\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 102, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/102\hfil The Fuchsian Cauchy problem]
{The Fuchsian Cauchy problem}

\author[M. Belarbi,  M. Mechab \hfil EJDE-2009/102\hfilneg]
{Malika Belarbi, Mustapha Mechab}  % in alphabetical order

\address{Laboratoire de Math\'ematiques \\
Universit\'e Djilali Liab\`es \\
B.P. 89, 22000 Sidi Bel Abb\`es, Algeria}
\email[M. Belarbi]{mkbelarbi@yahoo.fr}
\email[M. Mechab]{mechab@yahoo.com}

\thanks{Submitted July 23, 2009. Published August 21, 2009.}
\subjclass[2000]{32W50}
\keywords{Fuchsian operators; holomorphic functions}

\begin{abstract}
 This article presents a global version of the main theorem by
 Baouendi and  Goulaouic \cite{bago}, in the space of  differentiable
 functions with respect to Fuchsian variable,  and  holomorphic
 with respect to other variables. We have no assumptions
 on the characteristic exponents, and no hyperbolicity conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

 Baouendi and Goulaouic \cite{bago} generalized the Cauchy-Kowalevsky
and Holmgren theorems for the Cauchy problem
\begin{equation}
 \begin{gathered}
\mathcal{P} u(t,x) = f(t,x),\quad (t,x)\in \mathbb{R}\times\mathbb{R}^q \\
 D_t^l u(0, x) = w_l(x) , \quad 0\leq l \leq m-k-1
\end{gathered} \label{maII}
\end{equation}
with
\begin{equation}\small
\mathcal{P}=  t^{k} D_t^m +  \sum_{j=m-k}^{m-1} a_j(x) t^{j-m+k} D_t^j+
\sum_{j=0}^{m-1} \sum_{|\beta|\leq m-j} a_{j,\beta}(t,x)
t^{\max\{0, j-m+k+1 \}} D_t^j
 D_x^\beta
 \label{equation11}
\end{equation}
called Fuchsian operators with weight $m-k$ with respect to $t$.
Its associated characteristic
polynomial (or indicial polynomial) is defined by
\begin{align*}
\mathcal{C}(\lambda, x)
&= \lambda (\lambda-1)\dots (\lambda-m+1)+
a_{m-1}(x) \lambda (\lambda-1)\dots (\lambda-(m-1)+1)\\
&\quad+\dots+
a_{m-k}(x) \lambda (\lambda-1)\dots (\lambda-(m-k)+1)
\end{align*}
and its $m$ roots $\lambda_1(x),\dots, \lambda_k(x),\ \lambda_{k+1}=0, \ \dots, \lambda_m= m-k-1$ are called characteristic
exponents (or characteristic index) of $\mathcal{P}$ at $x$. Under the
condition
\begin{itemize}
\item[(C1)]  For any integer $\lambda \geq m-k$,
 $\mathcal{C}(\lambda, 0)\neq 0$,
\end{itemize}
they solve \eqref{maII} in the space of real-analytic functions in
a neighborhood of the origin. When the problem \eqref{maII} is
considered in the spaces of partial-analytic functions
($C_k^{m-k+h}$ class on the variable $t$ which will be defined in
Section 2.), the condition (C1) is not sufficient to solve this
problem. In order to show the well-posedness in these functional
spaces, they imposed an additional assumption of the form
\begin{itemize}
\item[(C2)] $\Re e \lambda_l(x) <m-k+h$,  for $x\in \mathcal{V}$
and  $1\leq l \leq m$,
\end{itemize}
where $\mathcal{V}$ is a connected neighborhood of $0$ in $\mathbb{C}^q$.

 Yamane \cite{yamane} gave a global version of
the main result of \cite{bago} for Fuchsian operators with
polynomial coefficients, by using  the condition (C1) and
the method of successive approximations.
 Mandai \cite{mandai} has been interested in the assumptions
of characteristic exponents. Applying the method of Frobenius,
he could omit them, in particular for hyperbolic equations considered
by  Tahara  in different functional spaces.

When $m=k$, (case of weight $0$),  Lope \cite{jose} multiplied
the coefficients $a_{j,\beta}(t,x)$ by $\mu(t)^{|\beta|}$
where $\mu(t)$ is a weight function introduced by  Tahara
\cite{tahara3}, in order to construct another class of Fuchsian
operators slightly more general than the ones given in \cite{bago}.
After specifying some properties of this function, without the
condition (C1), he showed  the very important role of the
condition (C2) to invert some operators for getting a unique
local solution of$\mathcal{C}_m^0$ class in $t$ and
analytic in $x$ for the above problem.
On the other hand,  Tahara \cite{tahara2} used the condition
(C1)  for any $x\in \mathbb{R}^q$  as an essential argument under suitable
hyperbolicity conditions on $\mathcal{P}$ to prove  $\mathcal{C}^{\infty}$
well-posedness of the Cauchy problem  \eqref{maII}.


The goal of our work is to establish the existence and uniqueness of
a solution of \eqref{maII} in spaces of partially holomorphic
functions for Fuchsian operators with the principal part whose
coefficients are polynomial in non-Fuchsian variables with a
particular degree. The consideration of this type of operators is
classic and natural for global solution of the Cauchy problem. They
have been considered by several authors, in characteristic and non
characteristic case,  as \cite{pers, yamane, pon1}.
Hamada \cite{hamada}  gave   an example of operators with
polynomial coefficients for which the associated Cauchy problem,
with some polynomial data, does not admit any
global solution.

The main result of this paper is obtained without the
hyperbolicity condition and other conditions related to the
characteristic exponents.  We use the same techniques as in
\cite{bemame3}. We construct some Banach spaces, following the
idea developed in \cite{wa1}, where we reduce our differential
problem into inversion of some operator.

Contrary to \cite{jose}, our techniques allow us to give result of
Nagumo's type (see Theorem  \ref{t03}) for Fuchsian operators without
the conditions (C1) and  (C2). On other hand, we use the
condition  (C1) in our main result.

