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\title[Parameterization method for invariant manifolds]
{The parameterization method for\\
invariant manifolds I: manifolds\\
associated to nonresonant subspaces}
\author[X. Cabr\'e]{
Xavier Cabr\'e
}
\address
{
Departament de Matem\`atica Aplicada I \\
Universitat Polit\`ecnica de Catalunya \\
Diagonal 647, 08028 Barcelona, Spain \\
}
\email[X. Cabr\'e]{cabre@ma1.upc.es}
\author[E. Fontich]{
Ernest Fontich
}
\address{
Departament de Matem\`atica Aplicada i An\`alisi \\
Universitat de Barcelona \\
Gran Via, 585, 08007 Barcelona, Spain
}
\email[E. Fontich]{fontich@mat.ub.es}
\author[R. de la Llave]{
Rafael de la Llave
}
\address{
Department of Mathematics \\
The University of Texas at Austin\\
Austin, TX 787121082, U.S.A \\
}
\email[R. de la Llave]{llave@math.utexas.edu}
\date{Oct 8 2001}
\begin{document}
%\bibliographystyle{alpha}
\begin{abstract}
We introduce a method to prove existence of
invariant manifolds and, at the same time study their dynamics.
As a first application,
we consider the dynamical system given by a $C^r$ map
$F$ in a Banach space $X$ close to
a fixed point: $F(x) = Ax + N(x)$, $A$ linear, $N(0)=0$, $DN(0)=0$.
We show that if $X_1$ is an invariant subspace of~$A$
and $A$ satisfies certain spectral properties,
then there exists a unique
$C^r$ manifold which is invariant under $F$ and tangent to $X_1$.
The method of proof also provides information about the dynamics on the
manifold.
In addition, we prove regularity results
on dependence on parameters.
When $X_1$ corresponds to spectral subspaces associated to
sets of the spectrum contained in disks around the origin
or their complement,
we recover the classical (strong) (un)stable manifold theorems.
Our theorems, however,
apply to other invariant spaces.
Indeed, we do not require $X_1$ to be an spectral subspace or even
to have a complement invariant under $A$.
\end{abstract}
\maketitle
\section{Introduction and statements of results}\label{sec:intro}
The main goal of this paper is to develop a method
to prove existence and regularity
of invariant manifolds for dynamical systems.
We call it the parameterization method,
and we use it to prove a variety of invariant manifold
results. We establish optimal regularity for
the invariant objects,
as well as regularity with respect to dependence on parameters.
As a particular case, our results generalize
the classical stable and unstable invariant manifold theorems in
the neighborhood of a fixed point.
We consider a $C^r$ map $F$ on a Banach space $X$ such that
$F(0) = 0$. We want to study aspects of the dynamics
in a neighborhood of the fixed point.
The heuristics is that, in small neighborhoods,
the map is very similar to its linear part.
One can hope that subspaces $X_1$ invariant under the linearization
have nonlinear counterparts: smooth manifolds
tangent to $X_1$ at 0
which are invariant under the map $F$.
Roughly speaking, the method consists on trying to find at the
same time a parameterization of the manifold and the dynamics
restricted to it. The parameterization has to
satisfy a functional equation that expresses that
its range is invariant and that it semiconjugates
the dynamics to the dynamics
of a simpler map (in some cases linear).
More precisely, we look for maps $K:U_1\subset X_1\rightarrow
X$ (the parameterization)
and $R:X_1\rightarrow X_1$ satisfying the functional equation
\[
F\circ K=K\circ R \qquad\mbox{in }U_1 \ .
\]
Recall that $X_1$ is a subspace of $X$ invariant under
$DF(0)$. Then, the previous equation guarantees that
$K(U_1)$ is an invariant manifold for $F$.
We will show that, provided that the spectrum of
$DF(0)$ satisfies certain nonresonance
conditions, one can indeed solve the functional equation
above and find these invariant manifolds.
This will extend previous results from \cite{delaLlave97}.
We point out that \cite{delaLlave97} contains examples
which show that the nonresonance conditions
are necessary for existence.
In \cite{delaLlave97},
the invariant spaces for $DF(0)$ for which the
corresponding manifolds invariant under $F$ were constructed
need not be spectral subspaces. Nevertheless,
it was required there that they had an invariant complement.
In this paper we only require that there is a complement,
but it need not be invariant under $DF(0)$.
In this way, we can associate, for example, invariant manifolds
to the spaces corresponding to eigendirections in
a nontrivial Jordan block.
As a technical improvement over
\cite{delaLlave97}, in the present paper we obtain
invariant manifolds of class $C^r$ whenever the map $F$ is $C^r$
rather than just $C^{r1 + \Lip}$ manifolds as
in \cite{delaLlave97}. This improvement
has also been obtained in \cite{ElBialy98}.
Moreover, we also study the regularity of the dependence with
respect to parameters and obtain rather sharp results.
Indeed, both technical improvements
rely on the same idea of recursively obtaining candidates
for the derivative and showing that they indeed are derivatives.
Even if our results are already novel for finite dimensions,
we work in the generality of Banach spaces since the use
of finite dimensions does not simplify
much the proofs (except the ones on sharp differentiability
results). Moreover, the use of infinite
dimensional systems leads to interesting applications in
dynamical systems.
A construction of \cite{HirschP70} shows that one can reduce
the existence of finite dimensional
invariant objects (such as foliations)
to the existence of invariant manifolds
for certain operators defined on infinite dimensional
spaces of sections; see Section~\ref{lifting} for these questions.
The parameterization method that we present has
several advantages (in contrast with
the graph transform method, for instance):
\smallskip
(i) We are lead to consider functional equations
given as fixed points of operators which are
differentiable in spaces of smooth functions.
This allows to obtain very simply smooth
dependence on parameters by applying the standard implicit
function theorem. The regularity obtained in this way
is almost optimal.
This is in contrast with the operators whose fixed points
give the invariant manifolds in the graph transform method.
These graph transform operators are only a contraction
(or differentiable) in $C^r $ norms when considered in subsets
of spaces of $C^r$ functions consisting of $C^{r+1}$
functions.
\smallskip
(ii) The method lends itself very well to numerical implementations.
The papers \cite{FranceschiniR81}
and \cite{FornaessG92} use this method to deal with
one dimensional manifolds.
Numerical work for higher dimensional maps has
been done in \cite{BeynK98}, that
undertook the task of systematically computing
Taylor expansions of invariant manifolds.
We note that the method of proof leads easily to
{\it a posteriori} estimates (see Remark~\ref{rkoutofthe}
in Section~3.4).
That is, if we can find functions
which solve our equation
approximately, we can assure that there are true solutions whose
distance form the computed solution is bounded by the residual of the
solution.
Preliminary implementations seem to show that the method is quite
stable.
Perhaps the numerical stability is related to good functionalanalytic
properties.
\smallskip
(iii) The method is geometrically natural.
That is, the equations we use can be expressed in a coordinate free way
(they are just semiconjugacy equations). Hence, they carry
through geometric structures such as symplectic forms,
volume forms, etc.
We have not explored these possibilities here, but we hope to
come to them in future work.
\smallskip
(iv) The method provides a global representation of the manifold.
The solutions of the equation that we study provide
a representation of the invariant manifold which covers the whole
manifold. When the map is entire (e.g. polynomial),
the functions representing the
manifold are entire.
This is in contrast with the graph transform methods, which
only provide
a representation of the part of the manifold which is a graph and,
hence they fail when the manifold starts to fold.
{From} the numerical point of view, this avoids
a complicated step of {\it globalization}.
\smallskip
(v) The method can be used to study a variety of invariant
objects, both in finite and in infinite dimensional
Banach spaces (see Section~2).
\smallskip
One of the applications of the nonresonant manifolds
that we produce is to make
sense of the so called {\it slow manifolds} near a fixed point.
That is, manifolds associated to the slowest eigenvalues.
They are important in applications since
one can argue indeed we justify to a certain
extent that the eigenvalues closest to the
unit circle are the ones which get suppressed the slowest
and, hence, dominate the asymptotics of the convergence.
This is particularly relevant in chemical kinetics since
often one has large sets of equations and the eigenvalues
of fixed points are widely different.
Indeed, we think that our results provide
a rigorous foundation for some geometric methods to
study long term asymptotics that have been used in the applied
literature.
We just quote some recent references in chemical kinetics:
\cite{Fraser98},
\cite{Smooke91},
\cite{PetersR93}, \cite{WarnatzMD96}.
For the use of slow manifolds in chemical reactions in a more
dynamical context, see \cite{Barkley88}.
The name slow manifolds seems to have been used for
related but perhaps similar objects in atmospheric
science (see \cite{Lorenz92} for a review).
It is interesting to note that there is
another possible construction of slow manifolds, namely
Irwin's pseudostable manifolds (see \cite{delaLlaveW95}
for a modern treatment). As it is discussed in
detail in \cite{delaLlave97}, it turns out that our
smooth manifolds and the Irwin's one may not coincide
even in systems in which both of them can be defined.
Irwin's manifolds are unique under conditions of
global behavior, while the ones considered in the present
paper are unique under smoothness conditions. There are
many indeed they are $C^1$ generic systems in
which it is impossible to satisfy at the same time
the global conditions and the smoothness conditions.
Hence, the Irwin method and the one here may produce different invariant
manifolds.
Roughly, the situation of slow manifolds can be summarized
as saying that we can find a number $r_0 \le r_1$ depending
only on the spectrum of the linearization such that:
\begin{itemize}
\item There are infinitely many manifolds which are $C^{r_0\delta}$.
\item There is at most one $C^{r_1 + \delta}$.
\item In case that there
is a $C^{r_1 + \delta}$ manifold, it is as smooth as the map.
\item There are nonresonance conditions that imply that there is
a $C^{r_1 + \delta}$ manifold.
\item The smooth manifold if it exits depends smoothly
on parameters.
\end{itemize}
Indeed, we think that this explains certain discrepancies
in the numerical literature in chemical
kinetics. We believe that the
method of Fraser computes the smooth manifolds described
here, while the method of MaasPope computes Irwin manifolds.
See
\cite{Fraser88}, \cite{MaasP92}, \cite{Fraser98}
and a more detailed discussion in
Appendix~\ref{sec:historical}.
We warn the reader interested only in finite dimensional
results and not interested in the
optimal regularity of the invariant manifold that, in this case,
the proof of our main result just requires
reading Lemmas~\ref{formal}, \ref{invertible} and~\ref{almostthere}.
The dependence on parameters is considered in detail in Section
\ref{sec:dependenceCr}, where optimal regularity is
obtained both for Theorem \ref{main1} and
\ref{main2}. See Theorems \ref{main1differentiable}
and \ref{main2differentiable} respectively.
Several of the spectral results
that we need are quite easy and well
known when our Banach spaces are finite dimensional.
In order to facilitate the exposition for those
readers interested only in finite dimensional
systems, we have relegated the
proofs in general Banach spaces to Appendix~\ref{sec:spectral}.
We finish the article with a historical section
(Appendix~\ref{sec:historical}),
where we describe some early works related
to ours. We also comment on some literature which deals with
computations of the invariant manifolds and applications to chemical
kinetics and fluid dynamics.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Some notation and conventions used}
\subsubsection{Complexification}
Since we use some spectral theory, we will
need that some operators and other objects are
defined on a complex Banach space. If the system we are interested in
is defined on a real Banach space, we use the
well known
device of complexification. That is, if $X$ is
a real space we can consider the space $\tilde X = X \oplus i X$
with the obvious multiplication by a complex number.
Given a multilinear (in particular, a linear)
operator $A$ in $X$, we can
extend it to $\tilde X$ in a canonical way
by defining $\tilde A( x_1 + i y_1,\cdots , x_n + i y_n)$
as the result of expanding all the sums and taking all the
$i$'s out (i.e., proceeding as if it was multilinear).
It is routine and well known how to check that the
resulting operator in $\tilde X$ is multilinear
when we consider it as an operator in
the complex space $\tilde X$.
Of course, a multilinear operator in $\tilde X$
which has range in the real subspace for
given data in the real subspace can also be considered as
an operator in $X$. It turns out that the extension
and the restriction are inverse operations.
We will make some remarks along the proofs
about this very general and well known procedure.
\subsubsection{Spectrum}
The spectrum of a linear operator $A$ in $X$ will be denoted by
$\Spec(A)$. We emphasize that even if $X$ is a real Banach space,
$\Spec(A)$ denotes indeed the spectrum of
the complex extension $\tilde A$
of $A$, and hence $\Spec(A)$ is a compact subset of $\complex$.
Given two sets $\Lambda$ and $\Sigma$ of complex numbers,
we use the notation
\[
\Lambda\Sigma = \{\lambda\sigma\mid \lambda\in\Lambda,
\sigma\in\Sigma\}\subset\complex
\]
and
\[
\Lambda^n = \{\lambda_1 \lambda_2 \cdots
\lambda_n\mid \lambda_1,\dots
,\lambda_n\in\Lambda\}\subset\complex \ .
\]
A polynomial $P$ defined on $X$ taking values in $Y$ is
a function from $X$ to $Y$ of the form $P=\sum_{i=1}^n P_i$,
where $P_i$ is the restriction to the diagonal of a multilinear map
of degree $i$ from $X\times \cdots\times X$ to $Y$.
We will then say that $P$ is a polynomial of degree not larger than $n$
(or simply of degree $n$).
\subsubsection{Spaces of functions}
Given $X,Y$ Banach spaces and $U\subset X$ an
open set,
$C^r(U,Y)$ is the set of functions $f:U\to Y$ which are $r$ times
continuously differentiable (in the strong sense) and
which have all derivatives up to order $r$ bounded on $U$.
It is a Banach space with the norm
\begin{equation}\label{norm}
\f\_{C^r(U,Y)} = \max \Big\{ \f\_{C^0(U)}, \Df\_{C^0(U)},\ldots,
\D^rf\_{C^0(U)} \Big\}\ ,
\end{equation}
where for functions $g$ taking values in a Banach space, we write
\[
\g\_{C^0(U)} = \sup_{x\in U} g(x)\ .
\]
The space $C^\infty(U,Y)$ consists of those functions which are $r$ times
continuously differentiable for every $r\in\natural$.
We also consider the space $C^\omega (U,Y)$
of bounded analytic functions defined in
a complex neighborhood $U$,
equipped with the supremum norm (see Section~\ref{sec:linearized}
for details). When $U,Y$ are clear from the context,
we will suppress them from the notation.
\subsection{Statement of results}
The first main result of the paper is the following:
\begin{thm}\label{main1}
Let $X$ be a real or complex Banach space,
$U$ an open set of $X$, $0\in U$,
and let $F:U\to X$, $F(0)=0$, be a $C^r$ map, with
$r\in \natural \cup \{\infty,\omega\}$.
Let $A= DF(0)$, $N(x) =F(x) Ax$, and $X= X_1\oplus X_2$ be a
direct sum decomposition into
closed subspaces.
Denote by $\pi_1,\pi_2$ the corresponding projections.
Assume$:$
\begin{itemize}
\item[0)] $F$ is a local diffeomorphism. In particular, $A$ is
invertible.
\item[1)] The space $X_1$ is invariant under $A$. That is
\[
A X_1\subset X_1\ .
\]
\smallskip
Let $A_1 = \pi_1A_{X_1}$, $A_2 = \pi_2 A_{X_2}$
and $B=\pi_1A_{X_2}$. Hence, we have
$A= \left( \begin{array}{cc} A_1&B \\ 0&A_2\end{array}\right)$
with respect to the above decomposition. Assume$:$
\smallskip
\item[2)] $\Spec (A_1) \subset \{z\in\complex \,\, z< 1\}$.
\item[3)] $0\notin \Spec (A_2)$.
\end{itemize}
Let $L\ge 1$ be an integer such that
\begin{equation}\label{condition}
\big( \Spec (A_1) \big)^{L+1} \Spec (A^{1}) \subset
\{z\in\complex \,\, z < 1\}\ ,
\end{equation}
and assume that$:$
\begin{itemize}
\item[4)]
$ \big( \Spec (A_1) \big)^i \cap\Spec (A_2) = \emptyset $
for every integer $i$ with $2\le i\le L$ $($in case that $L\ge 2)$.
\item[5)] $L+1 \le r$.
\end{itemize}
Then,
\begin{itemize}
\item[a)]
We can find a polynomial map $R:X_1\to X_1$ with
\begin{equation}\label{diffR}
R(0)=0\ ,\quad DR(0)=A_1\ ,
\end{equation}
of degree not larger than $L$,
and a $C^r$ map $K:U_1 \subset X_1\to X$,
where $U_1$ is an open neighborhood of~$0$, such that
\begin{equation}\label{conjugacy}
F\circ K = K\circ R
\end{equation}
holds in $U_1$, and
\begin{eqnarray}
&&K(0)=0\ , \label{originfixed} \\
&&\pi_1 DK(0)= \Id \ ,\quad \pi_2 DK(0)=0 \ . \label{tangency}
\end{eqnarray}
In particular, $K(U_1)$ is a $C^r$ manifold invariant under
$F$
and tangent to $X_1$ at $0$.
\item[b)] In case that
we further assume
\begin{equation}
\big(\Spec (A_1\big))^i \cap \Spec (A_1) = \emptyset \qquad
\hbox{for every integer $i$ with } 2\le i\le L\ ,
\label{extracondition}
\end{equation}
then we can choose $R$ in a) above to be linear.
More generally, if
\begin{equation}
\big(\Spec (A_1)\big)^i \cap \Spec (A_1) = \emptyset \qquad
\hbox{for every integer $i$ with } M \le i\le L\ ,
\label{extracondition2}
\end{equation}
then we can choose $R$ in a) above to be a polynomial of degree
not larger $M 1$.
\item[c)]
The $C^r$ manifold produced in a) is unique among
$C^{L+1}$ locally invariant manifolds
tangent to $X_1$ at $0$. That is, every two $C^{L+1}$
locally invariant manifolds will coincide
in a neighborhood of $0$ in $X$.
Note that the parameterization $K$
and the map $R$ need not be unique; it is the manifold $K(U_1)$
which is unique.
\item[d)]
The following parameter version of the result also holds.
Let $\Lambda $ be an open subset of a Banach space $\Lambda _0$
and let $F:\Lambda \times U\subset \Lambda_0 \times X \to X$,
that we will write as $F_\lambda (x)$.
Assume that $F\in C^r(\Lambda \times U,X)$ $($i.e., $F$ is
jointly $C^r$ in its two arguments$)$,
that the subspace $X_1$ $($which is independent of $\lambda)$
is invariant under $A_\lambda:=D_xF_\lambda(0)$ for every~$\lambda$, and that
hypotheses 0)5) above are satisfied uniformly in $\lambda$.
Then, we can find $K_\lambda$ and $R_\lambda$ satisfying a) and
b) for every $\lambda$, and such that $K_\lambda$ is
$C^{rL1}$ and
$R_\lambda$ is $C^{rL}$ jointly in their two arguments.
\end{itemize}
\end{thm}
As we will see in Remark~\ref{rknthatone},
a result on invariant manifolds for flows can be deduced rigorously
from Theorem~\ref{main1} using time $t$ maps and the
uniqueness result.
Equation \eqref{conjugacy}
ensures that the range
of $K$ is an invariant manifold under $F$. By \eqref{tangency},
it is tangent to $X_1$ at the origin.
We will see that the composition $K\circ R$ in
equation \eqref{conjugacy} is well defined, since $R$ will send
certain neighborhoods of 0 in $X_1$ into
themselves (indeed, balls centered at zero of sufficiently small radius
for an appropriate norm in which $A_1$ is a contraction).
Note that, in both finite and infinite dimensional cases,
assumption 2) on the spectrum of $A_1$ guarantees the existence
of an integer $L\ge 1$ for which \eqref{condition} holds.
We also point out that, by hypothesis 5), we always have $r\ge 2$
($r$ is the order of differentiability of the map~$F$).
It is appropriate to call condition 4)
a nonresonance condition since, in the finite
dimensional setting, it amounts to the fact that
every product of at most $L$ eigenvalues of $A_1$
is not an eigenvalue of $A_2$.
One situation in which condition 4) is satisfied automatically
is when $X_1$ corresponds to the spectral subspace
associated to a closed disk of radius $\rho <1$.
Then, if $A_1$ is invertible,
our theorem produces the strong stable manifolds
for invertible maps.
The differentiability assumptions required by the theorem are, however,
stronger than those in the classical proofs,
due to hypotheses \eqref{condition} and 5). This is to be expected
since we also obtain information on the dynamics (weaker
differentiability assumptions will be addressed in Theorem~\ref{main2}).
Indeed, we obtain that the dynamics on the invariant manifold is
semiconjugated to the dynamics of $R$ that is just a polynomial,
which under hypothesis \eqref{extracondition}, becomes a linear map.
Since $A$ is invertible, passing to the inverse, we
can establish the existence of strong unstable manifolds.
Theorem~\ref{main1} applies, however, to other invariant
subspaces. In this respect, note that the theorem does not require
$X_1$ to be an spectral subspace or even to have
a complement invariant under $A$ (see the examples
in Remark~\ref{rkweemph}).
\begin{rk}\label{rkorderL}
\rm{
The existence of the map $K$ in conclusion a) of Theorem~\ref{main1}
also holds if one assumes, instead of the nonresonances 4),
the existence of polynomials $R$ and $K^\le$ of degree not larger
than $L$ satisfying \eqref{diffR}, \eqref{originfixed},
\eqref{tangency} and $F\circ K^\le (x) =K^\le\circ R(x)+o(x^L)$.
It may happen that there exist
several $R$, $K^\le $ satisfying the above mentioned
conditions. Associated to each such solution $K^\le $ there will
be a unique $C^{L+1}$ invariant manifold $K(U)$ with
$K$ such that its $L$jet is $K^\le $, i.e. $D^j K(0) = D^jK^\le (0)$
for each $j\in \{ 0, \dots ,L \}$.
As we will see in Section~\ref{sec:formal},
the nonresonance conditions~4) ensure that certain linear
equations can be solved (see Lemma~\ref{formal}).
However, even if there exist resonances, it could happen
that these equations can nevertheless
be solved for the system at hand.
Indeed, we could use some conditions weaker than
4), but at the price of imposing restrictions on
the nonlinear part. When $X$ is finitedimensional,
these restrictions are submanifolds of finite codimension
and, if we have enough parameters to select, we can
get the invariant manifolds for some special values of the parameters.
When the map preserves a geometric structure
(e.g. volume, contact or symplectic) or
a symmetry, resonances often exist,
but at the same time, there are usually cancellations that allow
the equations to be solved.
}
\end{rk}
\begin{rk}\label{rkthedef}
\rm{
Recall that, since $A$ is invertible,
the equality
\[
A^{1}\lambda\Id =
\lambda A^{1}(A\lambda^{1}\Id)
\]
implies that
the spectrum of
$A^{1}$ is exactly $\{\lambda^{1}\ \ \lambda\in\Spec(A)\}$.
Therefore, \eqref{condition} and conditions 2) and~4) in Theorem~\ref{main1}
imply
\[
\big(\Spec (A_1) \big) ^i \cap \Spec (A_2)
= \emptyset, \qquad \forall \ i\ge 2\ .
\]
Similarly, if in addition \eqref{extracondition} holds then
\[
\big(\Spec (A_1) \big)^i \cap \Spec (A_1) =
\emptyset, \qquad \forall \ i\ge 2\ .
\]
Our formulation in the theorem
makes clear that these are really a finite number of
conditions since, for $i> L$, they are satisfied
automatically.
}
\end{rk}
\begin{rk}\label{rkweemph}
\rm{
We emphasize that the spectrum of $A_1$ and that of $A_2$
need not
be disjoint, since condition 4) is only required
for powers bigger or equal than 2.
For example, the theorem applies to
\[
A=\left( \begin{array}{cc} 1/2&0\\ 0&1/2\end{array}\right), \quad
X_1= \{(x^1,0)\}, \;\; X_2=\{(0,x^2)\}\ ,
\]
and also to
\[
A= \left( \begin{array}{cc} 1/2&1\\ 0&1/2\end{array}\right), \quad
X_1= \{(x^1,0)\}, \;\;
X_2= \{(0,x^2)\}\ .
\]
A more sophisticated example where our results apply is
\[
A = \left( \begin{array}{ccccc}
1/2&1&&&\\
0&1/2&&&\\
&&1/3&&\\
&&&1/5&1\\
&&&&1/5\end{array}\right)\ .
\]
Then, denoting by $E_i$ the $i^{th}$ coordinate axis, we could associate
invariant
manifolds to $E_1,E_3, E_4$,
$E_1\oplus E_2$, $E_4\oplus E_5$,
or to sums of these spaces, e.g., $E_1\oplus E_4$,
$E_1\oplus E_2 \oplus E_4$, etc.
}
\end{rk}
If $X$ is finite dimensional and $A$ is invertible
then $A_1$ and $A_2$ are also invertible.
Instead, when $X$ is infinite dimensional, the fact that
$A$ is invertible does not
ensure that $A_1$ and $A_2$ are also invertible
(see Example \ref{inverses}).
Nevertheless, from the invertibility of $A$, which
is equivalent to the unique solvability of the system
\begin{eqnarray*}
&&A_1 x^1 + Bx^2 = y^1\\
&&A_2 x^2 = y^2 \ ,
\end{eqnarray*}
we deduce that $A_1$ is injective
and $A_2$ is onto.
Moreover we have that if two of the linear maps $A_1$, $A_2$,
$A$ are invertible then the third one is also invertible.
In particular, under the hypotheses of Theorem~\ref{main1}
we have that $A_1$ is invertible, since $A$ and $A_2$
are assumed to be invertible.
On the other hand, we have that
\[
A^{1}= \left( \begin{array}{cc} A_1^{1}&A_1^{1}BA_2^{1}
\\ 0&A_2^{1}\end{array}\right)\ .
\]
Using the formula
\begin{equation}\label{specrad}
\rho(A) = \max \big(\rho(A_1), \rho(A_2) \big)
\end{equation}
which is proved in Appendix~\ref{sec:spectral},
applied to $A^{1}$ instead of $A$,
we deduce
\begin{equation}\label{specinv}
\rho(A^{1}) = \max \big(\rho(A_1^{1}), \rho(A_2^{1}) \big)
\ge \rho(A_2^{1}) \ .
\end{equation}
The last inequality leads to the following fact:
\begin{equation}\label{specimpl}
\begin{split}
\big( \Spec (A_1) \big)^{L+1} & \Spec (A^{1}) \subset
\{z < 1\}\ \\
& \implies \big( \Spec (A_1) \big)^{L+1} \Spec (A_2^{1}) \subset
\{z < 1\} \ .
\end{split}
\end{equation}
\begin{rk}
\rm{
Note that, in finite dimensions, \eqref{specimpl} is obvious
because then,
\begin{equation} \label{elementary}
\Spec (A_2^{1}) \subset \Spec(A^{1}) \ .
\end{equation}
In infinite dimensions, \eqref{elementary} could be
false. See examples in Section~\ref{sec:spectral}.
The argument presented for \eqref{specimpl} is
true both for finite and infinite dimensions.
}
\end{rk}
Theorem~\ref{main1} does not cover exactly
the main theorem in \cite{delaLlave97} since the definition of~$L$,
which in Theorem~\ref{main1} is
given by \eqref{condition},
in \cite{delaLlave97} is
\begin{equation}
\label{conditionold}
\big(\Spec (A_1)\big)^{L+1} \Spec (A_2^{1}) \subset
\{z\in \complex \mid z < 1\}\ .
\end{equation}
Therefore, by \eqref{specimpl},
the exponent~$L$ of Theorem~\ref{main1} is larger than that of
\cite{delaLlave97}. In particular, assumption
5) of the present paper requires more differentiability
for the map $F$ than the corresponding result in \cite{delaLlave97}.
This stronger assumption is reasonable
since the conclusions of Theorem~\ref{main1} are also stronger than those
in \cite{delaLlave97}
because they include information, through semiconjugacy,
about the dynamics on the invariant manifold (given by $R$).
The shortcoming mentioned above is remedied in the following
result. Theorem~\ref{main2} below is a strict generalization of
the main result in \cite{delaLlave97} since it
makes exactly the same differentiability assumptions,
obtains the same conclusions, and improves the
differentiability result for $K$.
Moreover, we obtain uniqueness conclusions.
Another way that Theorem~\ref{main2}
improves on the main result of
\cite{delaLlave97} is that it
does not require the decomposition
$X = X_1 \oplus X_2$ to be invariant under $A$
a result that seems not to appear
in any reference on invariant manifolds that we are aware of.
\begin{thm}\label{main2}
Assume hypotheses 0)  5) of Theorem~\ref{main1} except that
\eqref{condition} is replaced by
\begin{equation}\label{condition2}
\big(\Spec (A_1)\big)^{L+1} \Spec (A_2^{1}) \subset
\{z\in\complex \,\, z <1\}\ , \qquad
L\ge 1 \ .
\end{equation}
Then,
\begin{itemize}
\item[a)]
We can find a $C^r$ map $K$ and a $C^r$ map $R$ satisfying
\eqref{conjugacy}.
Moreover we can choose $K$ in such a way that
\begin{equation} \label{normalization}
\pi_1 K = \Id .
\end{equation}
\item[b)]
{\bf b1)} Furthermore, the $C^r$ manifold produced is the unique
$C^{L+1}$ locally
invariant manifold tangent to $X_1$ at $0$.
In fact, the following stronger result holds.
{\bf b2)} $K$ is the unique $($locally around the origin$)$
solution of \eqref{conjugacy}
in the class of Lipschitz functions
$K:U_1\subset X_1 \rightarrow X$
of the form $K =(\Id, w_L+h)$,
with $w_L$ being a polynomial of degree $L$ such that
$w_L (0)= 0 $ and $Dw_L(0) = 0$,
and with $\sup_{x\in U_1} (h(x)/x^{L+1}) < \infty$.
\item[c)]
Let $\Lambda $ be an open subset of a Banach space $\Lambda _0$
and let $F:\Lambda \times U\subset \Lambda_0 \times X \to X$,
that we will write as $F_\lambda (x)$.
Assume that $F\in C^r(\Lambda \times U,X)$ $($i.e., $F$ is
jointly $C^r$ in its two arguments$)$,
that the subspace $X_1$ $($which is independent of $\lambda)$
is invariant under $A_\lambda:=D_xF_\lambda(0)$ for every~$\lambda$, and that
hypotheses 0)5) above are satisfied uniformly in~$\lambda$.
Then the unique solution $K_\lambda$
is $C^{rL1}$ jointly in their two arguments.
\end{itemize}
\end{thm}
\begin{rk}\label{rknoinv}
\rm{
Following the proof of this theorem one sees that we only use
that $A$ is invertible (hypothesis 0) to get, from
the spectral conditions on $A$, the existence of a norm in $X$,
equivalent to the original one, such that the following
condition is satisfied:
$\A_2^{1}\\, \A_1\^{L+1} < 1$.
Actually, this norm condition is the technical condition we
use through the rest of the proof.
