数理解析研究所講究録 1212 巻 2001 年 116-132

116

CONVERGENCE OF FORMAL SOLUTIONS OF SINGULAR FIRSTORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OFTOTALLY CHARACTERISTIC TYPE
名古屋大学多元数理科学研究科

白井 朗 (AKIRA SHIRAI)GRADUATE SCHOOL OF MATHEMATICS, NAGOYA UNIVERSITY

1. INTRODUCTION
Let$(t, x)=(t_{1}, \ldots, t_{d}, x_{1}, \ldots, x_{n})\in \mathrm{C}^{d}\cross \mathrm{C}^{n}$$(d\geq 1, n\geq 1)$

be $(d+n)$ -dimensional complex variables.

We consider the following first order nonlinear partial differential equation:$\sum_{:,j=1}^{d}a_{ij}(x)t_{i}\partial_{t_{j}}u+\sum_{k=1}^{n}b_{k}(x)\partial_{x_{k}}u+c(x)u$

(1.1)

,

$\{$

$= \sum_{|l|=K}d_{l}(x)t^{l}+f_{K+1}(t, x, u, \{\partial_{t_{j}}u\}, \{\partial_{x_{k}}u\})$$u(t, x)=O(|t|^{K})$ ,, $c(x)$ ,where $|t|=t_{1}+\cdots+t_{d}$ , $K$ is afixed positive integer satisfying $K\geq 2$ and$c(x)$ andare holomorphic in aneighbourhood of the origin, and $f_{K+1}(t, x, u, \tau, \xi)$,is also holomorphic in aneighbourhood of the origin
with the following Taylor expansion:$a_{ij}(x)$$d_{l}(x)$$(\tau=\langle\tau_{j})\in \mathrm{C}^{d}$$\xi=(\xi_{k})\in \mathrm{C}^{n})$$f_{K+1}(t, x, u, \tau, \xi)=\sum_{|p|+Kq+(K-1)|r|+K|s|\geq K+1}f_{pq_{lS}}(x)t^{p}u^{q}\tau^{r}\xi^{s}$where $q\in \mathrm{Z}_{\geq 0}=\{0,1,2, \ldots\}$ , $p=$$s=(s_{1}, \ldots, s_{n})\in(\mathrm{Z}_{\geq 0})^{n}$$(p_{1}, \ldots,p_{d})\in(\mathrm{Z}_{\geq 0})^{d}$,, $r=(r_{1}, \ldots, r_{d})\in(\mathrm{Z}_{\geq 0})^{d}$ ,,$|p|=p_{1}+\cdots+p_{d}$,$|r|=r_{1}+\cdots+r_{d}$,$|s|=s_{1}+\cdots+s_{n}$,and$t^{p}= \prod_{j=1}^{d}t_{j}^{p_{j}}$,$\tau^{r}=\prod_{j=1}^{d}\tau_{j}^{r_{j}}$,$\xi^{\epsilon}=\prod_{k=1}^{n}\xi_{k}^{s_{k}}$.

This eqauation seems to be anatural extension of totally characteristic type studied byChen-Tahara ([CT]) to several time-space variables

117
Here we remark that the assumption $K\geq 2$ implies $\partial_{t_{j}}u(0,0)=0(j=1,2, \ldots, d)$which assures that $(0, 0, u(0,0), \{\partial_{t_{j}}u(0,0)\}, \{\partial_{x_{k}}u(0,0)\})$ belongs to the domain of definition of $f_{K+1}(t, x, u_{1}\tau, \xi)$ .
Now our first theorem is stated as follows:
Theorem 1. Let
be the eigenvalues of the matrix $(a_{ij}(0))$ . We assume thatand $b_{k}(0)=0$ for $k=1,2$ ,, , and let
be the eigenvalues of Jacobi
matrix of $(b_{1}(x), \ldots, b_{n}(x))$ at $x=0$ . Then the formal power series solution of (1.1)
exists uniquely and converges if the following conditions are satisfied:
There exists a positive constantsuch that$\{\lambda_{j}\}_{j=1}^{d}$$b_{k}(x)\not\equiv \mathrm{O}$$\ldots$$n$$\{\mu_{k}\}_{k=1}^{n}$$\sigma_{0}>0_{f}$(1.2)$| \sum_{j=1}^{d}\lambda_{j}l_{j}+\sum_{k=1}^{n}\mu_{k}m_{k}|\geq\sigma_{0}(|l|+|m|)$(Poincari condition),

and
(1.3)

hold for all$j \sum_{=1}^{d}\lambda_{j}l_{j}+\sum_{k=1}^{n}\mu_{k}m_{k}+c(0)\neq 0$$(l, m)\in(\mathrm{Z}_{\geq 0})^{d}\cross(\mathrm{Z}_{\geq 0})^{n}$(Non-resonance condition)with $|l|\geq K$ and $|m|\geq 0$ .

Remark 1. It is easy to show the following proposition.
The conditions (1.2) and (1.3) imply that
(1.4)$| \sum_{j=1}^{d}\lambda_{j}l_{j}+\sum_{k=1}^{n}\mu_{k}m_{k}+c(0)|\geq\sigma(|l|+|m|)$holds by some positive constant $\sigma>0$ for allwith $|l|\geq K$ and$|m|\geq 0$ . In the proof of Theorem 1this condition will be used instead of (1.2) and(1.3).
$(l, m)\in(\mathrm{Z}_{\geq 0})^{d}\cross(\mathrm{Z}_{\geq 0})^{n}$$\square$Next, we consider the following general equation:(1.5)$f(t, x, u(t, x), \{\partial_{t_{j}}u(t, x)\}, \{\partial_{x_{k}}u(t, x)\})=0$

,

$\{$

$u(0, x)\equiv 0$.