\section{Notations and  Main result}

 Let $ q\in\mathbb{N}^*$ and $\beta=(\beta_1, \dots, \beta_q) \in \mathbb{N}^q$,
we denote $|\beta|=\sum_{i=1}^q\beta_i$. In what follows,
$x=(x_1,\dots,x_q)\in \mathbb{C}^q$, for $1\leq i\leq q $,
$D_{x_i}=\frac{\partial}{\partial x_i}$ is
the  partial differentiation with respect to $x_i$,
$ D_x=(D_{x_1},\dots,D_{x_q})$  and we set
\begin{gather*}
 x^{\beta}=\prod_{i=1}^q x_i^{\beta_i}, \quad
D_x^{\beta}=  D_{x_1}^{\beta_1}\dots  D_{x_q}^{\beta_q} ,\\
 |x|=\max_{ 1\leq i\leq q }|x_i|,\quad
B^x_R=\{x\in \mathbb{C}^q: |x|< R\}\,.
\end{gather*}
For fixed positive integers $ s>m+ 1$
and $s'=s-1$, we set:
\begin{gather*}
  \Delta_{\rho,R}=\big\{(t,x)\in
\mathbb{R} \times \mathbb{C}^q: (\rho R)^{s'/s}(\rho |t|)^{1/s}+
 |x_1|+\dots+|x_q|< \rho R \big\}\\
 |\Delta_{\rho, R}|=\{(|t|, x): (t,x)  \in \Delta_{\rho,R}\}\,.
\end{gather*}
Let
$\mathbb{C}[[X]]$ be the  space of formal series on $X$ whose  coefficients
belong to  $\mathbb{C}$ and  $\mathcal{H}( \mathbb{C}^q)$ be the space of
entire functions in $\mathbb{C}^q$.
For $h\in \mathbb{N}$, $\mathcal{U} \subset\mathbb{R}$ and $\Omega\subset \mathbb{C}^q$ open sets,
$\mathcal{C}^{h,\,\omega}(\mathcal{U}\times\Omega)$ denote
the algebra of functions $f(t,x)$ of
class $\mathcal{C}^{h}$ in $t$ on $\mathcal{U}$ and holomorphic in $x$
on $\Omega$.

For $m \in \mathbb{N}$,  we consider  $ \mathcal{C}_m^{h,\omega}(\mathcal{U}\times\Omega)$
the set of functions $f\in \mathcal{C}^{h,\omega}(\mathcal{U}\times\Omega)$
such that for every $0 \leq  \gamma \leq m$,
$t^{\gamma} f\in  \mathcal{C}^{h+\gamma,\omega}(\mathcal{U}\times\Omega)$.

The expression  $f= O(t^{\nu})$ in  $\mathcal{C}^{0,\omega}_m(\mathbb{R}
\times\mathbb{C}^q)$ means
$$
\exists g\in \mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q): f(t,x)
=  t^{\nu} g(t,x)\,.
$$


Consider problem \eqref{maII} for a Fuchsian operator $\mathcal{P}$ given
in \eqref{equation11}. Assume that the coefficients $ a_j(x)=a_j$
is a constant in $\mathbb{C}$ and
$a_{j,\beta}(t,x) \in \mathcal{C}^{\infty,\,\,\omega}(\mathbb{R}\times\mathbb{C}^q)$
satisfy the following assumption
\begin{itemize}
\item[(H0)] For any $(j, \beta)$ such that $j+ |\beta|=m$,
 the functions $a_{j,\beta}(t,x )$ are  polynomial in $x$  with
 $\mathop{\rm ord}_x a_{j,\beta}(t,x ) < |\beta|$ and  their coefficients
 are of  $\mathcal{C}^{\infty}$ class in $t$ on $\mathbb{R}$.
\end{itemize}

\begin{remark}  \label{remark1} \rm
The Fuchsian characteristic polynomial associated with
operator $\mathcal{P}$ given in \eqref{equation11}, is defined by the
 identity:
$$
\forall \lambda\in \mathbb{N}, \; t^{-\lambda+ m-k} \mathcal{P}
t^{\lambda}\big|_{t=0}= \mathcal{C}(\lambda, x)\,.
$$
\end{remark}

\begin{remark} \rm
From the hypothesis $a_j(x)$ constant in $\mathbb{C}$, we can write
$\mathcal{C}(\lambda, x) =\mathcal{C}(\lambda)$.
\end{remark}

According to this remark, the condition (C1) becomes:
\begin{itemize}
\item[(C1)] For any integer$\lambda \geq m-k$,\quad
$\mathcal{C}(\lambda)\neq 0$.
\end{itemize}
Under the above hypotheses on coefficients of $\mathcal{P}$, our main result
is as follows.

\begin{theorem} \label{principal1}
Let  $\mathcal{P}$ be a  Fuchsian operator defined by
\eqref{equation11}.
 If  condition {\rm (C1)}  holds, then for any functions
 $f\in\mathcal{C}^{\infty,\omega}(\mathbb{R}\times \mathbb{C}^q)$ and
$w_1, \dots , w_{m-k-1}\in \mathcal{H}(\mathbb{C}^q)$,
there exists a unique solution
$u\in \mathcal{C}^{\infty,\omega} (\mathbb{R}\times \mathbb{C}^q)$
of Cauchy problem \eqref{maII}.
\end{theorem}

\begin{remark} \rm
Take  the expression of operator $\mathcal{P}$ given in \eqref{equation11} and
look for solution $u$ of our Fuchsian Cauchy problem
\eqref{maII} in the  form
$$
u(t,x)= \sum_{l=0}^{m-k-1} \frac{w_l(x)}{l!} t^l+ t^{m-k}v(t,x)
$$
which satisfies the initial conditions, then
problem \eqref{maII} is equivalent to
\begin{equation}
\mathcal{P} t^{m-k}v(t,x)= f(t,x)- \mathcal{P} \Big[\sum_{l=0}^{m-k-1}
\frac{w_l(x)}{l!} t^l\Big]\,. \label{problem}
\end{equation}
Note that the right hand side is a known function  belongs to
 $\mathcal{C}^{\infty,\,\omega}(\mathbb{R}\times\mathbb{C}^q)$, and the  operator
$\mathcal{P}_1$ defined by $ \mathcal{P}_1=  \mathcal{P} t^{m-k}$is also Fuchsian with
weight $0$ and its Fuchsian characteristic polynomial
$\mathcal{C}_1$ satisfies
$$
\forall \lambda \in \mathbb{N}, \quad  \mathcal{C}_1(\lambda)=t^{-\lambda}
\mathcal{P}_1 t^{\lambda}\Big|_{t=0}.
$$
Since
$$
\forall \lambda \in \mathbb{N}, \quad  t^{-\lambda}  \mathcal{P}_1
t^{\lambda}\Big|_{t=0}= t^{-\lambda-(m-k)} t^{m-k} \mathcal{P} t^{m-k}
t^{\lambda}\Big|_{t=0},
$$
from   remark \ref{remark1} we have
$$
\forall \lambda \in \mathbb{N}, \quad \mathcal{C}_1(\lambda) =  \mathcal{
C}(\lambda+ m-k)\,.
$$
Then the condition (C1) implies
\begin{equation} \forall \lambda \in
\mathbb{N}, \quad \mathcal{C}_1(\lambda) \neq 0\,. \label{condition}
\end{equation}
Hence, the transformation of problem \eqref{maII} to
\eqref{problem}, allows us to limit our studies for the case
weight $0$, with the condition \eqref{condition},
in the functional space $\mathcal{C}^{\infty,\omega}(\mathbb{R}\times\mathbb{C}^q)$.
\end{remark}