In finite dimensional spaces the spectral condition and the norm
condition are equivalent, but this property
does not seem to be true
in general. We prefer to put the spectral condition on the statement
because it is intrinsic. However the conclusions of Theorem~\ref{main2}
also hold if we do not assume that $A$ is invertible and instead
of \eqref{condition2} we assume that
$\A_2^{1}\\, \A_1\^{L+1} < 1$.
}
\end{rk}
\begin{rk}\label{rknthatone}
\rm{
Note that one of the consequences of
the uniqueness conclusions of Theorem~\ref{main1} and
Theorem~\ref{main2} is that these theorems apply also to
flows with a fixed point at zero.
That is, we can associate a manifold which is invariant
under all the elements of the flow to
every subspace that is invariant under all
the elements of the linearized flow and that satisfies
the nonresonance conditions for one element
of the flow.
For, if we denote by
$\{S_t\}_{t \in \real}$ a flow of class $C^r$, and $S_{t_0}$
satisfies either the conditions of
Theorem~\ref{main1} or the ones of
Theorem~\ref{main2} for some $t_0$, we know that there exists
a manifold~$W$ tangent to the given
subspace, say $X_1$, and such that
$S_{t_0}( W ) \subset W$.
We claim that, for any $s$, $S_s (W)$ is invariant by $S_{t_0}$ and
that $T_0 S_s(W) =X_1$.
Indeed,
\[
S_{t_0}\circ S_s( W) = S_s \circ S_{t_0}(W) \subset S_s (W)\ .
\]
Moreover, since by assumption we know that $D S_s (0) X_1 = X_1$, then
\[
T_0 S_s(W) = D S_s (0) T_0 W = X_1 \ .
\]
Hence, we obtain that the manifold $S_s(W)$
satisfies the conclusions of Theorem~\ref{main1} or~\ref{main2}
for the map $F=S_{t_0}$ and hence, by uniqueness, $S_s (W) = W$.
}
\end{rk}
In the following section we indicate how to obtain other types of
invariant objects using Theorems~\ref{main1} and
\ref{main2}. For the proofs of these two theorems (that start in Section~3),
one can skip Section~2, which is devoted to applications.
\section{Invariant manifolds for normally hyperbolic
manifolds and invariant foliations} \label{lifting}
The previous construction of invariant manifolds associated
to nonresonant subspaces can be lifted to construct
invariant manifolds near another invariant manifold
that generalize the stable manifolds
of normally hyperbolic invariant manifolds.
It can also be used to construct invariant foliations
or invariant prefoliations.
Even if the existence of these objects can be obtained
by following the steps of the proof in the present paper,
most of them can also be obtained immediately through
a device used in \cite{HirschP70} by applying
Theorems~\ref{main1} and \ref{main2} to an appropriate map
between certain infinite dimensional Banach spaces.
This was indeed one of the motivations to formulate our results
in general Banach spaces.
The main idea is,
given a diffeomorphism, to consider
its action on spaces of continuous sections.
In the following, we will detail the construction.
Let $M$ be a $C^\infty $ Riemannian manifold, $N\subset M$ a
$C^1$ submanifold (an
important case is $N=M$), and $f:M\longrightarrow M$ a $C^r$
diffeomorphism, $r\ge 1$.
Assume that $f(N) = N$.
Following \cite{HirschP70}, we define
an operator $\L_f$, acting on sufficiently
small sections $v$ of $TM$ defined on $N$ (i.e.,
$v\in C^0(N,TM) $), by
\begin{equation} \label{induced}
[\L_f v] (x) = \exp ^{1}_{x} f (\exp_{f^{1}(x)} v(f^{1}(x))),
\end{equation}
where $\exp$ is the exponential map of Riemannian geometry associated
to a $C^\infty $ metric.
This operator is well defined since $f(N)=N$.
In a more suggestive way, we write formula \eqref{induced} as
\begin{equation} \label{additive}
[\L_f v] (f(y)) = f(y+v(y))  f(y)
\end{equation}
where, of course, by the sum of a point and a vector, we mean the
exponential. In the particular case that $M$ is a torus
with the flat metric, \eqref{additive} agrees
with the usual sum of vectors in the torus.
We recall that
\[
\L_f :U\subset C^0(N, TM) \longrightarrow C^0(N, TM)
\]
is $C^{r1}$, even when $N$ is
a Banach manifold (here, $U$ is an open set formed by sufficiently
small sections).
If $N$ is a compact submanifold,
then $\L_f$ is $C^r$
(see \cite{delaLlaveO99} for these questions about
regularity of composition operators).
Throughout the rest of this section, we will assume that $M$
and, therefore, $N$ are finite dimensional manifolds.
Note that we also have
\begin{eqnarray*}
\L_f (0) & = & 0 \\
D\L_f (0) & = & f_*
\end{eqnarray*}
where $f_* \Delta \sigma = Tf\circ \Delta \sigma \circ f^{1}$.
We also recall the following result
from \cite{Mather68} (see also \cite{ChiconeL99}):
\begin{thm} \label{mather}
The map $f_* :C^0 (N,TM) \longrightarrow C^0 (N,TM) $ satisfies
\begin{equation} \label{decompositionspec}
\Spec (f_*, C^0(N,TM)) \subset \bigcup_{i=1}^{n}
\{ z\in \complex\ \ \lambda _i^ \leq z \leq \lambda _i^+ \}
\end{equation}
for some integer $n$, and some reals
$\lambda _i^ \leq \lambda _i^+ < \lambda _{i+1}^\leq \lambda _{i+1}^+$,
$i=1, \dots, n1$.
Moreover, we can find a continuous decomposition
\begin{equation} \label{decompositionbundle}
T_x M = \bigoplus_{i=1}^{n} E _x^i
\end{equation}
such that
\[
v\in E^i_x \quad \Leftrightarrow \quad
\left\{ \begin{array}{l}
Df^n (x) v  \leq c (\lambda _i^+ +\varepsilon)^n v, \quad n\ge 0, \\
Df^n (x) v  \leq c (\lambda _i^ \varepsilon)^n v, \quad n\le 0,
\end{array} \right.
\]
for some constant $c$ and some $\varepsilon >0$ sufficiently
small.
Finally, if we denote by $\Pi ^i$ the spectral projections of $f_*$
corresponding to
$\{ z\in \complex\ \ \lambda _i^ \leq z \leq \lambda _i^+ \}$
and by $P^i _x$ the projections corresponding to the bundles
above, we then have
\[
[\Pi ^i v ](x) = P^i_x v(x).
\]
\end{thm}
\begin{rk}\label{rkevifiis}
\rm{
Even if $C^0(N,M)$ is a real Banach space, there is
a canonical way to complexify it and to extend $f_*$ to this complex
space. Then,
the spectrum of the operator refers to the complexified operator.
Nevertheless, the spectral projections are real Banach spaces;
see \cite{Mather68} for more details.
}
\end{rk}
\begin{rk}\label{rkanother}
\rm{
Another theorem of \cite{Mather68}, which we will not use, in the
formulation of our results,
shows that if $f_{\mid N}$ is such that aperiodic orbits
are dense, then $\Spec (f_*, C^0(N,TM)) $ is indeed a union
of annuli as in \eqref{decompositionspec}.
Also, it is shown that, when $M = N$,
$f$ is Anosov if and only if $1\notin
\Spec (f_*, C^0(M,TM)) $.
Note that there
cannot be more spectral gaps than the dimension of
$TM$, since the existence of spectral
gaps implies the existence of subbundles.
}
\end{rk}
\begin{rk}\label{rkevifdecom}
\rm{
Even if the decomposition produced in Theorem \ref{mather}
is only claimed there to be continuous, using the invariant
section theorem \cite{HirschP70} it is possible to
show that it is actually more regular.
The regularity depends on the numbers $\lambda _i$.
Similarly, the theory of normally hyperbolic manifolds
shows that if $T_xN \subset \bigcup_{i=\alpha }^{\beta } E_x^i$,
then the manifold $N$ is actually more regular than $C^1$.
}
\end{rk}
If we apply our Theorem \ref{main1} to $\L_f $, we obtain:
\begin{thm} Let $f$ and $\L_f $ be as described previously.
Assume that $\L_f$ is $C^r$
and that $\Sigma \subset \{ 1, \dots ,n\}$ is a subset such that
\begin{itemize}
\item[$1)$] $\lambda _i ^+ < 1$ for every $i\in \Sigma$.
\item[$2)$] There exists $L\ge 1$ such that
$(\lambda _\Sigma^+)^{L+1} (\lambda _1^)^{1} < 1$,
where
$ \lambda_\Sigma^+ \equiv \max _{i\in \Sigma} \lambda ^+ _i$.
\item[$3)$]
\[
\Big(\bigcup _{i\in \Sigma} [\lambda _i ^ , \lambda _i ^+ ] \Big) ^j
\cap \Big( \bigcup _{i\notin \Sigma} [\lambda _i ^ , \lambda _i ^+ ] \Big)
= \emptyset \quad \mbox{ for every } 2\le j\le L\ .
\]
\item[$4)$] $L+1 \le r$.
\end{itemize}
Then, we can find a $C^{r}$ manifold $W^\Sigma \subset
C^0(N,TM) $ invariant under $\L_f$.
\end{thm}
We now define
\[
W^\Sigma_x = \{ x+v(x)\mid v\in W^\Sigma \}.
\]
The invariance of $W^\Sigma$ under $\L_f $,
and the fact that by definition of $\L_f $ we have
\begin{equation} \label{interpretation}
f(x+v(x)) = f(x) + \L_fv (f(x)),
\end{equation}
imply that
\begin{equation} \label{invariance}
f(W^\Sigma_x) \subset W^\Sigma_{f(x)}.
\end{equation}
Moreover, since $W^\Sigma_x$ is the image under the exponential map
of $W^\Sigma$, then it is a $C^{r}$ manifold.
The dependence $x\mapsto W^\Sigma_x$ is $C^0$.
Therefore, we have produced a continuous family
of $C^{r}$ manifolds which is invariant under $f$
in the sense of \eqref{invariance}.
We also note that if
$ \lambda_\Sigma^+ \equiv \max _{i \in \Sigma} \lambda ^+ _i $,
$\lambda _\Sigma^ \equiv \min _{i\in \Sigma} \lambda ^ _i
$
and we assume
\[
c (\lambda _\Sigma^ \varepsilon)^n v
\leq
 \L_f^n v 
\leq
c(\lambda _\Sigma^+ +\varepsilon)^n v,
\qquad \mbox{for all }n\ge 0, v\in W^\Sigma\ ,
\]
then
\begin{equation} \label{consequence}
c (\lambda_\Sigma^ \varepsilon)^n \leq d(f^n (x) , f^n(y))
\leq c(\lambda_\Sigma^+ +\varepsilon)^n,
\qquad \mbox{for all }n\ge 0, y\in W^\Sigma _x.
\end{equation}
In case that $N \subset M$ and that $\Sigma$ corresponds to the part
of the spectrum closest
to the origin, the previous construction
reduces to the strong stable
manifold for invariant submanifolds.
In this case, \eqref{consequence} is not
only consequence
of $y\in W^\Sigma _x$, but also
\[
d(f^n (x) , f^n(y) ) \leq c(\lambda _\Sigma^+
+\varepsilon)^n, \qquad \mbox{for all }n\geq 0
\]
is sufficient for $y \in W^\Sigma_x$.
Hence, when $\Sigma$ is the part of the spectrum closest to
the origin, the relation between $x$ and $y$
given by \eqref{consequence} is equivalent to
$x\in W^\Sigma_y$, and also to $y\in W^\Sigma_x$.
Since this is clearly an equivalence relation
between $x$ and $y$, we obtain that
$W^\Sigma_x$ constitute a foliation.
This is the strong stable foliation.
When we take $\Sigma$ to consist just of one
band, a similar result proved by
a different method can be found
in \cite{Pesin73}.
When $\Sigma$ contains intermediate components,
then \eqref{consequence} may not be equivalent to $x\in
W^\Sigma_y$.
Indeed, in \cite{JiangLP95} it is shown that the $W^\Sigma_x$ may fail
to be a foliation.
In some cases ($M=\torus ^d$) it is possible to use other
constructions \cite{delaLlaveW95} to associate invariant manifolds
to invariant subspaces. The Irwin construction does
indeed lead to a foliation. Nevertheless, the leaves are not very
smooth.
The above construction of invariant foliations
admits several extensions:
\begin{itemize}
\item
As shown in \cite{HirschP70}, the map $\L_f$ can be defined
even when $N$ is a much more general set than a manifold.
This allows to show that $W^\Sigma_x$ is a manifold even
when $N$ is not a manifold.
\item
The map $x\mapsto W^\Sigma_x$ is more regular than just
continuous. One can indeed show that it is H\"older
continuous for some exponent.
\end{itemize}
In spite of the fact that the regularity consequences of the above
method are not optimal, we hope that the
painless way to construct these geometric objects out
of our main
theorem may serve as motivation for the study of
these objects.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem \protect\ref{main1}} \label{sec:pmain1}
The proof will proceed by showing first that,
due to the nonresonance conditions, we can solve equation
\eqref{conjugacy},
\[
F\circ K = K\circ R\ ,
\]
order by order in the sense of power series.
We will then show that, once we have reached a high enough order
(actually $L$), we can fix the polynomials $K^\le$ and $R$ so obtained and
reduce the equation for the higher order
terms to a fixed point problem for $K^>$ (where $K=K^\le +K^>$),
which can be solved
by appealing to the standard contraction mapping theorem.
This procedure will lead to the loss of
one derivative in the conclusions, and we will
need a separate argument to recover it.
In numerical applications, the computation to high enough
order may provide sufficient precision and the iteration leading to a
fixed point may be readily implementable.
We emphasize the fact that one of the consequences of our analysis
is that if we start the iteration leading to the fixed point
with a polynomial
approximation of an order which is not high enough, then
the procedure may
converge to a different, less differentiable solution.
It will also follow from the analysis that we present
that if we compute functions which solve
$F\circ K = K \circ R$ with good enough accuracy, then there is a
true solution near by and the distance from the true solution
to the computed one is bounded by the error incurred in
solving the equation.
Hence, the contraction mapping fixed point theorem
leads to {\it a posteriori} error estimates for numerical methods
(see Remark~\ref{rkoutofthe}).
This section deals with the proof of the existence statements
a) and b) of Theorem~\ref{main1}.
The uniqueness statement c) of the theorem is a particular case
of the uniqueness result of Theorem~\ref{main2},
which is proved later in Section~\ref{subuniqueness} (recall
that if all assumptions of Theorem~\ref{main1} are satisfied
then the ones of Theorem~\ref{main2} are also satisfied
for the same $L$ and $r$).
The dependence on parameters, statement d) in Theorem~\ref{main1},
will be postponed to Section~\ref{sec:dependenceCr},
where we will prove a more general result.
Along the proofs we will need some results on
spectral properties of operators in Banach spaces, that we
present in Appendix~\ref{sec:spectral}.
In the finite dimensional case, these results are rather
simple and wellknown.
Sections~\ref{sec:formal} to \ref{sec:solution} establish the existence
of a solution $K\in C^{r1}$ of \eqref{conjugacy}.
The proof that $K$ is indeed $C^r$ is given in Section~\ref{sec:sharp1}
and consists in studying the equation satisfied by the first
derivative or differential $DK$. Such equation has strong analogies
with the one satisfied by $K$, \eqref{conjugacy}.
Knowing that $K$ is already $C^{r1}$, we will prove the existence of
a solution $G\in C^{r1}$ to the equation for $DK$.
Then we will have that $DK=G$, by a uniqueness property.
Hence, $DK=G\in C^{r1}$ and we will conclude $K\in C^r$.
To simplify the proofs, we scale the maps involved in the equations.
Following standard practice, given
a real number $\delta>0$ and a map $H$,
we consider $H^\delta (x) = \frac{1}{\delta} H(\delta x)$.
Note that \eqref{conjugacy} holds in the
ball of radius $\delta$ if and only if
\[
F^\delta \circ K^\delta = K^\delta \circ R^\delta
\]
holds in the ball of radius~1. Moreover,
\[
F^\delta = A+N^\delta\ ,
\]
where $N^\delta$ satisfies $N^\delta (0)=0$, $DN^\delta (0)=0$ and
that $\N^\delta\_{C^r}$ is arbitrarily small in the ball of radius~3
if we take $\delta$ sufficiently small.
We also note that this change of scale does not affect
conclusions \eqref{diffR}, \eqref{originfixed} and \eqref{tangency}.
Therefore, rather than considering
small balls, we will assume
that $\N\_{C^r}$ is sufficiently small in the ball of radius~1.
This has the advantage that it simplifies the spaces
and also some arguments used in the proofs.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Formal solution}
\label{sec:formal}
In this section we show that, under the nonresonan\ce hypotheses,
we can find
polynomials $K^\le$ and $R$ satisfying
\[
F\circ K^\le (x) = K^\le \circ R(x) + o(x^L)\ .
\]
For this, we will
use the device of complexification. The precise result is the following:
\begin{lem}\label{formal}
Assume that $X$ is a real or complex Banach space,
hypotheses 0), 1), 3) and 4) of Theorem~\ref{main1},
and $r\ge L$. Then,
\begin{itemize}
\item[a)] We can find polynomials $K^\le = \sum_{i=1}^L K_i$ and
$R = \sum_{i=1}^L R_i$ of degree not larger than $L$,
where $K_i$ and $R_i$ are multilinear maps of degree $i$, satisfying
\[
F\circ K^\le (x) = K^\le \circ R (x) + o(x^L)\ ,
\]
and \eqref{diffR}, \eqref{originfixed}, \eqref{tangency}, i.e.,
$R(0)=0$, $DR(0)=A_1$, $K^\le (0)=0$, $DK^\le (0)=(\Id,0)$.
\item[b)] If we further assume that
\begin{equation}
\label{conditionb}
\big(\Spec (A_1) \big)^j\cap \Spec (A_1) = \emptyset
\qquad\mbox{for some $j$ with }
2\leq j \leq L\ ,
\end{equation}
then we can choose $R_j=0$.
\item[c)]
Let $\Lambda$ be an open subset of a Banach space $\Lambda _0$
and let $F:\Lambda \times U\subset \Lambda_0 \times X \to X$,
that we will write as $F_\lambda (x)$.
Assume that the subspace $X_1$ $($which is independent of $\lambda)$
is invariant under $A_\lambda:=D_xF_\lambda(0)$ for every~$\lambda$,
and that
hypotheses 0), 1), 3), 4) above are satisfied uniformly
in $\lambda$.
Suppose that
the map $\lambda \mapsto D^i_xF_\lambda(0)$ is
$C^l$ for every $i\le L$.
Then it is possible to find mappings $\lambda \mapsto K_i $ and
$\lambda \mapsto R_i$ of class $C^l$.
In particular,
if $F_\lambda (x)$ is $C^r$ in $(\lambda, x)$ then it is possible to
find $K_i$ and $R_i$ to be $C^{ri}$ in $\lambda$.
\end{itemize}
\end{lem}
\begin{rk}\label{rkunderthe}
\rm{
In Remark~\ref{rkthedef} we have pointed out that,
under all hypotheses of Theorem~\ref{main1},
the nonresonances assumptions 4) of Lemma~\ref{formal} hold up to order
$r$, and not only $L$. Hence, by Lemma~\ref{formal} applied with $L$
replaced by $r$, we
can obtain polynomials $K$ of degree not larger than $r$ and
$R$ of degree not larger than $L$, such that
$F\circ K(x) = K\circ R(x) + o(x^r)$.
Therefore, under the nonresonance conditions
of Theorems~\ref{main1} and \ref{main2} and
provided that $F$ is $C^\infty $,
the calculations can be continued up to any order.
This is indeed a practical algorithm that has been implemented.
Using Lemma~\ref{almostthere} one can develop
aposteriori bounds for the computation (see Remark~\ref{rkoutofthe}).
}
\end{rk}
\begin{rk}\label{rkwewobtain}
\rm{
We will obtain $K_i$ and $R_i$ in a recursive way.
Since the solution of the recursive equations will involve a right hand
side that depends
on $N$ the nonlinear part of $F$ and we can assume that $N$
is sufficiently $C^L$ small (by the scaling procedure described above),
we can ensure that all the coefficients of
the polynomials
$K^\le$ and $R$ of degree $2$ or higher are arbitrarily close to
zero. In particular the polynomial
$K^\le$, considered as a function,
is arbitrarily close to the immersion of
$X_1$ into $X$ for any
smooth norm on functions defined in the unit ball.
Similarly, $R$ is arbitrarily close to $A_1$.
This remark plays an essential role in the calculations of the next
sections and will be used throughout.}
\end{rk}
The main ingredient in the proof of Lemma~\ref{formal} is the following:
\begin{prop}
\label{prop:induced}
Let $X$ and $Y$ be complex Banach spaces.
Denote by $\MM_n$ the space of
$n$multilinear maps on $X$ taking values in
$Y$, and by $\SS_n$ the space of symmetric
$n$multilinear maps on $X$ taking values in $Y$.
Given bounded linear operators $A: X \to X$
and $B: Y \to Y$,
consider the operators $\L _B$, $\R^k_A$ and $\L_{n,A,B}$ acting
on $\MM_n$ by
\begin{eqnarray}\label{scriptL}
(\L _B \, M)(x_1,\dots, x_n) & = & B\, M(x_1, \dots, x_n), \nonumber
\\
(\R^k_A\, M)(x_1,\dots, x_n) & = &
M(x_1,\dots x_{k1}, A x_k, x_{k+1}, \dots , x_n) , \\
(\L_{n,A,B} M)(x_1,\ldots, x_n) & = & B M( A x_1,\ldots, A x_n) \ .\nonumber
\end{eqnarray}
Note that $\L_{n,A,B}$ also acts on $\SS_n$ by the same formula.
Then,
\begin{eqnarray}\label{eq:spectra}
\Spec( \L_B, \MM_n) & \subset & \Spec(B, Y) \ ,\nonumber \\
\Spec( \R^k_A, \MM_n) & \subset & \Spec(A, X) \ , \\
\Spec( \L_{n,A,B} , \SS_n)
\subset \Spec( \L_{n,A,B} , \MM_n)
& \subset & \Spec(B, Y) \big(\Spec(A,X) \big)^n \ . \nonumber
\end{eqnarray}
In case that $X$ and $Y$ are finite dimensional
(or more generally, that the spectra of $A$ and $B$
are the closure of the set of their eigenvalues),
all inclusions in \eqref{eq:spectra} are equalities.
\end{prop}
We give a proof of Proposition~\ref{prop:induced} in
Appendix~\ref{sec:spectral},
which essentially follows the one in \cite{delaLlaveW97}.
The operators in
\eqref{scriptL} are called
Sylvester operators in
\cite{BeynK98}. Their solution
is crucial in the numerical study
of stable invariant manifolds.
We point out that there are examples arising naturally in dynamical
systems in which inclusions
\eqref{eq:spectra} are strict (see \cite{delaLlave98}).
\pf{Proof of Lemma~\ref{formal}}
We first present the proof for complex Banach spaces, where
we can use with ease spectral theory. At the end of the proof
we will discuss the changes needed to deal with real
Banach spaces.
Equating the derivatives of both sides of \eqref{conjugacy}
evaluated at zero, we obtain that
\[
AK_1 = K_1 R_1\ .
\]
We see that this equation is indeed satisfied if we choose $K_1$,
$R_1$ as in Theorem~\ref{main1},
i.e., $K_1=(\Id, 0)$ and $R_1=A_1$.
However, in general this is not the only possible choice!
Equating derivatives of order $i$ at zero in \eqref{conjugacy}
for $ i > 1$, we obtain
\begin{equation} \label{induction}
AK_i + \Gamma_i = K_i A_1^{\otimes i} + K_1 R_i\ ,
\end{equation}
where $\Gamma_i$ is a polynomial expression in
$K_j$, $R_j$ (with
$j \le i1$) and in the derivatives of $F$ at zero up to order $i$.
We study the system of equations \eqref{induction} by induction
on $i$, by considering \eqref{induction} as an equation to be solved for
$K_i$ and $R_i$ once $K_1,\dots, K_{i1}$,
$R_1,\dots, R_{i1}$ and therefore $\Gamma _i$
are known.
So, we turn our efforts into studying the solvability of \eqref{induction}
considered as an equation for $K_i$ and $R_i$
when all the other terms are known.
Taking projections into $X_1$ and $X_2$, and using the notation
$K_i^{1} = \pi_1 K_i$, $K_i^{2} = \pi_2 K_i$,
etc., we see that \eqref{induction} is equivalent to
\begin{eqnarray}
&&A_1K_i^{1}  K_i^{1} A_1^{\otimes i}  R_i
=  \Gamma_i^{1}  BK_i^{2}, \label{induction11}\\
&& A_2 K_i^{2}  K_i^{2} A_1^{\otimes i} =  \Gamma_i^{2} \ .
\label{induction12}
\end{eqnarray}
With the notation \eqref{scriptL}, equations
\eqref{induction11} and \eqref{induction12} can be written as
\begin{eqnarray}
&& \Big(\L_{A_1}  \L_{i,A_1,\Id} \Big) K_i^{1}  R_i
=  \Gamma_i^{1}  BK_i^{2} \ , \label{induction21}\\
&& \Big(\L_{A_2}  \L_{i,A_1,\Id} \Big)
K_i^{2} =  \Gamma_i^{2} \ .
\label{induction22}
\end{eqnarray}
The crux of the problem is the second equation \eqref{induction22}.
Once equation \eqref{induction22} is solved,
we see that the first one can be
solved, e.g., by taking $K_i^{1} =0$ and $R_i$ in such a way
that it matches all the rest.
The fact that \eqref{induction12}, that is
\eqref{induction22}, can be uniquely solved is a consequence of
Proposition~\ref{prop:induced}.
Indeed, since by hypothesis 3) $A_2$ is invertible,
we can write
\[
\L_{A_2}  \L_{i,A_1,\Id} =
\L_{A_2} \Big(\Id  \L_{i, A_1, A^{1}_2} \Big)
\]
Now, hypothesis 4) of Theorem~\ref{main1}
and Proposition~\ref{prop:induced} imply that both
$\L_{A_2}$ and $\Id  \L_{i, A_1, A^{1}_2}$ are invertible.
Now we turn to statement b) of the lemma.
If we add the condition
\[
\big(\Spec (A_1) \big)^j\cap \Spec (A_1) = \emptyset
\]
at the level
$i=j$, then we can choose
$R_j = 0$ and solve uniquely $K_j^{1}$ in the first equation,
\eqref{induction21}, by writing
\[
\L_{A_1}  \L_{j,A_1,\Id} =
\L_{A_1} \Big(\Id  \L_{j, A_1, A^{1}_1} \Big)
\]
and using Proposition~\ref{prop:induced} (recall that $A_1$ is invertible
since $A$ and $A_2$ are assumed to be invertible).
To prove c), we note that the above procedures ensure the
differentiability on parameters that we claimed
for the coefficients of $K^\le$ and $R$. Here we need to make the same
choices (for all the values of $\lambda$)
on which terms of equations \eqref{induction11} and
\eqref{induction12} are eliminated (see Remark~\ref{rkofcourse}).
It is clear that the mappings that
to $A_1$, $A_2$ associate
$\L_{i,A_1, A_2}$ are analytic (in fact polynomial) so
that if
$A_{1,\lambda}$, $A_{2,\lambda}$
are $C^l$ on $\lambda$ then
\[
\L_{A_{1,\lambda}},\
\L_{A_{2,\lambda}},\
\L_{i,A_{1,\lambda},A^{1}_{2,\lambda}} \mbox{ and }
\L_{i,A_{1,\lambda},A^{1}_{1,\lambda}}
\]
are also $C^l$ on parameters as maps from
$\Lambda$ to $L(\MM_i , \MM_i)$.
Therefore, if the right hand sides of
\eqref{induction11} and \eqref{induction12} are $C^l$ on parameters,
then, $K^{1}_i$, $K^{2}_i$ and $R_i$ are also $C^l$ on parameters.
Finally, in case that our Banach space is real, we can reduce
ourselves to the complex case by using the
well known device of complexification.
First note that the result will be
proved if we use, in place of $F$,
its $r$ Taylor approximation
\[F^{[\le r]}(x) =
\sum_{j = 0}^r \frac{1}{j!} D^j F(0) x^{\otimes j}\ .\]
Now, the space $X$
can be complexified
to $\tilde X = X + i X$
and the
function $F^{[\le r]}$,
since it is a polynomial,
can be complexified
to a function in $\tilde X$.
Note that the operators $\L_{i,A,B}$ behave well
under complexification, that is,
$\L_{i,\tilde A, \tilde B} =
\widetilde{ \L_{i,A,B}}$.
Then, it follows by induction that if
all the $\tilde R_i$, $\tilde K_i$ preserve the real
subspace $X$ for $i \le i_0$, then, the same is true for
$\Gamma^{1}_{i_0}$, $\Gamma^{2}_{i_0}$
and therefore is also true for $K_i$ and $R_i$
chosen according to the prescriptions we have made explicit.
\qed
As we have said in Remark~\ref{rkorderL},
even in the case that the nonresonance
assumptions were not satisfied, we could find
solutions of the recursive equations \eqref{induction11} and
\eqref{induction12} provided
that we assumed conditions on their right hand sides.
In the finite dimensional case, these conditions happen in
a set of finite codimension.
Note however that, in case that the nonresonance conditions are
not satisfied, the solution of $K_i$ that we find will not be unique.
In some cases, one is able to use this
freedom to ensure that the equations of higher order
will satisfy the solvability conditions. Hence, the
codimension of the maps possessing invariant manifolds may
be much smaller than what a naive count of parameters would give.
We will not pursue this line of research here since it seems that it
is best done for concrete examples.
\begin{rk}\label{rkofcourse}
\rm{
Concerning the dependence on parameters, since we have many
choices on which terms are eliminated with $R$ and which with $K^\le$,
we could make different choices for different
values of $\lambda$, and then obtain that
the resulting $K^\le$ and $R$ are quite irregular with respect to
$\lambda$. The regularity claimed in Lemma \ref{formal} is, of course,
for the procedure that we described in which the same
choices of which terms to eliminate are
made for all the values of $\lambda$.
}
\end{rk}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Formulation of the problem as a fixed point problem}
\label{sec:formulation}
{From} now on, $K^{\le}$ and $R$ are the polynomials of
degree not larger than $L$ obtained in Lemma~\ref{formal},
and we write
\[
K= K^{\le} + K^>\ .
\]
We will show that it is possible to find $K^>$ such that
$D^i K^> (0)=0$ for $i\le L$, and such that $K=K^{\le} +K^>$ satisfies
\eqref{conjugacy}, which can be written as
\[
AK^{\le} + AK^> + N\circ (K^{\le} +K^>) = K^{\le} \circ R + K^>
\circ R
\]
or in the equivalent form
\begin{equation}\label{transformed2}
AK^>  K^> \circ R
=  N\circ (K^{\le} + K^>)  AK^{\le} + K^{\le} \circ R \ .