Assumption 1. $f(t, x, u, \tau, \xi)(\tau=(\tau_{j})\in \mathrm{C}^{d}, \xi=(\xi_{k})\in \mathrm{C}^{n})$ is holomorphic in aneighbourhood of the origin, and is an entire function in variables for any fixed , ,and . Moreover we assume that$\tau$

$t$

$x$

$u$

$\xi$

(1.6)

$f(0, x, 0, \tau, 0)\equiv 0$

fornear the origin andsingular equations defined in [MS]$x\in \mathrm{C}^{n}$$\tau\in \mathrm{C}^{d}$, which is ageneralization of the definition of

118
For the equation (1.5), we do not know whether or not the equation has aformal
solution in general. Therefore, we assume the following:
Assumption 2. The equation (1.5) has aformal solution of the form
(1.7)

$u(t, x)= \sum_{j=1}^{d}\varphi_{j}(x)t_{j}+\sum_{|l|\geq 2,|m|\geq 0}u_{lm}t^{l}x^{m}\in \mathrm{C}[[\mathrm{t}, x]]$By the existence of aformal solution,(1.8)$\{\varphi_{j}(x)\}$$f(0, x, 0, \{\varphi_{j}(x)\}, 0)\equiv 0$.

satisfy the following system formally:(trivial relation),

and
(1.8)

$\frac{\partial}{\partial t_{i}}f(t, x, u(t, x), \{\partial_{t_{\mathrm{j}}}u(t, x)\}, \{\partial_{x_{k}}u(t, x)\})|_{t=0}$$= \frac{\partial f}{\partial t_{i}}(0, x, 0, \{\varphi_{j}(x)\}, 0)+\frac{\partial f}{\partial u}(0, x, 0, \{\varphi_{j}(x)\}, 0)\varphi_{i}(x)$$+ \sum_{k=1}^{n}\frac{\partial f}{\partial\xi_{k}}(0, x, 0, \{\varphi_{j}(x)\}, 0)\frac{\partial\varphi_{i}}{\partial x_{k}}(x)$$=0$ ,

for $i=1,2$ ,$\ldots$, .
$d$

The formal solution of this system is not convergent in general. Therefore, we assume
Assumption 3. The coefficients.
origin of$\{\varphi_{j}(x)\}$are all holomorphic in aneighbourhood of the$\mathrm{C}^{n}$Remark 2. In the case $d=1$ ( is the dimension of variables), asufficient condition
for the formal solution of (1.9) to converge has been already obtained by Miyake-Shirai[MS]. In the case $d\geq 2$ , we give asufficient condition for the formal solution of system(1.9) to be convergent, which will be given by Theorem 3in Section 5, but for awhile
we consider the problem under Assumption 3for simplicity of our arguments.
$d$

$t$

$\square$Now we put $\mathrm{a}(x)=(0, x, 0, \{\varphi_{j}(x)\}, 0)$ for simplicity, and define(1.10)$A_{ij}(x):= \frac{\partial^{2}f}{\partial t_{i}\partial\tau_{j}}(\mathrm{a}(x))+\frac{\partial^{2}f}{\partial u\partial\tau_{j}}(\mathrm{a}(x))\varphi_{i}(x)+\sum_{k=1}^{n}\frac{\partial^{2}f}{\partial\tau_{j}\partial\xi_{k}}(\mathrm{a}(x))\frac{\partial\varphi_{i}}{\partial x_{k}}(x)$for $i,j=1,2$ ,(1.11)$\ldots$,, . Moreover we define$d$$B_{k}(x):= \frac{\partial f}{\partial\xi_{k}}(\mathrm{a}(x))$

, for

$k$

$=1,2$ ,$\ldots$

Remark 3. The functions $A_{ij}(x)$ and $B_{k}(x)$ correspond to
1, respectively (see (1.13) below).

, .
$n$

$a_{ij}(x)$and$b_{k}(x)$Here we assume that the equation is of totally characteristic type, that isin Theorem$\square$119
Assumption 4.

$B_{k}(x)\not\equiv \mathrm{O}$and $B_{k}(0)=0$ , for $k=1,2$ ,$\ldots$, .
$n$

Now our second theorem which is our main result is stated as follows:
Theorem 2. Suppose Assumptions 1, 2, 3and 4. Let
be the eigenvalues of$(A_{ij}(0))$ , and let
be the eigenvalues of Jacobi matrix of the vector $(B_{k}(x))$ at$x=0$ . Then the formal solution (1.7) is convergent if the following condition is satisfied:
There exists a positive constant $\sigma>0$ , such that,$\{\lambda_{j}\}_{j=1}^{d}$$\{\mu_{k}\}_{k=1}^{n}$(1.12),$| \sum_{j=1}^{d}\lambda_{j}l_{j}+\sum_{k=1}^{n}\mu_{k}m_{k}+\frac{\partial f}{\partial u}(\mathrm{a}(0))|\geq\sigma(|l|+|m|)$holds for all$(l, m)\in(\mathrm{Z}_{\geq 0})^{d}\cross(\mathrm{Z}_{\geq 0})^{n}$with $|l|\geq 2$ , $|m|\geq 0$ .

Remark 4. We put $v(t, x)=u(t, x)- \sum_{j=1}^{d}\varphi_{j}(x)t_{j}$ as anew unknown function. By
Assumptions 1, 2, 3and 4, we can easily see that $v(t, x)$ satisfies the equation of thefollowing form:’$\sum_{i,j=1}^{d}A_{ij}(x)t_{i}\partial_{t_{j}}v+\sum_{k=1}^{n}Bk\{x)dXkv+\frac{\partial f}{\partial u}(\mathrm{a}(x))v$(1.13)$= \sum_{|l|=2}d_{l}(x)t^{l}+f_{3}(t, x, v, \{\partial_{t_{j}}v\}, \{\partial_{x_{k}}v\})$$\backslash v(t, x)=O(|t|^{2})$,.

This is an equation considered in Theorem 1in the case $K=2$ . Therefore, it is sufficient
to prove Theorem 1in order to prove Theorem 2.
$\square$

2. REDUCTION OF THE EQUATIONAs is mentioned in Remark 4, it is sufficient to study the equation (1.1).
By the assumption of Theorem 1,$\lambda_{1}$$\delta_{1}$$\lambda_{2}$$(a_{ij}(0))\sim\{$.. ..