\section{Fuchsian Cauchy problem with weight $0$}

Let  $\mathcal{P}$ be a Fuchsian operator with weight $0$,  expressed in the
form
\begin{equation}
 \mathcal{P} = t^m D_t^m+ \sum_{j =0}^{m-1} a_j(x)
t^j D_t^j   +\sum_{j=0}^{m-1} \sum_{ 0< |\beta| \leq m-j }  t
a_{j,\beta}(t,x)
  t^{j} D_t^{j}D_x^{\beta}\ .
 \label{k01}
 \end{equation}
We assume that $a_j(x)=a_j$ are constants  in $\mathbb{C}$ and
$a_{j,\beta} \in \mathcal{C}^{\infty,\omega}
(\mathbb{R}\times\mathbb{C}^q)$ satisfying (H0).

\begin{theorem}  \label{principal2}
 Let  $\mathcal{P}$ a Fuchsian  operator with weight  $0$defined in \eqref{k01}.
Under above hypotheses, if its Fuchsian characteristic
polynomial  $\mathcal{C}(\lambda)\not= 0$ for all $\lambda \in  \mathbb{N}$,
then for any functions  $f$ in
$ \mathcal{C}^{\infty,\omega} (\mathbb{R}\times \mathbb{C}^q)$, the following equation
\begin{equation}
\mathcal{P} u(t,x) =f(t,x) \label{j}
\end{equation}
admits a unique solution
$u \in \mathcal{C}^{\infty,\omega} (\mathbb{R}\times \mathbb{C}^q)$.
\end{theorem}

The sketch of the proof is follows:\\
$\bullet$ We first give  two decompositions of operator $\mathcal{P}$.
\\
$\bullet$ We solve \eqref{j} in the spaces
 $\mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$.
For that, we transform our problem to inversion of an operator
$(I+\mathcal{B})$.
\\
$\bullet$ We introduce Banach spaces which cover
$\mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$  and prove
that $\|\mathcal{B}\|<1$.
\\
$\bullet$ We complete the proof in the subsection \ref{fin}.


\subsection{Decomposition of operator $\mathcal{P}$}

We have the following properties:

\begin{lemma}[\cite{bago}]
 Let $j\in \mathbb{N}$. we have
\begin{enumerate}
\item   $  t^j  D_t^j = \displaystyle\Big(  D_t t\Big)^j+
\sum_{k =0}^{j-1} c_k \Big( D_t t\Big)^k$, where
$c_k\in \mathbb{Z} $.
\item  For every $u\in  \mathcal{C}^{\infty}(\mathbb{R})$,
there exists a set of functions
$\{ u_l\}_{0\leq l \leq j} \subset \mathcal{C}^{\infty}(\mathbb{R})$  such that
\begin{equation}
\forall \, v\in \mathcal{C}^{\infty}(\mathbb{R}),\quad
t^j u D_t^j v = \displaystyle\sum_{l=0}^j t^l D^l_t[u_l v]
 \label{quation10}
\end{equation}
\end{enumerate}
\end{lemma}

From this lemma, we can rewrite  $\mathcal{P}$ in the following forms
\begin{equation}
 \mathcal{P} = ( D_t t)^m+ \sum_{j =0}^{m-1}b_j
(D_t t)^j   +\sum_{j=0}^{m-1} \sum_{0<|\beta| \leq m-j }  t
\widetilde{b}_{j,\beta}(t,x)
   (D_t t)^{j}D_x^{\beta}
 \label{k02}
 \end{equation}
where $b_j \in \mathbb{C}$ and
$\widetilde{b}_{j,\beta} \in \mathcal{C}^{\infty,\omega}
(\mathbb{R}\times \mathbb{C}^q)$,
or
\begin{equation}
 \mathcal{P} = t^{m} D_t^{m} + \sum_{j=0}^{m-1} a_j
t^{j} D_t^{j}- \sum_{j=0}^{m-1} t^{j+1} D_t^{j} \mathcal{B}_{m-j}
 \label{k1}
 \end{equation}
 where
$ \mathcal{B}_{m-j}= \displaystyle \sum_{0< |\beta| \leq m-j } b_{j,\beta}(t,x) D_x^{\beta}$
 and $b_{j,\beta}\in  \mathcal{C}^{\infty,\omega} (\mathbb{R}\times \mathbb{C}^q)$.


\begin{remark}  \label{remarque1} \rm
According to (H0), for all $(j, \beta)$ such that  $j+ |\beta|=m$,
the coefficients $ \widetilde{b}_{j,\beta}(t,x ) $  and
$b_{j,\beta}(t,x) $ are also polynomials in $x$  with
$\mathop{\rm ord}_x \widetilde{b}_{j,\beta}(t,x ) < |\beta|$ and
$ \mathop{\rm ord}_x b_{j,\beta}(t,x ) < |\beta|$ whose  coefficients are of
class $\mathcal{C}^{\infty}$  in $t$ on $\mathbb{R}$.
\end{remark}

\subsection{Resolution of equation \eqref{j} in
$\mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$}

Let $\mathcal{P}$ a Fuchsian operator defined in \eqref{k01} with
the same assumptions.
Take a positive  integer $\nu$ enough large such that
\begin{equation}
 \sum_{j <m}
|b_{j}| \frac{1}{\nu ^{m-j}}<  \frac{1}{3}\,. \label{theta}
\end{equation}

\begin{theorem}\label{t03}
 For any  function  $f= O(t^{\nu})$ in
$\mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q),$there exists a
unique solution $u=O(t^{\nu}) $ in
$ \mathcal{C}_m^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$  of the equation \eqref{j}
\end{theorem}

\begin{proof}
We start by seeking an equivalent problem.
It is known that
$$
\forall u\in \mathcal{C}_1^{0,\omega}(\mathbb{R}\times\mathbb{C}^q), \quad
D_t t [ t^{\nu} u(t,x)]=  t^{\nu} (D_t t +\nu) u(t,x)
$$
then using  the expression \eqref{k02} of the operator $\mathcal{P} $
and a change of unknown $u(t,x)= t^{\nu} v(t,x)$ with
$v \in \mathcal{C}_m^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$,
we transform \eqref{j} to
\begin{equation}
\mathcal{P}_1 v(t, x) =g(t, x), \label{quation12}
\end{equation}
where
$$
\mathcal{P}_1= (D_t t +\nu )^m + \sum_{j=0}^{m-1} b_{j} (D_t t +\nu)^j
+ \sum_{j=0}^{m-1} \sum_{0<|\beta| \leq m-j  }
t \widetilde{b}_{j,\beta}(t,x) D_x^{\beta} (D_t t + \nu)^{j}
$$
and $g\in \mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$ such that
$f(t,x)=t^\nu g(t,x)$.\\

