x_0$ and $y'=g(t,y), y(t_0)=y_0$ then $$|x(t)-y(t)| \leq |x_0-y_0|\cdot e^{L\cdot |t-t_0|}+\frac{M}{L}\left(e^{L\cdot |t-t_0|}-1\right),$$ where $L=\sup \frac{|f(t,x)-f(t,y)|}{|x-y|}$ and $M=\sup|f(t,x)-g(t,x)|$.

This is a theorem in one of my books about ODEs . Now I was wondering, for what functions $f(t,x)=f(x)$ and $g(t,x)=g(x)$, the inequality becomes an equality?

The first supremum is over $(t,x)\neq (t,y)\in V$, the second supremum is over $(t,x)\in V$.

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Just curious, what book is this from? –  Christopher A. Wong Oct 30 '12 at 20:05
–  Montaigne Oct 30 '12 at 20:12