$\frac{\partial(b_{1},\ldots,b_{n})}{\partial(x_{1},\ldots,x_{n})}|_{x=0}\sim(\begin{array}{llll}\mu_{1} \nu_{1} \mu_{2} \ddots \ddots \nu_{n-1} \mu_{n}\end{array})$$\delta_{d-1}$$\lambda_{d}$where , $\nu_{k}=0$ or 1 $(1 \leq j\leq d-1,1\leq k\leq n-1)$ .
Then by transforming the variables, (1.1) is reduced to the following form:$\delta_{j}$(2.1)$(\Lambda+\Delta)v(t, x)$$=$$\sum_{|l|=K}\alpha_{l}(x)t^{l}+\sum_{i,j=1}^{d}\beta_{ij}(x)t_{i}\partial_{t_{j}}v+\gamma(x)v$$+ \sum_{k=1}^{n}\varphi_{k}(x)\partial_{x_{k}}v+\overline{f}_{K+1}.(t, x, v, \{\partial_{t_{j}}v\}, \{\partial_{x_{k}}v\})$

,

,

120
with $v(t, x)=O(|t|^{K})$ , where$\Lambda=\sum_{j=1}^{d}\lambda_{j}t_{j}\partial_{t_{j}}+\sum_{k=1}^{n}\mu_{k}x_{k}\partial_{x_{k}}+c(0)$$\Delta=\sum_{j=1}^{d-1}\delta_{j}t_{j}\partial_{t_{j+1}}+\sum_{k=1}^{n-1}\nu_{k}x_{k}\partial_{x_{k+1}}$

,

,

,
,
and
and
are holomorphic in aneighbourhood of the origin, andsatisfy $\beta_{ij}(x)=O(|x|)$ , $\gamma(x)=O(|x|)$ and $\varphi_{k}(x)=O(|x|^{2})$ , andis aholomorphic function which has asimilar Taylor expansion with $f_{K+1}(t, x, u, \tau, \xi)$ .
In the following sections, we shall prove the existence and convergence of the uniqueformal solution of (2.1).
$\alpha_{l}(x)$$\beta_{\dot{l}j}(x)$$\gamma(x)$$\varphi_{k}(x)$$\tilde{f}_{K+1}(t, x, u, \tau, \xi)$

3. PREPARATION TO prove THEOREM 1
Let
degree $M$ inbe the set of homogeneous polynomial of degree
variables, that is,$\mathrm{C}[t, x]_{L,M}$

$x$

$L$

in variables and of$t$$\mathrm{C}[t, x]_{L,M}=\{f_{LM}(t, x)=\sum_{|l|=L,|m|=M}f_{lm}t^{l}x^{m}|f_{lm}\in \mathrm{C}\}$For the operator$\Lambda+\Delta$.

, the following lemma holds:

Lemma 1. For all $L\geq K$ and $M\geq 0$ , the operator$\Lambda+\Delta$:$\mathrm{C}[t, x]_{L,M}arrow \mathrm{C}[\mathrm{t}, x]_{L,M}$

is invertible. Moreover, if the majorant relation $f_{LM}(t, x)<<F\cross(t_{1}+\cdots+t_{d})^{L}(x_{1}+$$\ldots+x_{n})^{M}(f_{LM}(x)\in \mathrm{C}[t, x]_{L,M}, F>0)$

holds: then we$\mathit{0}6ton$

the folloeving majorantrelation:(3.1)$( \Lambda+\Delta)^{-1}f_{LM}(t, x)\ll\frac{C}{L+M}F\cross(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$

where $C>0$ is a positive constant independent of

Proof. We define anorm of$L$$u_{LM}(t, x)\in \mathrm{C}[\mathrm{t}, x]_{L,M}$,

and $M$ .
by$||u_{LM}||:= \inf\{C>0|fLM(t, x)\ll C(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}\}$

We remark that
becomes aBanach space by this norm.
First, by (1.4) it is easily checked that Ais invertible on$\mathrm{C}[t, x]_{L,M}$$\mathrm{C}[t, x]_{L,M}$(3.2)

holds for the operator norm of$|| \Lambda^{-1}||\leq\frac{1}{\sigma(L+M)}$$\Lambda^{-1}$on$\mathrm{C}[t, x]_{L,M}$

.

and

.

121

Next, since $u_{LM}(t, x)<<||u_{LM}||(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$ , we have$\Delta u_{LM}(t, x)$$\sum_{j=1}^{d-1}L|\delta_{j}|\cdot||u_{LM}||(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$$\ll$$+ \sum_{k=1}^{n-1}M|\nu_{k}|\cdot||u_{LM}||(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$$\{L(d-1)\max_{=j1,\ldots,d-1}|\delta_{j}|+M(n-1)\max_{=k1,\ldots,n-1}|\nu_{k}|\}\cross$$\ll$$\cross||u_{LM}||(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$.

,
Here we make achange of variables by, thenand(thecomponents of nilpotent part of Jordan canonical form) turns to, respectively.
andTherefore, by choosing sufficiently small, we may assume that the components ofnilpotent part are small enough. Hence we may assume that$x_{k}=\epsilon^{k-1}y_{k}$$t_{j}=\epsilon^{j-1}\tau_{j}$$\delta_{j}$$\epsilon\delta_{j}$$\nu_{k}$$\epsilon\nu_{k}$$\epsilon$(3.3)$j=1, \ldots,d-1\max|\delta_{j}|<\frac{\sigma}{2(d-1)}$,$k=1, \ldots,n-1\max|\nu_{k}|<\frac{\sigma}{2(n-1)}$.

Then$\Delta u_{LM}(t, x)<<\frac{\sigma(L+M)}{2}||u_{LM}||(t_{1}+\cdots+t_{d})^{L}(x_{1}+\cdots+x_{n})^{M}$

holds, and we obtain$|| \Delta||\leq\frac{\sigma(L+M)}{2}$Therefore, the operator norm of$\Delta\Lambda^{-1}$.

is estimated by$|| \Delta\Lambda^{-1}||\leq\frac{1}{\sigma(L+M)}\frac{\sigma(L+M)}{2}=\frac{1}{2}<1$

By using the Neumann’s series, we can see that
inverse operator is estimated by$\Lambda+\Delta$.

is invertible and the norm of the$||( \Lambda+\Delta)^{-1}||\leq\frac{2}{\sigma}\frac{1}{L+M}$,

which we want to prove since $C=2/\sigma$ is independent of$L$and M.

$\square$Now, we define some notations, which are used in the proof of Theorem 1.

Definition (1) Let
power series $f(t, x)=\Sigma_{|l|\geq 0}$ ,$(t, x)\in \mathrm{C}^{d}\cross \mathrm{C}^{n}(d\geq 0, n\geq 0)$$|m|\geq 0f_{l,m}t^{l}x^{m}$be complex variables. For formal, we define$|f|(t, x)= \sum_{|l|\geq 0,|m|\geq 0}|f_{l,m}|t^{l}x^{m}$.

