Distance between fixed points of functions

Question

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

Answer 1

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.

Answer 2

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 $M$.