We also show the following proposition.

\begin{proposition} \label{proposition2}
 For all $j\in \mathbb{N}^*$, the operator $(D_t t + \nu)^{j}$ is an
isomorphism from  $ \mathcal{C}_j^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$ to
$ \mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$,
and its inverse  $H_{\nu}^{j} $  is defined by
 $$
 H_{\nu}^{j}=\underbrace{  H_{\nu}  \dots  H_{\nu}}_{j\text{ factors}}\,,
\quad\text{where }
H_{\nu} u(t, x) =\int_0^1 \sigma^{\nu} u(\sigma t, x ) d  \sigma\,.
$$
\end{proposition}

Then we can look for a solution  of  \eqref{quation12} in the  form
$$
v(t, x) =H^m_{\nu}\psi(t, x) \quad \hbox{and}\quad
\psi \in \mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)
$$
which establishes the equivalence of  \eqref{j}  with
\begin{equation}
(I+\mathcal{B}) \psi = g \label{III}
\end{equation}
where
$$
\mathcal{B}= \sum_{j =0}^{m-1} b_{j} H_{\nu}^{m-j}+ \sum_{j =0}^{m-1} \sum_{
0<|\beta| \leq m-j  } t \widetilde{b} _{j,\beta}(t,x)
  D_x^{\beta} H_{\nu}^{m-j}\,.
$$
In the next subsection  we construct a suitable Banach space,
on which we solve  problem \eqref{III}.


\subsubsection{The Banach spaces $E_{\rho, R, a }^{0,\omega}$}
Let
\[
\Phi(t, x)=\sum_{\delta \in \mathbb{N}^q}
\varphi_{\delta}(t) \frac{x^{\delta}}{\delta! },\quad
 \Psi(t, x)=\sum_{\delta \in \mathbb{N}^q}
\psi_{\delta}(t) \frac{x^{\delta}}{\delta! }
\]
be two formal series in $ \mathbb{R}_+[[x]]$. We denote
$\Psi(t,x) \ll \Phi(t,x)$ if
for all $\delta \in \mathbb{N}^q$,
$\psi_{\delta}(t) \leq \varphi_{\delta}(t)$.

For $u\in \mathcal{C}^{0,\omega}(\Delta_{\rho, R})$,
  $u(t,x) \lll \Phi(t, x )$  means
$$
  \forall |t|< R, \quad \forall \delta \in \mathbb{N}^q,\quad
\Big|D_x^{\delta}u(t,0) \Big| \leq
\varphi_{\delta} (t)\,.
$$

\begin{proposition} \label{itee}
For  $u,  v  \in \mathcal{C}^{0,\omega}(\Delta_{\rho, R})$,
we have the following:
\begin{enumerate}
\item
 $ \Big(u(t,x)\lll \Phi(t, x ) \quad \hbox{and}\quad
\Phi(t, x ) \ll \Psi(t, x )\Big)$ imply
$u (t,x)\lll \Psi (t, x )$;

\item
If $\lambda,\mu \in \mathbb{C}$, we have
$\Big(u(t,x)\lll \Phi(t, x )\ \hbox{and}\
v(t,x)\lll \Psi(t, x ) \Big)$ implies
$\lambda u(t,x)+ \mu v(t,x) \lll |\lambda|\Phi(t, x )+ |\mu|\Psi(t, x )$;

\item
$\Big(u(t,x)\lll \Phi(t, x )\ \hbox{and}\ v(t,x)\lll \Psi(t, x )\Big)$
implies \\
$u(t,x) v(t,x) \lll \Phi(t, x )\Psi(t, x )$;

\item
$ u(t,x)\lll\Phi(t, x )$ if and only if
$\forall \gamma \in  \mathbb{N}^q, \; D_x^{\gamma} u(t,x)
\lll  D_x^{\gamma} \Phi(t, x ) $.
\end{enumerate}
\end{proposition}


To construct our Banach
spaces, we use the same formal series given in \cite{pon1}.
Let  $\tau$ and $\xi$ be one-dimensional variables. For all
$R, a \in \mathbb{R}^*_+$, we set
$$
\varphi_R^a(\xi)= \frac{e^{a \xi}}{ R- \xi}\in \mathbb{R}_+[[\xi]]
 $$
which converges  in the open set $ \{\xi \in \mathbb{C}: |\xi| < R\}$ and
satisfies the following properties.

\begin{lemma}[{\cite[Lemma 1.4]{pon1}}] \label{Al1}
For all  $p, l \in \mathbb{N}$, we have
\begin{enumerate}
\item  $D^p\varphi_R^a \ll  a^{-l} D^{p+l}\varphi_R^a$;

\item  $D^p\varphi_R^a \ll \displaystyle \frac{p!}{(p+l)!} R^l D^{p+l}\varphi_R^a$.
\end{enumerate}
\end{lemma}

We also put
$$
 \Phi^{a}_{R} (\tau, \xi) =
 \sum_{p\in \mathbb{N}} \tau^p  R^{s'p} \frac{D^{s
 p}\varphi^a_{R }(\xi)}{ (s p) !} \in \mathbb{R}_+[[\tau,\xi]],
$$
which converges in the set $\{ (\tau, \xi)\in \mathbb{R}\times \mathbb{C} :
 R^{s'/s} |\tau|^{1/{s}} + |\xi| < R \}$ and satisfies the
following estimates
\begin{gather}
\frac{1}{R- (\tau+\xi)} \ll \Phi_{R}^{a}(\tau,\xi),
\label{Al3}\\
\forall\,\eta > 1,\quad \frac{\eta R}{\eta R-
(\tau+\xi)}\Phi_{R}^{a}(\tau,\xi)
 \ll \frac{\eta }{\eta -1}\Phi_{R}^{a}(\tau,\xi)\,.
 \label{App1}
\end{gather}
For parameters $\rho,  a  >1$,
for all $t\in \mathbb{R}$  and $x=(x_1, \dots, x_q)\in \mathbb{C}^q$,  we set:
\begin{itemize}
\item $\tau =\rho |t|$, $\xi=x_1+\dots+x_q$;