\end{equation}
Note that the way that $K^\leq $ and $R$ are determined ensures that,
since $K^>$ vanishes at 0 up to order $L$, all derivatives up to
order $L$ of the right hand side of \eqref{transformed2}
also vanish at the origin.
Hence,
we will consider \eqref{transformed2} as a functional equation for $C^r$
functions $K^>$ whose first $L$ derivatives vanish at $0$.
In Appendix~\ref{sec:spectral}
we prove that it is possible to substitute the norm in $X$
by an equivalent one for which
\[
\A_1\<1\ ,\quad \B\ \mbox{ is as small as necessary, }
\]
and
\[
\A^{1}\ \A_1\^{L+1} <1 \quad\mbox{for Theorem~\ref{main1}}
\]
or
\[
\A_2^{1}\ \A_1\^{L+1} <1 \quad\mbox{for Theorem~\ref{main2}.}
\]
We use this norm throughout the rest of the paper.
By Remark~\ref{rkwewobtain}
ue know that we can assume $K^\leq$ to be arbitrarily close to
the immersion of $X_1$ into $X$, and $R$ to be arbitrarily close to $A_1$.
Note also that, by taking the scaling parameter
$\delta $ sufficiently small, we can also assume that
$\R^\delta  DR^\delta (0)\_{C^r}$
is sufficiently small in the ball of radius~1.
Hence, in order to solve equation \eqref{transformed2},
we may assume that $R$ is
approximately linear and, since $DR(0)= A_1$,
$R$ may be assumed to be a contraction
which maps the unit ball into a ball of radius smaller than~1.
We also take the scaling parameter such that
\begin{equation} \label{boundDR}
\DR\_{C^0(B_1)} \leq \A_1 \ + \ep <1 \ ,
\end{equation}
with $\ep$ small enough such that
\[
\begin{split}
\DR\_{C^0(B_1)}^{L+1}& \A^{1}\ \\
& \le
(\A_1 \ + \ep)^{L+1} \A^{1}\ \\
& < 1\ .
\end{split}
\]
Finally, from the fact that $R$ is a polynomial
of degree $L$ arbitrarily close to $A_1$,
we easily deduce that
\begin{equation}\label{Rderivatives}
D^kR^j (x) \le C_k(\A_1\ +\ep)^j
\end{equation}
with $C_k$ independent on $j$
(these bounds were already considered in page 574 of \cite{delaLlaveMM86}
and in Lemma 5.4 of \cite{BanyagaLW96}).
If we have a parameter
family that satisfies the assumptions
with uniform constants, it is possible to choose a scaling that
reaches the above smallness conditions for
all values of the parameters.
In such a case, it is easy to see that if the parameter family
is $C^l$ when $N$ is given the $C^r$ topology, so is the scaled family.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Study of the linearized problem}
\label{sec:linearized}
Since $N$ may be assumed
to be small, to solve equation \eqref{transformed2} we first study
the linear operator
\begin{equation}\label{Loperator}
\S H = AH  H\circ R
\end{equation}
acting on functions $H:B_1 \subset X_1 \longrightarrow X$,
where $B_1$ is the unit ball and $H$
belongs to the Banach space $\Gamma_{s,l} $ defined as follows.
Given a Banach space $Y$ (that in this subsection
will coincide with $X$), given $s\in \natural \cup \{ \omega \}$
and $l \in \natural $ with $s\ge l$, we consider
\begin{eqnarray*}
\Gamma_{s,l} & = &
\{ H: B_1 \subset X_1 \longrightarrow Y \mid H\in
C^s(B_1), D^kH(0) = 0 \hbox{ for } 0 \le k\le l, \\
&& \qquad \sup _{x\in B_1} \big(D^{l} H(x) /x \big) < \infty \}
\end{eqnarray*}
equipped with the norm
\[
\H\_{\Gamma_{s,l}} := \max\Big\{
\H\_{C^0(B_1) }, \dots , \D^s H\_{C^0(B_1) },
\sup _{x\in B_1} \big(D^{l} H(x) /x \big)
\Big\}
\]
if $s\in \natural$, and with the norm
\[
\H\_{\Gamma_{\omega,l}} := \D^{l+1}H\_{C^0(B_1)}
\]
if $s=\omega$. We emphasize that, in the case $s=\omega$,
we take the supremum norm on $B_1$ where, if the space
$X$ is real then $B_1$ is the unit ball of $X_1\oplus iX_1$
in the complexified space $X\oplus iX$. Note here that if $X$ is real
and a function is (real) analytic in a neighborhood of $0$ in $X$,
then it can be extended to be (complex) analytic in a ball around
$0$ in the complexified space $X\oplus iX$. Then, choosing the scaling
parameter $\delta$ small enough, we may (and do) assume that such ball
in $X\oplus iX$ is the unit ball.
It is also clear that $\D^{l+1}H\_{C^0(B_1)}$
is a norm in $\Gamma_{\omega,l}$, since these functions have all
derivatives at $0$ up
to order $l$ equal to zero.
We have that $\Gamma_{s,l}$ is a Banach space.
Note that the term
\[ \sup _{x\in B_1} \big(D^{l} H(x) /x \big) \]
included in the definition of $\ \cdot \_{\Gamma_{s,l}} $
is relevant only whenever $s=l$ and it could be omitted (since
$D^{l} H(0) = 0$) when $s>l$.
The rest of this section is dedicated to proving the following result.
Here $Y=X$ in the definition of $\Gamma_{s,L}$.
\begin{lem}\label{invertible}
Under the assumptions of Theorem~\ref{main1} and under the standing
assumptions arranged by scaling at the beginning of
Section~\ref{sec:pmain1}, if
$s\in \natural \cup \{\omega\}$ then
$\S : \Gamma_{r1,L} \longrightarrow \Gamma_{r1,L} $
is a bounded invertible operator.
Moreover, $\S^{1}\$ can be bounded by a constant independent
of the scaling parameter. Obviously, if $r=\omega $ then $\Gamma _{r1,L}
= \Gamma _{\omega ,L}$.
\end{lem}
This result is a simplified version of Theorem 5.1 in
\cite{BanyagaLW96}.
\pf{Proof of Lemma \ref{invertible}}
Note that $r1\ge L$. It is easy to verify $\S$ is a bounded
operator from $\Gamma_{r1,L}$ into itself.
Next, we need to show that given $\eta\in \Gamma_{r1,L} $,
we can find a unique $H\in \Gamma_{r1,L}$ such that
\begin{equation}\label{Sequation}
\S H = \eta .
\end{equation}
We will also see that
$\H\ _{\Gamma _{r1,L}} \leq C \\eta \ _{\Gamma _{r1,L}} $.
The equation \eqref{Sequation} for $H$ is equivalent to
\begin{equation} \label{manipulation1}
H= A^{1} H\circ R + A^{1} \eta\ .
\end{equation}
We claim that the solution of \eqref{manipulation1} is given by
\begin{equation}\label{solution}
H= \sum_{j=0}^\infty A^{(j+1)} \eta\circ R^j \ .
\end{equation}
To establish the claim, we will show that the series
in \eqref{solution} converges absolutely in $\Gamma _{r1,L}$
and that
\begin{equation}
\label{rbounds}
\sum_{j=0}^\infty \ A^{(j+1)} \eta\circ R^j \_{\Gamma _{r1,L}}
\le C\\eta \_{\Gamma _{r1,L}}, \qquad
\forall \eta \in \Gamma _{r1,L}\ .
\end{equation}
In particular,
when substituted in \eqref{manipulation1}, we can rearrange
the terms and show that $H$ is indeed a solution.
The previous fact also establishes the uniqueness of solution.
Indeed, if $\eta =0$, by \eqref{manipulation1} we have
$H=A^{1} H \circ R $, and hence $H=A^{j} H\circ R^j $ for every
$j\ge 1$. But $\A^{j} H\circ R^j \_{\Gamma _{r1,L}} \to 0$ as
$j\to \infty $, by \eqref{rbounds} applied with $\eta =H$.
We conclude that $H=0$.
We now establish \eqref{rbounds} when $n\in \natural$. Since
$\eta \in \Gamma _{r1,L}$, we have
$D^L \eta (\xi)  \le \\eta \_{\Gamma _{r1,L}} \xi$ for
$\xi\in B_1$,
and $\eta (0) = 0, \dots , D^L\eta (0)=0$. Hence, by Taylor's
formula, we have that
$\eta (y) \le C \\eta \ _{\Gamma _{r1,L}} y^{L+1}$,
for $y\in B_1$. Moreover,
$\Lip (R) \le \A_1\ + \varepsilon $ where $\varepsilon >0$ is small
by the scaling argument. We conclude
that
\[
\eta \circ R^j (x)  \le C
\\eta \_{\Gamma _{r1,L}}
(\A_1\ +\ep)^{(L+1)j} x^{L+1}\ .
\]
Since
$\A^{1}\ (\A_1\+\ep)^{L+1} <1$, we deduce that
the \eqref{solution} converges absolutely in the $C^0$~norm.
Now we turn to the estimates in $\Gamma _{r1,L}$.
By the FaadiBruno formula, we have
\begin{equation} \label{faadibruno}
D^k (\eta \circ R^j)
= \sum_{i=0}^k
\sum_{1\le k_1,\ldots,k_i\le k \atop
k_1 +\dots + k_i = k}
\sigma_{k_1,\ldots,k_i}^{i,k} ([D^i\eta] \circ R^j) \,
D^{k_1} R^j \cdots D^{k_i} R^j ,
\end{equation}
where $\sigma_{k_1,\ldots,k_i}^{i,k}$ is an explicit
combinatorial coefficient.
We recall that, by \eqref{Rderivatives},
\[
D^k R^j (x)  \le C (\A_1\ + \varepsilon ) ^j, \qquad x\in B_1\ ,
\]
where $C$ is independent of $j$. Moreover, for $i\le r1$,
we have
\begin{eqnarray}\label{etaderivatives}
 [D^i \eta ] \circ R^j(x)
& \le & C\\eta \_{\Gamma _{r1,L} }
R^j(x)^{(L+1i)_+} \nonumber
\\
& \le &
C\\eta \_{\Gamma _{r1,L} }
(\A_1\ +\ep)^{j(L+1i)_+}
x^{(L+1i)_+}
\end{eqnarray}
where we have used the notation $t_+ = \max (t,0)$ to treat
simultaneously the cases $i\le L+1 $ and $i > L+1 $.
Using these bounds and \eqref{faadibruno}, we deduce that
\begin{equation*}
\begin{split}
D^k [A^{(j+1)} \eta \circ R^j](x)  & \le
C\\eta \_{\Gamma _{r1,L} } \sum_{i=1}^k
\A^{1}\^{j} \\
& \quad \cdot (\A_1\ + \ep)^{j[(L+1i)_+ +i]}
x^{(L+1i)_+} \\
& \le C\\eta \_{\Gamma _{r1,L} }
[\A^{1}\ (\A_1\ +\ep)^{L+1}]^j
x^{(L+1i)_+} \ .
\end{split}
\end{equation*}
Taking the supremum in $B_1$ of this expression for
$k\le r1$, and also the supremum of the Lderivative divided by
$x$, we conclude \eqref{rbounds}.
This establishes the result when $r\in \natural$.
In the analytic case $r=\omega$, we first note that,
using $\eta \in \Gamma _{\omega ,L}$ and Taylor's theorem,
we have
$\D^i\eta \_{C^0(B_1)}\le C\D^{L+1}\eta \_{C^0(B_1)}
=C\\eta \_{\Gamma_{\omega,L}}$ for every $0\le i\le L+1$.
Hence, by \eqref{faadibruno} with $k=L+1$,
\[
\D^{L+1}[A^{(j+1)}\eta\circ R^j\_{C^0(B_1)} \leq
C\\eta \_{\Gamma_{\omega,L}}
[\A^{1}\(\A_1\ +\varepsilon)^{(L+1)}]^{j}
\]
with $C$ independent of $j$. We therefore
conclude \eqref{rbounds}.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Solution of the fixed point problem}
\label{sec:solution}
We want to solve equation \eqref{transformed2}, that
can be rewritten using
the operator $\S$ introduced in
\eqref{Loperator} as
\[
\S K^> =  N \circ (K^{\le} + K^>)  AK^\leq + K^{\le} \circ R \ .
\]
We have shown in
Lemma~\ref{invertible} that
$\S$ is invertible in $\Gamma _{r1,L} $.
Note that the way we determined
$K^{\le}$ and $R$ in Lemma~\ref{formal} ensures
that
$ N \circ (K^{\le} + K^>)  AK^\leq + K^{\le}\circ R$
vanishes up to order $L$ at the origin whenever
$K^>\in\Gamma _{r1,L}$.
Hence, we can rewrite $F\circ K = K \circ R$ as
\[
K^> = \Tau (K^>)\ ,
\]
where $\Tau$ is defined by
\begin{equation}\label{fixedpoint}
\Tau (K^>)=\S^{1} \big[ N\circ (K^{\le} + K^>)  AK^\leq + K^ \leq
\circ R\ \big] \ .
\end{equation}
Since we are assuming
that $N$ is $C^r$ small, it will be easy to show (this is the
content of Lemma \ref{almostthere} below) that $\Tau$ is a contraction when
$K^>$ is given the $\Gamma _{r1,L} $ topology.
This will establish Theorem~\ref{main1} for
$r\in\natural\cup \{\omega\}$, but with one less derivative in the
conclusions when $r\in\natural$. The $C^\infty$
case is treated at the end of this subsection. Finally,
for $r\in\natural$ a separate argument (given in Section~\ref{sec:sharp1})
will allow us to recover the last derivative and finish the proof of
parts a) and b) of Theorem~\ref{main1} as stated.
\begin{lem}\label{almostthere}
Under the hypotheses of Theorem~\ref{main1} and under the standing
assumptions arranged by scaling at the beginning of
Section~\ref{sec:pmain1}, if
$s\in \natural \cup \{\omega\}$ then $\Tau$
sends the closed unit ball $\bar{B}_1^{r1}$ of $\Gamma _{r1, L}$ into
itself, and it is a contraction in $\bar{B}_1^{r1}$ with the
$\Gamma _{r1,L}$ norm. Therefore $\Tau$ has a fixed point
$K^>$ in the closed unit ball of $\Gamma _{r1, L}$.
\end{lem}
Equation $K^>=\Tau (K^>)$ can also be studied
in a very concise way by appealing to the standard
implicit function theorem in Banach spaces. Even if
this leads to a shorter proof
(see, e.g., the expository work \cite{CFL00}) it gives less
differentiability for the solution. Of course, the standard proof of the
implicit function theorem reduces to a contraction mapping theorem.
\begin{rk}\label{rkoutofthe}
\rm{
Out of the proof Lemma \ref{almostthere}, we
will obtain rather explicit bounds for
the contraction constant which are rather good,
especially for analytic functions.
We can use these bounds to obtain {\it a posteriori}
estimates for the validity of numerical
calculations.
Note that from
\[
\Lip(\Tau) \le \kappa < 1
\quad\mbox{and}\quad \ \Tau(K^>_0)  K^>_0 \ \le \delta
\]
for some computed $K^>_0$, one can deduce that there exists a fixed point
$K^>$ satisfying
\[
\ K^>  K^>_0 \ \le \delta/(1 \kappa) \ .
\]
}
\end{rk}
\pf{Proof of Lemma \ref{almostthere}}
By Remark \ref{rkwewobtain}
we can assume that $K^ \leq $
is arbitrarily close to the identity and $R$ is
arbitrarily close to $A_1$.
Therefore, if $K^>$ is in the closed unit ball of
$\Gamma _{r1, L}$, the image of the unit ball $B_1$ in $X_1$ by
$K^{\le} + K^>$ is contained in the ball of radius~3.
Hence, the composition $N\circ (K^{\le} + K^>)$ is well defined
and of class~$C^{r1}$.
We first show the contraction property for $\Tau$. For this,
note that
the derivative of $\Tau$ guessed by manipulating the system formally is
\begin{equation}
\label{formalderivative}
D\Tau (K^>) \Delta = \S^{1} DN \circ (K^{\le} + K^>) \Delta \ ,
\end{equation}
but the notation $D\Tau$ is only formal
and should not be
taken to imply that $D\Tau$ is a derivative in the Fr\'echet sense
(see Remark~\ref{rkthereason} below).
Nevertheless, for $K^>$ and $K^>+\Delta$ in the closed unit ball
of $\Gamma _{r1,L}$,
we trivially have the finite increments formula
\begin{eqnarray}
\label{finiteincrement}
\Tau( K^> + \Delta)  \Tau( K^>)& = &
\int_0^1 \frac{d}{ds}\Big[ \Tau (K^> + s \Delta) \Big] \, ds
\nonumber \\
& = &  \int_0^1 \S^{1}
DN \circ (K^{\le} + K^> + s \Delta) \Delta \, ds \ .
\end{eqnarray}
Taking derivatives of this expression up to order $r$
and using that $\S^{1}$ maps $\Gamma _{r1,L}$ into itself, we deduce
that $\Tau( K^> + \Delta)  \Tau( K^>)\in \Gamma _{r1,L}$. Next,
for $r\in\natural$ we take derivatives of \eqref{finiteincrement}
up to order $r1$
and their supremum in $B_1$, and also the supremum of the $L$derivative
at $x$ divided by $x$. When $r=\omega$, we simply take
the supremum of \eqref{finiteincrement} on $B_1$.
In both cases, we deduce
\[
\\Tau (K^>+ \Delta )  \Tau (K^>) \_{\Gamma _{r1,L}}
\le C\N\_{C^r} \ \Delta \_{\Gamma _{r1,L}}\ .
\]
Using that
$\N\_{C^r}$ is small, this proves the contraction property for
$\Tau$ with the $\Gamma _{r1,L}$ norm.
It remains to prove that $\Tau$ sends the closed unit ball
of $\Gamma _{r1,L}$ into itself.
For this, note that if $\K^>\_{\Gamma_{r1,L}}\le 1$ then
\[
\begin{split}
\Tau (K^>)& =\Tau (0) +\{\Tau (K^>)\Tau (0)\} \\
& =\{\S^{1}[K^\le\circ R
F\circ K^\le]\}+ \{\Tau (K^>)\Tau (0)\}\ .
\end{split}
\]
By Lemmas~\ref{formal} and \ref{invertible}, we know that
the first term in the last expression
has derivatives at~$0$ up to order $L$
equal to zero, and that it has small $\Gamma_{r1,L}$
norm after scaling. On the other hand,
in the previous proof of the contraction property we have seen that
$\Tau (K^>)\Tau (0)$ has derivatives at $0$ up to order $L$
equal to zero, and $\\Tau (K^>)\Tau (0)\_{\Gamma_{r1,L}}\le
\nu \K^>\_{\Gamma_{r1,L}}\le \nu <1$.
We conclude that $\Tau$ maps the closed unit ball of
$\Gamma _{r1, L}$ into itself.
\qed
\begin{rk}\label{rkthereason}
\rm{
The reason why
\eqref{finiteincrement} does not imply that
$\Tau$ is differentiable is because the map $K^> \mapsto
DN\circ (K^{\le} + K^>) $ in \eqref{formalderivative} could fail to be
continuous as a function of $K^>$ (some examples of this are
constructed in \cite{delaLlaveO99}).
In finite dimensional spaces, or more generally
if $D^r N$ is uniformly continuous
(e.g., if it is $C^{r + \delta}$),
it can be shown that $\Tau$ is $C^1$ on $\Gamma _{r1,L}$.
We also emphasize that the operator
$\Tau$ in \eqref{fixedpoint} is better behaved than the operator
appearing
in the usual graph transform proofs
(see Section \ref{sect:proofthm2} below and also the pedagogical
expositions in \cite{Lanford73} and \cite{Lanford83}),
which usually
involves the composition of a function with an expression
that depends on the function itself.
The graph transform operator cannot be differentiable in any $C^r$
space whenever $r\in\natural$,
the essential reason being that the model map
$K\in C^r \mapsto K\circ K\in C^r$ is not differentiable
when $r\in\natural$.
Several problems will be overcome in the proof of
Theorem \ref{main2}.
}
\end{rk}
Finally, we deal with the case $F\in C^{\infty }$. According to Lemma
\ref{almostthere}, for any $r\in \natural $ with $r\ge L+1$,
if $\N\_{C^{r}}$ is sufficiently
small there is a $C^{r1}$ invariant manifold parameterized by~$K$.
Note that
given $r$ and $r'$, if the required
smallness conditions are simultaneously verified then
the parameterization $K$ coincides
for both values (since both parameterizations
are a fixed point of the same
contraction). Looking at $F$ before the scaling is made, this
amounts to saying
that there exists $\rho _r$ such that $K$ is $C^{r1}$ in
the ball of radius $\rho_r$ of $X_1$.
In particular $K$ is defined and $C^1$ in the ball of radius $\rho _1$.
Now, the key point is that equation $F\circ K=K\circ R$ leads to
\begin{equation} \label{globalize}
K = F^{j}\circ K\circ R^j \qquad \mbox{for every }j\geq 1\ .
\end{equation}
Therefore, since $R$ is a contraction,
this equality (with $j$ large) allows to recover $K$ in the ball of
radius $\rho _1$ from $K$ restricted to the ball of radius $\rho _r$.
Therefore $K$ is $C^{r1}$ in a fixed ball, for every $r\in
\natural$. That is, $K$ is in $C^\infty$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sharp regularity}\label{sec:sharp1}
Here we improve the previous result to obtain the
$C^r$ not just $C^{r1}$
differentiability for $K^>$ claimed in Theorem \ref{main1}.
Therefore, in this section we always have $r\in \natural$.
\begin{prop}\label{differentiable}
The function $K^>$ produced in Lemma~\ref{almostthere} (under perhaps
stronger smallness conditions on $\Vert N\Vert_{C^r}$) is $C^r$.
\end{prop}
The proof of this result is based on differentiating the equation
for $K^>\in \Gamma_{r1,L}$,
\[
AK^>  K^> \circ R
=  N\circ (K^{\le} + K^>)  AK^{\le} + K^{\le} \circ R \ ,
\]
to obtain
\begin{equation} \label{lin1}
\begin{split}
ADK^>  DK^> \circ R \ DR
=  DN\circ & (K^{\le} + K^>)(DK^{\le} + DK^>) \\
&  ADK^{\le} + DK^{\le} \circ R \ DR \ ,
\end{split}
\end{equation}
with $DK^>:B_1\subset X_1\longrightarrow {\mathcal L}(X_1,X)$.
Taking $Y={\mathcal L}(X_1,X)$ in the definition of the spaces
$\Gamma_{s,l}$ in Section~\ref{sec:linearized},
we study equation \eqref{lin1} for $G:=DK^>\in \Gamma_{r2,L1}$,
that we can rewrite as
\begin{equation}
\label{lin1op}
\tilde{\S} DK^> = \tilde{\Tau} DK^> +U \ ,
\end{equation}
where
\[
\tilde{\S}G =AGG\circ R \ DR \ ,
\]
\[
\tilde{\Tau} G= DN\circ (K^{\le} + K^>) \ G
\]
and
\begin{equation}
\label{defU}
U = DN\circ (K^{\le} + K^>)DK^{\le}
 ADK^{\le} + DK^{\le} \circ R \ DR \ .
\end{equation}
Note that in the definitions of $\tilde{\Tau}$ and $U$
we take $K^>\in \Gamma_{r1,L}$ to be the solution found in the
previous section and, in this way, we look at $\tilde{\Tau}$
as a linear operator acting on $G$.
We will show that equation $\tilde{\S} G_0= \tilde{\Tau} G_0 +U$
admits a solution $G_0\in C^{r1}$. Then, by a uniqueness property
and \eqref{lin1op}, we will deduce that $DK^>=G_0\in C^{r1}$ and hence
$K^>\in C^{r}$.
The argument will be based on the following:
\begin{lem}\label{invlin}
Under the hypotheses of Theorem~\ref{main1} and under the standing
assumptions arranged by scaling at the beginning of
Section~\ref{sec:pmain1},
if $s\in \natural$ and $L1\le s \le r1$ then
$\tilde{\S}$ and $\tilde{\Tau}$ are bounded linear operators from
$\Gamma_{s,L1}$ into itself.
Moreover, taking $\Vert N\Vert_{C^r}$ sufficiently small,
$\tilde{\S}$ is invertible and
$\Vert {\tilde{\S}}^{1} \Vert\Vert\tilde{\Tau}\Vert <1$ .
\end{lem}
Using this lemma with $s=r2 \ge L1$ and with $s=r1$,
we can finish the
\pf{Proof of Proposition~\ref{differentiable}}
Since $DK^>\in \Gamma_{r2,L1}$, Lemma~\ref{invlin}
applied with $s=r2$ gives that $\tilde{\S}DK^>$ and
$\tilde{\Tau}DK^>$ also belong to $\Gamma_{r2,L1}$.
Hence, since $DK^>$ is a solution of \eqref{lin1op},
i.e. $\tilde{\S}DK^> \tilde{\Tau}DK^>=U$,
we deduce that $U\in\Gamma_{r2,L1}$ and
\[
(\Id  {\tilde{\S}}^{1}\tilde{\Tau})DK^>={\tilde{\S}}^{1}U\ .
\]
Recall that $\Vert {\tilde{\S}}^{1} \tilde{\Tau}\Vert \le
\Vert {\tilde{\S}}^{1} \Vert\Vert\tilde{\Tau}\Vert <1$
by the lemma. Hence
\begin{equation}
\label{seriesd}
DK^>=\sum_{j=0}^{\infty} ({\tilde{\S}}^{1} \tilde{\Tau})^j
{\tilde{\S}}^{1} U \qquad \mbox{in }\Gamma_{r2,L1} \ .
\end{equation}
Now, by the definition \eqref{defU} of $U$ we have that $U\in
\Gamma_{r1,L1}$. Moreover, $\Gamma_{r1,L1}\subset \Gamma_{r2,L1}$
and, by Lemma~\ref{invlin} applied now with $s=r1$,
the operators ${\tilde{\S}}^{1}$ and $\tilde{\Tau}$
send $\Gamma_{r1,L1}$ into itself. Since the series \eqref{seriesd}
is convergent in $\Gamma_{r1,L1}$, we conclude that
$DK^>\in \Gamma_{r1,L1}\subset C^{r1}$ and hence that
$K^>\in C^{r}$.
\qed
Finally we give the
\pf{Proof of Lemma~\ref{invlin}}
The statements about the operator
$\tilde{\Tau} G= DN\circ (K^{\le} + K^>)\ G$ are easily proved. Indeed,
if $G\in\Gamma_{s,L1}$ with $L1\le s\le r1$, then $\tilde{\Tau}
G\in C^s$ and its derivatives at the origin up to order $L1$ vanish.
In addition $D^{L1}(\tilde{\Tau} G)(x)/\vert x\vert$ is bounded,
and hence $\tilde{\Tau} G\in \Gamma_{s,L1}$. Moreover,
$\Vert \tilde{\Tau}\Vert$ is small if
$\Vert N\Vert_{C^r}$ is sufficiently small (just note that,
as pointed out in the previous section,
$K^\le + K^>$ remains in the ball of radius $3$ of $\Gamma_{r1,L}$
independently of the scaling).
We now study the operator $\tilde{\S}G=AGG\circ R\ DR$, which is clearly
bounded from $\Gamma_{s,L1}$ into itself. Given $\eta\in
\Gamma_{s,L1}$ we need to establish the existence and uniqueness
of a solution in $\Gamma_{s,L1}$ for the equation
\begin{equation}
\label{eqStil}
AGG\circ R\ DR=\eta\ .
\end{equation}
For this, we proceed as in the proof of Lemma~\ref{invertible}
for the operator $\S$. It is clear that the series
\[
G=\sum_{j=0}^{\infty} A^{(j+1)}\eta\circ R^j \ DR^j
\]
is formally the solution of \eqref{eqStil}. To finish the proof we only
need to show that
\begin{equation}
\label{serabs}
\sum_{j=0}^{\infty} \Vert A^{(j+1)}\eta\circ R^j
\ DR^j\Vert_{\Gamma_{s,L1}}
\le C\Vert\eta\Vert_{\Gamma_{s,L1}}
\end{equation}
for some constant $C$ independent of the scaling.
For $0\le l\le s$, we have
\begin{eqnarray}\label{sumder}
&&\vert D^l (A^{(j+1)}\eta\circ R^j \ DR^j)(x)\vert\nonumber\\
&& \qquad\le C\Vert A^{1}\Vert^{j}
\sum_{k=0}^l \vert D^k(\eta\circ
R^j)(x)\vert \vert D^{(lk)}DR^j(x)\vert \nonumber\\
&& \qquad\le C\Vert A^{1}\Vert^{j}
(\Vert A_1\Vert +\ep)^j
\sum_{k=0}^l \vert D^k(\eta \circ R^j)(x)\vert \ ,
\end{eqnarray}
by \eqref{Rderivatives}.
Since $\eta\in\Gamma_{s,L1}$, for $0\le i\le s$
we have $\vert D^i\eta (y)\vert
\le C\Vert\eta\Vert_{\Gamma_{s,L1}} \vert y\vert^{(Li)_+}$,
and hence
\[
\vert (D^i\eta)\circ R^j(x)\vert\le C\Vert\eta\Vert_{\Gamma_{s,L1}}
(\Vert A_1\Vert +\ep)^{j(Li)_+} \vert x\vert^{(Li)_+} \ .
\]
Combined with the FaadiBruno formula \eqref{faadibruno}
and the bound \eqref{Rderivatives} , this leads to
\begin{equation*}
\begin{split}
\vert D^k(\eta\circ R^j)(x)\vert & \le C\Vert\eta\Vert_{\Gamma_{s,L1}}
\sum_{i=0}^k (\Vert A_1\Vert +\ep)^{j[(Li)_++i]}\vert x\vert^{(Li)_+} \\
& \le C\Vert\eta\Vert_{\Gamma_{s,L1}} (\Vert A_1\Vert +\ep)^{jL}
\vert x\vert^{(Lk)_+} \ . \nonumber
\end{split}
\end{equation*}
Using \eqref{sumder}, we finally arrive at
\[
\vert D^l (A^{(j+1)}\eta\circ R^j \ DR^j)(x)\vert
\le C\Vert\eta\Vert_{\Gamma_{s,L1}} \left[ \Vert A^{1}\Vert
(\Vert A_1\Vert +\ep)^{(L+1)}\right]^{j} \vert x\vert^{(Ll)_+} \ .
\]
Taking the supremum of this expression in $B_1$ for $0\le l\le s$,
and also the supremum of the $(L1)$derivative divided by
$x$, we conclude \eqref{serabs}.
\qed
\section{Proof of Theorem \protect{\ref{main2}}} \label{sect:proofthm2}
Equation $F\circ K=K\circ R$ can be written in components as
\begin{eqnarray*}
A_1 K^1 + N_1 \circ K + BK^2 & = & K^1\circ R\ , \\
A_2 K^2 + N_2 \circ K & = & K^2\circ R \ .