122(2) Let $(t, X)\in \mathrm{C}^{d}\cross \mathrm{C}(d\geq 0)$ be complex variables. For formal power series, we define the shift operator by$S$$\mathrm{g}(\mathrm{t}, X)=\Sigma_{|l|\geq 0,M\geq 0}f_{l,M}t^{l}X^{M}$$S(f)(t, X)= \sum_{|l|\geq 0,M\geq 0}f_{l,M+1}t^{l}X^{M}=\frac{f(t,X)-f(t,0)}{X}$.

Remark 5. The following facts are easily shown:$\bullet f(t, x)<<|f|(t, x)$;

If $f(t, x)$ and $g(t, X)$ are convergent power series, then $|f|(t, x)$ and $S(g)(t, X)$ are also
convergent.
$\bullet$$\square$

4. Proof OF THEOREM 1
First, we prove aunique existence of formal power series solution.

Let$u(t, x)= \sum_{|l|\geq K,|m|\geq 0}u_{lm}t^{l}x^{m}=\sum_{L\geq K}u_{L}(t, x)=\sum_{L\geq K,M\geq 0}u_{LM}(t, x)$be aformal solution of (2.1), where$u_{LM}(t, x)= \sum_{|l|=L,|m|=M}u_{lm}t^{l}x^{m}\in \mathrm{C}[t, x]_{L,M}$$u_{L}(t, x)= \sum_{|l|=L}u\iota(x)t^{l}=\sum_{M\geq 0}u_{LM}(t, x)$,.

We put $P=\Lambda+\Delta$ for simplicity. We substitute $u(t, x)= \sum_{L\geq K}u_{L}(t, x)$ into (2.1),
then we have the following recursion formula:$Pu_{K}(t, x)= \sum_{|l|=K}\alpha_{l}(x)t^{l}+\sum_{i,j=1}^{d}\beta_{ij}(x)t_{i}\partial_{t_{j}}u_{K}(t, x)$,$+ \gamma(x)u_{K}(t, x)+\sum_{k=1}^{n}\varphi_{k}(x)\partial_{x_{k}}u_{K}(t, x)$$\{$$Pu_{L}(t, x)= \sum_{i,j=1}^{d}\sqrt ij(x)t_{i}\partial_{t_{j}}u_{L}(t, x)+\gamma(x)u_{L}(t, x)+\sum_{k=1}^{n}\varphi_{k}(x)\partial_{x_{k}}u_{L}(t, x)$$+G_{L}(t, x, \{u_{p}\}_{K\leq p<L}, \{\partial_{t_{j}}u_{p}\}_{K\leq p<L}, \{\partial_{x_{k}}u_{p}\}_{K\leq p<L})$

where$G_{L}(t, x, \zeta, \tau, \xi)$is apolynomial of$(t, \zeta, \tau, \xi)$.

, for $L>K$ ,123First, we consider the case $L=K$ . We substitute $u_{K}(t, x)= \sum_{M\geq 0}u_{KM}(t, x)$ into theabove recursion formula, we have
$Pu_{K0}(t,x)= \sum_{|l|=K}\alpha_{l}(0)t^{l}$

,

$\{$

$Pu_{KM}(t,x)= \sum_{|l|=K}\alpha_{l}^{M}(x)t^{l}+\sum_{i,j=1}^{d}\sum_{p=1}^{M}\beta_{i}+\sum_{\mathrm{p}=1}^{M}\gamma^{p}(x)u_{K,M-p}(t,x)pj(x)t_{i}\partial+\sum_{k=1p}^{n}\sum_{=2}^{M}\varphi_{k}^{p}(x)\partial_{x_{k}}u_{K,M-p+1}(t,x)t_{j^{u_{K,M-p}(t,x)}}$,where we put$\alpha_{l}(x)=\sum_{M\geq 0}\alpha_{l}^{M}(x)$,$\alpha_{l}^{M}(x)=\sum_{|m|=M}\alpha_{lm}x^{m}$$\beta_{ij}(x)=\sum_{M\geq 1}\beta_{ij}^{M}(x)$,$\beta_{ij}^{M}(x)=\sum_{|m|=M}\beta_{ijm}x^{m}$$\gamma(x)=\sum_{M\geq 1}\gamma^{M}(x)$$\varphi_{k}(x)=\sum_{M\geq 2}\varphi_{k}^{M}(x)$,,, $\gamma^{M}(x)=\sum_{|m|=M}\gamma_{m}x^{m}$ ,,$\varphi_{k}^{M}(x)=\sum_{|m|=M}\varphi_{km}x^{m}$.

By Lemma 1, we know that the solution sequence $\{u_{KM}(t, x)\}_{M\geq 0}$ exists uniquely. Moreover, by the same argument, we see that $\{u_{LM}(t, x)\}(L>K)$ exist uniquely. Theseshow that the formal solution exists uniquely.
Next, we prove the convergence of the formal solution. We put $U(t, x)=Pu(t, x)$ as

anew unknown function. By Lemma 1, the equation (2.1) is reduced to the following
equation:(4.1)$U(t, x)$$=$$\sum_{|l|=K}\alpha_{l}(x)t^{l}+\sum_{i,j=1}^{d}\beta_{ij}(x)t_{i}\partial_{t_{j}}P^{-1}U(t, x)$$+ \gamma(x)P^{-1}U(t, x)+\sum_{k=1}^{n}\varphi_{k}(x)\partial_{x_{k}}P^{-1}U(t, x)$$+\overline{f}_{K+1}(t,$$x,$$P^{-1}U(t, x),$ $\{\partial_{t_{j}}P^{-1}U(t, x)\},$ $\{\partial_{x_{k}}P^{-1}U(t, \mathrm{x})\},$

We know that (4.1) has aunique formal solution of the form$U(t, x)= \sum_{|l|\geq K,|m|\geq 0}U_{lm}t^{l}x^{m}=\sum_{L\geq K}U_{L}(t, x)=\sum_{L\geq K,M\geq 0}U_{LM}(t, x)$

.

.