\item $ \Phi^{a}_{\rho, R} (t,x)
 = \Phi_{\rho R}^a (\rho |t| , \xi)
 = \displaystyle \sum_{p\in \mathbb{N}} \tau^p (\rho R)^{s'p}
\dfrac{D^{s p}\varphi^a_{\rho R } (\xi)}{ (s p) !}$;

\item $ E_{\rho, R, a }^{0,\omega}
=\{u\in \mathcal{C}^{0,\omega}
(\Delta_{\rho, R}),\;\exists C\geq 0: u(t,x)\lll C
\Phi^{a}_{\rho, R} (t,x)\}$;

\item $ \|u\|=\min\{C\geq 0 : u(t,x)\lll C  \Phi^{a}_{\rho, R} (t,x)\} $
\end{itemize}

\begin{proposition}
$ \big( E_{\rho, R, a }^{0,\omega}, \|\cdot\|\big)$ is a Banach
space.
\end{proposition}

The proof of the above proposition follow the steps in
 \cite[Proposition 6.1]{wa1}.

\begin{remark}[\cite{beme2}] \label{propertx1} \rm
Let  $ \rho_1 \geq   \rho_2$  and  $R_1 \geq R_2$, for all
$a_1,a_2\in\mathbb{R}_+ $. Then we have
$E_{\rho_1, R_1, a_1 }^{0,\omega} \subset
E_{\rho_2, R_2, a_2 }^{0,\omega}$ with continuous injection.
\end{remark}

\begin{proposition} \label{malika.3}
 Let $ R_0, \rho  \in \mathbb{R}_+^*$  and
$ f\in \mathcal{C}^{0,\omega}([-R_0,+R_0] \times
\overline{ B_{\rho R_0}^x)}$, then for all $ R\in ]0,R_0[$,
$$
f(t,x) \lll  C \frac{ \rho R }{ \rho R - (\tau+\xi) }\,,
$$
where  $C= \sup_{[-R_0,+R_0] \times \overline{B^x_{  \rho R_0}}}|f(t,x)|$.
\end{proposition}

\begin{proof}
 For $R\in]0,R_0[$, applying the Cauchy's Estimate to holomorphic
function $x\to f(t,x)$ in $B_{\rho R}^x$ and bounded by $C$, we obtain
\begin{equation}
\forall |t|< R, \; \forall \delta \in \mathbb{N} ^q,\quad
|D_x^{\delta}f(t,0) | \leq C \frac{|\delta|!}{(\rho R)^{|\delta|}}\,.
\label{E1}
\end{equation}
Since $\displaystyle\Big(\frac{\rho R}{\rho R - \tau}\Big)^{|\delta|+1}\geq 1 $
and
$$
D_x^{\delta}\Big(\frac{\rho R}{\rho R - (\tau+\xi)}\Big)_{\Big|x=0}=
D_\tau^{|\delta|}\Big(\frac{\rho R}{\rho R - \tau}\Big)=
\frac{|\delta|!}{(\rho R)^{|\delta|}} \Big(\frac{\rho R}{\rho R -
\tau}\Big)^{|\delta|+1}\,,
$$
from \eqref{E1}, we have
$$
\forall |t|< R, \; \forall \delta \in \mathbb{N} ^q,\quad
|D_x^{\delta}f(t,0) | \leq C\,
 D_x^{\delta}\Big(\frac{\rho R}{\rho R - (\tau+\xi)}\Big)\big|_{x=0}
$$
which implies
\begin{equation}
f(t,x)\lll C \frac{\rho  R}{ \rho  R- (\tau+\xi)}\,. \label{emino21}
\end{equation}
\end{proof}

According  to (\ref{Al3}), we have the following result.

\begin{remark} \label{Aoublies} \rm
If  $ f\in \mathcal{C}^{0,\omega}(\mathbb{R} \times \mathbb{C}^q)$, then for all
$R, \rho$
and  $ a, f\in E_{\rho, R, a }^{0,\omega}$ and $\|f\|$ \ is
independent of  parameter $a$.
See \cite[Proposition 3.6]{beme2}.
\end{remark}


\begin{corollary}  \label{kika}
Let $p\in \mathbb{N}$ and let a polynomial
$ F(t,x)= \sum_{|\gamma|\leq p} f_{\gamma}(t) x^{\gamma}$
where $f_{\gamma}\in\mathcal{C}^{0}(\mathbb{R})$, then for all $\rho, R>0$
such that $\rho R >1,$we have
$$
 F(t,x) \ll M(R) (\rho R)^p  \frac{ \rho R}{ \rho R-   (\tau+\xi)},
$$
where  $ M(R)= \mathop{\rm card} \{\gamma \in \mathbb{N}^q:
|\gamma|\leq p\}\max_{|\gamma|\leq p }
\big\{\sup_{|t|<R}|f_{\gamma}(t)|\big\}$,
 with $\mathop{\rm card} A$ denoting the cardinality of the set $A$.
\end{corollary}

\subsubsection{Resolution of equation \eqref{III} in
$E_{\rho, R, a }^{0,\omega}$}

\begin{proposition} \label{ma40}
 Let $\Phi(t,x)= \sum_{\delta}\varphi_\delta(t)\frac{x^\delta}{\delta!}\in\mathbb{R}_+[[x]]$
and
$u  \in \mathcal{C}^{0,\omega}(\Delta_{\rho, R})$ such that
$u(t,x)\lll \Phi(t, x)$, then
$$
\forall j \in \mathbb{N},\quad \mathcal{H}_{\nu}^{j}u (t,x) \lll
\mathcal{H}_{\nu}^{j}\Phi(t, x )
$$
\end{proposition}