\end{eqnarray*}
If we now decide to solve this equation by setting $K^1 = \Id$ and
then determine $R$ from the first equation, we obtain that the
second one becomes
\[
A_2 K^2 + N_2 \circ (\Id,K^2) = K^2 \circ
\Big( A_1 + N_1 \circ (\Id, K^2) + BK^2\Big)
\]
which, since $A_2$ is invertible, we can rewrite as a fixed
point problem
\begin{equation}\label{graphtransform}
K^2 (x)
= A_2^{1} \Big[ K^2 \Big(A_1 x + N_1 (x,K^2 (x)) + BK^2 (x) \Big)
 N_2 (x,K^2 (x)) \Big] \ .
\end{equation}
The reader will recognize immediately that, when $B=0$, this is the customary
functional equation that appears in the graph transform methods
(see e.g. \cite{Lanford83}).
In our case, besides including the term with $B\ne 0$,
in addition we are not
assuming that the decomposition $X= X_1\oplus X_2$ corresponds to
spectral projections in a disk and its complement.
Note that if there is any manifold tangent to $X_1$, by the implicit
function theorem it can be written in a unique way as the graph of a
function from $X_1$ to $X_2$.
Then, the usual manipulations in the graph transform method
lead to the fact that this function must satisfy
equation \eqref{graphtransform}
(see the uniqueness argument at the end of Section~\ref{subuniqueness}
below).
The study of the equation under nonresonance conditions were undertaken
for $B=0$ in \cite{delaLlave97} by reducing the nonlinear terms to a
particularly simple form.
In this paper we will follow a different route which follows closely
both numerical implementations and
the proof of Theorem \ref{main1}.
We will use, therefore, some lemmas of the previous section.
In a first step, we use Lemma \ref{formal} to show
that, under the nonresonance conditions included in the
theorem and having prescribed $K^2(0)=0$ and $DK^2(0)=0$, it is possible to
determine uniquely $D^i K^2(0)$ for
$2 \le i\le L$ from the requirement that the first $L$ derivatives at
zero of \eqref{graphtransform} have to match.
The second step shows that the fixed point equation
$K^{2} = \NN (K^{2}) $, with $\NN$ defined by
\begin{equation}\label{Noperator}
[\NN K^2 ] (x)
= A_2^{1}\Big[ K^2 \Big(A_1 x + N_1 (x,K^2(x)) + BK^2(x)\Big)
 N_2 (x,K^2 (x))\Big]\ ,
\end{equation}
has a unique solution in the class of Lipschitz functions
claimed in Theorem~\ref{main2}.
In a third step, we will show that there is a fixed point of
the operator $\NN$ of class $C^{r1}$.
We point out that, unfortunately, the operator $\NN$ defined in
\eqref{Noperator} is not differentiable acting on $C^l$ spaces.
Nevertheless, as in the methods based on
graph transform, we will be able to show that
the operator leaves invariant a set of
functions with bounded derivatives of order up to $r$
and that it is a contraction on the $\Gamma_{r1,L}$ norm there
for this we will consider the associated operator
acting on $K^{>,2}=K^{2}K^{\le,2}$, where $K^{\le,2}$ will be the polynomial
of degree $L$ given by Lemma~\ref{formal}.
A fourth step will improve the regularity of the function to
conclude that it is $C^r$.
A fifth step, that we postpone to Section~\ref{sec:dependenceCr},
will establish differentiability with respect to parameters.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Uniqueness of the solution of the fixed point equation}
\label{subuniqueness}
To simplify notation, we write $K^{2}=w:B_1\subset X_1\longrightarrow X_2$.
We need to solve equation $w=\NO (w)$,
where $\NO$ is given by \eqref{Noperator} and that,
with this notation, becomes
\begin{equation}\label{defiNO}
\NO (w)=A_2^{1}\Big[ w\circ\psi_w  N_2 \circ (\Id ,w)\Big]
\end{equation}
with
\begin{equation}\label{defipsi}
\psi_w =A_1 + N_1 \circ (\Id ,w) + Bw \ .
\end{equation}
We start solving $w=\NO (w)$ up to order $L$. In the proof
of Lemma~\ref{formal} we have seen that there exists a unique
polynomial $K^\le=\sum_{i=1}^LK_i$ of degree $L$ such that $K_1=(\Id,0)$,
$\pi_1K^\le = K^{\le ,1} =\Id$ (that is, $K^1_i=0$ for $2\le i\le L$)
and such that $F\circ K^\le (x)=K^\le\circ R(x)+o(x^L)$
for some polynomial $R$ of degree $L$ with $R_1=A_1$.
We know that, having set
$K^{\le ,1}=\Id$, $F\circ K^\le (x)=K^\le\circ R(x)+o(x^L)$
is equivalent to
$K^{\le ,2}(x)=\NO (K^{\le ,2})(x)+o(x^L)$.
Hence, setting
\[
w^\le =\pi_2K^\le = K^{\le ,2}\ ,
\]
we have:
\begin{lem}\label{formalgraph}
Assume that the nonresonance conditions of Theorem~\ref{main2} are
satisfied.
Then, there exists a unique polynomial $w^\le$ of degree $L$ such that
\begin{eqnarray}
& & w^\le (x) = \NO (w^\le) (x) + o(x^{L}) \ ,
\label{fixedorder}
\\
& & w^\le(0) = 0\ , \qquad Dw^\le (0) = 0 \ .\label{fixedorder2}
\end{eqnarray}
Moreover,
with a suitable scaling of $F$ we can get $w^\le$ as small as we want.
Finally, if $F_\lambda $ is $C^r$ in $(\lambda ,x)$, then $w^\le_\lambda $
is $C^{rL}$.
\end{lem}
Note that, in particular, all the $C^L$ functions $w$ that satisfy
\[
w(x)=\NO (w)(x)+ o(x^{L})
\] as well as
\eqref{fixedorder2} must have
their derivatives at zero up to order
$L$ equal to those of $w^\le$. Hence, defining $w^>$ by
\[
w=w^\le+w^>
\]
(where $w^\le$ is the polynomial produced in Lemma \ref{formalgraph} above),
we have that $w^>$ and its derivatives up to order $L$ vanish at $0$,
and the fixed point equation $w=\NO (w)$ becomes
\begin{equation}\label{eqwbig}
w^>=\MO (w^>) \ ,
\end{equation}
where
\begin{equation}\label{defiMO}
\MO (w^>)=w^\le + \NO(w^\le +w^>) \ .
\end{equation}
Before dealing with the existence of invariant manifolds,
we prove in this section the uniqueness statement of Theorem~\ref{main2}.
As a consequence we will also
obtain the uniqueness result of Theorem~\ref{main1}.
The key ingredient is the following:
\begin{lem} \label{lemuniqueness}
Under the hypotheses of Theorem~\ref{main2}
(and with $\B\$ and $\N\_{C^r}$ small enough after scaling),
equation $w^>=\MO (w^>)$ has at most one (in a local sense
around the origin) Lipschitz solution
$w^>:B_1\subset X_1\rightarrow X_2$ such that
$\sup_{x\in B_1} (w^>(x)/x^{L+1})< \infty $.
\end{lem}
\Proof
Consider two solutions $w_i=w^\le +w_i^>$, $i=1,2$.
Note that $w_2^>w_1^> =w_2w_1$ and $\MO (w_2^>) \MO (w_1^>) =
\NO (w_2)\NO (w_1)$. We introduce the quantity
\[
[w^>]_{L+1} = \sup _{x\in B_1}\frac{w^>(x)}{x^{L+1}} \ .
\]
Note that, if we take $\B\$ and $\N\_{C^1}$ small enough
(depending here on $[w_2^>]_{L+1}$), we have that
\begin{equation}\label{contraction}
\psi_{w_2}(x) \le (\A_1 \ + \varepsilon )x,
\end{equation}
with $\varepsilon$ small such that
$\A_2^{1}\ (\A_1\ + \varepsilon )^{L+1} <1$.
Using the bound \eqref{contraction},
and \eqref{defiNO} \eqref{defipsi},
we have:
\begin{equation*}
\begin{split}
[ \MO (w_2^>) &\MO (w_1^>) ]_{L+1} =
[\NO (w_2)\NO (w_1) ]_{L+1} \\
& \leq
\A_2^{1}\ \,\sup_{x\in B_1}
w^\le (\psi_{w_2}(x))  w^\le (\psi_{w_1}(x)) /x^{L+1} \\
&\quad + \A_2^{1}\ \,
\sup_{x\in B_1} \Big\{
w^>_2 (\psi_{w_2}(x)) w^>_1 (\psi_{w_2}(x)) \\
&\quad \quad \quad+w^>_1 (\psi_{w_2}(x))
w^>_1 (\psi_{w_1}(x)) \Big\} /x^{L+1} \\
& \quad + \A_2^{1}\ \,
\sup _{x\in B_1}
\big N_2 (x,w^\le (x) + w^>_2(x)) \\
&\quad \quad \quad N_2 (x,w^\le (x) + w^>_1(x)) \big
/x^{L+1} \\
& \le
\A_2^{1}\ (\Lip w^\le ) (\Lip N_1 + \B\ )
[w_2^> w_1^> ]_{L+1} \\
& \quad +
\A_2^{1}\ \Big( [w_2^> w_1^> ]_{L+1} \psi_{w_2}(x)^{L+1} /x^{L+1}\\
& \quad\quad + (\Lip w^>_1) (\Lip N_1 + \B\ ) [w_2^> w_1^> ]_{L+1} \Big) \\
& \quad + \A_2^{1}\ (\Lip N_2) [w_2^> w_1^> ]_{L+1} \\
& \leq \Big(\ep+\A_2^{1}\ (\A_1\+\ep)^{L+1}\Big) [w_2^> w_1^> ]_{L+1} \\
& \leq \nu [w_2^> w_1^> ]_{L+1}
\end{split}
\end{equation*}
for some constant $\nu <1$, under the smallness assumptions
(that here may depend on $\Lip w^>_1$ and also, as
already pointed out, on
$[w_2^>]_{L+1}$). This proves that $[w_2^> w_1^> ]_{L+1}=0$
and therefore that $w^>_1=w^>_2$.
\qed
\pf{Proof of the uniqueness statements in Theorems
\ref{main1} and \ref{main2}}
We start with the second part b2) of Theorem~\ref{main2}.
After a scaling, we may assume that the solution $K=(\Id,w_L+h)$ of
\eqref{conjugacy} in
the statement is defined in the unit ball $B_1$ of $X_1$.
Since $\sup _{x\in B_1}(h(x)/x^{L+1})<\infty$ and $w_L$ is
a polynomial of degree $L$, Lemma~\ref{formalgraph} and the
remarks preceding it imply that $w_L$ must coincide with the
polynomial $w^\le$ of Lemma~\ref{formalgraph}. We deduce
that $h$ is a solution of $h=\MO (h)$ recall also that
equation \eqref{conjugacy} is equivalent to the graph transform equation
when $\pi_1K=\Id$.
Now the conclusion follows from Lemma~\ref{lemuniqueness}.
We can now deduce easily that there is a unique (locally around 0)
$C^{L+1}$ invariant manifold tangent to $X_1$ at $0$, as stated
in Theorems~\ref{main1} and \ref{main2}. Indeed, let
$G=(G^1, G^2)$ be a $C^{L+1}$ parameterization of the manifold.
By the tangency condition, we must have $G= (\Id, 0) + O(x^2)$.
Therefore, we can represent uniquely (locally near zero)
the invariant manifold as a graph of a
$C^{L+1}$ function $H : U_1 \subset X_1 \longrightarrow X_2$.
For this, we simply take $H= G^2 \circ (G^1)^{1}$.
Now, the graph transform equation
\[
F^2\circ (\Id,H) = H \circ F^1 \circ (\Id,H)
\]
is a consequence of the invariance assumption. In particular,
\eqref{conjugacy} also holds for $K=(\Id, H)$ and $R=F^1\circ K$.
Hence, uniqueness of $H$, and therefore uniqueness
of the manifold, follows from the proof of b2) in
Theorem~\ref{main2}
given above. For this, we take
$w_L$ to be the Taylor expansion of degree $L$ of $H$ at $0$,
and $h=Hw_L$ which clearly satisfies
$\sup_{x\in B_1} (h(x)/x^{L+1})< \infty$.
To end the proof of uniqueness,
we note that from local uniqueness we get uniqueness,
since by formula \eqref{globalize} we can recover the whole manifold.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Existence of solution of
the fixed point equation}
\label{sec:graphtransform}
The proof will be similar to that in \cite{delaLlave97},
which in turn is somewhat related to the proofs based on
graph transform methods. The main difference with the
standard graph transform methods is that we
consider our operator acting on Banach spaces of
functions whose first derivatives are prescribed.
Another difference with \cite{delaLlave97} is
that we obtain the optimal regularity by following a
method very similar to the one that we have used in the proof of
Proposition~\ref{differentiable}.
Recall that $L\geq 1$, $r\geq L+1$ and
consider $w^>\in \Gamma_{r,L}$, the Banach space defined with
$Y=X_2$ in Section~\ref{sec:linearized}. We then have:
\begin{prop}\label{mapM}
Let $\MO$ be defined as in $\eqref{defiMO}$ and
$r\in\natural\cup\{\omega\}$.
Under appropriate smallness conditions on $\B\$
and $\N\_{C^{r}}$, the map $\MO$ sends the closed unit ball ${\bar B}_1^r$
of $\Gamma_{r,L}$ into itself,
and it is a contraction in ${\bar B}_1^r$ with
the $\Gamma_{r1,L}$ norm. That is$:$
\[
\\MO (w^>)\_{\Gamma_{r,L}}\le 1 \quad\mbox{if }
\w^>\_{\Gamma_{r,L}}\le 1
\]
and, for some constant $\nu <1$,
\begin{equation}\label{contrMO}
\begin{split}
\\MO (w^>_2) &\MO (w^>_1)\_{\Gamma_{r1,L}} \\
&\le
\nu \w^>_2w^>_1\_{\Gamma_{r1,L}}\quad\mbox{if }
\w^>_2\_{\Gamma_{r,L}}\le 1\mbox{ and }\w^>_1\_{\Gamma_{r,L}}\le 1\ .
\end{split}
\end{equation}
In particular, equation $w^>=\MO (w^>)$ admits a
$C^{r1}$ solution which belongs to the closed
unit ball of $\Gamma_{r1,L}$.
\end{prop}
{From} standard results (see \cite{Lanford73}, Lemma 2.5),
the fixed point in $\Gamma_{r1,L}$ of the previous proposition
is, indeed, $C^{r 1 + {\rm Lipschitz}}$ (due to the uniform
$\Gamma_{r,L}$ bound on the sequence of iterates). In
the next section, we will prove the $C^r$ regularity result.
\pf{Proof of Proposition \ref{mapM}}
The last statement of the proposition follows easily from the rest.
Indeed, starting with $w^>=0$, all the iterations $\MO^k(0)$ remain in
the closed unit ball of $\Gamma_{r,L}$ and hence, by \eqref{contrMO},
they converge in the $\Gamma_{r1,L}$ norm to a solution
(which belongs to the closed unit ball of $\Gamma_{r1,L}$ since
all iterations $\MO^k(0)$ stay in such ball).
Next, note that
\begin{equation}\label{Nderivative}
\begin{split}
D[\NO (w)] = & A_2^{1} (Dw \circ\psi_w) D\psi_w
 A_2^{1}D_1N_2 \circ (\Id,w)\\
& A_2^{1}[D_2 N_2\circ (\Id,w)] Dw
\end{split}
\end{equation}
and
\begin{equation} \label{psiderivative}
D\psi_w (x)= A_1 + D_1N_1(x,w(x)) + [D_2 N_1(x,w(x))+B]Dw(x) \ .
\nonumber
\end{equation}
Proceeding by induction, it is not hard to show that, for
$2\le i\le r$, we have
\begin{eqnarray}\label{derexpression}
D^i [\NO (w)] & = &
A_2^{1} (D^iw\circ\psi_w) D\psi_w^{\otimes i}
\nonumber\\
&& + A_2^{1} (Dw\circ\psi_w) [ D_2 N_1\circ(\Id,w) + B] D^i w \\
&&  A_2^{1} [D_2 N_2 \circ(\Id,w)] D^i w + V_i \nonumber
\end{eqnarray}
where $V_i$ is a polynomial expression of $w,Dw, \dots,
D^{i1}w$ involving as coefficients $A,B$ and derivatives of $N$.
All terms in $V_i$ contain at least one derivative $D^jw$
and at least one factor $D^jN$, $1\le j\le i$, or
$B$ (to see this, note that for $j\ge 2$, every term in
the expression for $D^j\psi_w$ contains at least one of such factors).
We start showing the contraction property \eqref{contrMO}.
For $w^>=w^>_1$ and $w^>+\Delta=w^>_2$ in the closed
unit ball of $\Gamma_{r,L}$,
we have
\[
\MO (w^>+\Delta)  \MO (w^>) = \NO (w+\Delta)  \NO (w)
= \int_0^1 \ \frac{d}{ds} [\NO (w+s\Delta)] \ ds \ .
\]
Since
\[
\psi_{w+s\Delta}(x)=A_1x+N_1(x,(w+s\Delta)(x))+
B[w+s\Delta](x)
\]
and
\[
\NO (w+s\Delta)(x) = A_2^{1}\Big\{
(w+s\Delta)(\psi_{w+s\Delta}(x))N_2(x,(w+s\Delta)(x))\Big\} \ ,
\]
we deduce that
\begin{equation*}
\begin{split}
\MO (w^>+\Delta) &  \MO (w^>)\\
= \int_0^1 & \, ds\ A_2^{1} \Big\{
\Delta \circ\psi_{w+s\Delta} \\
& +[D(w+s\Delta) \circ\psi_{w+s\Delta}]
[ D_2 N_1 \circ(\Id ,w+s\Delta) +B]\Delta \\
& D_2 N_2 \circ(\Id ,w+s\Delta)\Delta \Big\} \ .
\end{split}
\end{equation*}
Therefore, for $0\le i\le r1$, we have
\begin{equation}\label{derdifint}
D^i[\MO (w^>+\Delta)  \MO (w^>)] =
\int_0^1 \, ds\left\{ A_2^{1}
[(D^i\Delta) \circ\psi_{w+s\Delta}] D\psi_{w+s\Delta}^{\otimes i}
+W_i\right\}\ ,
\end{equation}
where $W_i$ is a polynomial expression in the derivatives of $\Delta$
up to order $i$.
Every term in this polynomial contains at least one derivative
of $\Delta$ up to order $i\le r1$, and at least one factor which is $B$
or a derivative of $N$ up to order $i+1\le r$. The terms also include
factors involving the derivatives of $w+s\Delta$
up to order $i+1\le r$, which are all bounded by $\w^\le\_{C^r}+1\le
\ep+1\le 2$ since
we are assuming $\w^>\_{\Gamma_{r,L}}\le 1$ and
$\w^>+\Delta\_{\Gamma_{r,L}}\le 1$.
We use the notation $\I=\MO (w^>+\Delta)  \MO (w^>)$.
For $r\in\natural$, from \eqref{derdifint} and
$D^L\Delta(y)\le \\Delta\_{\Gamma_{r1,L}}y$ if $y\in B_1$,
we deduce for $x\in B_1$
\begin{equation}\label{derLbound}
D^L\I(x)/x\le [\A_2^{1}\(\A_1\+\ep)^{L+1}+\ep]
\\Delta\_{\Gamma_{r1,L}}
\le \nu\\Delta\_{\Gamma_{r1,L}}
\end{equation}
for some constant $\nu <1$. For $L+1\le i\le r1$ (in case that $L+1\_{\Gamma_{r,L}}\le 1$ then
\[
\begin{split}
\MO (w^>) & =\MO (0) +[\MO (w^>)\MO (0)] \\
& =[\NO(w^\le)w^\le]+ [\MO (w^>)\MO (0)]\ .
\end{split}
\]
By Lemma~\ref{formalgraph}, we know that $\NO(w^\le)w^\le$
has derivatives at $0$ up to order $L$
equal to zero, and that $\\NO(w^\le)w^\le\_{\Gamma_{r1,L}}$
is small after scaling. On the other hand,
in the previous proof of the contraction property we have seen that
$\MO (w^>)\MO (0)$ has derivatives at $0$ up to order $L$
equal to zero, and that $\\MO (w^>)\MO (0)\_{\Gamma_{r1,L}}\le
\nu \w^>\_{\Gamma_{r1,L}}\le \nu$.
Adding these two bounds, we have
$\\MO (w^>)\_{\Gamma_{r1,L}}\le \ep +\nu$.
Finally, when $r\in\natural$, \eqref{derexpression} leads to
\begin{equation*}
\begin{split}
\D^r \MO (w^>)\_{C^0} &\le \D^rw^\le\_{C^0} + \D^r \NO (w)\_{C^0} \\
& \le\ep+[\A_2^{1}\(\A_1\+\ep)^{L+1}+\ep]\D^rw\_{C^0} \\
& \le \ep+\nu(\D^rw^\le\_{C^0}+\w^>\_{\Gamma_{r,L}}) \\
&\le \ep +\nu (\ep+1) \ .
\end{split}
\end{equation*}
Taking $\ep$ small enough, we conclude that
$\MO (w^>)\in\Gamma_{r,L}$ and
$\\MO (w^>)\_{\Gamma_{r,L}} \le 1$.
\qed
When $r=\omega$, the previous proposition establishes
statement a) of Theorem~\ref{main2}. The proposition also leads to
the result for $r=\infty$, using the argument presented
for Theorem~\ref{main1} at the end
of Section~\ref{sec:solution}. Finally, next section recovers the
last derivative when $r\in\natural$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Sharp regularity}\label{sec:sharp2}
To establish the optimal $C^r$ regularity
we will use a method very similar to the one of
Section~\ref{sec:sharp1} for Theorem~\ref{main1}.
In this section we always have $r\in \natural$.
\begin{prop}\label{Cr2}
The solution $w^>$ produced in
Proposition~\ref{mapM} (under perhaps stronger
smallness conditions on $\B\$ and $\Vert N\Vert_{C^r}$) is $C^r$.
\end{prop}
As in Section~\ref{sec:sharp1}, to establish this result we
differentiate the equation $w^>=\MO (w^>)$ to find a fixed
point equation for the unknown $H^>:=Dw^>:B_1\subset X_1\longrightarrow
{\mathcal L}(X_1,X_2)$, in which we consider $w^>\in C^{r1}$
(the solution of Proposition~\ref{mapM}) as a given function
(and not as an unknown) whenever it appears in the equation for $H^>$
without being differentiated.
This ``freezing'' technique will make the fixed point equation for
$H^>$ simpler than the equation of the preceding
section for $w^>$. Indeed, here we will obtain a bilinear equation for
$H^>$.
Differentiating the equation $w=\NO (w)$ satisfied by the solution
$w=w^\le+w^>\in C^{r1}$ of Proposition~\ref{mapM} and using
\eqref{Nderivative}, we obtain
\[
Dw= A_2^{1} \Big\{ (Dw \circ\psi_w) D\psi_w
 D_1N_2 \circ (\Id,w)  [D_2 N_2\circ (\Id,w)] Dw \Big\} \ ,
\]
where
\begin{equation} \label{psidef}
\psi_w (x)= A_1x + N_1(x,w(x)) + Bw(x)
\end{equation}
and
\[
D\psi_w (x)= A_1 + D_1N_1(x,w(x)) + [D_2 N_1(x,w(x))+B]Dw(x) \ .
\]
We obtain that $H_0:=Dw$ is a solution of
\[
H=\NT (H)\ ,
\]
where we define, with $\psi_w$ given by \eqref{psidef},
\begin{equation}\label{defNT}
\begin{split}
\NT (H)(x) = & A_2^{1} \Big\{ H(\psi_w(x)) \Big[ A_1 + D_1N_1(x,w(x))
\\ &+ \Big( D_2 N_1(x,w(x))+B\Big) H(x) \Big] \\
&  D_1N_2(x,w(x))  D_2 N_2 (x,w(x)) H(x) \Big\} \ ,
\end{split}
\end{equation}
where we emphasize that here $w$, and hence $\psi_w$, are fixed
(hence independent of $H$) and given by the solution $w=w^\le+w^>$
of Proposition~\ref{mapM}.
Let us consider $H^\le=Dw^\le$ fixed,
and write $H=H^\le+H^>$. We need to solve the
fixed point equation $H^>=\MT (H^>)$, where
\begin{equation}\label{definnt}
\MT (H^>)=H^\le +\NT (H^\le+H^>)\ .
\end{equation}
We take $H^>:B_1\subset X_1\longrightarrow {\mathcal L}(X_1,X_2)=Y$
in the space $\Gamma_{s,L1}$ defined in Section~\ref{sec:linearized},
with either $s=r2$ or $s=r1$.
We know that $H^>_0:=Dw^>\in\Gamma_{r2,L1}$ is a solution of
$H^>=\MT (H^>)$. It is easy to check that $\H^>_0\_{\Gamma_{r2,L1}}
\le 1$, since we know that $\w^>\_{\Gamma_{r1,L}}\le 1$.
Proposition \ref{Cr2} will follow easily combining this fact and the
following result.
\begin{lem}\label{contrMT}
Under appropriate smallness conditions on $\B\$ and $\N\_{C^{r}}$,
if $r2\le s\le r1$ then $\MT$ maps the closed unit ball
${\bar B}_1^s$ of
$\Gamma_{s,L1}$ into itself,
and it is a contraction in ${\bar B}_1^s$
with the $\Gamma_{s,L1}$ norm.
\end{lem}
Applying this lemma with $s=r1$ we obtain a solution
$H^>_1\in\Gamma_{r1,L1}$ such that $\H^>_1\_{\Gamma_{r1,L1}}\le 1$.
In particular, $\H^>_1\_{\Gamma_{r2,L1}}\le 1$ and therefore,
by the contraction property of the lemma with $s=r2$,
both solutions $H^>_0$ and $H^>_1$ in the closed unit ball of
$\Gamma_{r2,L1}$ must agree. Hence $Dw^>=H^>_0=H^>_1\in C^{r1}$,
and we conclude that $w^>\in C^{r}$.
\pf{Proof of Lemma~\ref{contrMT}}
First, note that $\NT$ and $\MT$ map functions of class $C^s$
into functions of class $C^s$ whenever $s\le r1$.
We start showing the contraction property.
For $H^>$ and $H^>+\Delta$ in the closed unit ball of $\Gamma_{s,L1}$,
we have
\begin{equation}\label{difint2}
\begin{split}
[\MT & (H^>+\Delta)  \MT (H^>)](x) = [\NT (H+\Delta)  \NT (H)](x) \\
& = \int_0^1 \frac{d}{ds} [\NT (H+s\Delta)(x)]\, ds \\
& = \int_0^1 \, ds\ A_2^{1}
\Big\{ \Delta(\psi_w(x)) \\
& \qquad \cdot \Big[ A_1 + D_1N_1(x,w(x))
+ \Big( D_2 N_1(x,w(x))+B\Big) (H+s\Delta) (x) \Big] \\
& \quad +(H+s\Delta)(\psi_w(x)) \Big( D_2N_1(x,w(x))+B\Big) \Delta(x) \\
& \quad D_2 N_2 (x,w(x)) \Delta(x) \Big\} \ .
\end{split}
\end{equation}
Therefore, for $0\le i\le s$, we have
\begin{equation}\label{derdifint2}
D^i[\MT (H^>+\Delta)  \MT (H^>)] =
\int_0^1 \, ds\left\{ A_2^{1}
[(D^i\Delta) \circ\psi_{w}] D\psi_{w}^{\otimes i}\otimes A_1 +U_i\right\}\ ,
\end{equation}
where $U_i$ is a polynomial expression in the derivatives of $\Delta$
up to order $i$.
Every term in $U_i$ contain at least one derivative $D^j\Delta$
and at least one factor which is $B$
or a derivative of $N$ up to order $i+1\le r$.
Using the notation $\J=\MT (H^>+\Delta)  \MT (H^>)$,
\eqref{derdifint2} and
$D^{L1}\Delta(y)\le \\Delta\_{\Gamma_{s,L1}}y$ for $y\in B_1$,
we deduce, for some constant $\nu <1$,
\begin{eqnarray}\label{derLbound2}
& D^{L1}\J(x)/x\le \A_2^{1}\(\A_1\+\ep)^{L}D^{L1}
\Delta(\psi_w(x))
+\ep\\Delta\_{\Gamma_{s,L1}} \nonumber \\
&\qquad
\le [\A_2^{1}\(\A_1\+\ep)^{L+1}+\ep] \\Delta\_{\Gamma_{s,L1}}
\le \nu\\Delta\_{\Gamma_{s,L1}} \quad \mbox{for }x\in B_1
\end{eqnarray}
and, for $L\le i\le s$ (in case $L1\_{\Gamma_{s,L1}}\le 1$ then
\begin{equation}\label{threesums}
\MT (H^>)=Dw^>+[\MT (0)\MT(Dw^>)]+[\MT (H^>)\MT (0)]\ ,
\end{equation}
since, by definition \eqref{definnt}, $Dw^>\MT(Dw^>)=Dw\NT (Dw)=0$.
Expression \eqref{threesums} is useful in order to check that
$\MT (H^>)$ has all derivatives at 0 up to order $L1$ equal to zero.
Indeed, we know that the term $Dw^>$ in \eqref{threesums}
belongs to $\Gamma_{r2,L1}$ and has small norm after scaling
(and after freezing $K^>$).
On the other hand,
the previous proof of the contraction property gives that
the two last terms of \eqref{threesums}
also belong to $\Gamma_{r2,L1}$, and have $\Gamma_{r2,L1}$norm
bounded by $\nu \Dw^>\_{\Gamma_{r2,L1}}\le \ep$
and $\nu \H^>\_{\Gamma_{r2,L1}}\le \nu$, respectively.
Adding the three bounds, we conclude
$\\MT (H^>)\_{\Gamma_{r2,L1}}\le \ep +\nu$.
This concludes the proof for the case $s=r2$.
In the case $s=r1$, since $\G\_{\Gamma_{r1,L1}}=
\max (\G\_{\Gamma_{r2,L1}},\D^{r1}G\_{C^0})$,
it only remains to prove the bound
\[
\D^{r1}[\MT (H^>)]\_{C^0}\le 1\qquad\mbox{if }
\H^>\_{\Gamma_{r1,L1}}\le 1 \ .
\]
To establish it, we use definition \eqref{definnt} and
that $H^\le =Dw^\le$ is a polynomial of degree $L1)]=D^{r1}[\NT (H)]$,
where $H=H^\le+H^>$. We also note that $\H\_{C^{r1}}\le \ep
+ \H^>\_{C^{r1}}\le \ep +1$. Hence, differentiating \eqref{defNT}
$r1$ times and using the smallness of $\B\$ and $\N\_{C^r}$,
we conclude
\begin{equation}
\begin{split}
\D^{r1}[\MT (H^>)]\_{C^0}&=\D^{r1}[\NT (H)] \_{C^0} \\
& \le \Big( \A_2^{1}\(\A_1\+\ep)^r +\ep\Big) \H\_{C^{r1}}\\
& \le \Big(\A_2^{1}\(\A_1\+\ep)^r +\ep\Big) (\ep +1) \\
&\le 1
\end{split}
\end{equation}
if we take $\ep$ small enough.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Dependence on parameters} \label{sec:dependenceCr}
The main results of this section are
Theorem~\ref{main1differentiable} and
Theorem~\ref{main2differentiable},
which imply the conclusions about smooth dependence on
parameters stated in the last parts of Theorem~\ref{main1} and of
Theorem \ref{main2}, respectively.
