Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y\_{1},y\_{2}\in Y$ one has
$d(\Sigma(x,y\_{1}),\Sigma(x,y\_{2}))\leq Cd(y\_{1},y\_{2})$. The fixed point theorem for complete metric spaces then implies that for all $x\in X$, there exists a unique $y(x)\in Y$ such that $\Sigma(x,y(x))=y(x)$. It is possible to prove that the correspondence $x\mapsto y(x)$ is a continuous map of metric spaces $X\rightarrow Y$.


So far I stated what I know about general metric spaces. Now furtherly assume that in particular, $X,Y$ are Fréchet spaces, and that $\Sigma$ is a smooth map of Fréchet manifolds.


My question is this: can we conclude under these additional assumptions that the correspondence $x\mapsto y(x)$ is actually a smooth map between Fréchet manifolds? I haven't found any reference for this, nor could I find a way to extend the continuous-metric proof to the smooth setting.


I will be extremely thankful for guidance or reference.



Smooth dependence in the fixed point theorem between complete Fréchet manifolds
Deane Yang's comment shows that the premise of my question naively fails even in finite dimension. However, thanks to his insight, I think that I managed to figure it out in the Fréchet setting as well. For reference's sake, here is how I think my question falls into the setting of Theorem 3.3.1 in Richard Hamilton's article on the Nash-Moser inverse function theorem (1982):


Suppose that $X,Y$ are tame Fréchet spaces (which they are in my original setting, since they are both section spaces of vector bundles over a compact Riemannian manifold with boundary, equipped with the $C^{\infty}$-topology. More details are in Hamilton's article) and $\Sigma$ is a tame map (in my original setting, a nonlinear differential operator between these spaces). Consider the map $A=\Sigma-I\_{Y}$, where $I\_{Y}(x,y)=y$. Then the correspondence $x\mapsto y(x)$ in the above becomes the condition $A(x,y(x))=0$. Here is where Deane's comment about the assumption on the differential of $\Sigma$ enters into play: we need to assume that the partial directional derivative (also known as partial Gateaux derivative) $D\_{x}A(x,y)=D\_{x}\Sigma(x,y):X\rightarrow X$ is a surjective linear map whenever $A(x,y)=0$. Under this assumption, and a certain "good guess" for an inverse which Hamilton explains about in his article, if $A(x\_{0},y\_{0})=0$ for some $(x\_{0},y\_{0})$, we can find a neighbourhood of $(x\_{0},y\_{0})$ for which every $x$ has a $y$ such that $A(x,y)=0$, and the solution map $x\mapsto y(x)$ restricted to this neighbourhood is a smooth tame map.


Of course the situation in the case where $X,Y$ are Banach spaces (which includes the finite dimensional case) is much simpler as we only need to use the standard implicit function theorem. Thank you Deane!


