Distance between fixed points of functions I'm trying to bound the distance of fixed points of two functions assuming there's some bound on the distance between the functions.
Specifically, assume $f_1, f_2:[0,1] \rightarrow [0,1]$ are two continuous functions (and assume any regularity conditions you would like) with unique fixed points on $[0,1]$.
Also assume that $\forall x \in [0,1]$, $|f_1(x)- f_2(x)| \le M$.
Denote the fixed points of $f_1$ and $f_2$ as $x^*_1$ and $x^*_2$ respectively.
I'm interested in results about the distance $|x^*_1 - x^*_2|$ and about $|f_1(x^*_1)- f_2(x^*_2)|$. Any measure for distance ($L_1$, $L_2$ and others) are fine.
I would appreciate references to anything similar, or to how this problem will be called.
Thanks,


*

*Ron

 A: I don't now, if by assumptions on $X$ something can be said, but I doubt the for general $X$ you will get an interesting bound: Consider $X = [0,1]$, and for $\epsilon > 0$ the functions
$f_1, f_2 \colon [0,1] \to [0,1]$ given by 
\begin{align*}
  f_1(x) &= \begin{cases} 0 & x \le \epsilon\\
                          x-\epsilon & x \ge \epsilon
            \end{cases}\\
  f_2(x) &= \begin{cases} x+\epsilon & x \le 1-\epsilon\\
                      1 & x \ge 1 - \epsilon
\end{cases}
\end{align*}
Then $f_1$ and $f_2$ are continuous, fulfill $\|f_1 - f_2\|_{\infty} \le 2\epsilon$, and have unique fixed points $x_1^* = 0$, $x_2^* = 1$. So the distance between the fixed points is as large as possible, but the distance between the functions can be made arbitrary small.
A: Consider $f_1(x) = x + \epsilon \tanh(x - x_1)$ and $f_2(x) = x + \epsilon \tanh(x - x_2)$ on $\mathbb R$, where $\epsilon > 0$ is arbitrary. These have unique fixed points $x_1$ and $x_2$ and $|f_1(x) - f_2(x)| < 2 \epsilon$.  So you'll need to base any bound on more than just $M$.