They provide optimal differentiability results
in certain classes of regularities to be defined below and
do not require any extra nonresonance
assumptions.
The method of proof of
Theorem~\ref{main1differentiable}
will be very similar to the one of the proof of
optimal regularity in Theorems \ref{main1}
and \ref{main2}, that is, first we prove that the $K^> $
produced in Section~\ref{sec:pmain1} has almost optimal regularity
with respect to the variables $(\lambda ,x)$, and then,
studying the equation which the derivative
$D_xK^>$ satisfies, we get the optimal regularity for $K^>$
stated in the Theorems.
This kind of proof does not work for
Theorem~\ref{main2differentiable}
because the parameter version of the operators
$\NO$ and $\MO$ used in the proof of Theorem~\ref{main2}
are not contractive in the usual norms of the spaces
of differentiable functions. However, it is possible
that a clever choice of a norm with different weights
for the partial derivatives
$D^i_\lambda D^j_x$
would make $\MO$ a contraction, but we have not explored
systematically this possibility.
The proof of Theorem~\ref{main2differentiable} follows a
different route than the one outlined above.
We hope that having two methods of proof available
may help to study other theorems.
In the proof of Theorem~\ref{main2differentiable} that
we present,
we inductively prove that the
derivatives in the parameter and in the variable exist
and are continuous and bounded,
provided that derivatives of {\em lower order} exist
and are continuous and bounded.
The precise definition of derivatives of
lower order is somewhat intricate
(they are derivatives on two variables, so the ordering
has to involve two indices)
and will be motivated by the induction proof and the structure
of the operator $\MO$, which involves composition in the $x$ variable.
That is, we will define a suitable order on the sets of
indices of derivatives for functions of two variables, in
such a way that we can use the functional
equation to express derivatives of
a certain index in terms of derivatives with indices
which are smaller in the indicated order.
First we will show that
whenever $N$ has enough partial derivatives,
we can express
$D^i_\lambda D^j_x\MO(w^>)$
as
\[
D^i_\lambda D^j_x\MO(w^>)
=
\A(w^>) D^i_\lambda D^j_x w^> + \B(w^>)\ ,
\]
where $\A(w^>)$ is a linear operator and $\B(w^>)$ is an expression
that involves only derivatives of lower order (indeed, this calculation
will motivate the definition of what lower order means).
Starting with $w^>_0=0$, we consider the iteration
$w^>_n = \MO^n(0)$.
We will show
that $\A_n:=\A(w^>_n)$ is a contractive operator.
{From} this it will follow that,
when $D^{k}_\lambda D^{l}_x\MO^n(0)$
converge uniformly on compact sets as $n\to \infty $
for all $(k,l)$ of lower order
than $(i,j)$, then
$D^i_\lambda D^j_x\MO^n(0)$
also converges uniformly on compact sets.
{From} the finite increment formula
for the derivatives of the form
$D^{i1}_\lambda D^j_x w^>_n $ or
$D^{i}_\lambda D^{j1}_x w^>_n $,
we will recover the finite increment formula for their limits
as $n\to \infty $. This will give us
that the limit of $D^{i}_\lambda D^j_x w^>_n $
as $n$ tends
to infinity is indeed a
true derivative.
We will encounter the difficulty that
in infinite dimensions, continuity on compact sets
does not imply continuity and boundedness on
a ball, and we will need to use an
extra argument. We will show
that the derivative which we have obtained as a limit
satisfies a certain functional equation.
By studying this equation,
it will be possible to show that, when the derivatives of
lower order are continuous and bounded, then the limit of
$D^i_\lambda D^j_x\MO^n(0)$
is also continuous.
The induction starts with the regularity results that
we have already
proved, and stops when the conditions requiring enough
derivatives for $N$ are violated.
The initial step of the induction is
the fact (already established) that
$\MO^n(0)$ and $D_x\MO^n(0)$ converge to $w^>$ and
$D_xw^>$, respectively.
To carry on the induction process, it is important to
define spaces of regular functions
which are adapted to the argument.
These classes must require the existence of
those derivatives
needed to make the inductive argument work.
We now detail the classes which we have found useful.
We consider functions $f: \Lambda \times U \rightarrow Y$,
where $\Lambda$ and $U$ are open sets in two Banach spaces and
$Y$ is another Banach space.
We say that $f$ is $C^{l,m}$ when
$D^i_\lambda D^j_x f(\lambda,x)$
exist, are continuous and bounded for
$0 \le i \le l$, $0 \le j \le m$.
We consider the space of such functions endowed with the topology given
by the supremum of all the derivatives above.
We say that $f$ is jointly $C^r$ when it is
a $C^r$ mapping from $\Lambda \times U \rightarrow Y$
with bounded derivatives up to order $r$.
This is equivalent to the existence, continuity and
boundedness of
$D^i_\lambda D^j_x f(\lambda,x)$
for $0 \le i +j \le r$.
More generally,
if $\Sigma \subset (\Z^+)^2$ is such that
$(i,j) \in \Sigma $
and $ \tilde \imath \le i, \tilde \jmath \le j $ implies
$ (\tilde \imath, \tilde \jmath) \in \Sigma$,
we denote $C^\Sigma$ the set of
functions $f$ for which $D^i_\lambda D^j_x f $ exists, is
continuous and bounded for every $(i,j) \in \Sigma$.
We consider $C^\Sigma$ endowed with the norm
\[
 f _{C^\Sigma(B)} = \sup_{(i,j) \in \Sigma, (\lambda,x) \in B}
D^i_\lambda D^j_x f(\lambda,x)  \ ,
\]
which makes it a Banach space.
When the set $B$ is understood from the context, we
will suppress it from the notation.
It is clear that
if $\Sigma' \subset \Sigma$ then
$C^{\Sigma} \subset C^{\Sigma'}$.
In particular
\begin{eqnarray*}
C^{l+r} \subset C^{l,r} \ , \\
C^{r,r} \subset C^r\ , \\
C^{l+1,r} \subset C^{l,r}\ , \\
C^{l,r+1} \subset C^{l,r}\ ,
\end{eqnarray*}
and, except in trivial cases such as
zero dimensional spaces or $r =0 $ in the second formula,
these inclusions are strict.
Now we turn to the definition of derivatives of {\em lower order}.
The operators $\Tau$ and $\MO$ involve compositions. If $f_\lambda $
and $g_\lambda $ are families of maps, when we
take derivatives of the composition $h_\lambda = f_\lambda \circ
g_\lambda $ with respect to parameters,
we are forced to take derivatives of $f_\lambda$
with respect to the variable
(and not only with respect to the parameter).
Hence, more derivatives with respect to
the parameter require also more derivatives with respect to the
space variables. Quantitative versions of this heuristic argument
are made precise in Lemma~\ref{inverseS} and
Lemma~\ref{lem:derivadaNIW}.
This motivates the following definition:
\begin{defi}\label{Sigma}
Given $(i,j) \in (\Z^+)^2$, we define
\[
\Sigma_{i,j} = \{ (a, b) \in (\integer^+)^2 \mid \;
a+b \le i + j , \, a \le i \} \ .
\]
We say that the derivative $D_\lambda ^kD_x^l$ is of lower order
than the derivative $D_\lambda ^iD_x^j$ if $(k,l) \in \Sigma_{i,j} $.
\end{defi}
We will see that the classes
$C^{\Sigma_{i,j}}$
are well adapted to the
study of regularity with respect to parameters,
since one can use them to make induction arguments.
\begin{rk}
\rm{
As we have seen in Section~\ref{sec:pmain1}, in general
$R$ depends on $\lambda $. When we consider the equation
\[
F_\lambda\circ(K^\leq_\lambda + K^> )
= (K^\leq_\lambda + K^> ) \circ R_\lambda
\]
and we try to solve it applying the implicit function theorem,
we are led to consider the operator
$\PO : \Lambda \times \Gamma _{r,L} \longrightarrow \Gamma _{r,L}
$ defined by
\[
\PO (\lambda , K^>) =
F_\lambda\circ(K^\leq_\lambda + K^> )
 (K^\leq_\lambda + K^> ) \circ R_\lambda \ .
\]
We already know that for each $\lambda _0$ such that the hypotheses of
Theorem \ref{main1} are satisfied, there exists $K^>_{\lambda_0} $
such that $\PO (\lambda_0 , K^>_{\lambda _0}) = 0$.
Here
we are interested in the regularity of $K^>_{\lambda}$ with
respect to $(\lambda ,x)$. The following technical difficulty appears.
The map $K^> \mapsto K^>\circ R_\lambda $ can not be differentiable
from $C^r$ to $C^r$ spaces for any $r< \omega $. Therefore it is not
possible to apply the implicit function theorem in a straightforward
way.
The situation changes if it happens that $R$ does not depend on
$\lambda $. For instance this is the case
when $A_1$ does not depend on $\lambda$ and the nonresonance
assumption
\[
\big(\Spec (A_1\big))^i \cap \Spec (A_1) = \emptyset \qquad
\hbox{for every integer $i$ with } 2\le i\le L\ ,
\]
which appeared in
condition b) of Theorem~\ref{main1} (and, also in
b) of Lemma~\ref{formal}), is satisfied.
This situation occurs when we consider
only the effect of nonlinear terms in the results of
the problem.
Of course, there could be other situations when the
dependence of the coefficients of $R$ in $\lambda$ just happens to
vanish.
In this case $\PO$ can be made differentiable, of course
assuming that $F_\lambda $ is $C^s$ and $K^>$ is $C^r$ with $s>r$.
In this respect, we call attention that
\cite{delaLlaveO99} contains a
classification of the cases where the composition operator among
H\"older functions (and differentiable functions)
is differentiable.
In these cases, one can obtain regularity results very
quickly
using the implicit function theorem, but these results are
one derivative short of optimal.
}
\end{rk}
In the following we use the convention to
represent functions $G$ defined in subsets of
$\Lambda \times X_1$ or $\Lambda \times X$ by
$G_\lambda (x) = G(\lambda ,x)$.
Therefore, we will write $G$, $G_\lambda $ or $G_\lambda (x)$
according to the context. In particular, in
formulas involving composition in the $x$ variable, where
$\lambda $ acts as a parameter,
we will write $G_{\lambda }\circ H_{\lambda } $ for
$G(\lambda , H(\lambda ,\cdot))$.
For typographical reasons, when a map already has a subindex we will not
write
explicitly the dependence on $\lambda $.
That is, we will write $A_1$, $A_2$, $N_1$ and $N_2$ for
$A_{1,\lambda }$, $A_{2,\lambda }$, $N_{1,\lambda }$
and $N_{2,\lambda }$
respectively.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dependence on parameters for
Theorem \protect{\ref{main1}} } \label{sec:main1parameters}
The following is the main result of this section.
We assume that $F$ has derivatives
$D_\lambda^a D^b _x F $ with $(a,b) $ in a set $\Sigma _{i,j}$
and we get that the parameterization $K $
given by Theorem~\ref{main1}
has derivatives
$D_\lambda^a D^b _x K $ with $(a,b) $ in the same set.
Below
we will show that this immediately
implies the result on differentiability with
respect to parameters claimed in Theorem~\ref{main1}.
\begin{thm}\label{main1differentiable}
Let $\Lambda _0$ and $X$ be Banach spaces,
$\Lambda $ an open set of $\Lambda _0$, and $U$ an open set of $X$,
with $0\in U$.
Let $F:\Lambda \times U\longrightarrow X $ be a family of maps.
Assume that$:$
\begin{itemize}
\item[$1)$] For some $(i,j) \in (\Z^+)^2$,
\[
F \in C^{\Sigma_{i,j}}
\]
where $ \Sigma_{i,j}$ is the set introduced in
Definition~\ref{Sigma} and $C^{ \Sigma_{i,j}}$
denotes the set of functions
which have all the
derivatives of
orders in $ \Sigma_{i,j}$
defined, continuous and bounded.
\item[$2)$]
\begin{itemize}
\item[$2.1)$]
For all $\lambda \in \Lambda$,
$D_xF_\lambda(0)$ preserves
the space $X_1$, which is independent of $\lambda $.
\item[$2.2)$] For all values $\lambda \in \Lambda$,
conditions 0)4) of Theorem~\ref{main1}
are satisfied with uniform constants.
\end{itemize}
\item[$3)$] $j \ge L+1$.
\end{itemize}
Then, the family of maps
$K: \Lambda \times U_1 \longrightarrow X$ given by
Theorem~\ref{main1} belongs to $C^{ \Sigma_{i,j}}$.
\end{thm}
We now show that
the regularity with respect
to parameters claimed in d) of Theorem~\ref{main1}
follows from Theorem~\ref{main1differentiable}.
Indeed,
first observe that $C^r \subset C^{ \Sigma_{rL1,L+1}}
\subset C^{rL1}$.
Hence, the hypothesis of regularity on parameters in
Theorem~\ref{main1} imply that
$F \in
C^{ \Sigma_{rL1,L+1}}$ and, applying
Theorem~\ref{main1differentiable}, we obtain that
$K \in C^{ \Sigma_{rL1,L+1}} \subset C^{rL1} $.
\begin{rk}\label{rksecondcond}
\rm{
Condition 2.1) is not a serious restriction.
If $X_{1,\lambda }$ depends smoothly on $\lambda $,
one can make a
change of coordinates in such a way that $X_{1,\lambda }$
becomes independent of $\lambda $ and then apply
Theorem~\ref{main1differentiable}
to the transformed family of maps.
}
\end{rk}
To prove Theorem \ref{main1differentiable} we need a series of lemmas.
We begin by recalling that for every given value $\lambda \in \Lambda $,
the parameterization $K^>_\lambda $ given by Theorem~\ref{main1}
is the fixed point of
the operator $\Tau $ defined by
\[
(\Tau K^>)_\lambda = (\S^{1} H)_\lambda
= \sum_{k=0}^\infty
A_\lambda ^{(k+1)} H_\lambda \circ R^k_\lambda\ ,
\]
where
\[
H_\lambda = N_\lambda \circ (K^{\le}_\lambda + K^>_\lambda )
 A_\lambda K^\leq
_\lambda + K^\leq_\lambda \circ R_\lambda
\]
and $K^{\le }_\lambda $ and $R_\lambda $ are the polynomials
of degree $L$ given by Section \ref{sec:formal}.
Under the assumptions of the theorem, these polynomials
are $C^{i}$ with respect to $\lambda $.
We assume that we have scaled the map in such a way that
$\N \_{C^{\Sigma _{i,j}}}$, $\K^\le \Id\_{C^{i,j}}$ and
$\R A_1\_{C^{i,j}}$
are as small as we need, uniformly in $\lambda \in \Lambda $.
The scheme of the proof of Theorem~\ref{main1differentiable}
will be the same as the one of the proofs of existence
and sharp regularity of $K$ in Section~\ref{sec:pmain1}.
The main difference is that here we will work in spaces
of differentiable functions jointly with respect
to the space variable and the parameter.
To describe in some detail the structure of the
expression of the derivatives
of the composition in terms of the derivatives of the functions,
we introduce the following auxiliary sets of indices,
closely related to $\Sigma _{i,j}$,
\begin{equation} \label{defSigmai0}
\begin{split}
\Sigma^* _{i,0} & =
\{ (a,b) \in (\integer^+)^2 \mid \; a+b\leq i,\, b\ge 1 \}
\cup \{ (i,0) \} \ ,\\
\Sigma ^*_{i,j} & = \{ (a,b) \in (\integer^+)^2 \mid
\; a+b\leq i+j,\, a \le i,\, b\ge 1 \}
\qquad \mbox{ if } j\ge 1 \ ,
\end{split}
\end{equation}
\begin{equation}\label{defSigmaij}
\begin{split}
\tilde \Sigma _{i,0} & = \{ (a,b) \in (\integer^+)^2
\mid \; a+b\leq i \}\ , \\
\tilde \Sigma _{i,j} & = \Sigma ^*_{i,j}
\qquad \qquad \mbox{ if } j\ge 1 \ ,
\end{split}
\end{equation}
For $s\ge l$ let
\begin{equation}
\begin{split}
\H \_{\Gamma_{i,s,l}} := \max\Big\{
\max_{a,b\in \Sigma _{i,s}} &
\D_\lambda^a D^b _x H \_{C^0(\Lambda \times B_1) },
\,\\
& \max_{0\le a\le i}
\sup _{(\lambda ,x)\in \Lambda \times B_1} \big(D_\lambda^a D_x^{l}
H_\lambda (x) /x \big) \Big\}
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
\Gamma_{i,s,l} = &
\{ H : \Lambda \times B_1 \subset \Lambda _0\times X_1
\longrightarrow Y \mid H \in C^{\Sigma_{i,s}} (\Lambda \times
B_1), \\
& D_\lambda ^aD^b_x H_\lambda (0) =
0 \hbox{ for } 0 \le a\le i,\, 0 \le b\le l, \
\H \_{\Gamma_{i,s,l}} < \infty \}\ . \label{defgammabig}
\end{split}
\end{equation*}
The norm $\\cdot \_{\Gamma_{i,j,l}}$
makes
$\Gamma_{i,j,l}$ a Banach space.
As we remarked for the space $\Gamma _{i,l}$ in
Section~\ref{sec:linearized},
here the terms
$\sup _{(\lambda ,x)\in \Lambda \times B_1}
\big(D_\lambda ^aD_x^{l}
H_\lambda (x) /x \big) $ are
only relevant whenever $s=l$ and they could be omitted (since
$D^a_\lambda D_x^{l} H_\lambda (0) = 0$) when $s> l$.
Note that if $H \in \Gamma_{i,s,l} $ and $(a,b)\in \Sigma
_{i,s}$, by Taylor's theorem and the definition of
$\\cdot\_{\Gamma_{i,j,l}}$ we have
\[
D_\lambda ^aD^b_xH_\lambda(x)  \le \frac{1}{(lb)_+!}
\H \ _{\Gamma_{i,s,l}} x ^{(l b+1)_+},
\qquad (\lambda ,x) \in \Lambda \times B_1
\ .
\]
\begin{lem}
\label{inverseS} With the notation above, assuming that
$H \in C^{\Sigma _{i,j}}$,
$R \in C^{i,j}$ and $A_\lambda \in C^{i,\infty }$,
we have that
\[
D_\lambda^i D_x^j \Big[A_\lambda^{(k+1)} H_\lambda \circ
R_\lambda^k \Big]
\]
is a polynomial expression that involves only derivatives $D^m_\lambda
A_\lambda^{(k+1)} $ with $0\le m\le i$, derivatives $D_\lambda^{a} D_x^{b}
H_\lambda$ evaluated
at $R_\lambda ^k$ with $(a,b) \in \tilde \Sigma _{i,j} $,
and derivatives
$D_\lambda^{\alpha } D_x^{\beta } R^k_\lambda $
with $0\leq \alpha \leq i$, $0\leq \beta \leq j$, except
for $(\alpha
,\beta ) = (0,0)$.
Moreover, each term in the polynomial expression contains:
\begin{itemize}
\item A factor
$D^m_\lambda A_\lambda^{(k+1)} $
\item A
factor
$D_\lambda^{a} D_x^{b} H_\lambda \circ
R^k_\lambda$
\item A factor
\[D^{i_1}_\lambda D^{j_1}_x R^k_\lambda \cdots
D^{i_b}_\lambda D^{j_b} _x R^k_\lambda\]
with $j_1 + \dots + j_b = j$,
$i_1 + \dots + i_b = ma$ and
$(i_l, j_l) \ne (0,0)$ for $1\le l \le b$.
\end{itemize}
\end{lem}
\Proof
We start by computing the $m^{\rm th}$ derivative of
$H_\lambda \circ R^k_\lambda$
with respect
to $\lambda$.
Let $T_m = D_\lambda ^m [H_\lambda \circ R^k_\lambda] $.
It is easily checked by
induction that
\begin{equation}\label{highderivative}
T_m =
\sum_{(a,b) \in \Sigma ^*_{m,0}} \sum_{i_1, \dots, i_b}
C D^a_\lambda D_x^b H_\lambda \circ R^k_\lambda \,
D^{i_1}_\lambda R^k_\lambda \cdots D^{i_b}_\lambda
R^k_\lambda\ ,
\end{equation}
where $i_1 + \dots + i_b = ma$ and $C$ is a coefficient which
depends on $a,b,i_1, \dots, i_b$.
Then, we differentiate the expression
\eqref{highderivative} $j$ times with respect to $x$.
It is also easy to see by induction that
\begin{equation} \label{derjtm}
D^j_x T_m =
\sum_{(a,b) \in \Sigma ^*_{m,j}}^{} \sum_{i_1, j_1, \dots, i_b, j_b} C
D^a_\lambda D_x^b H_\lambda \circ R^k_\lambda \,
D^{i_1}_\lambda D^{j_1}_x
R^k_\lambda \cdots D^{i_b}_\lambda D^{j_b}_x R^k_\lambda\ ,
\end{equation}
where $i_1 + \dots + i_b = ma$,
$j_1 + \dots + j_b = j$ and $C$ is a combinatorial coefficient depending
on the indices.
Finally,
\begin{equation} \label{derijatm}
D^i_\lambda D_x^{j} [ A^{(k+1)}_\lambda H_\lambda \circ R^k_\lambda]
= \sum_{m=0}^{i} C [D^{im}_\lambda A^{(k+1)}_\lambda] D^j_x T_m \ .
\end{equation}
Substituting \eqref{derjtm} into \eqref{derijatm}, the resulting
expression contains
derivatives $D^a_\lambda D^b_xH_\lambda $ of orders
$(a,b)\in \Sigma ^*_{0,j} \cup \Sigma ^*_{1,j} \cup \dots \cup
\Sigma ^*_{i,j}
= \Sigma ^*_{i,j} = \tilde \Sigma _{i,j}$
if $j > 0$, or $\{ (a,b) \mid \, a+b \leq i \}
= \tilde \Sigma _{i,0}$ if $j=0$.
\qed
\begin{lem}\label{invertibleparam}
Let $R $ be the map given in Lemma~\ref{formal}.
Assuming that $R \in C^{i, \infty }$,
$A_\lambda \in C^{i, \infty }$ and that $j\ge L$, then
$\S: \Gamma_{i,j,L} \longrightarrow \Gamma_{i,j,L} $
defined by
$(\S H)_\lambda = A_\lambda H_\lambda
H_\lambda \circ R_\lambda $
is an invertible operator,
and $\\S^{1}\$ can be bounded by a constant independent
of the scaling parameter.
\end{lem}
\Proof
The formal
expression for $\S^{1}$ is the same to the one
we have obtained in
Lemma~\ref{invertible}:
\begin{equation} \label{inversaparam}
(\S^{1} \eta) _\lambda = \sum_{k=0}^{\infty }
A_\lambda^{(k+1)} \eta _\lambda \circ R_\lambda^k \ .
\end{equation}
Now we have
to see that the series \eqref{inversaparam}
converges in the norm of $\Gamma_{i,j,L} $ and that
\[
\ \sum_{k=0}^{\infty }
A_\lambda^{(k+1)} \eta _\lambda \circ R_\lambda^k \ _{\Gamma_{i,j,L} }
\le C \\eta \ _{\Gamma_{i,j,L} }\ .
\]
We begin by showing that if
$\eta \in \Gamma_{i,j,L}$ and
$(a,b)\in \Sigma _{i,j}$ then
\begin{equation} \label{sumaderivades}
\sum_{k=0}^{\infty }
D_\lambda^a D_x^b \Big[A_\lambda^{(k+1)} \eta _\lambda \circ
R_\lambda^k \Big] (x)
\end{equation}
converges absolutely.
It is not hard to prove that
\begin{equation} \label{boundAparam}
\ D^m_\lambda A_\lambda ^{(k+1)} \
\le C_m (k+1)^m \ A_\lambda ^{1}\^{k+1}
\end{equation}
and that
\begin{equation} \label{boundRparam}
D^m_\lambda D^l_x R^k_\lambda (x)  \leq C_{m,l} (\A_1\ +
\varepsilon )^k x^{(1l)_+}\ .
\end{equation}
By Lemma \ref{inverseS}
every term in the sum \eqref{sumaderivades} is bounded by a sum
of terms of the form
\begin{equation} \label{longbound}
C \ D_\lambda ^{am} A^{(k+1)}_\lambda \\,
 D^\alpha _\lambda D_x^\beta \eta _\lambda (R^k_\lambda (x))  \,
 D^{i_1}_\lambda D^{j_1}_x
R^k_\lambda (x)  \cdots  D^{i_\beta }_\lambda D^{j_\beta }_x
R^k_\lambda (x) 
\end{equation}
with $0\le m\le a$,
$(\alpha ,\beta ) \in \tilde \Sigma _{m,b}$,
$i_1 + \dots + i_\beta = m\alpha $ and $j_1 + \dots + j_\beta = b$.
Hence,
every term in the sum \eqref{sumaderivades}
is bounded by
\begin{equation*}
\begin{split}
C (k+1)^{am} &
\ A^{1}_\lambda \^{k+1} \ \eta \_{\Gamma _{i,j,L}}
 R^k_\lambda (x)  ^{(L\beta + 1)_+ } (\ A_1\ +
\varepsilon )^{k\beta } x^{(\beta b)_+} \\
&\leq
C \ \eta \_{\Gamma _{i,j,L}}
(k+1)^{a} \ A^{1}_\lambda \^{k+1}
(\ A_1\ + \varepsilon )^{k(L\beta +1)_+}  x ^{(L\beta +1)_+}
\\
& \quad \quad \cdot
(\ A_1\ + \varepsilon )^{k\beta } x^{(\beta b)_+}
\\
& \leq
C \ \eta \_{\Gamma _{i,j,L}}
(k+1)^{a}
\Big( \ A^{1}_\lambda \ (\ A_1\ + \varepsilon )^{L+1} \Big)^k
 x ^{(Lb +1)_+}
\end{split}
\end{equation*}
with $C$ independent of $k$,
where we have used
\begin{equation*}
\begin{split}
(1j_1)_+ + (1j_2)_+ + \dots + (1j_\beta )_+
&\ge (\beta ( j_1 + \dots + j_\beta )) _+ = (\beta b)_+ \ ,\\
(L\beta +1)_+ + \beta & \ge L +1,\\
(L\beta +1)_+ + (\betab)_+ &\ge (Lb +1)_+
\end{split}
\end{equation*}
which hold because the function $(\cdot)_+$ is subadditive.
Since
$\sum_{k=1}^{\infty } (k+1)^a
\Big( \ A^{1}_\lambda \ (\ A_1\ + \varepsilon )^{L+1} \Big)^k
$
converges, we deduce that \eqref{sumaderivades} converges absolutely in
$\Lambda \times B_1$.
Hence, in the same way as in Lemma~\ref{invertible},
we conclude that
the expression \eqref{inversaparam} defines a function $H $
in $C^{\Sigma_{i,j}} $ which satisfies
$\S H = \eta $.
Moreover
\[
D^a_\lambda D^b _x (\S^{1} \eta) _\lambda (x)  \le
C \ \eta \_{\Gamma_{i,j,L}} x^{(Lb+1)_+} \ ,
\]
and then, clearly, $\\S^{1} \eta \_{\Gamma_{i,j,L}} \le
C \ \eta \_{\Gamma_{i,j,L}} $. \qed
\begin{lem}\label{almostthereparam} Under the hypotheses
of Theorem~\ref{main1differentiable} and
assuming that the norm $\N\_{C^{\Sigma _{i,j}}}$ is sufficiently
small in the ball of radius~$3$, we have that
$\Tau: \Gamma_{i,j1,L} \longrightarrow \Gamma_{i,j1,L}$
sends the closed unit ball $\overline{B}^{j1}_1$
of $\Gamma_{i,j1,L} $
into itself, and it is a contraction there.
\end{lem}
\Proof
As we remarked before, we can
assume that $K^ \leq $
is arbitrarily close to the identity and $R $ is
arbitrarily close to $A_1$, uniformly in
$\lambda \in \Lambda $.
In the same way as we did in the proof of Lemma~\ref{almostthere},
one checks that
$\Tau$ maps the unit ball of
$\Gamma _{i,j1, L}$ into itself.
Indeed, let $K^> \in \overline{B}^{j1}_1$. We claim that
$\H \_{\Gamma _{i,j1,L}}$ can be made arbitrarily small,
taking $\N \_{C^{\Sigma _{i,j}}}$ sufficiently small
and therefore $\\Tau (K^>)\_{\Gamma _{i,j1,L}} \le 1$.
Indeed, we already know that for all $\lambda \in \Lambda $,
$D_x^bH_\lambda (0) = 0$ for $0\le b\le L$
and then the $D^a_\lambda D^b _x$ derivative of $H$ at $x=0$
will be zero for
$0\le a \le i$, $0\le b \le L$.
We are going to bound $D^a_\lambda D^b _x H_\lambda(x) $ for $(a,b)\in
\Sigma _{i,j}$. In the case $b\le L$, by Taylor's theorem we have
\[
D^a_\lambda D^b _x H_\lambda (x) \le
C \D^a_\lambda D^L _x H \_{C^0} \, x^{Lb} \ .
\]
We decompose $H_\lambda $ as $H_\lambda = H^1_\lambda + H^2_\lambda $
with
\[
H^1_\lambda =
N_\lambda \circ K^{\le}_\lambda  A_\lambda K^\leq
_\lambda + K^\leq_\lambda \circ R_\lambda
\]
and
\[
H^2_\lambda =
N_\lambda \circ (K^{\le}_\lambda + K^> _\lambda )
+ N_\lambda \circ K^{\le}_\lambda \ .
\]
Using the previous expressions and that $R $ is a polynomial
of degree $L$ in $x$, we have
\begin{equation} \label{closedball1}
D^a_\lambda D^L _x H^1_\lambda (x)  \le
\ D^a_\lambda D^{L+1} _x
[
N_\lambda \circ K^{\le}_\lambda
+ (K^\leq_\lambda \Id) \circ R_\lambda
] \_{C^0} \, x
\end{equation}
and
\begin{eqnarray}
D^a_\lambda D^L _x H^2_\lambda & = &  \int_{0}^{1}
\frac{d\, }{ds}\Big\{
D^a_\lambda D^L _x [N_\lambda \circ
(K^{\le}_\lambda + sK^> _\lambda ) ]
\Big\} \, ds \nonumber \\
& = & 
\int_{0}^{1}
D^a_\lambda D^L _x [D_xN_\lambda \circ
(K^{\le}_\lambda + s K^> _\lambda ) \, K^> _\lambda )]
\, ds \ . \label{closedball2}
\end{eqnarray}
In the case $b>L$ we write
\begin{equation} \label{closedball3}
D^a_\lambda D^b _x H_\lambda
= D^a_\lambda D^b _x
[N_\lambda \circ (K^{\le}_\lambda + K^> _\lambda )
+ (K^\leq_\lambda \Id) \circ R_\lambda ] \ .