124
In order to get amajorant series of $U(t, x)$ , we prepare some majorant series for thecoefficients of (4.1). We put $T=t_{1}+\cdots+t_{d}$ , $X=x_{1}+\cdots+x_{n}$ , and choose$\sum_{|l|=K}\alpha_{l}(x)t^{l}<<A(X)T^{K}$,$\gamma(x)<<|\gamma|(X, \ldots, X)=:XG(X)$ ,$\overline{f}_{K+1}(t, x, u, \tau, \xi)$

$<<$

$=$

:

$=$

,

$\beta_{ij}(x)<<|\beta_{ij}|(X, \ldots, X)=:XB_{ij}(X)$$\varphi_{k}(x)<<|\varphi_{k}|(X, \ldots, X)=:X^{2}\Phi_{k}(X)$,$|\tilde{f}_{K+1}|(T, \ldots, T, X, \ldots,X, u, \tau, \xi)$$F_{K+1}(T, X, u, \tau, \xi)$$\sum_{|p|+Kq+(K-1)|r|+K|s|\geq K+1}F_{pqrs}(X)T^{|p|}u^{q}\tau^{r}\xi^{s}$,are holomorphic in aneighbourhood of $X=0$ ,, $G(X)$ andwhere $A(X)$ ,and $F_{K+1}(T, X, u, \tau, \xi)$ is also holomorphic near $(T, X, u, \tau, \xi)=(0,0,0,0,0)$ .
Now, we consider the following equation:$\Phi_{k}(X)$$B_{\dot{\iota}j}(X)$(4.2)$w(T, X)$$=$

$A(X)T^{K}+C \sum_{i.j=1}^{d}XB_{\dot{t}j}(X)w(T, X)$$+CXG(X)w(T, X)+C \sum_{k=1}^{n}X^{2}\Phi_{k}(x)(t, x)S(w)(T, X)$$+F_{K+1}$(, ,$T$ $X$$Cw$,$\{\frac{Cw}{T}\}$, $\{CS(w)\}$),where is apositive constant appeared in Lemma 1.
Let $w(T, X)=\Sigma_{L\geq K,M\geq 0}w_{LM}(T, X)$ be the formal solution of (4.2). By the construction of (4.2), we can easily check that $U(t, x)<<w(T, X)$ by the next lemma.
$C$

Lemma 2. For two formal power series $U(t,$x) and $w(T,$X) satisfying,$U(t, x)= \sum_{L\geq K,M\geq 0}U_{LM}(t, x)\ll w(T, X)=\sum_{L\geq K,M\geq 0}w_{LM}T^{L}X^{M}$

the following majorant relations hold:(1) $P^{-1}U(t, x)\ll Cw(T, X)$ ,(2) $t_{:}\partial_{t_{j}}P^{-1}U(t, x)\ll Cw(T, X)$ ,(3) $\partial_{t_{j}}P^{-1}U(t, x)\ll\frac{Cw(T,X)}{T}$ ,(4) $\partial_{x_{k}}P^{-1}U(t, x)<<CS(w)(T, X)$ .

Proof. By using Lemma 1, we can prove this lemma easily. First, (1) is proved asfollows:$P^{-1}U(t, x)= \sum_{L\geq K,M\geq 0}P^{-1}U_{LM}(t, x)\ll\sum_{L\geq K,M\geq 0}\frac{C}{L+M}w_{LM}T^{L}X^{M}\ll Cw(T, X)$.

125Secondly, (2) and (3) is proved as follows:$=$$t_{i}\partial_{t_{j}}P^{-1}U(t, x)$$\sum_{L\geq K,M\geq 0}t_{i}\partial_{t_{j}}P^{-1}U_{LM}(t, x)$$\sum_{L\geq K,M\geq 0}\frac{CL}{L+M}w_{LM}T^{L}X^{M}\ll Cw(T, X)$

$\ll$

$=$

$\partial_{t_{j}}P^{-1}U(t, x)$;$\sum_{L\geq K,M\geq 0}\partial_{t_{j}}P^{-1}U_{LM}(t, x)$$\sum_{L\geq K,M\geq 0}\frac{CL}{L+M}w_{LM}T^{L-1}X^{M}\ll\frac{Cw(T,X)}{T}$

$\ll$

.

Finally, (4) is proved as follows:$\partial_{x_{k}}P^{-1}U(t,x)=L\geq\geq 0\Sigma\partial_{x_{k}}P^{-1}U_{LM}(t,x)$$\sum_{L\geq K,M\geq 1}\frac{CM}{L+M}w_{LM}T^{L}X^{M-1}<<CS(w)(T, X)$$<<$

This completes the proof.
Since $w(T, X)>>0$ , we have
(4.3).
$\square$$XS(w)(T, X)=w(T, X)-\mathrm{w}(\mathrm{T}, 0)\ll w(T, X)$ .

Let us consider the following equation:(4.4)$v(T, X)$$=A(X)T^{K}+CXh(X)v(T, X)$$+F_{K+1}$

(, ,
$T$

$X$

$Cv$

,

$\{\frac{Cv}{T}\}$, $\{CS(v)\}$):

with $v(T, X)=O(T^{K})$ , where $h(X)=\Sigma_{i,j=1}^{d}B_{ij}(X)+G(X)+\Sigma_{k=1}^{n}\Phi_{k}(X)$ . Then thefollowing majorant relation is obvious:$w(T, X)<<v(T, X)$ .

We put $y(T, X)=v(T, X)/T$ as anew unknown function. By substituting this into
(4.4), we see that $y(T, X)$ satisfies(4.5)$y(T, X)$$=$

$A(X)T^{K-1}+CXh(X)y(T, X)$$+ \frac{1}{T}F_{K+1}(T, X, CTy, \{Cy\}, \{CTS(y)\})$

with $y(T, X)=O(T^{K-1})$ .
We decompose the formal solution $y(T, X)$ as follows:$y(T, X)=y_{1}(X)T^{K-1}+y_{2}(X)T^{K}+T^{K}z(T, X)$ .

,126
We remark that $y_{1}(X)$ and $y_{2}(X)$ are holomorphic functions in aneighbourhood of$X=0$ . Indeed, $y_{1}(X)$ and $y_{2}(X)$ are given by$y_{1}(X)= \frac{A(X)}{1-CXh(X)}$,$y_{2}(X)= \frac{1}{1-CXh(X)}\sum_{|p|+Kq+(K-1)|r|+K|s|=K+1}F_{pqrs}(X)\{Cy_{1}(X)\}^{q+|r|}\{CS(y_{1})(X)\}^{|s|}$ .

