165

q ‐analogue of a system of equations

from geometry

By

JUN YAMAMOTO * and HIROSHI YAMAZAWA **Abstract

R.Bielawski [1] studied some systems of partial‐differential equations and gave the uniqueexistence theorem of holomorphic solutions. In this paper we investigate that a system of q‐

difference‐differential equations under some conditions which Yamazawa have given in [3]. Our
purpose in this paper is to obtain the same results in [3]. We show for a general equations anexistence of holomorphic solutions and for an example case we consider an existence of singularsolutions.

§1.

Introduction and Result

Throughout this paper, let suppose q>1 and define |x|= \max_{1\leq i\leq n}|x_{i}| for(x_{1}, \cdots : x_{n})\in \mathbb{C}_{x}^{n} . We denotex=\mathbb{N}=\{0,1, \cdots\}, \mathbb{N}^{*}=\mathbb{N}\backslash \{0\},D_{R}=\{x= (x_{1}, \cdots , x_{n})\in \mathbb{C}_{x}^{n}|x|<R\},\bullet D_{T}=\{t\in \mathbb{C}_{t};|t|<T\},
D_{T}\cross D_{R}=\{(t, x)\in \mathbb{C}_{t}\cross \mathbb{C}_{x}^{n};t\in D_{T}, x\in D_{R}\},\cdot\mathcal{O}(D_{R}) be the set of all holomorphic functions on D_{R},\mathcal{O}(D_{T}) be the set of all holomorphic functions on D_{T},\mathcal{O}(D_{R}\cross D_{T}) be the set of all holomorphic functions on D_{R}\cross D_{T}.

\bullet\bullet\bullet\bullet\bullet\bulletFirst we define the q ‐difference operator D_{q} for a function f(t, x) by

D_{q}f(t, x)= \frac{f(qt,x)-f(t,x)}{qt-t}.
2010 Mathematics Subject Classification(s): Primary 39A13 ; Secondary 35C10
Key Words: Formal solutions, Holomorphic solutions, Singular solutions, q‐difference‐differential
equations*

Graduate School of Engineering and Science, Shibaura Institute of Technology. Saitama 337‐8570,
Japan.

”College of Engineering and Design, Shibaura Institute of Technology, Saitama 337‐8570, Japan.

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JUN YAMAMOTO, HIROSHI YAMAZAWA

Let m_{1} and m_{2} be positive integers. In this paper we investigate the following systemof q‐difference‐differential equations:

(1.1)
where

\{ begin{ar y}{l tD_{q}u=F(t,xtD_{q}u,\{ parti l_{x}^\alpha}u\_{|\alpha|\leqm_{1} ,v) D_{q}v=c(t,x)+\sum_{|\alpha|\leqm_{2}d_{\alpha}(x)\parti l_{x}^\alpha}u \end{ar y}(t, x)=(t, x_{1}, \cdots , x_{n})\in \mathbb{C}_{t}\cross \mathbb{C}_{x}^{n}, \alpha=(\alpha_{1}, \cdots , \alpha_{n})\in \mathbb{N}^{n}, |\alpha|=\alpha_{1}+\cdots+\alpha_{n},\partial_{x}^{\alpha}=\partial_{x_{1} ^{\alpha_{1} \cdots\partial_{x_{n} ^{\alpha_{n} . c(t, x)\in \mathcal{O}(D_{T}\cross D_{R_{0}}) for some T, R_{0}>0, F(t, x, u, V, W) with
V=\{V_{\alpha}\in \mathbb{C};|\alpha|\leq m_{1}\} is a function defined in some polydisk \triangle centered at the origin\mathbb{C}_{t}\cros \mathbb{C}_{x}^{n}\cros \mathbb{C}_{u}\cros \mathbb{C} _{V}^{\delta}\cros \mathbb{C}_{W} and \delta is the cardinal of \{\alpha\in \mathbb{N}^{n};|\alpha|\leq m_{1}\} . Let us denote\triangle_{0}=\triangle\cap\{t=0, u=0, V=0, W=0\}.
Yamazawa [3] considered the following nonlinear q‐difference‐differential equation

of

(1.2)

tD_{q}u=F(t, x, \{\partial_{x}^{\alpha}u\}_{|\alpha|\leq m}) ,defined the q‐difference‐differential equations of Briot‐Bouquet type and constructed theholomorphic and singular solutions for the equation (1.2). In [3], Yamazawa gave thedefinition of q‐Briot‐Bouquet equation as below:

Definition 1.1 (H. Yamazawa [3]). The equation (1.2) is called a q ‐analogue of the
Briot‐Bouquet type with respect to t or simply q‐Briot‐Bouquet type with respect to t

if the equation (1.2) satisfies the following three conditions:
A_{1}:F(t, x. V) is holomorphic in \triangle,
A_{2}:F(0, x, 0)=0 in \triangle 0,
A_{3}:\partial_{V_{\alpha}}F(0. x, 0)=0 in \triangle_{0} for all 1\leq|\alpha|\leq m,where \triangle is a polydisk centered at the origin of \mathbb{C}_{t}\cros \mathbb{C}_{x}^{n}\cros \mathbb{C}_{V}^{\delta}, \delta is the cardinal of\{\alpha\in \mathbb{N}^{n};|\alpha|\leq m\} and \triangle_{0}=\triangle\cap\{t=0, V=0\}.