\end{equation}
The smallness assumptions on $N $ and
$K^{\le }  \Id$ together with the fact
that $K^> \in \Gamma _{i,j1,L}$ give that
\eqref{closedball1}, \eqref{closedball2}
and \eqref{closedball3} are small and hence
$D^a_\lambda D^b _x H_\lambda (x) $ and
$\sup _{(\lambda ,x)} D^a_\lambda D^L _x H_\lambda (x)/x$
are also small.
As in Lemma~\ref{almostthere}, to compute the Lipschitz constant
of $\Tau$ we write
\begin{equation} \label{finiteincrementparam}
\begin{split}
(\Tau( K^> + \Delta) & \Tau( K^>))_\lambda \\
& =
 \int_0^1 \S^{1}
DN_\lambda \circ
(K_\lambda ^{\le} + K^>_\lambda + s \Delta_\lambda ) \Delta _\lambda \,
ds \ .
\end{split}
\end{equation}
Taking the $D^a_\lambda D^b _x$ derivative of the previous
expression with indices $(a,b)$ in $\Sigma _{i,j1}$, we easily see that
\[
\D^a_\lambda D^b _x [\Tau( K^> + \Delta)  \Tau( K^>) ] \_{C^0}
\le C \\S^{1} \ \D N\_{C^{\Sigma _{i,j1}}} \cdot
\\Delta \_{C^{\Sigma _{i,j1}}} \ .
\]
When $b=L$, note that the $D^a_\lambda D^b_x $ derivative of
$DN_\lambda \circ (K_\lambda ^{\le} + K^>_\lambda
+ s \Delta_\lambda ) \Delta_\lambda $
is a sum of terms, each one having a derivative of $\Delta $
with respect to $x $ of order $\beta \le L$, which is bounded
by $Cx^{L\beta +1}$.
Then the supremum of the $D^a_\lambda D^b_x $ derivative
divided by $x$ is less than
$C\D_xN \_{C^{\Sigma _{i,j1}}} $.
By the smallness requirement on $N $
the map $\Tau$ can be made a contraction in
$\overline{B}^{j1}_1$.
\qed
As a consequence of Lemma \ref{almostthereparam}, $\Tau $ has a unique
fixed point $K^> $ in the unit ball
of $\Gamma _{i,j1,L}$. Since, for any $\lambda \in \Lambda $ fixed,
the $K^>_\lambda $ obtained in Lemma~\ref{almostthereparam} belongs
to the closed unit ball of $\Gamma _{i+j1, L}$, by uniqueness
it must
coincide with the $K^>$ obtained in
Section~\ref{sec:formulation}.
Now we are going to improve the differentiability
conclusions from $C^{\Sigma _{i,j1}}$ to $C^{\Sigma _{i,j}}$.
We follow essentially the same arguments that we used in
Section~\ref{sec:sharp1}.
We have that $D_xK^> $ satisfies the following equation, which
is the parameter version of \eqref{lin1}:
\begin{equation} \label{lin1param}
\begin{split}
A_\lambda D_xK^>_\lambda & D_xK^>_\lambda \circ R_\lambda \
D_xR_\lambda \\
= &  D_xN_\lambda \circ (K^{\le}_\lambda + K_\lambda ^>)
(D_xK^{\le}_\lambda + D_xK_\lambda ^>) \\
& A_\lambda D_xK^{\le}_\lambda +
D_xK^{\le}_\lambda \circ R_\lambda \ D_xR_\lambda \ .
\end{split}
\end{equation}
We also need the parameter version of the operators $\tilde
\S$, $\tilde \Tau$ and $U$:
\[
(\tilde \S G)_\lambda = A_\lambda G _\lambda
 G_\lambda \circ R_\lambda \ D_xR_\lambda \
, \]
\[
(\tilde \Tau G)_\lambda =
 D_xN_\lambda \circ (K^{\le}_\lambda + K_\lambda ^>) G_\lambda \ ,
\]
and
\[
U_\lambda =  D_xN_\lambda \circ (K^{\le}_\lambda + K_\lambda ^>)
D_xK^{\le}_\lambda
 A_\lambda D_xK^{\le}_\lambda + D_xK^{\le}_\lambda \circ R
_\lambda \ D_xR_\lambda \ ,
\]
with $K^> $ being the function obtained in
Lemma~\ref{almostthereparam}, which belongs to
$\Gamma _{i,j1, L}$.
We consider $\tilde \S$ and $\tilde \Tau$ acting on the space $\Gamma
_{i,s,L1}$ defined in \eqref{defgammabig} with $Y=\L(X_1, X)$.
\begin{lem} \label{lastlemparam1}
Under the assumptions of
Theorem~\ref{main1differentiable} and under the standing
assumptions arranged by scaling at the beginning of
Section~\ref{sec:formulation},
if $s\in \natural$ and $L1\le s \le j1$ then
$\tilde{\S}$ and $\tilde{\Tau}$ are bounded linear operators from
$\Gamma_{i,s,L1}$ into itself.
Moreover, taking $\Vert N\Vert_{C^{\Sigma _{i,j}}}$ sufficiently small,
$\tilde{\S}$ is invertible and
$\Vert {\tilde{\S}}^{1} \Vert\Vert\tilde{\Tau}\Vert <1$ .
\end{lem}
\Proof
Let $G \in \Gamma _{i,s,L1}$ and $(a,b)\in \Sigma _{i,s}$.
We have that
\begin{equation} \label{Leibparam}
\begin{split}
D^a_\lambda D_x^b [N_\lambda \circ & (K^{\le}_\lambda + K^>_\lambda )
G_\lambda] \\
& = \sum_{0\le l\le a,\atop 0\le m\le b} C D^l_\lambda
D_x^m [N_\lambda \circ (K^{\le}_\lambda + K^>_\lambda )] D^{al}_\lambda
D_x^{bm} G_\lambda
\end{split}
\end{equation}
and
\begin{equation*}
\begin{split}
D^l_\lambda D_x^m [N
_\lambda \circ (K^{\le}_\lambda &+ K^>_\lambda )] \\
= \sum_{(\alpha ,\beta )\in \Sigma ^*_{l,m}}
\sum_{i_1+ \dots + i_\beta
= l\alpha \atop
j_1+ \dots + j_\beta = m } & C
D^\alpha _\lambda D^\beta _x N_\lambda
\circ (K^{\le}_\lambda + K^>_\lambda ) \\
& \cdot \quad D_\lambda ^{i_1} D_x^{j_1}
(K^{\le}_\lambda + K^>_\lambda ) \cdots
\quad D_\lambda ^{i_\beta } D_x^{j_\beta } (K^{\le}_\lambda + K^>_\lambda )
\end{split}
\end{equation*}
as a consequence of Faa di Bruno's formula.
Therefore, each term in \eqref{Leibparam} has a derivative
$D^\alpha _\lambda D_x^\beta N $ with $(\alpha ,\beta )\in
\Sigma _{a,b} \subset \Sigma _{i,s}$ and a derivative
$ D^\alpha _\lambda D_x^\beta G $ with
$0\le \alpha \le a $, $0\le \beta \le b$.
Note that
$
D^\alpha _\lambda D_x^\beta G_\lambda(x)  \le \frac{1}{(L\beta1 )_+!}
\G\_{\Gamma _{i,s,L1}} x^{(L \beta )_+}
$.
Then
\[
[D^a_\lambda D_x^b [N_\lambda \circ (K^{\le}_\lambda + K^>_\lambda)
G_\lambda](x)
\le C \N \_{C^{\Sigma _{a,b}}}
\G \_{\Gamma_{i,s,L1}} x^{(L b+m)_+}\ .
\]
Hence
\[
\D^a_\lambda D_x^L [N_\lambda \circ (K^{\le}_\lambda + K^>_\lambda )
G_\lambda]\_{\Gamma_{i,s,L1}}
\le C \N \_{C^{\Sigma _{i,j}}} \G \_{\Gamma_{i,s,L1}}
\]
and $\tilde \Tau$ is bounded and $\\tilde \Tau \$ is as small as we
need.
To bound
$\tilde \S$, we argue as in the proof of Lemma~\ref{invlin}.
It is clear that
the series
\[
\sum_{k=0}^{\infty} A_\lambda ^{(k+1)}
\eta_\lambda \circ R_\lambda ^k \ DR_\lambda ^k
\]
provide the formal solution of $\tilde \S H = \eta $.
To finish the proof, it is enough to check that
\begin{equation}
%\label{serabs}
\sum_{k=0}^{\infty} \Vert A_\lambda ^{(k+1)}\eta_\lambda \circ
R^k_\lambda \ D_xR_\lambda ^k\Vert_{\Gamma_{i,s,L1}}
\le C\Vert\eta \Vert_{\Gamma_{i,s,L1}}
\end{equation}
for some constant $C$ independent of the scaling.
For $(a,b)\in \Sigma _{i,s}$, we have
\begin{equation} \label{abderofcarro}
\begin{split}
D^a_\lambda D_x^b
[A_\lambda ^{(k+1)}\eta _\lambda \circ & R^k_\lambda
\ D_xR_\lambda ^k ] \\
& = \sum_{m=0}^{a} C [D^{am}_\lambda A^{(k+1)}_\lambda ]\,
D^m_\lambda D_x^b [
\eta_\lambda \circ R^k_\lambda \ D_x R_\lambda ^k ]\ .
\end{split}
\end{equation}
Using the FaadiBruno formula,
\eqref{boundAparam}, \eqref{boundRparam} and
\[
D^\alpha_\lambda D_x^\beta \eta_\lambda (R^k_\lambda (x))
\le C \\eta \_{ \Gamma_{i,s,L1} } R^k_\lambda
(x)^{(L\beta )_+}\ ,
\]
we deduce that the sum in \eqref{abderofcarro} is bounded by
\[
C \\eta \_{ \Gamma_{i,s,L1} }
\Big( \A^{1}_\lambda \ (\A_1\+\varepsilon )^{L+1} \Big)^k \ .
\]
Moreover, when $b=L1$, either $\beta \le L1$ and then
$R^k_\lambda (x)^{(L\beta )_+}$ provides a factor
$x^{(L\beta )_+}$, or else $\beta > L1$ and then,
since $j_1+\dots + j_\beta =b=L1$, at least one index $j$
of $\{j_1,\dots , j_\beta \}$
must
be zero and then
$
D^{i_1}_\lambda D^{j_1}_x R^k_\lambda \cdots
D^{i_\beta }_\lambda D^{j_\beta } _x R^k_\lambda
$
provides a factor $x$. This means that
\[
\begin{split}
D^a_\lambda D_x^{L1}
[A_\lambda ^{(k+1)}\eta\circ & R^k_\lambda
\ DR_\lambda ^k ](x) / x \\
& \le C \\eta \_{ \Gamma_{i,s,L1} }
\Big( \A^{1}_\lambda \ (\A_1\+\varepsilon )^{L+1}
\Big)^k \ ,
\end{split}
\]
which proves Lemma \ref{lastlemparam1}.
\qed
To finish the proof of Theorem~\ref{main1differentiable}, that is,
to prove that $K^{>} \in C^{\Sigma _{i,j}}$,
we use
Lemma~\ref{lastlemparam1}
exactly in the same way as
we have used Lemma~\ref{differentiable}
to prove of Proposition~\ref{sec:sharp1}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Dependence on parameters for
Theorem \protect{\ref{main2} } }
The following result
implies the result on differentiability with respect to
parameters claimed in Theorem~\ref{main2}.
\begin{thm} \label{main2differentiable}
Let $F$ be a family of maps.
Assume that$:$
\begin{itemize}
\item[$1)$] For some $(i,j) \in (\Z^+)^2$,
\[
F \in C^{\Sigma_{i,j}}
\]
where $ \Sigma_{i,j}$ is the set introduced in
Definition~\ref{Sigma} and $C^{ \Sigma_{i,j}}$
denotes the set of functions
which have all the
derivatives of indices in $ \Sigma_{i,j}$
defined, continuous and bounded.
\item[$2)$]
\begin{itemize}
\item[$2.1)$]
For all $\lambda \in \Lambda$,
$D_xF_\lambda(0)$ preserves
the space $X_1$, which is independent of $\lambda $.
\item[$2.2)$] For all values $\lambda \in \Lambda$,
conditions 0)4) of Theorem~\ref{main2}
are satisfied with uniform constants.
\end{itemize}
\item[$3)$] $j \ge L+1$.
\end{itemize}
Then, the family of maps
$K_\lambda$ given by Theorem~\ref{main2} belongs to
$C^{ \Sigma_{i,j}}$.
\end{thm}
We call attention to Remark~\ref{rksecondcond}
after
Theorem~\ref{main1differentiable} about the meaning of
2.1).
The proof of the regularity with respect
to parameters claimed in Theorem~\ref{main2}
using Theorem~\ref{main2differentiable}
is the same as the one corresponding to Theorem~\ref{main1}
using Theorem~\ref{main1differentiable} and that was
given in the previous subsection.
We consider the parameter version of the operators
$\NO$ and $\MO$ introduced in Section~\ref{sect:proofthm2},
\[
(\NO (w))_\lambda = A_2^{1} w_\lambda \circ \psi _{w_\lambda}
 A_2^{1} N_2\circ (\Id, w_\lambda )
\]
where
\[
\psi _{w_\lambda} = A_1 + N_1\circ (\Id, w_\lambda )
+ B_\lambda w_\lambda\ ,
\]
\begin{equation} \label{defNO}
(\MO (w^>))_\lambda = (\NO
(w^\le + w^>))_\lambda  w^\le _\lambda
\end{equation}
and $w^\le $ is the polynomial (in $x$) given by
Lemma~\ref{formalgraph}, which belongs to $C^{i,\infty }$.
We will write
$w_\lambda = w^\le _\lambda + w^> _\lambda$,
and $\psi _{\lambda} $ for $\psi _{w_\lambda}$.
\begin{prop}\label{Aoperatormain2}
Assume that $w^\le$ is $C^{i}$ with respect to
$\lambda $, and that $F$ and $w ^>$ belong to $C^{\Sigma _{i,j} }$.
Then,
the following formula holds$:$
\begin{eqnarray}
\lefteqn{
[D_\lambda^{i} D_x^{j} \MO ( w^>) ]_\lambda }
\nonumber \\
& = &
A_2^{1} D_\lambda^{i} D_x^{j} w^> _\lambda \circ \psi _{\lambda}
(D_x \psi _\lambda )^{\otimes j}
\nonumber \\
&& +
A_2^{1} \Big[D_x w_\lambda \circ \psi _\lambda
(D_2N_1 \circ (\Id, w_\lambda ) + B_\lambda )
 D_2N_2 \circ (\Id, w_\lambda )
\Big] D_\lambda^{i} D_x^{j} w_\lambda^>
\nonumber \\
&& + V_{i,j} (w_\lambda ^>)\ ,
\end{eqnarray}
where $V_{i,j}$ is a multilinear operator in
$D^{a}_\lambda D^{b}_x w^> $
with $(a,b) \in \Sigma _{i,j} \{ (i,j) \}$,
whose coefficients are derivatives of $N$
of orders in $\Sigma _{i,j} $, or derivatives of $A_1$ or
$B$
with respect to $\lambda $ of orders in
$\{ 0, \dots , i \}$.
\end{prop}
\Proof
The proof is an immediate consequence of the following lemmas,
which provide a quite detailed structure of the derivatives of the
composition. \qed
As in the previous subsection, here we see
that the rules for computing higher
order derivatives with respect to
parameters of a composition suggest the introduction of special sets
of indices.
We introduce two more auxiliary sets of indices
for $i \in \natural, j \ge 1$:
\begin{equation*}
\begin{split}
\Sigma ^{**}_{i,0} & = \{ (a,b,0)\in (\Z^+)^3 \mid \; a+b\leq i,\,
b\ge 1 \} \cup \{ (i,0,0) \}, \\
\Sigma ^{**}_{i,j} & =
\{ (a,b,c)
\in (\Z^+)^3 \mid
\; a+b+c \leq i+j,\, a \le i, \, c\le j, \, b\ge 1 \} \ .
\end{split}
\end{equation*}
In the rest of the section,
$C$ means a generic constant which
depends only on the map $F$. Hence, the same letter may denote
different constants in different places.
\begin{lem} \label{inverseSmain2}
Under the hypotheses of Proposition~\ref{Aoperatormain2},
with the notation above,
we have
that $ \MO (w^> )\in C^{\Sigma _{i,j}}$
and that
\[
D_\lambda^i D_x^j
[ \MO (w^> )]
\]
is an expression of the form
\begin{equation} \label{formuladerivadesN}
\begin{split}
&A_2^{1}
\sum_{(a,b)\in \Sigma^*_{i,j} }^{}
\sum_{i_1 + \dots + i_b = ia,\atop j_1 + \dots + j_b = j}
C D_\lambda ^{a} D_x ^b w_\lambda \circ \psi _\lambda
D_\lambda ^{i_1} D_x ^{j_1} \psi _\lambda
\cdots
D_\lambda ^{i_b} D_x ^{j_b}\psi _\lambda
\\
&+
(D_\lambda A_2^{1} ) D_\lambda^{i1} D_x^j [w_\lambda \circ \psi
_\lambda ]
+ \dots +
(D^i_\lambda A_2^{1} ) D_x^j [w_\lambda \circ \psi _\lambda ]
\\
& 
A_2^{1}
\sum_{(a,b,c) \in \Sigma ^{**}_{i,j}}^{}
\sum_{i_1 + \dots + i_b = ia, \atop j_1 + \dots + j_b = jc }
C D^a_\lambda D_1^c D_2^b N_2 \circ (\Id, w_\lambda )\,
D^{i_1}_\lambda D^{j_1}_x w_\lambda
\cdots D^{i_b}_\lambda D^{j_b}_x w_\lambda \nonumber \\
&  (D_\lambda A_2^{1} ) D_\lambda^{i1} D_x^j
[N_2 \circ (\Id, w_\lambda )]
 \dots
 (D^i_\lambda A_2^{1} ) D_x^j
[N_2 \circ (\Id, w_\lambda )] \\
&  D_\lambda^i D_x^j w^{\le }_\lambda \ .
\end{split}
\end{equation}
\end{lem}
\Proof
The proof of Lemma~\ref{inverseSmain2} follows
directly from the next two lemmas.
\qed
\begin{lem} \label{derivadescomposta}
Let $w \in C^{\Sigma _{i,j}} $ and $\psi \in C^{i,j}$.
Then the derivatives of
$w_\lambda \circ \psi _\lambda (x) =
w(\lambda , \psi (\lambda,x)) $ have the form
\begin{equation*}
\begin{split}
D_x^j D_\lambda ^i [w_\lambda \circ \psi _\lambda ]
= &
\sum_{(a,b)\in \Sigma ^*_{i,j}}^{}
\sum_{i_1 + \dots + i_b = ia,
\atop j_1 + \dots + j_b = j}^{} C
D_\lambda ^a D_x ^b w_\lambda \circ \psi _\lambda \, \\
& \cdot D_\lambda ^{i_1} D_x^{j_1} \psi _\lambda
\cdots
D_\lambda ^{i_b} D_x^{j_b} \psi _\lambda\ ,
\end{split}
\end{equation*}
where
$C$ is a combinatorial coefficient which depends on
$a,b$, $i_1, \dots , i_b$, $j_1, \dots, j_b$, and $\Sigma^*_{i,j} $
are defined in $\eqref{defSigmai0}$ and $\eqref{defSigmaij}$.
\end{lem}
\noindent{\bf Proof.}
The proof of this lemma is contained in the proof
of Lemma~\ref{inverseS} when one identifies
$w $ and $\psi $ with $H $ and
$R^k $ of that lemma respectively. \qed
\begin{lem} \label{lem:derivadaNIW}
Let $N \in C^{\Sigma_{i_0,j_0}}$ and
$w \in C^{\Sigma_{i_0,j_0}}$. Then we have that
$N_\lambda \circ (\Id, w_\lambda )\in C^{\Sigma_{i_0,j_0}}$, and
if $(i,j)\in \Sigma_{i_0,j_0}$
\begin{eqnarray}
\lefteqn{
D^i_\lambda D^j_x [N_\lambda \circ (\Id, w_\lambda )] } \nonumber \\
& = &
\sum_{(a,b,c) \in \Sigma ^{**}_{i,j}}^{} \sum_{i_1, j_1, \dots, i_b,
j_b} C D^a_\lambda D_1^c D_2^b N_\lambda \circ (\Id, w_\lambda )\,
D^{i_1}_\lambda D^{j_1}_x w_\lambda
\cdots D^{i_b}_\lambda D^{j_b}_x w_\lambda
\label{derivadaNIW}
\end{eqnarray}
where $i_1 + \dots + i_b = ia$,
$j_1 + \dots + j_b = jc$ and $C$ is a
coefficient depending on the indices.
The only term in the sum $\eqref{derivadaNIW}$ that contains
$D^i_\lambda D^j_xw_\lambda $ is
$D_2 N_\lambda \circ (\Id, w_\lambda )\,
D^{i}_\lambda D^{j}_x w_\lambda $.
\end{lem}
\Proof
We begin by computing the $i^{\rm th}$ derivative
of $N_\lambda \circ (\Id, w_\lambda )$ with respect
to $\lambda$, which we denote
by
$
T_i = D^i_\lambda [N_\lambda \circ (\Id, w_\lambda )]
$.
We have
\[
D^i_\lambda \Big[ N_\lambda \circ (\Id, w_\lambda ) \Big]
= \sum_{(a,b)\in \Sigma ^*_{i,0} } \sum_{}^{}
C D^a_\lambda D_2^b N_\lambda \circ (\Id , w_\lambda )
D^{i_1}_\lambda w_\lambda \cdots
D^{i_b}_\lambda w_\lambda\ ,
\]
where the second sum is taken for
$i_1 + \dots + i_b = ia$ and $C$ is a coefficient which
depends on $a,b,i_1, \dots, i_b$.
This follows from Lemma \ref{derivadescomposta}
identifying $N_\lambda (x,.)$ with
$w _\lambda $, and $w _\lambda $ with $\psi _\lambda $.
Note that $ D^{j}_x T_i =
D^i_\lambda D^j_x [N_\lambda \circ (\Id, w_\lambda )]$.
The rest of the proof is by induction in $j$.
When $j=0$ the formula \eqref{derivadaNIW} holds.
The induction step
incrementing $j$ corresponds to taking one more derivative with respect
to $x$.
Assuming that \eqref{derivadaNIW} is true for $j$,
\begin{equation*}
\begin{split}
D^{j+1}_x T_i
= &
\sum_{(a,b,c) \in \Sigma ^{**}_{i,j}}^{} \sum_{i_1, j_1, \dots, i_b,
j_b} C \Big[
D^a_\lambda D_1 ^{c+1} D_2^{b} N_\lambda \circ (\Id, w_\lambda ) \\
& \quad \quad + D^a_\lambda D_1 ^{c} D_2^{b+1} N_\lambda \circ (\Id, w_\lambda )
\, D_x w_\lambda \Big] \cdot
[D^{i_1}_\lambda D^{j_1}_x
w_\lambda \cdots D^{i_b}_\lambda D^{j_b}_x w_\lambda ] \\
& +
\sum_{(a,b,c) \in \Sigma ^{**}_{i,j}}^{} \sum_{i_1, j_1, \dots, i_b,
j_b} C D^a_\lambda D_1^c D_2^{b} N_\lambda \circ (\Id, w_\lambda )\cdot \\
& \qquad \qquad D_x [D^{i_1}_\lambda D^{j_1}_x
w_\lambda \cdots D^{i_b}_\lambda D^{j_b}_x w_\lambda ] \ .
\end{split}
\end{equation*}
{From} the previous formula we see that taking one more derivative has
the effect that each term labeled with indices $(a,b,c)$ generates
three new terms labeled with indices $(a,b,c)$,
$(a,b+1,c)$ and $(a,b,c+1)$,
except for the term with $b=0$ which only appears when $j=0$ and it
must have indices $(i,0,0)$. Such term only generates two terms
with indices $(i+1,0,0)$ and $(i,1,0)$.
For $j> 0$ we decompose
$\Sigma ^{**}_{i,j} = \tilde \Sigma ^{0}_{i,j} \cup \dots \cup
\tilde \Sigma ^{m}_{i,j} \cup \dots \cup \tilde \Sigma ^{i}_{i,j}$,
where
\[
\tilde \Sigma ^{m}_{i,j} =
\{ (m,b,c) \mid \; b+c \leq i+jm,\, c\le j, \, b\ge 1 \} \ .
\]
In this way, it is easy to see that the symbolic process
\[
(a,b,c) \mapsto (a,b,c) + (a,b+1,c) + (a,b,c+1)
\]
is exhaustive from
$\tilde \Sigma ^{m}_{i,j} $ to $\tilde \Sigma ^{m}_{i,j+1} $.
For the case $j=0$ the same argument works, but
one has to consider the term $(i,0,0)$
separately.
We note that the highest derivative of $w$ appears accompanying
$D_2 N$ (this means $a=c=0$, $b=1$)
and it is $D^{i}_\lambda D^{j}_x w $.
\qed
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
Given a subset $S\subset \Lambda \times B_1$,
we introduce the space
\begin{equation*}
\begin{split}
& \Gamma ^*_{i,j}(S) \\
& =
\{ v \in C^0( S , L^i(\Lambda _0; L^j(X_1;X_2)))\mid
\;
\sup _{(\lambda ,x)\in S, \, x\ne 0}
 v_\lambda (x) /x^{(Lj+1)_+} <\infty \}
\end{split}
\end{equation*}
with $
\ v \_ {\Gamma ^*_{i,j}(S)}
= \sup _{(\lambda ,x)\in S, \, x\ne 0}
 v _\lambda (x) /x^{(Lj+1)_+}
$
and, given a set of indices $\Sigma $, the space
\[
\begin{split}
& \Gamma (\Sigma , S) \\
&=
\{ w \in C^\Sigma(S)
\mid
\;
\sup _{(\lambda ,x)\in S, \, x\ne 0}
 D^a_\lambda D^b_x w_\lambda (x) /x^{(Lb+1)_+} <\infty,
\forall (a,b)\in \Sigma \}
\end{split}
\]
with $
\ w \_ {\Gamma (\Sigma , S)}
= \max_{(a,b)\in \Sigma}\sup _{(\lambda ,x)\in S, \, x\ne 0}
 D^a_\lambda D^b_x w_\lambda (x) /x^{(Lb+1)_+}
$.
\begin{prop}\label{inductiondifferentiablemain2}
Assume that $F \in C^{\Sigma _{i_0,j_0}} (\Lambda \times B_1)$,
with $j_0\ge L+1$, and
that $N $ has small enough $C^{\Sigma _{i_0,j_0}}$ norm.
Assume that $w^{\le} \in C^{i,\infty }$.
Let $(i,j) \in \Sigma _{i_0,j_0}$.
Let $w^>_{n}$ be the sequence of
families of functions defined by
\begin{equation} \label{defwlambdan}
w^>_{n} = \MO^n(0).
\end{equation}
Assume that
for every
$(a,b) \in \Sigma'_{i,j} \equiv \Sigma_{i,j}\{ (i,j) \}$,
\begin{itemize}
\item[$1)$] $w^>_{n}$ converges to $w^>_{\infty }$
in $\Gamma (\Sigma_{0,1} , \Lambda \times B_1) $
and
$w^>_{ \infty}\in
C^{\Sigma'_{i,j}}$.
\item[$2)$]
For each given compact set $G\subset \Lambda \times B_1$,
$w^>_{n} $ converges to
$w^>_{\infty}$
in $\Gamma (\Sigma'_{i,j} , G) $.
\end{itemize}
Then,
\begin{itemize}
\item[$a)$]
$w^>_{ \infty} \in C^{ \Sigma_{i,j}}(\Lambda \times B_1)$.
\item[$b)$]
For each given compact set $G\subset \Lambda \times B_1$,
$D^i_\lambda D^j_x w^>_{n}$ converges to
$D^i_\lambda D^j_\lambda w^>_{ \infty}$
in
$\Gamma ^*_{i,j}(G)$, and therefore
$w^>_{n} $ converges to
$w^>_{\infty}$
in $\Gamma (\Sigma_{i,j} , G) $.
\end{itemize}
\end{prop}
Before proving Proposition \ref{inductiondifferentiablemain2}
we will introduce some notation and we will establish some
preliminary lemmas.
First we remark that, since $j_0\ge 1$, if $F \in
C^{\Sigma _{i_0,j_0}}$ then $N = F  D_xF (0)
\in C^{\Sigma _{i_0,j_0}}$.
We write \eqref{defwlambdan} in inductive form,
\begin{equation} \label{inductiveformw}
w^>_{n+1} = \MO (w^>_{n} ),
\quad n\in \Z^+ ,
\qquad w^>_{0} = 0
\ .
\end{equation}
We introduce
\[
w_{n}= w^{\le} + w^>_{n}
\]
and
\[
\psi _{\lambda,n} (x) = A_1 x
+ N_1(x, w_{\lambda,n}(x) )
+ B_\lambda w_{\lambda,n}(x),
\qquad n\in \Z^+ \cup \{ \infty \}.
\]
Differentiating in \eqref{inductiveformw} we have
\begin{equation} \label{derivadeswlambdan}
D_\lambda^i D_x^j w^>_{\lambda , n+1}
=
D_\lambda^i D_x^j [ \MO (w^>_{\lambda , n}) ] \ .
\end{equation}
By Proposition \ref{Aoperatormain2}
and Lemmas \ref{inverseSmain2} to \ref{lem:derivadaNIW}
we can expand the right hand side of \eqref{derivadeswlambdan}.
We
define the operator $\A_n$ in such a way that
$\A_n D^i_\lambda D_x^j w_{n}^> $ contains the only terms
in such expansion
with
$D^i_\lambda D_x^j w_{n}^ > $.
We take $\B_n$ as the sum of
all the remaining terms of the expansion.
We allow $n$ to be $\infty $.
More explicitly, we define
\begin{eqnarray} \label{definitionAn}
\lefteqn{
[\A_n v] _\lambda } \nonumber \\
& = &
A_2^{1} v _\lambda \circ \psi _{\lambda,n}
(D_x \psi _{\lambda,n} )^{\otimes j}
\nonumber \\
&& + \,
A_2^{1} \Big[D_x w _{\lambda,n} \circ \psi _{\lambda,n} \
(D_2N_1 \circ (\Id ,w_{\lambda,n} ) + B_\lambda )
 D_2N_2 \circ (\Id ,w_{\lambda,n} ) \Big]
v_\lambda
\end{eqnarray}
for
$ n\in \Z^+ \cup \{ \infty \} $ and
\[
\B_n =
D_\lambda^i D_x^j [ \MO (w^>_{ n}) ]
 \A_n D^i_\lambda D_x^j w_{n}^> ,
\qquad n\in \Z^+ \cup \{ \infty \},
\]
that is,
$\B_n$
is the multilinear expression
in the derivatives $D^a_\lambda D^b_x w^>_{n}$ for
$(a,b) \in \Sigma'_{i,j}$, $(a,b) \ne (0,0)$,
which contains all terms in
the expansion of the right hand side of
\eqref{formuladerivadesN} except the only two ones
which contain
$D^i_\lambda D_x^j w^>_{n} $, which are included in $\A_n$.
