\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 $$\begin{gathered} \mathcal{P} u(t,x) = f(t,x),\quad (t,x)\in \mathbb{R}\times\mathbb{R}^q \\ D_t^l u(0, x) = w_l(x) , \quad 0\leq l \leq m-k-1 \end{gathered} \label{maII}$$ with $$\small \mathcal{P}= t^{k} D_t^m + \sum_{j=m-k}^{m-1} a_j(x) t^{j-m+k} D_t^j+ \sum_{j=0}^{m-1} \sum_{|\beta|\leq m-j} a_{j,\beta}(t,x) t^{\max\{0, j-m+k+1 \}} D_t^j D_x^\beta \label{equation11}$$ 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)] $\Re e \lambda_l(x) m+ 1$ 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 $$\mathcal{P} t^{m-k}v(t,x)= f(t,x)- \mathcal{P} \Big[\sum_{l=0}^{m-k-1} \frac{w_l(x)}{l!} t^l\Big]\,. \label{problem}$$ 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 $$\forall \lambda \in \mathbb{N}, \quad \mathcal{C}_1(\lambda) \neq 0\,. \label{condition}$$ 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 $$\mathcal{P} = t^m D_t^m+ \sum_{j =0}^{m-1} a_j(x) t^j D_t^j +\sum_{j=0}^{m-1} \sum_{ 0< |\beta| \leq m-j } t a_{j,\beta}(t,x) t^{j} D_t^{j}D_x^{\beta}\ . \label{k01}$$ 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 $$\mathcal{P} u(t,x) =f(t,x) \label{j}$$ 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 $$\forall \, v\in \mathcal{C}^{\infty}(\mathbb{R}),\quad t^j u D_t^j v = \displaystyle\sum_{l=0}^j t^l D^l_t[u_l v] \label{quation10}$$ \end{enumerate} \end{lemma} From this lemma, we can rewrite $\mathcal{P}$ in the following forms $$\mathcal{P} = ( D_t t)^m+ \sum_{j =0}^{m-1}b_j (D_t t)^j +\sum_{j=0}^{m-1} \sum_{0<|\beta| \leq m-j } t \widetilde{b}_{j,\beta}(t,x) (D_t t)^{j}D_x^{\beta} \label{k02}$$ where $b_j \in \mathbb{C}$ and $\widetilde{b}_{j,\beta} \in \mathcal{C}^{\infty,\omega} (\mathbb{R}\times \mathbb{C}^q)$, or $$\mathcal{P} = t^{m} D_t^{m} + \sum_{j=0}^{m-1} a_j t^{j} D_t^{j}- \sum_{j=0}^{m-1} t^{j+1} D_t^{j} \mathcal{B}_{m-j} \label{k1}$$ 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 \sum_{j 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 $$\forall |t|< R, \; \forall \delta \in \mathbb{N} ^q,\quad |D_x^{\delta}f(t,0) | \leq C \frac{|\delta|!}{(\rho R)^{|\delta|}}\,. \label{E1}$$ 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 $$f(t,x)\lll C \frac{\rho R}{ \rho R- (\tau+\xi)}\,. \label{emino21}$$ \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 $M(R)= \mathop{\rm card} \{\gamma \in \mathbb{N}^q: |\gamma|\leq p\}\max_{|\gamma|\leq p } \big\{\sup_{|t|0$, 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 $$D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x) \lll \|u\| \sum_{ p \in \mathbb{N}} \frac{1}{(\nu+p+1)^{m-j}} \tau^p (\rho R)^{s'p} \frac{D^{sp+|\beta|} \varphi^a_{\rho R }(\xi)}{(s p)!}\,.$$ \end{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 $$\label{ma10} D_x^{\beta} \mathcal{H}_{\nu}^{m-j}u(t,x) \lll \|u\| D_x^{\beta} \mathcal{H}_{\nu}^{m-j}\Phi_{\rho, R}^a (t, x)\,.$$ 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 $j+|\beta|0$ 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| 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{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}$$ \textbf{(a)} Since \mathcal{C}(l)\neq 0, for all l\in \mathbb{N} , then the functions D^l_t U^{(\nu)}(0,x) (l=0, \dots ,\nu-1) are determined uniquely. \textbf{(b)} We have f_{\nu} \in \mathcal{C}_{\nu}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q). By remark \ref{porque} there exists a unique solution u_{\nu} \in \mathcal{C}_{m+\nu}^{0,\omega}(\mathbb{R}\times\mathbb{C}^q) of \eqref{eq0}, then t^\nu u_{\nu} \in \mathcal{C}^{\nu,\omega}_m(\mathbb{R}\times\mathbb{C}^q). From (a) and (b), we deduce there exists a unique solution U^{(\nu)}\in \mathcal{C}^{\nu,\omega}_m(\mathbb{R}\times\mathbb{C}^q) of (\ref{j}). Since for any \nu_1>\nu_0, we have \mathcal{C}^{\nu_1,\,\omega}_m(\mathbb{R}\times\mathbb{C}^q)\subset \mathcal{ C}^{\nu_0,\omega}_m(\mathbb{R}\times\mathbb{C}^q), by the existence and uniqueness of the solution of (\ref{j}) in \mathcal{C}^{\nu_1,\omega}_m(\mathbb{R}\times\mathbb{C}^q) and \mathcal{C}^{\nu_0,\omega}_m(\mathbb{R}\times\mathbb{C}^q) we deduce that$$ \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. \begin{thebibliography}{00} \bibitem{bago} Baouendi, M. S.; Goulaouic, C.; Cauchy Problems with Characteristic Initial Hypersurface, \textit{Comm.\ Pure Appl.\ Math.}, \textbf{26} (1973), 455-475. \bibitem{beme2} Belarbi, M.; Mechab, M.; Holomorphic global solutions of the Goursat-Fuchs problem. 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