These are holomorphic functions in aneighbourhood of $X=0$ .
In this case, $z(T, X)$ satisfies the following equation:(4.6)$z(T, X)=CXh(X)z(T, X)+\mathrm{z}(\mathrm{T}, X, Tz(T, X), TS(z)(T, X))$ ,$\{$$z(0, X)\equiv 0$

,

where
$\mathrm{z}(\mathrm{T}, X,\eta_{1}, \eta_{2})$$=$$\frac{1}{T^{K+1}}[F_{K+1}(T,$ $X$, $Cy_{1}(X)T^{K}+Cy_{2}(X)T^{K+1}+CT^{K}\eta_{1}$ ,$\{Cy_{1}(X)T^{K-1}+Cy_{2}(X)T^{K}+CT^{K-1}\eta_{1}\}$,$\{CS(y_{1})(X)T^{K}+CS(y_{2})(X)T^{K+1}+CT^{K}\eta_{2}\})]$-$\sum_{|p|+Kq+(K-1)|r|+K|s|=K+1}F_{pqrs}(X)(Cy_{1}(X))^{q+|t|}(CS(y_{1})(X))^{|s|}$Remark 6. The order of zeros ingreater than or equal to 1.

$T$.

variable of $H(T, X, CTz(T, X), CTS(z)(T, X))$ is$\square$

In order to prove the convergence of $z(T, X)$ , it is sufficient to show the following:
Lemma 3. There exists a small positive constantconvergent in a neighbourhood of $\rho=0$ .

Proof. We substitute$T=\epsilon\rho$$\epsilon$$>0$ such that $z_{\epsilon}(\rho)=z(\epsilon\rho, \rho)$

and $X=\rho$ into (4.6) and by using the relation (4.3), we

have$\rho S(z)(\epsilon\rho, \rho)<<z_{\epsilon}(\rho)$.

By this relation, the following majorant relation also holds,$TS(z)(T, X)|_{T=\epsilon\rho,X=\rho}=\epsilon\rho S(z)(\epsilon\rho, \rho)<<\epsilon z_{\epsilon}(\rho)$

Here we consider
(4.7)

is

$\psi(\rho)=C\rho h(\rho)\psi(\rho)+H(\epsilon\rho, \rho, \epsilon\rho\psi(\rho), \epsilon\psi(\rho))$

.

.

127
In the right hand side of (4.7), we decompose
and otherwise as follows:

into the term of$H(\epsilon\rho, \rho, \epsilon\rho\psi(\rho), \epsilon\psi(\rho))$$\psi(\rho)$$H( \epsilon\rho, \rho, \epsilon\rho\psi(\rho), \epsilon\psi(\rho))=\epsilon\frac{\partial H}{\partial\eta_{2}}(0,0,0,0)\psi(\rho)+\overline{H}(\epsilon\rho, \rho, \epsilon\rho\psi(\rho), \epsilon\psi(\rho))$.

We remark that the following fact holds:.

$\frac{\partial\overline{H}}{\partial\psi}(\epsilon\rho, \rho, \epsilon\rho\psi, \epsilon\psi)|_{\rho=0,\psi=0}=0$

We put $(\partial H/\partial\eta_{2})(0,0,0,0)=K_{0}\geq 0$, then (4.7) is rewritten by(4.8)$(1-\epsilon K_{0})\psi(\rho)=C\rho h(\rho)\psi(\rho)+\overline{H}(\epsilon\rho, \rho, \epsilon\rho\psi(\rho), \epsilon\psi(\rho))$.

We choose $>0$ with $1-\epsilon K_{0}>0$ . Then by using the implicit function theorem, we cansee that (5.8) has aunique holomorphic solution
with $\psi(0)=0$ in aneighbourhoodof $\rho=0$ . Moreover we can see.
Thus we complete the proof of Lemma 3.
$\epsilon$$\psi(\rho)$$z_{\epsilon}(\rho)\ll\psi(\rho)$$\square$5. SOLVABILITY OF THE SYSTEM (1.9)
In this section, we give asufficient condition for the formal solution of the system (1.9)
to be convergent. Recall that (1.9) is(1.9)$\frac{\partial f}{\partial t_{i}}(0, x, 0, \{\varphi_{j}(x)\}, 0)+\frac{\partial f}{\partial u}(0, x, 0, \{\varphi_{j}(x)\}, 0)\varphi_{i}(x)$, $i=1,2$ ,$+ \sum_{k=1}^{n}\frac{\partial f}{\partial\xi_{k}}(0, x, 0, \{\varphi_{j}(x)\}, 0)\frac{\partial\varphi_{i}(x)}{\partial x_{k}}=0$$\ldots$, .
$d$

By Assumption 4of Theorem 2, the condition$\frac{\partial f}{\partial\xi_{k}}(0,0,0, \{\varphi_{j}(0)\}, 0)=0$,$k=1,2$ ,$\ldots$,$n$

was assumed.
Let

be the unknown functions. We put,,
as the constant term of. We substituteinto the system(1.9), and by restricting at $x=0$ , we see thatsatisfies the following system:$\varphi(x)={}^{t}(\varphi_{1}(x), \ldots, \varphi_{d}(x))$$\varphi_{d}^{0})\in \mathrm{C}^{d}$$\varphi(0)={}^{t}(\varphi_{1}^{0}$$\varphi(x)$$\ldots$$\varphi_{j}(x)=\varphi_{j}^{0}+\psi_{j}(x)$$\{\varphi_{j}^{0}\}$(5.1)$\frac{\partial f}{\partial t_{i}}(0,0,0, \{\varphi_{j}^{0}\}, 0)+\frac{\partial f}{\partial u}(0,0,0, \{\varphi_{j}^{0}\}, 0)\varphi_{i}^{0}=0$,$i=1,2$ ,$\ldots$, .
$d$This system has some roots by the assumption of the existence of aformal solution, andwe fix such.
$\{\varphi_{j}^{0}\}$128
For such fixed(5.2)$\{\varphi_{j}^{0}\}$I, we see that$\{\psi_{j}(x)\}$satisfies the system of the followir$\sum_{l=1}^{n}\frac{\partial^{2}f}{\partial\xi_{k}\partial x_{l}}(0,0,0,\{\varphi_{j}^{0}\},0)x_{l}\frac{\partial\psi_{i}}{\partial x_{k}}(x)$$+ \sum_{k=1p}^{n}\sum_{=1}^{d}\frac{\partial^{2}f}{\partial\xi_{k}\partial\tau_{p}}(0,0,0, \{\varphi_{j}^{0}\}, 0)\psi_{p}(x)\frac{\partial\psi_{i}}{\partial x_{k}}(x)$$+ \frac{\partial f}{\partial u}(0,0,0,\{\varphi_{j}^{0}\},0)\psi_{i}(x)$$+ \sum_{p=1}^{d}\{\frac{\partial^{2}f}{\partial t_{i}\partial\tau_{p}}(0,0,0,\{\varphi_{j}^{0}\},0)+\frac{\partial^{2}f}{\partial u\partial\tau_{p}}(0,0,0,\{\varphi_{j}^{0}\},0)\varphi_{i}^{0}\}\psi_{p}(x)$$+ \sum_{l=1}^{n}\{\frac{\partial^{2}f}{\partial t_{i}\partial x_{l}}(0,0,0,\{\varphi_{j}^{0}\},0)+\frac{\partial^{2}f}{\partial u\partial x_{l}}(0,0,0,\{\varphi_{j}^{0}\},0)\varphi_{i}^{0}\}x_{l}$(degree in