We remark that $\A_n$ and $\B_n$ actually depend on $i$ and $j$,
but we suppress them from the notation.
To study the convergence
of the sequence $D^i_\lambda D_x^j w_{n}^ >$
on compact sets, to each compact set
$G \subset \Lambda \times B_1$ we associate to it the
larger compact set
\[
G^* =
\bigcup _{n\in \natural \cup \{ \infty \} , \; l\in \Z^+}
\{ (\lambda , \psi ^l_{\lambda, n} (x) ) \mid \; (\lambda
,x) \in G \}\ ,
\]
which has the important property of being invariant by
$(\pi _\lambda , \psi _{\lambda , n})$ for every $n\in
\natural \cup \{ \infty \}
$, where $\pi _\lambda (\lambda , x) = \lambda $.
\begin{lem} \label{lemmaAnBn}
Assume the hypotheses and notation of Proposition
\ref{inductiondifferentiablemain2}.
Then, for
$n\in \Z^+ \cup \{ \infty \}$,
\begin{itemize}
\item[$a)$] Given a compact set $G\subset \Lambda \times B_1$,
$\A_n: \Gamma ^*_{i,j}(G^*) \longrightarrow
\Gamma ^*_{i,j} (G^*)
$ is a well defined operator and, denoting
$Y=L(\Gamma ^*_{i,j}(G^*) ,\Gamma _{i,j}^* (G^*) ) $,
its norm
satisfies
$\ \A_n \_Y \le
\gamma < 1$, for some $\gamma $ independent on $n$.
\item[$b)$] $\B_n \in \Gamma ^*_{i,j}(G^*) $ and
$\ \B_n \_{\Gamma ^*_{i,j}(G^*)} \le M$ for some $M> 0$ independent
of $n$.
\item[$c)$] For every $n\in \natural$ and
$(\lambda ,x) \in \Lambda \times B_1$,
$ D^i_\lambda D_x^j w_{\lambda,n}^> (x)
\le M_{i,j}  x ^{(Lj+1)_+}$ for some $M_{i,j}$ independent
of $n$.
\item[$d)$] $\B_n $ converges uniformly on compact sets to $\B_\infty $.
\end{itemize}
\end{lem}
\Proof
We begin by noting that
from hypothesis 1) of Proposition
\ref{inductiondifferentiablemain2} and
\eqref{derivadaNIW},
we have that for $n \in \Z^+ \cup \{ \infty \}$
\begin{eqnarray}
 \psi_{\lambda ,n} (x)  &\le &(\A_1\ + \varepsilon ) x,
\label{psibound00} \\
 D_x \psi _{\lambda ,n} (x)  &\le &\A_1\ + \varepsilon ,
\label{psibound01}
\end{eqnarray}
for $(\lambda,x )\in \Lambda \times B_1$, with $\varepsilon $ as
small as we need,
and
\begin{eqnarray} \label{psiboundj0}
 D^a_\lambda \psi _{\lambda,n} (x)  &\le & C x, \quad
a\le i \mbox{ if } j\ge 1, \mbox{ or } a\le i1 \mbox{ if } j = 0,
\end{eqnarray}
in a compact subset of $\Lambda \times B_1$ with $C$ being a constant
independent of $n$.
a)
$\A_n$ is a linear operator. We estimate its norm:
\begin{equation*}
\begin{split}
\ &\A_n v \_{\Gamma ^*_{i,j}(G^*)}
\\
& =
\sup _{(\lambda ,x) \in G^*}
\frac{1}{x ^{(Lj+1)_+}}
\ A_2^{1} \ \Big\{  v _\lambda (\psi _{\lambda,n} (x) )  \,
 D_x \psi _{\lambda,n} (x) ^j \\
& \quad +
\Big[ D_x w_{\lambda,n} (\psi _{\lambda,n}
(x) )  \, D_2N_1 (x,w _{\lambda,n} (x))
+ B_\lambda  \\
&\qquad \qquad +  D_2N_2 (x,w _{\lambda,n} (x))
 \Big]
 v_\lambda (x) \Big\} \\
& \leq
\sup _{(\lambda ,x) \in G^*}
\ A_2^{1} \ \Big\{ \ v \_{\Gamma ^*_{i,j}(G^*) } \,
\frac{\psi _{\lambda,n} (x) ^{(Lj+1)_+}}{x ^{(Lj+1)_+}}
 D_x \psi _{\lambda,n} (x) ^j \\
& \quad +
\Big[ D_x w_{\lambda,n} (\psi _{\lambda,n}
(x) )  \, D_2N_1 (x,w_{\lambda,n} (x))
+ B_\lambda  \\
& \qquad \qquad +  D_2N_2 (x,w_{\lambda,n} (x))  \Big]
\ v \_{\Gamma ^*_{i,j}(G^*) } \Big\} \\
& \leq
\ A_2^{1} \ \Big[\Big(\ A_1 \+ \varepsilon \Big)^{L+1}
+ \varepsilon \Big]
\ v \_{\Gamma ^*_{i,j}(G^*) }
\end{split}
\end{equation*}
by \eqref{psibound00}, \eqref{psibound01} and the
fact that
$\N\_{C^1}+\B_\lambda \ $ is as small as we want.
b)
Note that if $F \in C^{\Sigma _{i_0, j_0}}$
then $N \in C^{\Sigma _{i_0, j_0}}$,
if $w \in C^{\Sigma _{i,j}}$ then
$N_\lambda \circ (\Id, w_\lambda ) \in C^{\Sigma _{i,j}}$,
and if $w^> \in C^{\Sigma _{i,j}}$ then
$ \MO (w^> ) \in C^{\Sigma _{i,j}}$.
By the way that $\NO$ was constructed (see Section 3.1),
\[
[D^\l _x \MO (0 )]_\lambda (0)
=
D^\l _x [(\NO(w^{\le })_\lambda  w^{\le }_\lambda ] (0)
= 0, \qquad 0\le \l \le L\ .
\]
Then
\[
[D^a_\lambda D^b_x
\MO(0)]_\lambda (0) = 0, \qquad 0\le a\le i,
\quad 0\le b\le L\ ,
\]
and hence
\begin{equation} \label{mobelongsgamma}
\MO(0) \in \Gamma (\Sigma _{i,j}, \Lambda \times B_1) \ .
\end{equation}
To prove that $\B_n (\lambda ,x) \le M  x ^{(Lj+1)_+}$
for $(\lambda ,x)\in G^*$,
we write $\B_n $ as:
\[
D_\lambda^i D_x^j
\MO (0)
+
D_\lambda^i D_x^j \Big[ \MO(w^{>}_{n} )
 \MO(0) \Big]
 \A_n D^i_\lambda D_x^j w_{n}^> \ .
\]
We are going to decompose the terms in the expansion of
\[
D_\lambda^i D_x^j \Big[ \MO(w^{>}_{n} )
 \tilde \MO(0) \Big]
\]
given by \eqref{formuladerivadesN}
in telescopic form,
and see that all of them have the factor
$ x ^{(Lj+1)_+}$ and are bounded uniformly in $n$.
Since $ \MO(0) = \MO(w^{>}_{0} ) $, we study
the more general expression
\begin{equation} \label{compareMnm}
D_\lambda^i D_x^j \Big[ \MO(w^{>}_{n} )
 \tilde \MO(w^{>}_{m}) \Big]
\A_n D^i_\lambda D_x^j w_{n}^>
+ \A_m D^i_\lambda D_x^j w_{m}^>
\end{equation}
to be able to use the conclusions in the case $m=\infty $.
The following Lemma \ref{lemmaboundstel}
estimates
the different kinds of terms that will appear
in the telescopic decomposition of \eqref{compareMnm}.
These terms are of the following forms:
\begin{equation} \label{Ndiff}
\begin{split}
T_{1,\lambda } &= \Big[ D^a_\lambda D_1^c D_2^b N_1 \circ (\Id,
w_{\lambda,n}
) \\
&\quad  D^a_\lambda D_1^c D_2^b N_1 \circ (\Id, w_{\lambda,m} ) \Big] \,
D^{i_1}_\lambda D^{j_1}_x w_{\lambda, m}
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda, m}
\end{split}
\end{equation}
with $(a,b,c)\in \Sigma^{**}_{i,j} $ and
$i_1+ \dots +i_b=ima $, $0\le m\le ia$, $j_1+ \dots +j_b=jc $,
\begin{equation} \label{wdiff}
\begin{split}
T_{2,\lambda } =
D^a_\lambda & D_1^c D_2^b N_1 \circ (\Id, w_{\lambda,n} ) \,
\Big[ D^{i_1}_\lambda D^{j_1}_x w_{\lambda, n}
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda,n} \\
& 
D^{i_1}_\lambda D^{j_1}_x w_{\lambda, m}
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda, m} \Big]
\end{split}
\end{equation}
with $(a,b,c)\in \Sigma^{**}_{i,j} $ and
$i_1+ \dots +i_b=ima $, $0\le m\le ia$, $j_1+ \dots +j_b=jc $,
\begin{equation}
T_{3,\lambda } (x)=
D_\lambda ^{\tilde \imath} D_x ^{\tilde \jmath
} \psi _{\lambda,n} (x) 
D_\lambda ^{\tilde \imath}
D_x ^{\tilde \jmath} \psi _{\lambda,m} (x)
\end{equation}
with $\tilde \imath \le i$, ${\tilde \jmath} \le j$,
$(\tilde \imath, \tilde \jmath) \ne (i,j)$,
\begin{equation}
T_{4,\lambda } =
\Big[ D_\lambda ^{a} D_x ^b w_{\lambda,n} \circ \psi _{\lambda,n} 
D_\lambda ^{a} D_x ^b w_{\lambda,m} \circ \psi _{\lambda,m} \Big]
\,
D^{i_1}_\lambda D^{j_1}_x \psi _{\lambda,m}
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{\lambda,m}
\end{equation}
with $(a,b) \in \Sigma' _{i,j} $,
$i_1 + \dots + i_b = ima$, $0\le m\le ia$,
and $j_1 + \dots + j_b = j$,
\begin{equation}
\begin{split}
T_{5,\lambda } =
D_\lambda ^{a} & D_x ^b w^>_{\lambda,n} \circ \psi _{\lambda,n}
\, \\
& \Big[ D^{i_1}_\lambda D^{j_1}_x \psi _{\lambda, n}
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{\lambda,n} 
D^{i_1}_\lambda D^{j_1}_x \psi _{\lambda,m}
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{\lambda,m} \Big]
\end{split}
\end{equation}
with $(a,b) \in \Sigma '_{i,j} $,
$i_1 + \dots + i_b = ima$, $0\le m\le ia$,
and $j_1 + \dots + j_b = j$.
Before finishing the proof of Lemma~\ref{lemmaAnBn},
we state and prove the following lemma:
\begin{lem}\label{lemmaboundstel}
Assume the hypotheses of Lemma~\ref{lemmaAnBn}. Then,
there exists a constant $C$ independent on $n$ such that
\begin{eqnarray*}
 T_{1,\lambda }(x)  &\le & Cx^{L+1}, \\
 T_{2,\lambda }(x)  &\le & Cx^{(Lj+1)_+}, \\
 T_{3,\lambda }(x)  &\le &
C x^{(L \tilde \jmath+1)_+}, \\
 T_{4,\lambda }(x)  &\le &
C x ^{(L j + 1)_+ } \quad \mbox{ in case that } m=0, \\
 T_{5,\lambda }(x)  &\le &
C x ^{(L j + 1)_+ } \ .
\end{eqnarray*}
\end{lem}
\Proof
For \eqref{Ndiff}
we distinguish two cases.
When $a+b+c < i_0+j_0$, then we use the mean value theorem and
we bound $T_{1 }$ by
\[
\ D^a_\lambda D_1^c D_2^{b+1} N_1 \ _{C^0}
 w_{\lambda,n}(x)  w_{\lambda,m} (x)  \,
D^{i_1}_\lambda D^{j_1}_x w_{\lambda,m}(x)
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda,m}(x) \ ,
\]
and we recall that
\[
 w_{\lambda,n}(x)  w_{\lambda,m} (x)  =
 w^{>}_{\lambda,n}(x)  w^{>}_{\lambda,m}(x)
\le \ w^{>}_{n}  w^{>}_{m}\
_{\Gamma(\Sigma '_{i,j}, G^*) } x^{L+1}.
\]
In the second case
when $a+b+c = i_0+j_0$, since
$a+b+c \le i+ j$ then
$ j \ge i_0+j_0  i\ge j_0 \ge L+1 $
and thus $(L j + 1)_+ = 0 $. Hence it is enough to see that
$T_{1 }$
is bounded, which follows immediately form 2) of
Proposition~\ref{inductiondifferentiablemain2}.
Decomposing the differences of \eqref{wdiff} in telescopic form,
each difference will have a factor of the form
\begin{equation} \label{differencebound}
 D^{i_l}_\lambda D^{j_l}_x w_{\lambda,n} (x) 
D^{i_l}_\lambda D^{j_l}_x w_{\lambda,m} (x) 
\le
\w_{n}  w_{m} \_{\Gamma(\Sigma '_{i,j}, G^*) }
x^{(L j_l + 1)_+ },
\end{equation}
with $j_l \le j $.
The factor \eqref{differencebound} is bounded by
$C x^{(L j + 1)_+ }$
because $j_l \le j $.
For $T_{3 }$ we write
\begin{equation*}
\begin{split}
D_\lambda ^{\tilde \imath} & D_x ^{\tilde \jmath} \psi _{\lambda,n} 
D_\lambda ^{\tilde \imath} D_x ^{\tilde \jmath} \psi _{\lambda,m} \\
& =
\sum_{(a,b,c) \in \Sigma ^{**}_{\tilde \imath, \tilde \jmath }}
\sum_{i_1 + \dots + i_b = \tilde \imatha,
\atop j_1 + \dots + j_b = \tilde \jmathc } \\
&C
\Big[ D^a_\lambda D_1^c D_2^b N_1 \circ (\Id, w_{\lambda,n} )\,
D^{i_1}_\lambda D^{j_1}_x w_{\lambda, n}
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda,n} \\
& \qquad
 D^a_\lambda D_1^c D_2^b N_1 \circ (\Id, w_{\lambda,m} )\,
D^{i_1}_\lambda D^{j_1}_x w_{\lambda,m}
\cdots D^{i_b}_\lambda D^{j_b}_x w_{\lambda,m} \Big] \\
& +
\sum_{l=0}^{\tilde \imath} C D^l_\lambda B\lambda \,
\Big[ D_\lambda^{\tilde \imathl} D_x^{\tilde \jmath} w_{\lambda,n}
 D_\lambda^{\tilde \imathl} D_x^{\tilde \jmath} w_{\lambda,m} \Big]\ .
\end{split}
\end{equation*}
We deduce the bound for $T_3$
claimed in Lemma~\ref{lemmaboundstel} by applying the bounds for $T_{1 }$ and
$T_{2 }$ with $i= \tilde \imath $ and $j=\tilde \jmath$.
To establishe the bound for $T_{4 }$ we take into account that $m=0$
and hence $w_{0} = w^\le_{0}$.
Noting that $w_0$ is a polynomial
and, therefore, has derivatives,
we bound the first factor in the definition of $T_{4 }$ by
\begin{equation*}
\begin{split}
&  D_\lambda ^{a} D_x ^b w^>_{\lambda,n} \circ \psi _{\lambda,n}
(x) +
 D_\lambda ^{a} D_x ^b w_{\lambda,0} \circ \psi _{\lambda,n}(x) 
D_\lambda ^{a} D_x ^b w_{\lambda,0} \circ \psi _{\lambda,m}(x)
 \\
& \qquad \le
\ D_\lambda ^{a} D_x ^b w^>_{n} \
_{\Gamma(\Sigma '_{i,j}, G^*) }
 \psi _{\lambda,n} (x)  ^{(L b+1)_+} \\
& \qquad \qquad + \ D_\lambda ^{a} D_x ^{b+1} w_{0} \_{C^0}
 \psi _{\lambda,n}(x)  \psi _{\lambda,m}(x)  \\
& \qquad
(\A_1\ + \varepsilon ) ^{(L b+1)_+}
\ D_\lambda ^{a} D_x ^b w^>_{n} \
_{\Gamma(\Sigma '_{i,j}, G^*) }
x ^{(L b+1)_+}
+ C x ^{L+1}
\end{split}
\end{equation*}
where we have used the bound for $T_{3 }$.
Moreover the second factor of $T_{4 }$,
$
D^{i_1}_\lambda D^{j_1}_x \psi _{m}
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{m}
$,
is such that $j_1+ \dots + j_b = j$ and hence
it must have at least
$(jb)_+$ derivatives with index $j_l = 0$. Therefore
by \eqref{psibound01} this factor is bounded by
$Cx ^{(jb)_+}$.
For $T_{5}$,
decomposing the difference
\[
D^{i_1}_\lambda D^{j_1}_x \psi _{ n}
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{n}

D^{i_1}_\lambda D^{j_1}_x \psi_{m }
\cdots D^{i_b}_\lambda D^{j_b}_x \psi _{m}
\]
in a telescopic sum, each term
will have a factor
\[
 D^{i_l}_\lambda D^{j_l}_x \psi _{\lambda, n} (x)
 D^{i_l}_\lambda D^{j_l}_x \psi _{\lambda,m } (x)
\]
which is bounded by
$Cx^{(L j_l+1)_+} \le C x^{(L j+1)_+}$ by the bounds
of $T_{3}$.
\qed
\pf{Continuation of the proof of Lemma~\ref{lemmaAnBn}}
With the estimates of
Lemma \ref{lemmaboundstel}
and \eqref{mobelongsgamma}
one easily checks
that $\B_n\in \Gamma ^*_{i,j}(G^*)$ and that
there exists a
constant
$M$ independent on $n$ such that
\[
\B_n(\lambda ,x)  \le Mx^{(L j
+1)_+}.
\]
The same estimates work in the case $n=\infty $.
To establish c) of Lemma~\ref{lemmaAnBn},
from the definition of $\A_n$ and $\B_n$ we write
\[
D_\lambda ^{i} D_x ^j w^>_{n+1}
= \A_n D_\lambda ^{i} D_x ^j w^>_{n} + \B_n \ .
\]
Then
\[
\ D_\lambda ^{i} D_x ^j w^>_{n+1} \ _{\Gamma ^*_{i,j}(G^*)}
\le \ \A_n \_Y \, \ D_\lambda ^{i} D_x ^j w^>_{n} \
_{\Gamma ^*_{i,j}(G^*)}
+ \ \B_n \ _{\Gamma ^*_{i,j}(G^*)}
\]
and hence c) follows
taking $M_{i,j } = \frac{M}{1\gamma }$.
Now we will prove d) of Lemma~\ref{lemmaAnBn}.
Given a compact set $G$ of $\Lambda \times B_1$, we have to prove
that $\B_n \to \B_\infty $
converges uniformly on $G$.
For that we have to consider the differences \eqref{compareMnm}
with $m=\infty $ in the larger compact set $G^*$.
Examining the terms $T_{1}$
to $T_{5 }$ in the analogous way as
in Lemma~\ref{lemmaboundstel}, we see that all terms
are bounded by bounded factors
times one of the following terms:
\begin{equation} \label{difconvbn1}
\ w^{>}_{n}  w^{>}_{m}\
_{\Gamma(\Sigma '_{i,j}, G^*) }
\end{equation}
\begin{equation} \label{difconvbn2}
 D^a_\lambda D_1^c D_2^b N_j \circ (x, w_{\lambda,n}(x) )
 D^a_\lambda D_1^c D_2^b N_j \circ (x, w_{\lambda, \infty } (x) ) 
\end{equation}
\begin{equation} \label{difconvbn3}
\D_\lambda ^{a} D_x ^b w_{\lambda,\infty } \circ \psi _{\lambda,n} 
D_\lambda ^{a} D_x ^b w_{\lambda,\infty } \circ \psi _{\lambda,
\infty }\
\end{equation}
and
\begin{equation} \label{difconvbn4}
\D_\lambda ^{a} D_x ^b w_{\lambda,n } \circ \psi _{\lambda,n} 
D_\lambda ^{a} D_x ^b w_{\lambda,\infty } \circ \psi _{\lambda, n} \\ .
\end{equation}
The terms
\eqref{difconvbn2}, \eqref{difconvbn3} and \eqref{difconvbn4}
go to zero because $D^a_\lambda D_1^c D_2^b N_j $,
$D_\lambda ^{a} D_x ^b w_{\infty }$
are continuous and
$\{ (\lambda ,x,w_{\lambda ,\infty }(x)) \mid \;(\lambda ,x)\in G^* \}$
and
\[
\{ (\lambda ,\psi _{\lambda ,m }(x)) \mid \;(\lambda ,x)\in G^*,
\; m\in \Z^+\cup \{ \infty \}
\}
\]
are compact.
\qed
\noindent {\bf Proof of Proposition \ref{inductiondifferentiablemain2}.}
First we establish part b).
If we call
$v_{n} = D_\lambda ^{i} D_x ^j w^>_{n} $,
from the definition of
$w^>_{n}$, \eqref{derivadeswlambdan} and the
definitions of $\A_n$ and $\B_n$ we have
\begin{eqnarray*}
v_{0} & = & 0, \\
v_{n+1} & = & \A_n v_{n} + \B_n, \qquad n\geq 0.
\end{eqnarray*}
Hence we can write
\[
v_{n+1} =
\B_n + \A_n \B_{n1} +\dots + \A_n \A_{n1} \cdots
\A_1\B_0 \ .
\]
We claim that $v_{n}$ converges to
\[
\B_\infty + \A_ \infty \B_{\infty } +\A_\infty ^2 \B_\infty + \dots
\]
uniformly on compact sets.
Indeed
\begin{eqnarray*}
\lefteqn{
\A_n \A_{n1} \cdots \A_{nk}\B_{nk1} 
\A_\infty ^{k+1} \B_\infty } \\
&=&
\A_n \A_{n1} \cdots \A_{nk}(\B_{nk1} \B_{\infty } )
+
\A_n \A_{n1} \cdots \A_{nk}\B_{\infty }  \A_\infty ^{k+1} \B_\infty
\\
& = &
\A_n \A_{n1} \cdots \A_{nk}(\B_{nk1} \B_{\infty } ) \\
&& +
\sum_{i=1}^{k} \A_n \cdots \A_{ni+1}(\A_{ni} \A_{\infty } )
\A_\infty ^{ki} \B_\infty \\
&& +
(\A_{n} \A_{\infty } )
\A_\infty ^{k} \B_\infty
\end{eqnarray*}
and since given $g$ defined on $\Lambda \times B_1$,
$\A_n g \to \A_\infty g $
uniformly on compact sets $G^*$ of $\Lambda \times B_1$
then
\[
\A_n \A_{n1} \cdots \A_{nk}\B_{nk1} \to
\A_\infty ^{k+1} \B_\infty
\]
uniformly on compact sets.
For $k < n$ we have
\begin{eqnarray*}
&&
v _{n+1} (x) 
\Big(\B_n (x) + \A_n \B_{n1} (x) +\dots + \A_n \A_{n1} \cdots
\A_{nk}\B_{nk1}(x) \Big) \\
&& \qquad =
\A_n \A_{n1} \cdots \A_{nk1}\B_{nk2} (x) + \dots
+ \A_n \A_{n1} \cdots \A_1\B_0 (x)
\end{eqnarray*}
and we can bound
\begin{equation*}
\begin{split}
&  v _{n+1} (x) 
\Big(\B_n (x) + \A_n \B_{n1} (x) +\dots + \A_n \A_{n1} \cdots
\A_{nk}\B_{nk1} (x) \Big)  \\
& \qquad \le
\sum_{l=2}^{nk} \nu ^{k+l} \ \B_{nkl} \_{\Gamma^* _{i,j}(G^*)}
x ^{(Lj+1)_+} \\
& \qquad \leq \frac{\nu ^{k+2} }{1\nu }
x ^{(Lj+1)_+}
\sup_{0\leq l\leq nk} \ \B_{l} \_{\Gamma ^*_{i,j}(G^*)} \ .
\end{split}
\end{equation*}
Taking the limit $n \to \infty $, we obtain
\[
\begin{split}
 v _{\infty }(x) &
(\B_\infty (x) + \A_\infty \B_{\infty }(x) +\dots + \A_\infty ^{k+1}
\B_{\infty } (x) ) \\
& \leq \frac{\nu ^{k+2} }{1\nu } M
x ^{(Lj+1)_+} \sup_{l \geq 0} \ \B_{l} \_{\Gamma ^*_{i,j}(G^*)} \ .
\end{split}
\]
Then $D^i_\lambda D^j_x w^>_{n}$
converges to
\begin{equation} \label{funceqparametersmain2}
v_{\infty} = \B_\infty +\A_\infty \B_\infty +
\A_\infty^2 \B_\infty + \dots
\end{equation}
uniformly on compact sets $G^*$ as well as with the norm
$\\cdot\ _{\Gamma ^*_{i,j}(G^*)}$.
Note that, since $\A_\infty$ is a contraction in $\Gamma ^*_{i,j}(G^*)$
and $\B_\infty$ is
in $\Gamma ^*_{i,j}(G^*)$, then the resulting function
$ v_{ \infty} $ is in $\Gamma ^*_{i,j}(G^*)$.
Moreover
$\B_\infty $ is continuous. Then
$v_{\infty} $ is
the only solution of
\[
u
= \A_\infty u + \B_\infty
\]
and hence it is continuous.
If $1\le j \le i_0i+j_0$
we will show that the $D_x $ derivative of
$ D^{i}_\lambda D^{j1}_x w^>_{n} $ converges
uniformly to $v_{\infty}$ on compact sets of
$\Lambda \times B_1$.
Since for any $n$ we have
\[
D^{i}_\lambda D^{j1}_x w^>_{\lambda ,n} (y) 
D^{i}_\lambda D^{j1}_x w^>_{\lambda,n} (x)
= \int_0^1 dt\, D^{i}_\lambda D^j_x w^>
_{\lambda ,n} (x+ t(yx)) (yx ),
\]
passing to the limit, using that
$
D^{i}_\lambda D^j_x w^> _{n}
$
converges to $v_{\infty } $ in the compact set $G^*$
with $G=\{x+ t(yx) \mid\; t\in [0,1] \}$
\[
D^{i}_\lambda D^{j1}_x w^>_{\mu ,\infty } (x) 
D^{i}_\lambda D^{j1}_x w^>_{\lambda,\infty } (x)
= \int_0^1 dt\, v
_{\lambda ,\infty } (x+ t(yx)) (yx ),
\]
and using that $v_{\infty}$ is continuous,
we obtain that $v_{\infty}$ is indeed the
$D^{i}_\lambda D^j_x $ derivative
of $w^>$.
If $j=0$ and $i \ge 1$
we show that the $D_\lambda $ derivative of
$ D^{i1}_\lambda D^j_x w^>_{n} $ converges
uniformly to $v_{\infty}$ on compact sets of
$\Lambda \times B_1$.
{From} the identity
\[
D^{i1}_\lambda D^j_x w^>_{\mu ,n} (x) 
D^{i1}_\lambda D^j_x w^>_{\lambda,n} (x)
= \int_0^1 dt\, D^{i}_\lambda D^j_x w^>
_{\lambda + t(\mu\lambda ) ,n} (x) (\mu\lambda ),
\]
passing to the limit
\[
D^{i1}_\lambda D^j_x w^>_{\mu ,\infty } (x) 
D^{i1}_\lambda D^j_x w^>_{\lambda,\infty } (x)
= \int_0^1 dt\, v
_{\lambda + t(\mu\lambda ),\infty } (x) (\mu\lambda )
\]
and the argument finishes as before.
This ends the proof of Proposition~\ref{inductiondifferentiablemain2}.
\qed
Now, we are in a position to complete the proof
of Theorem~\ref{main2differentiable}.
\pf{Proof of Theorem~\ref{main2differentiable}}
Since $F \in C^{\Sigma _{i,j}} $
for fixed $\lambda$,
as a function of $x$ we have that
$F_\lambda(\cdot) \in C^{i+j}$.
{From} Proposition \ref{mapM}
we know that $w^>_\lambda = \lim_n \MO^n(0)$,
where the limit is obtained with $\lambda $ fixed and in the topology
given in the space $ \Gamma _{r1,L} $, with $r=i+j$,
and that this limit is $C^{i+j}$.
This means that
$\ D_x^{r1} w^> _{\lambda ,n}
 D_x^{r1} w^> _{\lambda ,\infty } \_{C^0} \to 0
$
and therefore,
for $0 \le j\le r1$,
\[
\sup \frac{
D_x^{j} w^> _{\lambda ,n}(x)
 D_x^{j} w^> _{\lambda ,\infty }(x) 
}{x^{(Lj+1)_+} } \le \frac{1}{(Lj)_+!}
\ w^> _{\lambda ,n}
 w^> _{\lambda ,\infty } \_{C^{r1}} \ .
\]
Then, we apply the induction
Proposition~\ref{inductiondifferentiablemain2}
to $i_0=0$ starting with $j_0=i+j1$, and
then we apply
Proposition~\ref{inductiondifferentiablemain2} repeatedly
from $i_0 = 1$ to $i_0 = i$ and for each
$i_0$ from $j_0=0$ to $j_0= i+ji_0$.
\qed
\section{Acknowledgments}
This work has been supported by the {\it Comisi\'{o}n Conjunta
Hispano Norte\americana de Cooperaci\'{o}n Cient\'{\i}fica y
Tecnol\'{o}gica}. The final version was prepared while Rafael de la
Llave was enjoying a {\it C\'{a}tedra de la Fundaci\'{o}n FBBV}.
X.C. was partially supported by CICYT (Spain),
grant PB980932C0201.
E.F. acknowledges the partial support of the
the Spanish Grant
DGICYT BFM20000805, the Catalan grant CIRIT 2000SGR00027
and the INTAS project 00221.
The work of R.L. has been supported by NSF grants. R.L. thanks
Universitat de Barcelona and Universitat Polit\`ecnica de Catalunya for
hospitality during several visits.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\section{Some results in spectral theory} \label{sec:spectral}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The results of this Appendix are well known in finite dimensional spaces
(see, for instance, \cite{Nelson69}). Here we present
proofs which are also valid in infinite dimensions and,
at the same time, we simplify some of the proofs in the finite
dimensional case.
We recall that given a bounded linear operator $A$ on a Banach space $X$,
we say that
$\lambda \in \Res (A)$ (the resolvent set of $A$) if and only if
$A\lambda:=A\lambda\Id$ is invertible, i.e., $A\lambda$
is one to one and onto. By the open mapping theorem, the inverse
$(A\lambda )^{1}$ is automatically a bounded operator.
We also define the spectrum of $A$ by $\Spec (A) = \complex  \Res
(A)$, a compact subset of $\complex$.