Definition 1.2 (R. Gérard and H. Tahara [2]). Set\rho(x)=\frac{\partial F}{\partial u}(0, x, 0)
then the holomorphic function \rho(x) is called the characteristic exponent of the equation(1.2).
On the other hand, R.Bielawski [1] studied the conditions that the given Kähler
metric h extends to a Ricci‐flat Kähler metric on a line bundle L in a manifold M suchthat the canonical S^{1} ‐action on

L

is Hamiltonian. The necessary condition which hegave is to solve the following Cauchy problem:\{begin{ar y}{l t\partil_{}v=-1+ce^{-V}\det[g_{ij}] \partil_{x 1}_{jv+\partil_{y \tau}y_{fv+c\partil_{}g \dot{i}j=0 (g_{ij})| t=0}h_{ij},(e^{v})|_t=0}c\deth. \end{ar y}167

q ‐ANALOGUE OF A SYSTEM OF EQUATIONS FROM GEOMETRY

In order to study this solvability for this Cauchy problem, R.Bielawski considered thefollowing system of partial differential equations:\{ begin{ar y}{l t\partial_{t}u=F(t,x_{:}u,\partial_{x}u,t\partial_{t}u_{\dot{c}\{v_i}\_{i= 1}^{N}) \partial_{t}v_{i}=c_{i}(t,x)+\sum_{|\alpha|\leq2}d_{i,\alpha}(x)\partial_{x}^{ \alpha}ufori=1 .,N \end{ar y}(1.3)

where c_{i}(t, x)\in \mathcal{O}(D_{T}\cross D_{R_{0}}) and d_{i.\alpha}(x)\in \mathcal{O}(D_{R_{0}}) . In [1], R.Bielawski gave thefollowing assumptions for (1.3):
F(t, x_{4}.u, V, W, Z) is holomorphic in \triangle_{:}F(0, x, 0_{:}0_{:}0,0)=0 in \triangle_{0},B_{3} : \partial_{V_{\tau}}F(0, x, 0,0,0,0)=0 in \triangle_{0}for i=1

B_{1} :
B_{2} :

. , n,\triangle is a polydisk centered at the origin of \mathbb{C}_{t}\cros \mathbb{C}_{x}^{n}\cros \mathbb{C}_{u}\cros \mathbb{C} _{V}^{n}\cros \mathbb{C}_{W}\cros \mathbb{C}_{Z}^{N}

where

and

\triangle 0=\triangle\cap\{t=0, u=0, V=0, W=0, Z=0\}.
Under these assumptions, R.Bielawski gave the unique exicetence theorem of holo‐

morphic solutions of (1.3) provided that the characteristic exponent \rho(x) satisfies \rho(0)\not\in\mathbb{N}^{*}In this paper we assume the following assumptions for (1.1):
q-B_{1} :q-B_{2}:q-B_{3} :F(t, x, u, V, W) is holomorphic in \triangle_{\backslash}F(0, x, 0,0,0)=0in\triangle_{0},\partial_{V_{\alpha}}F(0, x, 0,0,0)=0 in \triangle 0 for all 1\leq|\alpha|\leq m_{1}.

We concern the following Yamazawa;s result [3]:Theorem 1.3 (H. Yamazawa [3]). If (1.2) is of the q‐Briot‐Bouquet type and \rho(0)\neq(q^{i}-1)/(q-1) for i=1,2,\cdot\cdot\cdot, then we have:(1) (Holomorphic solutions) The equation (12) has a unique solution u_{0}(t, x) holo‐
morphic near the origin of \mathbb{C}_{t}\cros \mathbb{C}_{x}^{n} satisfying u_{0}(t_{:}x)\equiv 0.