\begin{proof}
 Let  $u  \in \mathcal{C}^{0,\omega}(\Delta_{\rho, R})$ such that
$u(t,x)\lll \Phi(t, x )$,
then
\begin{equation}
\forall  |t|<R,\; \forall \delta \in \mathbb{N}^q,\quad
\Big|D_x^{\delta}u(t,0) \Big| \leq \varphi_{\delta} (t)\,.
\label{hxpothese}
\end{equation}
By applying operator $\mathcal{H}_{\nu}^j$  to the function $u$,  we
obtain
\begin{equation}
\mathcal{H}_{\nu}^{j}u (t,x) = \int_{[0,1]^{j}}\prod_{l=1}^{j}
(\sigma_l)^{\nu} u(\prod_{l=1}^{j}\sigma_l\ t,x) \prod_{l=1}^{j}
d\sigma_l\,. \label{ma20}
\end{equation}
which implies
\begin{equation}
\forall \delta \in \mathbb{N}^q,\quad  \Big|D_x^{\delta} \mathcal{H}_{\nu}^{j}u
(t,0)\Big| \leq \int_{[0,1]^{j}}\prod_{l=1}^{j} (\sigma_l)^{\nu}
\Big| D_x^{\delta} u(\prod_{l=1}^{j}\sigma_l\ t,0) \Big|
\prod_{l=1}^{j} d\sigma_l \,.\label{equation01}
\end{equation}
 From (\ref{hxpothese}) we obtain:
$\forall  |t|<R,\; \forall \delta \in \mathbb{N}^q$,
\[
\Big|D_x^{\delta}\mathcal{H}_{\nu}^{j} u (t,0)\Big|
\leq \int_{[0,1]^j } \prod_{l=1}^j (\sigma_l)^{\nu} \varphi_{\delta}
(\prod_{l=1}^{j} \sigma_l  t) \prod_{l=1}^{j} d\sigma_l
=\mathcal{H}_{\nu}^{j}\varphi_{\delta} (t)\,.
\]
Hence, we have
$$
\mathcal{H}_{\nu}^{j} u (t,x)\lll \mathcal{H}_{\nu}^{j} \Phi(t,x)\,.
$$
\end{proof}

\begin{proposition} \label{propo01}
For all  $R>0$,  there exists  $\rho_0$ such that,
 for any $\rho > \rho_0$, there exists $a_{\rho}>0$  such that
for any $a> a_{\rho}$,  equation \eqref{III} admits a unique   solution
$ \psi \in E_{\rho, R, a }^{0,\omega}$.
\end{proposition}

To prove this proposition, we show that $\|\mathcal{B}\|< 1$ in
$ E_{\rho, R, a }^{0,\omega}$.
For that we establish the intermediate results.

\begin{lemma} \label{ma11}
For all $u\in E_{\rho, R, a }^{0,\omega}$, we have
\begin{equation}
 D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x) \lll
\|u\|   \sum_{ p \in \mathbb{N}} \frac{1}{(\nu+p+1)^{m-j}}
 \tau^p (\rho R)^{s'p} \frac{D^{sp+|\beta|}
\varphi^a_{\rho R }(\xi)}{(s p)!}\,.
\end{equation}
\end{lemma}

\begin{proof}
Let  $u\in E_{\rho, R, a }^{0,\omega}$, then
$u(t,x) \lll \|u\| \Phi_{\rho, R}^a (t, x )$.
From Proposition \ref{ma40} and the fourth
majoration of Proposition \ref{itee}, we obtain
 \begin{equation} \label{ma10}
D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x) \lll \|u\|  D_x^{\beta}
\mathcal{H}_{\nu}^{m-j}\Phi_{\rho, R}^a (t, x)\,.
\end{equation}
Using definitions of the formal  series  $\Phi_{\rho, R}^a$  and
the  operator $ \mathcal{H}_{\nu}^{m-j}$, we get
\begin{align*}
 D_x^{\beta}\mathcal{H}_{\nu}^{m-j}\Phi_{\rho,  R}^a (t, x)
&= \sum_{ p \in \mathbb{N}}   \prod_{l=1}^{m-j}
\int_0^1  (\sigma_l)^{\nu+p} d\,\sigma_l (\rho |t|)^p
 (\rho R)^{s' p} \frac{D^{s p+|\beta|} \varphi^a_{\rho R }
(\xi)}{(sp)!}  \\
& =
 \sum_{ p \in \mathbb{N}} \frac{1}{(\nu+p+1)^{m-j}}
 \rho^p |t| ^p (\rho R)^{s' p} \frac{D^{s p+|\beta|}
\varphi^a_{\rho R }(\xi)}{(sp)!}\,.
\end{align*}
By substituting this  right hand side  in  (\ref{ma10}),
we complete the proof.
\end{proof}

\begin{lemma} \label{lemme02}
 Let  $u\in E_{\rho, R, a }^{0,\omega}$, then
\begin{enumerate}
\item   If $j+|\beta|<m$, then
 \[
t\ D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x) \lll \|u\|  C_{j,
\beta}(R, \rho )   a^{-1}\Phi_{\rho, R}^a(t, x)
\]

\item  For all$j+|\beta|=m$, we have
$$
t\ D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x)
\lll C_0 \|u\| \rho ^{-1} (\rho R)^{1-|\beta|}
\Phi_{\rho, R}^a(t, x),
$$
where $C_0\in \mathbb{R}^*_+ $  and $C_{j, \beta}(R, \rho)$ is a positive
constant independent of parameter $a$.
\end{enumerate}
\end{lemma}

\begin{proof}
From Lemma  \ref{Al1}, we have
 \[
 D^{sp+|\beta|}\varphi^a_{\rho R }(\xi)
\ll
 a^{-1}(\rho R)^{s-(|\beta|+1)} \frac{(s p+|\beta|+1)!}{(s(p+1))!}
 D^{s(p+1)} \varphi^a_{\rho R } (\xi)\,.
\]
When we substitute this result in (\ref{ma11}), we get
\begin{equation} \label{ma12}
\begin{aligned}
t \ D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x)
&\lll  |t|\  \|u \| a^{-1}  (\rho R)^{- |\beta|}
\sum_{ p \in \mathbb{N}} \frac{1 }{(\nu+ p+1)^{m-j}}\tau^p (\rho R)^{s' p}
(\rho R)^{(s-1)}\\
&\quad\times \frac{(s p+|\beta|+1)!}{(s p)!}
 \frac{D^{s(p+1)} \varphi^a_{\rho R } (\xi)}{(s(p+1))!}
\end{aligned}
\end{equation}
Since   $ \frac{(s p+|\beta|+1)!}{(s p)!} \leq (s
p+|\beta|+1)^{|\beta|+1} $   and the sequence
\[
\Big(\frac{(s p+|\beta|+1)^{|\beta|+1} }{(\nu+ p+1)^{m-j}}\Big)_p
\]
 converges for all $ (j, \beta): j+|\beta|< m $,
 then there exists $C_0>0$ such that
$$
\forall p\in \mathbb{N}, \quad
\frac{(s p+|\beta|+1)!}{(s p)!}\frac{1}{(\nu+ p+1)^{m-j}}\leq C_0
$$
 From \eqref{ma12}, we deduce
\begin{align*}
t\ D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x)
& \lll  \|u \| C_0 \ a^{-1} \rho^{-1} ( \rho R)^{-|\beta|}
 \sum_{ p \in \mathbb{N}} ( \rho |t| )\tau^p  (\rho R)^{s' (p+1)}
 \frac{D^{s(p+1)} \varphi^a_{\rho R }(\xi)}{(s(p+1))!}\\
& \lll  \|u \| C_0 \ a^{-1} \rho^{-1} ( \rho R)^{-|\beta|}
\Phi_{\rho,  R}^a(t, x)
\end{align*}
Similarly,  using the second estimate of Lemma \ref{Al1}, we
prove the second part  of Lemma  \ref{lemme02}.
\end{proof}

