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.