# Smooth dependence in the fixed point theorem between complete Fréchet manifolds

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.

• This is not enough to conclude that the map is smooth. You need an assumption on the differential of $\Sigma$. You can see this when the dimension is finite or even just $1$. The smooth dependence is in general proved using an implicit function theorem. For Fréchet maps, the appropriate one is the Nash implicit function theorem. Jul 18 at 15:56

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!