An important subset of the spectrum of $A$
is given by the approximate point spectrum of $A$,
denoted by $\Spec_{ap}(A)$. By definition,
\begin{equation}\label{apspec}
\lambda\in\Spec_{ap} (A) \Leftrightarrow
\exists \{x_n\} \hbox{ with}\
\x_n\\geq\alpha >0 \ \hbox{and} \ \(A  \lambda )x_n\ \to 0 \ .
\end{equation}
The sequence $\{x_n\}$ is called an approximate eigenvector of $A$
for $\lambda$.
A well known result states that
the boundary of the spectrum is contained in the approximate
point spectrum, i.e.,
\begin{equation}\label{weyl}
\partial(\Spec (A)) \subset \Spec_{ap}(A) .
\end{equation}
This is easily verified as follows. Let $\lambda_n\to\lambda$,
with $\lambda\in\Spec (A)$ and $\lambda_n\in\Res (A)$ for all $n$.
Recall that, for every operator $B$ with $\Vert B \Vert <1$,
$\Id +B$ is invertible (the inverse of $\Id +B$ is just given by the
standard Neumann power series). This fact and the identity
$A\lambda = (A\lambda_n)\{\Id + (\lambda_n
\lambda)(A\lambda_n)^{1}\}$ lead to
$\vert \lambda \lambda_n\vert \Vert
(A\lambda_n)^{1} \Vert \ge 1$.
Hence, there exists $y_n$ such that
\[
\Vert (A\lambda_n)^{1}y_n \Vert \ge \frac{\Vert y_n\Vert}
{2 \vert \lambda \lambda_n\vert} .
\]
By scaling $y_n$, we may also assume that $x_n:= (A\lambda_n)^{1}y_n$
satisfies $\Vert x_n\Vert =1$. Since $\Vert (A\lambda_n)x_n\Vert
=\Vert y_n\Vert \le 2\vert\lambda \lambda_n\vert$, we have that
$\Vert (A\lambda)x_n\Vert \le 3\vert\lambda \lambda_n\vert
\to 0$, and hence that $\lambda
\in \Spec_{ap}(A)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Adapted norms}
Let $X$ be a Banach space and $A$ a linear map on $X$.
\label{sec:standing}
It is well known that given any $\ep>0$, we can find a norm in $X$
equivalent to the original one and such that
\[
\A\ \le \rho (A) +\ep ,
\]
where $\rho (A) = \sup_{z\in \Spec (A)}z$ is the spectral radius
of $A$.
We shall say that such norm is $\ep$adapted to $A$.
Moreover, if $A$ is invertible
we can find a norm in $X$
equivalent to the original one and $\ep$adapted to $A$ and
to $A^{1}$ simultaneously, i.e.,
\[
\A\ \le \rho (A) +\ep ,\qquad \A^{1}\ \le \rho (A^{1}) +\ep\ .
\]
An example of norm $\ep$adapted to $A$ and $A^{1}$
is given by
\[
 x = \sum_{i=0}^\infty (\rho(A)+\ep)^{i} \ A^i x \
+ \sum_{i=1}^\infty (\rho(A^{1})+\ep)^{i} \ A^{i} x \ \ .
\]
To verify this, it suffices to use the well known fact that
\[\rho(A)=\lim_{i\rightarrow +\infty} \Vert A^i\Vert^{1/i}.\]
A more general result is the following:
\begin{prop}\label{norms}
Let $X$ be a Banach space, $X= X_1\oplus X_2$ be a direct sum
decomposition into closed subspaces, and let
\[
A = \left( \begin{array}{cc} A_1&B\\ 0&A_2\end{array}\right)
\]
with respect to the above decomposition. Then,
\begin{itemize}
\item[a)]
$\rho(A) = \max \big(\rho(A_1), \rho(A_2) \big)$.
\item[b)]
Assume further that $A_1$ and $A_2$ are invertible.
Then, for every $\ep>0$, there exists a norm in $X$ which is
equivalent to the original one and such that
\[
\begin{array}{rclrcl}
\A_1\ & \le & \rho (A_1) +\ep\ , & \qquad
\A_1^{1}\ & \le & \rho (A_1^{1}) +\ep \ , \\
\A_2\ & \le & \rho (A_2) +\ep\ , & \qquad
\A_2^{1}\ & \le & \rho (A_2^{1}) +\ep \ , \\
\A\ & \le & \rho (A) +\ep\ , & \qquad
\A^{1}\ & \le & \rho (A^{1}) +\ep \ ,\qquad \hbox{and}\\
\B\ & \le & \ep \ .
\end{array}
\]
\end{itemize}
\end{prop}
\pf{Proof}
To prove part b) once we have established a),
it suffices to construct norms $\\cdot \_{X_1}$, $\\cdot \_{X_2}$
in $X_1$ and $X_2$ which are $(\ep/2)$adapted to $A_1$ and
$A_1^{1}$, and to $A_2$ and $A_2^{1}$, respectively.
Note that
\[
A^{1} = \left( \begin{array}{cc} A_1^{1}& A_1^{1}BA_2^{1}\\
0&A_2^{1} \end{array}\right)\ .
\]
Hence, by part a) applied to $A$ and to $A^{1}$, we have
\[
\rho(A) = \max \big(\rho(A_1), \rho(A_2) \big) \qquad\mbox{and}\qquad
\rho(A^{1}) = \max \big(\rho(A_1^{1}), \rho(A_2^{1}) \big)\ .
\]
Using these equalities and defining, for $\delta>0$ sufficiently small,
\[
\(x^1,x^2)\_X = \max (\delta \x^1 \
_{X_1},\x^2\_{X_2})\ \qquad\hbox{for } x= (x^1,x^2)\ ,
\]
it is easy to verify all the statements of part b).
Next, we prove a). Since $(A\lambda)x = y$ is equivalent to
\begin{eqnarray*}
&&(A_1\lambda) x^1 + Bx^2 = y^1\\
&&(A_2 \lambda) x^2 = y^2 \ ,
\end{eqnarray*}
we see that
\[
\lambda \in \Res (A_1) \cap \Res (A_2) \Rightarrow \lambda\in \Res
(A)\ . \]
Hence
\begin{equation}\label{inclusion1}
\Spec (A)\subset \Spec (A_1)\cup \Spec (A_2)
\end{equation}
and, in particular, $\rho(A) \le \max \big(\rho(A_1), \rho(A_2) \big)$.
Therefore, we only need to show that
\begin{equation}\label{ineqspec}
\rho(A) \ge \max \big(\rho(A_1), \rho(A_2) \big) \ .
\end{equation}
To prove this, we first claim that
\begin{equation}\label{inclusion2}
\Spec_{ap}(A) \supset \Spec_{ap} (A_1) \cup
\Big( \Spec_{ap} (A_2) \cap \Res (A_1) \Big) \ .
\end{equation}
Then, using that
$\partial(\Spec (A_i)) \subset \Spec_{ap}(A_i)$ for $i=1,2$
(i.e., property \eqref{weyl} applied to $A_1$ and $A_2$),
we conclude inequality \eqref{ineqspec} and the proposition.
To establish \eqref{inclusion2}, let first
$\lambda\in \Spec_{ap} (A_1)$ and $\{x^1_n\}$ be an approximate
eigenvector of $A_1$ for $\lambda$. Then,
$\{x_n\} = \{(x^1_n,0)\}$ clearly is an approximate
eigenvector of $A$ for $\lambda$.
Finally, if $\lambda\in \Spec_{ap}(A_2) \cap \Res (A_1)$ and
$\{x^2_n\}$
is an approximate eigenvector of $A_2$ for $\lambda$, then
\[
\{x_n\} = \{ ( (A_1\lambda)^{1} Bx^2_n\ ,\ x^2_n )\}
\]
is an approximate eigenvector of $A$ for $\lambda$.
This establishes \eqref{inclusion2}.
\qed
Of course, in finite dimensions we
have
\[
\Spec(A) = \Spec(A_1) \cup \Spec(A_2) \ .
\]
However, in infinite dimensions the
inclusion \eqref{inclusion1} could be strict,
as the following example shows.
\begin{xmpl}\label{inverses}
{\rm
Let $\Sigma = \{ \sigma :\natural \longrightarrow X \}$ be
the Banach space of sequences
into a Banach space
$X$ equipped with the sup norm
$($an $\l^2$ norm would also produce similar effects$)$.
Let $A: \Sigma \times \Sigma \longrightarrow
\Sigma \times \Sigma $ be defined by
\[
A(\sigma , \tau )
= (\tilde \sigma , \tilde \tau )\]
with
$\tilde \sigma (1) = \tau (1)$,
$\tilde \sigma (i) = \sigma (i1)$ for $i\ge 2$, and
$\tilde \tau (j) = \tau (j+1)$ for $j\ge 1$.
Then, it is easy to verify that $A$ is invertible, but
$A_1$ and $A_2$ are not.
}
\end{xmpl}
In \cite{delaLlave98} one can find examples arising in dynamical systems
where both \eqref{inclusion1} and \eqref{inclusion2} are strict.
This is due to the presence of residual spectrum.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{ Proof of Proposition \protect{\ref{prop:induced} } }
\label{sec:spectralinduced}
The proof of the equalities in \eqref{eq:spectra}
when $X$ and $Y$ are finite dimensional
is very easy, and we do it first.
Since the spectrum
depends continuously on
the matrix, it suffices to
establish the equalities
when $A$ and $B$ are diagonalizable
matrices (a dense subset in the
space of matrices).
Let us first consider the operator $\L_{n,A,B}$. If
$A u_{j} = \mu_{j} u_{j} $,
$B v_i = \lambda_i v_i$,
consider the form $M_{i;j_1,\ldots, j_n}$
defined by the conditions
\[
\begin{array}{rcl}
M_{i;j_1,\ldots, j_n}( u_{j_1}, \dots, u_{j_n}) & = & v_i, \\
M_{i;j_1,\ldots, j_n}( u_{s_1}, \dots, u_{s_n}) & = & 0, \qquad
\mbox{ when } ( s_1,\ldots, s_n ) \ne ( j_1,\ldots, j_n )\ .
\end{array}
\]
Clearly,
\[
\L_{n,A,B} M_{i;j_1,\ldots, j_n}
= \lambda_i \mu_{j_1} \cdots \mu_{j_n}
M_{i;j_1,\ldots, j_n} \ .
\]
Moreover, the set formed by the $M_{i;j_1,\ldots, j_n}$
is linearly independent and,
under the assumption that $A$,$B$ are diagonalizable,
its cardinal is equal to the dimension of $\MM_n$.
Hence, the spectrum of $\L_{n,A,B}$ is
indeed the set of numbers
$\lambda_i \mu_{j_1} \cdots \mu_{j_n}$,
as claimed.
The equality for the operator $\L_B$ is a particular case of
the previous one, since $\L_B=\L_{n,\Id ,B}$. A similar argument
also proves the equality for the operator $\R^k_A$.
For the case of symmetric forms,
a very similar argument
works.
We can consider the form
$S_{i;j_1,\dots, j_n} =
\sum_{ \pi } M_{i; j_{\pi(1)},\dots,j_{\pi(n)} }$
where the variable $\pi$ in the sum runs over the permutations
of $\{ 1,\dots,n\}$.
Each form $S_{i;j_1,\dots, j_n}$
is an eigenvector of $\L_{n,A,B}$ with
eigenvalue
$\lambda_i \mu_{j_1}\cdots \mu_{j_n}$.
Moreover, they are
linearly independent, and
there are as many of them
as the dimension of the space $\SS_n$. This proves the equality
for the spectrum of $\L_{n,A,B}$ in $\SS_n$.
\bigskip
Now, we turn to the proof of
Proposition \ref{prop:induced} for
general Banach spaces.
First note that
the operators $\L_B, \R^1_A, \ldots ,\R^n_A$ commute,
and that
\begin{equation}\label{defopl}
\L_{n,A,B} =
\L_B \R^1_A\cdots \R^n_A \ .
\end{equation}
Moreover, we have:
\begin{prop}
\label{prop:easyinclusion}
With the notations of Proposition \ref{prop:induced}, one has
\begin{eqnarray}
\Spec( \L_B, \MM_n) \subset \Spec(B, Y) \ , \label{specL}
\\
\Spec( \R^k_A, \MM_n) \subset \Spec(A, X) \label{specM} \ .
\end{eqnarray}
\end{prop}
\Proof
Note that
$\L_{B B'} = \L_B \L_{B'}$ and
$\L_{B + B'} = \L_B + \L_{B'}$, so that $\L$ can be considered
as a representation of the
Banach algebra of bounded operators in
$Y$ into the Banach algebra of operators
in $\MM_n$. Therefore, the
spectrum is smaller.
Similarly for all the $\R^k_A$.
\qed
We also recall the following
well known result in Banach algebras (see e.g., Theorem~11.23
in \cite{Rudin91}).
\begin{thm} \label{commutation}
Let $\A$ and $\B$
be two commuting elements
in a Banach algebra. Then
\[
\Spec(\A \B)
\subset \Spec( \A) \Spec(\B) \ .
\]
\end{thm}
The result of Proposition \ref{prop:induced} for $\MM_n$
follows immediately from
the fact that the operators $\L_B, \R^1_A,\ldots ,\R^n_A$ commute,
Theorem \ref{commutation}, \eqref{defopl},
\eqref{specL} and \eqref{specM}.
The result for $\SS_n$
follows from the following:
\begin{prop}
\label{prop:symmetric}
With the notations of Proposition \ref{prop:induced}, we have
\begin{equation}
\Spec(\L_{n,A,B} , \SS_n )
\subset
\Spec(\L_{n,A,B} , \MM_n ) \ .
\end{equation}
\end{prop}
\Proof
If $\lambda \notin \Spec(\L_{n,A,B} , \MM_n )$,
given any $\eta \in \MM_n$
there is a unique
$\Gamma\in \MM_n$ such that
\begin{equation}
\label{tosolve}
(\L_{n,A,B}  \lambda) \Gamma = \eta \ .
\end{equation}
Such $\Gamma$ satisfies
$\ \Gamma \ \le K \\eta\$.
If
$\Pi( x_1, \dots, x_n) =
(x_{\pi(1)}, \dots, x_{\pi(n)} )$
is a permutation of the
$n$ vectors, it is easy
to check that
\begin{equation}
\label{symmetry}
(\L_{n,A,B}  \lambda) (\Gamma \Pi)
=
[(\L_{n,A,B}  \lambda) \Gamma ] \Pi \ .
\end{equation}
Therefore, if $\eta \Pi = \eta$ we obtain,
using the uniqueness of $\Gamma$
and \eqref{symmetry}, that
$\Gamma \Pi = \Gamma$.
Hence, if equation
\eqref{tosolve}
can be solved uniquely and
boundedly in
$\MM_n$,
it can be similarly solved in
$\SS_n$.
\qed
Finally, we note that the construction of the forms
$M_{i;j_1,\dots, j_n}$ can be carried out
in general Banach spaces by using the
HahnBanach theorem to extend from the space
spanned by eigenvectors to the whole space.
This construction shows that
\begin{eqnarray}
\Spec_{pp}( \L_{n,A,B} , \MM_n)
\supset \Spec_{pp}(B) \big( \Spec_{pp}(A) \big)^n \ ,
\nonumber \\
\Spec_{pp}( \L_{n,A,B} , \SS_n)
\supset \Spec_{pp}(B) \big( \Spec_{pp}(A) \big)^n \, ,
\label{eq:spectrareverse}
\end{eqnarray}
where $\Spec_{pp}$ denotes the closure of the set of eigenvalues.
Therefore, if we assume
$\Spec_{pp}(B) = \Spec(B)$ and
$\Spec_{pp}(A) = \Spec(A)$ then, combining
\eqref{eq:spectrareverse} and
\eqref{eq:spectra}, we obtain equalities in
\eqref{eq:spectra}.
\section{Historical remarks and information on the literature} \label{sec:historical}
Invariant manifold theory seems to have
a not very well known history that abounds with rediscoveries
this has been eloquently mentioned in an often
repeated quote in the fourth paragraph of
\cite{Anosov69}, page 23.
We have made an effort in finding
works were results related to ours appeared. This
cannot be considered a definitive effort
(for instance, we have not been able to
trace the work of Darboux, which is mentioned
by Poincar\'e and Lyapunov).
We can only hope that our modest search can inspire others.
\subsection{Early history}
It seems to us that onedimensional invariant submanifolds
were more or less known in the analytic case,
and with resonance conditions somewhat
stronger than those considered in the present paper.
It seems well accepted that some versions of
invariant manifold theory, at least for the analytic case,
were known to Darboux, Poincar\'e and
Lyapunov. Unfortunately, we have not been able to locate
the works of Darboux, but we will comment on some works of
Poincar\'e and Lyapunov.
\subsection{Two results of Poincar\'e}
One of us (R.L.) learned about the existence of
\cite{Poincare90} (reproduced in \cite{Poincare50}) from conversations
with D. Ruelle in the early 80's.
The motivation for \cite{Poincare90}
was the theory of special functions.
When $F$ is a polynomial and
$X_1 = \complex$,
the equation
\begin{equation}\label{multiplicationrule}
F\circ K(t) = K(\lambda t)
\end{equation}
can be interpreted as saying that the
system of functions given by the components
of $K$ admits a multiplication
rule ({\it th\'eoreme de multiplication}).
Examples of such systems of functions
(or systems satisfying the closely related addition rules) are
the trigonometric functions and the elliptic functions.
For instance,$K(\theta ) = ( \sin \theta , \cos \theta ) $
satisfies
$K(2\theta ) = F(K(\theta )) $, where
$F(x,y) = (2xy, y^2  x^2)$. Note that $F(0,1) = (0,1)$.
Similar formulas for the duplication of the argument are
known for elliptic integrals. The fact that there
are duplication formulas is related to
the solvability of the quintic using
elliptic functions and their inverses.
The paper \cite{Poincare90} shows that, given a map $F$ and provided
that $\lambda$, $\lambda > 1$, is a simple
eigenvalue of $DF(0)$ and that
there are no
eigenvalues of $DF(0)$ which are
powers of $\lambda$,
one can find a formal series
for $K$. Moreover, using the
majorant method, one can show
that the formal
series for $K$ converges.
This paper also contains the interesting observation
see page 541 in \cite{Poincare50}
that when $F$ is a polynomial, every function
$K$ satisfying \eqref{multiplicationrule}
is entire. The reason is that, when
$F$ is a polynomial, the
functional equation \eqref{multiplicationrule}
forces the domain of definition of $K$ to be
invariant under multiplication by $\lambda$.
Hence, if it contains a ball, it is the
whole complex plane.
We note that this observation generalizes without difficulty
to the situation when $F$ is an entire function and
we are working on a Banach space.
In \cite{Poincare90}, Poincar\'e also
studies
the case when $F^{1} $ is
a rational transformation, that he calls
{\it Cremona.}
In this case, he makes some dynamical observations.
For instance, in the bottom half of page 561 of \cite{Poincare50},
he relates the question of
existence of solution to whether the iterates of the transformation
converges to a fixed point this is indeed the dynamical
characterization of invariant manifold.
{From} a more dynamical point of view, similar series
were considered in \cite{Poincare92}, where all
chapter VII is devoted to asymptotic expansions
around periodic solutions of periodic
vector fields. Taking time$T$ maps, this problem
reduces to the setting about maps that we have considered in this paper.
The logarithms of the eigenvalues of the time$T$ map are called
{\it exposants charact\'eristiques}. In modern language,
they are the Floquet exponents. Note that
what we would call today Lyapunov exponents (which
can be considered in more general settings than periodic systems)
are, in
the case of periodic systems,
the real part of Poincar\'e's {\it exposants charact\'eristiques}.
More confusingly, in the translation of Lyapunov
that we have used, the name {\it characteristic
exponent} refers to the negative of what
we call now Lyapunov exponent. This would be,
of course, the negative of the real part of
the {\it exposant charact\'eristique} for the particular case of
periodic systems.
In \cite{Poincare92}
the crucial chapters dealing with stable
and unstable manifolds are 104 and 105.
In paragraph 104, under the assumption
that there are no resonances
(the nonresonance condition is the last
formula of paragraph 104), it is
shown that one can obtain a formal power series expansion
of exponentials with arbitrary constants.
The convergence of the series is studied in 105.
The first paragraph asserts the convergence of the series
of expansions in powers of the exponential under
the assumption that the eigenvalues belong to
what we now call the Poincar\'e domain
(i.e., when the convex hull of the eigenvalues does not include
zero).
Of course, the reason why this condition
enters is that, for eigenvalues satisfying these conditions,
the small divisors that appear are bounded away from zero.
We note that, even if it is not said explicitly,
the condition that the eigenvalues are different is
indeed assumed. The proof of convergence is rather
succinct. Nevertheless, it should have
been quite clear to Poincar\'e and his
contemporaries since it is very similar
to arguments that had been done in detail
in his thesis \cite{Poincare79b} (reproduced in \cite{Poincare16}).
{From} the point of view of invariant manifold theory,
the last paragraph of page 339 is quite interesting. Here,
Poincar\'e discusses the case when there
are stable and unstable
characteristic exponents at the same time.
He observes that the series for $K$
remains convergent if one sets to zero the constants
corresponding to coordinates along the
expanding or neutral eigendirections.
The arguments here are somewhat skimpy, but
a modern mathematician can supply them without too much
trouble.
One is left with a set of solutions
which tend to zero parameterized by
as many constants as stable directions.
This is, of course, our modern stable manifold.
A similar construction works for the unstable solutions.
Poincar\'e called these solutions
{\it solutions asymptotiques.}
The rest of chapter VII contains a
variety of expansions of these sets of
solutions. It includes, quite notably, the
expansions in terms of a slow parameter, which
are then shown to be divergent. Of course,
much modern work is still being done in
these slow perturbations and related areas.
\subsection{The work of Lyapunov}
In chapters 1133 \cite{Lyapunov92}
(see also the summary in chapter 3 and
the proofs of convergence in chapter 23),
Lyapunov introduces the method
of arbitrary constants, which consists
in finding exponential solutions with
arbitrary constants. Since the constants
do not evolve in time, this is closely related
to the problem of linearization.
(Compare the expansions of the system studied
and those of the linear systems.)
Invariant manifolds can be obtained by setting
some of the constants to zero.
One important difference between
\cite{Lyapunov92} and \cite{Poincare90}
\cite{Poincare92} is that \cite{Lyapunov92} considers
systems which are {\it regular} (roughly, the definition is
that the forward and backward Lyapunov exponents agree).
This is a more general setting than that of periodic systems.
In the case of regular systems, \cite{Lyapunov92} contains
expansions of the solutions in terms of arbitrary constants.
The derivation of the formal expansions
in \cite{Lyapunov92} does not need nonresonance conditions.
In Chapter 23 of \cite{Lyapunov92}, the question of convergence
of these formal expansions
is studied. This is done
under the condition that there
are no resonances
and no repeated eigenvalues, and that
all the eigenvalues are stable or unstable.
Here one can find a note giving credit
to \cite{Poincare79b} for dealing with the more
general case of the Poincar\'e domain.
In particular, we call attention to Theorem II of
Section 24, which is a complete statement of
the strong stable manifold theorem for analytic systems
(see also Theorem II of chapter 13).
One interesting remark of Lyapunov in chapter 11
is that one can consider families that correspond
to any subset of eigenvalues. This amounts to setting to zero
a subset of the arbitrary constants used in the expansion.
This is hard to interpret from the dynamical point of
view since the arbitrary constants do not have a
dynamical interpretation. In particular,
the set obtained setting them to zero does not need to be
invariant. Of course, setting to zero all the
nondecreasing modes is an invariant set, as pointed out by
Poincar\'e. With modern hindsight, setting to zero
all the modes that are nondecreasing or decreasing
more slowly than a certain rate is invariant. Indeed, it
is the strongly stable manifold.
As it was shown in examples in \cite{delaLlave97}, in general, one
cannot get invariant manifolds tangent to
a subspace if there are resonances of the type we have
excluded in the present paper.
Overall, one cannot be but surprised by the enormous
similarities in the problems and
in the results between the contents of these chapters and
the corresponding ones of the book by
Poincar\'e, which appeared in the same year.
Of course, there are big differences in style
and in the methods as well as
in the way that proofs are presented.
A modern exposition of some of
the convergence results of Lyapunov can be found in
\cite{Lefschetz77} V.4. It contains
a statement and a proof of the expansion in
arbitrary constants under nonresonance
assumptions and provided that all
eigenvalues are stable, and that the linearization is
a constant (we remark that using Floquet theory,
one can reduce the periodic case to the constant case).
We have not been able to locate in any of these classical works the
consideration
of resonant terms.
\subsection{Modern work}
It seems that the particular case of one
dimensional stable invariant manifolds
(when there are no resonances)
has appeared several times in the modern literature.
The papers \cite{FranceschiniR81}
and \cite{FornaessG92} use the parameterization method
for one dimensional manifolds, specially in
conjunction with numerical analysis.
They establish not only convergence of
the series involved, but they also estimate
the errors incurred when using a numerical
approximation.
Indeed, both papers have taken care of
estimating actually the roundoff error so
that a finite calculation can establish
facts about transversality of intersections, etc.
It seems to us that similar results could be obtained
using the functional equations
\eqref{conjugacy} and the theory developed in the present
article.
Numerical work for higher dimensional maps has
been studied in \cite{BeynK98}, which
undertook the task of systematically computing
Taylor expansions of invariant manifolds.
This could be considered one
implementation of our result in Lemma~\ref{formal}
for finite dimensional systems.
The authors of \cite{BeynK98} indeed made the observation that
the calculations can be carried out to
any order provided that there are no resonances.
We note that our formalism could be used to
provide an {\it a posteriori} estimate of
the error of these numerical calculations.
Once a polynomial satisfies
\eqref{conjugacy} quite accurately, then
it is close to being a fixed point of
$\Tau$. Since $\Tau$ is a contraction,
there is a fixed point at a distance
that can be estimated by the error
of the numerical approximation.
This is the usual {\it a posteriori} estimates of numerical
analysis.
The work \cite{Poschel86} considers invariant manifolds
associated to nonresonant eigenspaces.
It also studies the equation of semiconjugacy
of the motion on the manifold
to a linear motion.
In contrast to the classical works,
it can deal with situations when
the eigenvalues are not stable and all of
the same sign. Small divisors appear,
but they can be overcome using the majorant
method and improvements of estimates in \cite{Siegel42}.
It is interesting to note that the nonresonance
conditions obtained in \cite{Poschel86}
are more general
than those required by the straightforward application of
usual KAM method. For more modern developments,
see \cite{Stolovitch94a}.
The paper \cite{CabreF94} took up the task of making
sense of the onedimensional manifolds of Poincar\'e
and Lyapunov, in the case of finite differentiability.
It reduced the problem to a fixed point equation that was
solved by the contraction method.
In this paper there was some consideration given to the resonant case,
and the result proved was that the
equation \eqref{conjugacy} could be solved if and only if
our equations for $K$ and $R$ provide a formal solution.
We note that all papers mentioned above seem to
require the dynamics in the invariant manifold
considered to be linearizable.
The work whose results
are most closely related to
those of the present paper is
\cite{delaLlave97}. It contains a study of invariant
manifolds which does not require the
motion on the invariant manifold to be linearizable.
In the present paper, we improve the results of
\cite{delaLlave97} by
not requiring the splitting to be invariant.
Hence, we can associate invariant manifolds
to eigenspaces in a nontrivial Jordan form.
Also, we can improve the regularity
obtained from $C^{r1+ {\rm Lip}}$ to
$C^r$ (this later improvement in the
regularity was obtained also in
\cite{ElBialy98}). The interest of
this sharp regularity is that the same method
that we use to optimize the regularity also allows
us to obtain rather sharp results on the dependence
on parameters.
The methods of the present article and of \cite{delaLlave97} are
quite different.
There are some generalizations of the nonresonant manifolds
considered in the present paper to some rather general
semigroups which include many parabolic equations
in \cite{delaLlaveW97}.
The main difference is that in \cite{delaLlaveW97},
the assumption on invertibility of the linearized operator
used in the present paper
is dropped, since the linearization of the
evolution of an elliptic equation is a compact operator,
hence, not invertible.
\subsection{Slow manifolds in applications.}
One of the cases which is most useful in applications
is when the invariant manifolds
correspond to the spaces that have eigenvalues closest to
the unit circle.
These are called {\it slow manifolds}. The reason why they are
useful is because they correspond to the modes
that converge the slowest and, hence, they can be used to
control the long term behavior.
It should be noted that, for slow manifolds, one could also
apply the Irwin theory of pseudostable manifolds.
These two methods, as it can be seen in examples in
\cite{delaLlave97}, do not lead to the same results.
In \cite{LiMS95}, it is argued that
beta function calculations of renormalization groups
correspond to computation of the jets of smooth invariant
manifolds as the ones we consider
in Lemma~\ref{formal} see specially
Remark \ref{rkunderthe}.
Nevertheless, it is also argued that
the ones which should be considered in
several problems of phase transitions
are other slow manifolds.
This is a very nice idea, and it would be quite
interesting to make it precise.
Of course, the mathematically precise
definition of renormalization group
as a well defined differentiable operator in a Banach
space, has only been achieved in a few cases
(mainly dynamical systems and hierarchical models).
Even in these cases where the renormalization operator
is a bona fide differentiable operator in Banach spaces,
its linearization is a compact operator, hence not
invertible, and the theorems of this paper do
not apply. One has to use those of
\cite{delaLlaveW97}.
In \cite{delaLlave97} it is shown that the
noncoincidence of these manifolds can be used
to prove that the invariant circles predicted
by the Hopf bifurcation for maps are often not $C^\infty$.
A field where slow manifolds have played an important role is
chemical kinetics.
Here, slow manifolds correspond to
the so called {\it quasisteady state approximations}.
There is a very extensive literature on these approximations.
We quote \cite{Fraser88}, \cite{MaasP92} for a geometric interpretation of
them.
We believe that the methods suggested in these two papers lead,
respectively
to the smooth nonresonance manifolds
discussed here and to the Irwin manifolds,
which are, in general very different.
Therefore, speaking of {\it the} slow invariant manifold
may bee misleading since there are several different objects that
can be given that name and should not be identified.
This, we think, would explain some of the discrepancies
reported in \cite{Davis97}. Some more recent references in
the chemical literature are \cite{Fraser98},
\cite{Smooke91},
\cite{PetersR93}, \cite{WarnatzMD96}.
For some specialized semigroups including the very
important NavierStokes equations some slow invariant manifolds
have been constructed in the remarkable articles
\cite{FoiasS84a},
\cite{FoiasS84b},\cite{FoiasS87}
(see also the surveys \cite{FoiasS86a}, \cite{FoiasS86b}).
In these works, slow manifolds are constructed
under nonresonance conditions. We note in particular
that \cite{FoiasS86b} establishes that these
invariant manifolds are analytic. This is in contrast
with examples in \cite{delaLlaveW97} which show
that, for semigroups, the slow manifolds may not be
analytic. We thank E. Titi for bringing these
papers to our attention.
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\end{document}