Does a fixed point depend smoothly on the parameters? Let $(X,d)$ be a complete metric space. A well-known theorem states that, for any map $G: X \to X$ satifying $d(G(x), G(y)) < Ld(x,y)$ for some fixed constant $L < 1$ and arbitrary $x, y \in X$, there exists a unique fixed point, i.e. a unique $x \in X$ such that $G(x) = x$.
Let us now restrict to the case where $X$ is a closed subset of $\mathbb{R}$. Suppose that $G$ depends on some parameter $r \in \mathbb{R}$. Formally, suppose there exists an open set $D \subseteq \mathbb{R}$ such that
$$
G : X \times D \to X : (x, r) \mapsto G_r(x)
$$
is a map satisfying $\vert G_r(x) - G_r(y) \vert < L\vert x - y\vert$ for some fixed constant $L < 1$ and arbitrary $x, y \in X$, $r \in D$. Then $G_r$ has a unique fixed point $F_r \in X$ for each $r \in D$.
We thus obtain a map
$$
F : D \to X : r \mapsto F_r
$$
which maps an $r \in D$ to the fixed point of $G_r$. I was wondering whether this $F$ inherits (some of) the smoothness of $G$, that is, suppose that
$$
G^x : D \to X : r \mapsto G_r(x)
$$
is $n \in \mathbb{N}$ times continuously differentiable, what can be said about $F$?

Using the Implicit Function Theorem, if $G$ is $C^n$ and
$$\frac{\partial (x-G_r(x))}{\partial x}\ne0,$$
in $x = x_0$, then $r\mapsto F_r$ exists locally and is $C^n$.
