Differentiability of the fixed points of a family of contraction maps

Question

Given a general Banach space $B$ and a one-parameter family of contractions $F_t:B\to B$ which is defined for all $t \in (a,b)$. $F_t$ depends continuously on $t$ (in the sense $\lim_{\varepsilon\to 0} \parallel F_{t+\varepsilon}(x)-F_t(x)\parallel = 0$). I want to study how the fixed points $x_t$ of $F_t$ depend on $t$.

For instance I can show that $x_t$ depends continuously on $t$. Now I would like to go further and think about differentiability.

Intuitively the following results should hold. Are they correct and if so, have they been stated in the literature?

1. Given that $F_t$ is differentiable in $t$ and $F_t$ is Fréchet differentiable, can we infer that $x_t$ is differentiable in $t$ as well? If I assume that this is true I can derive the following formula: $$ \tfrac{d}{dt}x_t = (1-DF(x_t))^{-1}\tfrac{d}{dt}F_t(x_t)$$ where $DF(x)$ is the Fréchet derivative. Is that result correct?

2. Given that $F_t$ is $k$ times differentiable in $t$ and also that $F_t$ is $k$ times Fréchet differentiable, is $x_t$ $k$ times differentiable in $t$ as well?

Answer 1

I found the answer myself: One can simply apply the Banach space version of the implicit function theorem to the function $G(t,x) = x-F_t(x)$. The implicit function theorem shows that, given $G$ is in $C^k$, that $x_t$ is in $C^k$ as well. The precise statement can be found for instance in the book 'Analysis Tools with Application' from Bruce K. Driver (section Banach space calculus).