$=$

$x$

is greater than or equal to 2),$i=1,2$ ,, .
$d$$\ldots$

This system is written as follows for simplicity,$n$$(5.3)$$d$$\sum\sum a_{kl}x_{l}\frac{\partial\psi_{i}}{\partial x_{k}}(x)nn+\sum\sum b_{kp}\psi_{p}(x)$$k=1l=1$$\frac{\partial\psi_{i}}{\partial x_{k}}(x)$$k=1p=1$$+c \psi_{i}(x)+\sum_{p=1}^{d}d_{ip}\psi_{p}(x)+\sum_{l=1}^{n}e:\iota x_{l}$$=$(degree in$x$

is greater than or equal to 2),$i=1,2$ ,, ,$d$$\ldots$where$a_{kl}:= \frac{\partial^{2}f}{\partial\xi_{k}\partial x_{l}}(0,0,0,\{\varphi_{j}^{0}\},0)$,$b_{kp}:= \frac{\partial^{2}f}{\partial\xi_{k}\partial\tau_{p}}(0,0,0,\{\varphi_{j}^{0}\},0)$$c:= \frac{\partial f}{\partial u}(0,0,0, \{\varphi_{j}^{0}\}, 0)$,$d_{\dot{l}}:= \frac{\partial^{2}f}{\partial t_{i}\partial\tau_{p}}p(0,0,0, \{\varphi_{j}^{0}\}, 0)+\frac{\partial^{2}f}{\partial u\partial\tau_{p}}(0,0,0, \{\varphi_{j}^{0}\}, 0)\varphi_{i}^{0}$,$e_{il}:= \frac{\partial^{2}f}{\partial t_{i}\partial x_{\mathrm{t}}}(0,0,0, \{\varphi_{j}^{0}\}, 0)+\frac{\partial^{2}f}{\partial u\partial x_{l}}(0,0,0, \{\varphi_{j}^{0}\}, 0)\varphi_{i}^{0}$

.

,

129
Here we decompose
intoVi. We substitute this into the system (5.3) and obtain$\mathrm{e}_{\mathrm{i}}(\mathrm{r})$$\mathrm{r}\mathrm{j}\mathrm{i}(\mathrm{x})\ovalbox{\tt\small REJECT}$$(\mathrm{x})+\mathrm{r}\mathrm{j}\mathrm{i}(\mathrm{x})\mathrm{i}\mathrm{p}\mathrm{i}(\mathrm{x})\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}=\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT}}rp_{it}x_{tt},$

V$\mathrm{i}(x)\ovalbox{\tt\small REJECT}$$O(\mathrm{D}|^{2}))$(5.4)$\sum_{k=1}^{n}\sum_{l=1}^{n}a_{kl}x_{l}(\frac{\partial\overline{\psi}_{i}}{\partial x_{k}}(x)+\frac{\partial\eta_{i}}{\partial x_{k}}(x))$$+ \sum_{k=1}^{n}\sum_{p=1}^{d}b_{kp}(\tilde{\psi}_{p}(x)+\eta_{p}(x))(\frac{\partial\overline{\psi_{i}}}{\partial x_{k}}(x)+\frac{\partial\eta_{i}}{\partial x_{k}}(x))$$+c( \tilde{\psi}_{i}(x)+\eta_{i}(x))+\sum_{p=1}^{d}d_{ip}(\overline{\psi}_{p}(x)+\eta_{p}(x))+\sum_{l=1}^{n}e_{il}x_{l}$(degree in

$=$

$x$

is greater than or equal to 2), $i=1,2$ ,By picking up the degree 1part on the both sides, we see that
following system:
(5.5)

, .
$d$

$\ldots$$\{\overline{\psi}_{i}(x)\}$satisfy the$\sum_{k=1}^{n}\sum_{l=1}^{n}a_{kl}x_{l}\frac{\partial\tilde{\psi}_{i}}{\partial x_{k}}(x)+\sum_{k=1}^{n}\sum_{p=1}^{d}b_{kp}\overline{\psi}_{p}(x)\frac{\partial\overline{\psi_{i}}}{\partial x_{k}}(x)$,$+c \overline{\psi}_{i}(x)+\sum_{p=1}^{d}d_{ip}\tilde{\psi}_{p}(x)+\sum_{l=1}^{n}e_{il}x_{l}=0$, .
for $i=1,2$ ,By the existence of aformal solution, (5.5) has some solutionsof linear functions. and we fix such.
For fixed, we see thatandsatisfy the following system:$\ldots$$d$$\{\overline{\psi}_{i}(x)\}$$\{\overline{\psi}_{i}(x)\}$$\{\overline{\psi}_{i}(x)\}$$\{\varphi_{i}^{0}\}$(5.6)$\{\eta_{i}(x)\}$$\sum_{k=1}^{n}\sum_{l=1}^{n}(a_{kl}+\sum_{p=1}^{d}b_{kp}\psi_{pl})x_{l}\frac{\partial\eta_{i}}{\partial x_{k}}(x)+c\eta_{i}(x)+\sum_{p=1}^{d}(d_{ip}+\sum_{k=1}^{n}b_{kp}\psi_{ik})\eta_{p}(x)$$=$(degree in$x$

is greater than or equal to 2.),$i=1,2$ ,$\ldots$, .
$d$

We remark that the degree 2part in the right hand side of this system does not include$\{\eta_{i}(x)\}$.