\begin{lemma} \label{normB}
For all $R$ and $\rho>0$, there exists positive constants
$M_1(\rho,R)$, dependent on $\rho$ and $R$, and
$M_2(R)$, dependent only on $R$, such that
$$
\|\mathcal{B}\| \leq  \frac{1}{3} +  a^{-1} M_1(\rho, R) + \rho ^{-1}
M_2(R)\quad \hbox{in}\quad
 E_{\rho, R, a }^{0,\omega}\,.
$$
\end{lemma}

\begin{proof}
 We choose $\eta >1$, then dependence
on $\eta$  will not  be  mentioned in all constants of the below
estimations.
  We denote by $M_1(\rho, R) $ all positive constants dependent on
$\rho$ and $R$,  and by $M_2(R) $ all positive constant dependent
of $R$ and independent on $\rho$. These constants are
independent of the parameter $a$.

 From Proposition \ref{malika.3}, remark \ref{remarque1} and
corollary \ref{kika}, there exists
positive constants $M_1(\rho,R) $ and $M_2(R) $ such that:
\begin{gather*}
 \text{if $j+|\beta|<m$, then }
   \widetilde{b}_{j,\beta}(t,x) \lll
M_1(\rho, R) \frac{\eta \rho R}{\eta \rho R - (\tau+\xi)}\,,\\
\text{if $j+|\beta|=m$, then }
 \widetilde{b}_{j,\beta}(t,x) \lll
M_2(R) \  (\rho R)^{|\beta|-1} \  \frac{\eta \rho R}{\eta
\rho R - (\tau+\xi)}\,.
\end{gather*}
Substituting these results in  Lemma  \ref{lemme02},  we obtain:
if $j+|\beta|<m$, then
\[
 t \widetilde{b}_{j,\beta}(t,x)  D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x) \lll
M_1(\rho, R)  \|u\| a^{-1} \frac{\eta \rho R}{\eta \rho R -
(\tau+\xi)}
 \Phi_{\rho,  R}^a(t, x)\,;
\]
if $j+|\beta|=m$, then
\[
 t \widetilde{b}_{j,\beta}(t,x)
D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x) \lll M_2(R) \|u\|  \rho
^{-1}   \frac{\eta \rho R}{\eta \rho R - (\tau+\xi)}
 \Phi_{\rho,  R}^a(t, x)
\]
By \eqref{App1}, we get
\begin{gather*}
\text{if $j+|\beta|<m$, then}
t \widetilde{b}_{j,\beta}(t,x)
D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x)
\lll M_1(\rho, R) \|u\| a^{-1}  \Phi_{\rho,  R}^a(t, x);\\
\text{if $j+|\beta|=m$, then }
t \widetilde{b}_{j,\beta}(t,x)
D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x) \lll M_2(R) \|u\|   \rho
^{-1}   \Phi_{\rho,  R}^a(t, x)\,.
\end{gather*}
Which in turn gives
\begin{gather}
\sum_{j<m}\sum_{0<|\beta|< m-j} t \widetilde{b}_{j,\beta}(t,x)
D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x) \lll
 \|u\| a^{-1}  M_1(\rho, R)  \Phi_{\rho, R}^a(t, x), \label{equation2}\\
\sum_{j<m}\sum_{|\beta|= m-j} t \widetilde{b}_{j,\beta}(t,x)
D_x^{\beta} \mathcal{H}_{\nu}^{m-j} u(t,x) \lll
  \|u\| \rho ^{-1}  M_2(R)  \Phi_{\rho, R}^a(t, x)\,. \label{equation3}
\end{gather}
According to the estimations (\ref{ma11}) and \eqref{theta},
the case $\beta=0$ gives
\begin{equation}
\sum_{j<m} b_j \mathcal{H}_{\nu}^{m-j}u(t,x) \lll
 \frac{\|u\|}{3}  \Phi_{\rho, R}^a(t, x)\,.
\label{equation4}
\end{equation}
Using the definition of the operator  $\mathcal{B}$ and the estimations
(\ref{equation2}), (\ref{equation3}) and (\ref{equation4}), we
deduce
$$
\mathcal{B} u(t,x) \lll  \|u\| \Big(\frac{1}{3} + a^{-1} M_1(\rho, R) + \rho
^{-1} M_2(R)\Big)
  \Phi_{\rho, R}^a(t, x)
$$
that means
$$
\|\mathcal{B}\| \leq  \frac{1}{3} +  a^{-1} M_1(\rho, R) + \rho ^{-1}
M_2(R)\quad \hbox{in }  E_{\rho, R, a }^{0,\omega}\,.
$$
\end{proof}

\begin{proof}[Proof of Proposition \em\ref{propo01}]
If we take, in the Lemma  \ref{normB},  parameters $a $ and
$\rho$  large enough such that
$$
a > 3 M_1(\rho, R), \quad  \rho > 3 M_2(R),
$$
we get $\|\mathcal{B}\|< 1$,  from which we deduce that operator
$(I+\mathcal{B})$ is invertible
in $E_{\rho, R, a }^{0,\omega}$ then for all
$g\in E_{\rho, R, a }^{0,\omega}$, equation \eqref{III}
admits a unique solution $u\in E_{\rho, R, a }^{0,\omega}$.
\end{proof}

\subsubsection{Final part in  the proof of Theorem \em\ref{t03}} \label{fin}
\quad \\

\noindent
\textbf{I. Existence of solution in
$\mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$.}
Let $f(t,x)=t^\nu g(t,x)$ with
$g\in \mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$. By
property \ref{Aoublies},$g \in  E_{\rho, R, a }^{0,\omega} $
for all $R, \rho,  a \in \mathbb{R}_+^*$.
Applying Proposition \ref{propo01},  we can
choose  two increasing sequences $(\rho_n)_n$ and $( a_n)_n$ such that
for all $n\in \mathbb{N}^*$ there exists a unique solution
$\psi_{n}$ of \eqref{III}  in  $E_{\rho_n, n, a_n }^{0,\omega}$.
By property \ref{propertx1}, we have
$$
\psi_{n+1} \in E_{\rho_{n+1},(n+1) , a_{n+1} }^{0,\omega} \subset
E_{\rho_n, n, a_n }^{0,\omega}\,.
$$
Since the  uniqueness of solutions  holds  in
$E_{\rho_n, n, a_n}^{0,\omega} $,  we deduce
$$
\psi_{n+1}=\psi_{n} \quad \text{on } \Delta_{\rho_n, n}
$$
which  allows us to define a solution  $u$ of  (\ref{III})
in  $\mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$by
$$
\psi= \psi_n,\quad  \hbox{on}\quad  \Delta_{\rho_n, n} , \quad \forall
n\in \mathbb{N}^*\,.
$$

\noindent \textbf{II. Uniqueness of this solution.}
Let $\psi_1$,$\psi_2$ be two solutions in
$\mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$  of (\ref{III}), then by
property \ref{Aoublies}, for all $R,\rho, a>0$, the functions
${\psi_1}/\Delta_{\rho, R}$ and
$ {\psi_2}/\Delta_{\rho, R} $
are also solutions of \eqref{III} in $ E_{\rho, R, a }^{0,\omega}$.
By the uniqueness of the solution of  the problem in this Banach space,
we obtain
$$
 \forall |t|< \frac{R}{2^s}, \quad \psi_1(t, \cdot)
= \psi_2(t, \cdot)\quad \hbox{on } B^x_{\rho R/ 2 q}\,.
$$
Using  analytic extension  theorem we get
   $$
\forall R>0, \;    \forall |t|< \frac{R}{2^s}, \quad
\psi_1(t,\cdot)= \psi_2(t, \cdot)\quad \hbox{on } \mathbb{C}^q
$$
which implies the uniqueness of the solution of Problem  \eqref{III} in
the functional space  $\mathcal{C}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$.


To complete the proof of the Theorem \ref{t03}, we note that
the solution of \eqref{j} is the form $u(t,x)=t^{\nu}H_\nu^m \psi(t,x)$,
 where $\psi$ is the unique solution of  (\ref{III}), then
$u=O(t^\nu)$ in $\mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$, because
 $H_\nu^m \psi\in \mathcal{C}^{0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$.
\end{proof}

\begin{remark}  \label{porque} \rm
Theorem  \ref{t03} holds for any Fuchsian operator $\mathcal{P}$
with weight $0$. Hence it also holds
for operator $(D_t t)^h\mathcal{P}$ for all $h\in\mathbb{N}$,
which allows us to deduce that:
For all $f=O(t^\nu)$  in $\mathcal{C}^{0,\omega}_{h}(\mathbb{R}\times\mathbb{C}^q)$,
there exists a unique solution $u=O(t^\nu)$  in
$\mathcal{C}^{0,\omega}_{m+h}(\mathbb{R}\times\mathbb{C}^q)$ of \eqref{j}\,.
\end{remark}

\subsection*{Proof of Theorem  \ref{principal2}}
Let $f\in \mathcal{C}^{\infty,\,\omega}(\mathbb{R}\times\mathbb{C}^q)$ and
$\nu_0>0$ satisfies \eqref{theta}.
For $\nu\geq\nu_0$,
the Mac-Laurin expansion of $f$ of order $\nu$ gives
$$
f(t,x)= \sum_{l=0}^{\nu-1} D^l_t f(0,x) \frac{t^l}{l!}+
t^{\nu}f_{\nu}(t,x)\,.
$$
By simple calculations we prove that
$f_{\nu} \in \mathcal{C}_{\nu}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$.
We look for  solution of  (\ref{j}) in  the  form
$$
U^{(\nu)} (t,x)= \sum_{l=0}^{\nu-1} D^l_t U^{(\nu)}(0,x) \frac{t^l}{l!}+
t^{\nu}u_{\nu}(t,x)\,.
$$
By the decomposition \eqref{k1}, problem \eqref{j}) is equivalent
to  system:
\begin{equation}
\begin{gathered}
 0\leq l \leq \nu-1,\\
\mathcal{C}(l) \frac{D^l_t U^{(\nu)}(0,x)}{l!}
=  \frac{D^l_t f(0,x)}{l!}+
\sum_{j=0}^{m-1} \frac{1}{ (l-j-1)!} \big[ D_t^{l-1} B_{m-j} U^{(\nu)}(t,x)
\big]\big|_{t=0}, \\
\mathcal{P}\big[t^{\nu}u_{\nu}(t,x)\big]= t^{\nu}f_{\nu}(t,x)\,.
\end{gathered} \label{eq0}
\end{equation}

\textbf{(a)} Since
$ \mathcal{C}(l)\neq 0$, for all $l\in \mathbb{N} $,
then the functions $D^l_t U^{(\nu)}(0,x)$
$(l=0, \dots ,\nu-1)$ are determined uniquely.

\textbf{(b)} We have  $f_{\nu} \in
\mathcal{C}_{\nu}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$.
By remark \ref{porque} there exists a unique solution
$u_{\nu} \in \mathcal{C}_{m+\nu}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q)$
of \eqref{eq0}, then
$t^\nu u_{\nu} \in \mathcal{C}^{\nu,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$.


 From (a) and (b), we deduce there exists a unique solution
$U^{(\nu)}\in \mathcal{C}^{\nu,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$ of (\ref{j}).
Since for any $\nu_1>\nu_0$, we have
$\mathcal{C}^{\nu_1,\,\omega}_m(\mathbb{R}\times\mathbb{C}^q)\subset \mathcal{
C}^{\nu_0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$,
by the existence and uniqueness of the solution of (\ref{j}) in
$\mathcal{C}^{\nu_1,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$ and
$\mathcal{C}^{\nu_0,\omega}_m(\mathbb{R}\times\mathbb{C}^q)$ we deduce that
$$
\forall\, \nu_1>\nu_0,\quad U^{(\nu_0)}\in\mathcal{C}^{\nu_1,
\omega}_m(\mathbb{R}\times\mathbb{C}^q);
$$
hence $U^{(\nu_0)}\in \mathcal{
C}^{\infty,\,\omega}(\mathbb{R}\times\mathbb{C}^q)$.

For uniqueness, we remark that
$\mathcal{C}^{\infty,\omega}(\mathbb{R}\times\mathbb{C}^q)\subset
\mathcal{C}^{\nu_0,\omega}(\mathbb{R}\times\mathbb{C}^q)$ where the
uniqueness of solutions holds, then
if  $U$ is solution of  \eqref{j}) in
$\mathcal{C}^{\infty,\omega}(\mathbb{R}\times\mathbb{C}^q)$,
we deduce that $U=U^{(\nu_0)}$, which completes the proof
of  Theorem  \ref{principal2}.\hfill $\Box$



\subsection*{Acknowledgements}
The authors are grateful to Professor
T. Mandai (Osaka Electro-Communication University) for his various
remarks and encouragement; also to  the anonymous referee
for his/her comments that improved the final version of this article.

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