The following theorem holds:
Theorem 3. Let

be a matrix defined by$(A_{kl})_{k,l=1,2,\ldots,n}$$(A_{kl})_{k,l=1,2},$

.

’

$n$

$=(a_{kl}+ \sum_{p=1}^{d}b_{kp}\psi_{pl})_{k,l=1,2,\ldots,n}$

130be the eigenvalues ofLetsuch that the condition$\{\kappa_{k}\}_{k=1}^{n}$$(A_{kl})_{k,l=1,2,\ldots,n}$$| \sum_{k=1}^{n}\kappa_{k}m_{k}|\geq\sigma_{0}|m|$,.

If there exists a positive constant$\sigma_{0}$(Poincar\’e condition)with $|m|\geq 2$ , then the formal solution of theholds for all $m=$system (1.9) is convergent in a neighbourhood of the origin.
$(m_{1}, \ldots, m_{n})\in(\mathrm{Z}_{\geq 0})^{n}$Remark 7. Let$(B_{ip})_{i,p=1,2,\ldots.d}$be amatrix defined by$(B_{ip})_{i,p=1,2,\ldots,d}=(d_{ip}+ \sum_{k=1}^{n}b_{kp}\psi_{ik})_{i,p=1,2,\ldots,d}$be the eigenvalues ofand letBy the same argument in Remark 1, we have$(B_{ip})_{i,p=1,2,\ldots,d}$$\{\omega_{j}\}_{j=1}^{d}$(5.7)$| \sum_{k=1}^{n}\kappa_{k}m_{k}+c+\omega_{j}|\geq\sigma|m|$,.

, by some $\sigma>0$ , and $j=1,2$ ,$\ldots$, ,for large $m$ , which will be used in the proof.

$d$$\square$

6. Proof OF THEOREM 3
The proof of Theorem 3is the same method of the proof of Theorem 1in case that theunknown function is avector values. However, there are some difference in the detail.
Therefore, we introduce only the outline of the proof of Theorem 3in this section.

Step 1. By taking alinear transformation of the independent variables and alinear
transformation of the unknown functions, (5.6) is reduced to the following form:(6.1)$(\Lambda+\Delta+\mathrm{B})$$:=\{$$(\begin{array}{l}w_{\mathrm{l}}(x)\vdots w_{d}(x)\end{array})$$(\begin{array}{lll}\Lambda_{1} \ddots \Lambda_{d}\end{array})$$+$$(\begin{array}{lll}\mathrm{A} \ddots \Delta\end{array})$$+\mathrm{B}\}$$=(\begin{array}{l}\sum_{|m|=2}a_{1,m}x^{m}+g_{3_{\prime}1}(x,w(x),\partial_{x}w(x))\vdots\sum_{|m|=2}a_{d,m}x^{m}+g_{3_{\prime}d}(x,w(x),\partial_{x}w(x))\end{array})$,$(\begin{array}{l}w_{1}(x)\vdots w_{d}(x)\end{array})$

131
where $w_{j}(x)(j=1,2, \ldots, d)$ denote new unknown functions after linear transformations
and$\Lambda_{j}=\sum_{k=1}^{n}\kappa_{k}x_{k}\partial_{x_{k}}+c+\omega_{j}.$,$\Delta=\sum_{k=1}^{n-1}\epsilon_{k}x_{k}\partial_{x_{k+1}}$,$\mathrm{B}=(\begin{array}{llll} \end{array})$

,

where
and denote the nilpotent components of the Jordan canonical forms of thematrices, respectively, andand$e_{j}$$\epsilon_{j}$$(A_{kl})$$(B_{ip})$.

$g_{3,i}(x, \eta, \zeta)=\sum_{|\alpha|+2|\beta|+|\gamma|\geq 3}g_{\alpha\beta\gamma}^{(i)}x^{\alpha}\eta^{\beta}\zeta^{\gamma}$

Step 2. We define$\mathrm{C}[x]_{M}$by $\mathrm{C}[x]_{M}=\{\Sigma_{|m|=M}u_{m}x^{m} ; u_{m}\in \mathrm{C}\}$ , and define anorm of$u(x)={}^{t}(u_{1}(x), \ldots, u_{d}(x))\in(\mathrm{C}[x]_{M})^{d}$by$||u||:= \inf\{C>0 ; Wj(x)\ll C(x_{1}+\cdots+x_{n})^{M}, i=1,2, \ldots, d\}$ .

By the same argument in the proof of Lemma 1and by Remark 7, we can prove thesame results of Lemma 1for the operator$\Lambda+\Delta+\mathrm{B}$.

Step 3. By the same method in the previous sections, we can construct amajorantequation whose formal solution is amajorant function of the all unknown functions of
the system. Finally, by the implicit function theorem, we prove the convergence of theformal solution of the majorant equation.

REFERENCES[CL]

Chen H. and Luo Z., On the Holomorphic Solution of Non-linear Totally Characteristic Equations with Several Space Variables, Preprint 99/23 November 1999, Institut fur Mathematik,
Universit\"at Potsdam.

[CLT] Chen H. and Luo Z. and Tahara H., Formal solutions of nonlinear first order totally characteristic
type PDE with irregular singularity, to appear.

[CT]

Chen H. and Tahara H., On Totally Characteristic Type Non-linear Partial Differential Equations
in Complex Domain, Publ. RIMS, Kyoto Univ. 35 (1999), 621-636.

[GT]

Gerard R. and Tahara H., Singular Nonlinear Partial Differential Equations, Vieweg, 1996.

[MS]

Miyake M. and Shirai A., Convergence of Formal Solutions of First Order Singular Nonlinear Partial Differential Equations in Complex Domain, Annales Polonic Mathmatici (volume dedicatedto the memory of B. Ziemian), 74 (2000), 215-228

132
Ouchi S., Genuine solutions and formal solutions with Gevrey type estimates of nonlinear partial
differential equations, J. Math. Sci. Univ. Tokyo, 2(1995), 375-417.
Shirai A., Maillet Type Theorem for Nonlinear Partial Differential Equations and Newton Polygons, J. Math. Soc. Japan. Vol53,3 (2001).

[O]
[S]

GRADUATE SCHOOL OF MATHEMATICS, NAGOYA UNIVERSITY, HurO-cho, CHIKUSA-KU, NAGOYA
464-8602, JApAN-mail address: m96034qQmath. nagoya-u.ac. jp$E$