(2) any\varphi(x)\in \mathb {C}\{x\}utions)exists
there(Singu1arso1 12)h.n_{ganexpanSi
a n\mathcal{oO}_{+}^{nof}S}et-sol\urt_{\dot{iho_{q}}onU(\varphi}(x)=\log\{) 1of+(q-1()\rho(x)\}/.\log q_{avi} When\Re\rho(0)>0forthe following form:

U( \varphi)=\sum_{i=1}^{\infty}u_{i}(x)t^{i}+\sum_{k\leq x+2m(j-1),j\geq 1} \varphi_{i,j k}(x)t^{i+\rho_{q}(x)j}(\log t)^{k},where the coefficients \{u_{?}(x)\in \mathbb{C}\{x\}:i\geq 1\} and \{\varphi_{i,j,k}(x)\in \mathbb{C}\{x\};j\geq 1, k\leq
i+2m(j-1)\} are determined by \varphi(x) .
In the above theorem, \mathbb{C}\{x\} is the ring of gcrms of holomorphic functions at theorigin of \mathbb{C}^{n} and for the definition of \overline{\mathcal{O} _{+} , see Definition 5.2. In relation to this result,

we aim to get the structure of holomorphic and singular solutions for (1.1). Our main
result is as follows:

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JUN YAMAMOTO, HIROSHI YAMAZAWA

Theorem 1.4. If the first equation of (1.1) satisfies the conditions q-B_{1} , q-B_{2} andq-B_{3} and \rho(0)\neq(q^{i}-1)/(q-1) for i\in \mathbb{N}^{*} , then the system (1.ı) has a pair of uniqueholomorphic solutions (u, v) satisfying u(0, x)=u(0_{:}x)\equiv 0.

This paper is organized as follows. In section 2 and 3 we prepare lemmas in orderto show Theorem 1.4. In section 4 we show Theorem 1,4 and in section 5 give a proofof an existence of formal solutions as singular solutions for the particular case of (1.1).
§2.

Lemma

In this section we give some lemmas. Set
and 0<R_{0}<R.

\Vert u\Vert_{R}=\sup_{x\in D}. |u(x)| for t4(x)\in \mathcal{O}(D_{R_{O}})

Lemma 2.1 ( Nagumo^{:}s lemma). Assume u(x) be holomorphic on D_{R} . If for any
0<r<R,\Vert u\Vert_{r}\leq\frac{C}{(R-r)^{p}holds for some p\geq 0 . then we have

\Vert\partial_{x_{f} u\Vert_{r}\leq\frac{Ce(p+1)}{(R-r)^{p+1}

for any

0<r<R

and

j=1,

\cdot\cdot\cdot

in.

Lemma 2.2. There exists a constant M such that for any i\in \mathbb{N}^{*}(m^{*}ie)^{m^{*}}\overline{q^{\dot{\iota} }\leq M
wherem^{*}= \max\{m_{1}, m_{2}\},

Proof. Let consider a function f(x)=(m^{*}xe)^{m^{*}}/q^{x}=(m^{*}e)^{m^{*}}x^{rn^{*}}q^{-x} , then wehavef'(x)=x^{7n}\{(m^{*}e)^{M^{*}}(m^{*}x^{-1}-\log q)\}.
Therefore, this function f(x) takes the maximum value at x=m^{*}/\log q . Setting x_{0}=m^{*}/\log q then we havef(x)\leq f(x_{0})and thisf(x_{0}) means the constant M of the statement in Lemma 2.2.

Lemma 2.3 (H. Yamazawa [3]). Let q>1 . Suppose\lambda(x)\neq q^{i}

for x\in D_{r}Then there exists a constant \sigma>0 such that(2.1)

|q^{i}-\lambda(x)|\geq\sigma q^{i}

for x\in D_{r}.

\square169

q ‐ANALOGUE OF A SYSTEM OF EQUATIONS FROM GEOMETRY§3.

Reduction equations

In this \sec ion we reduct the system (1.ı) into the following system of q‐difference‐differential equations under the assumptions q-B_{1},q-B_{2} and q-B_{3} to show Theorem 1.4.

(3.1)\{ begin{ar y}{l (\sigma_{q}-\lambda(x)u=a(x)t+b(x)v+G_{2}(x)t,\sigma_{q}u,\{ partial_{x} ^{\alpha}u\}_{|\alpha|\leqm_{1},v) \sigma_{q}v=\sum_{i\geq1}c_{i}(x)t^{i}+\sum_{|\alpha|\leqm_{2}d_{\alpha}(x) t\partial_{x}^{\alpha}u \end{ar y}where \sigma_{q}u(t, x)=u(qt_{:}x), a(x)_{\ovalbox{\t \small REJECT}}.c_{i}(x), d_{\alpha}(x)\in \mathcal{O}(D_{R_{0}}) and the function

G_{2}(x)(t, u, V, W)

has the following expansion:

(3.2)
where

G_{2}(x)(t_{\ovalbox{\t \smal REJECT} .u, V, W)= \sum_{p+\mu+|\nu|+\xi\geq 2} g_{p.\mu.\nu,\xi}(x)t^{p}u^{\mu}V^{\nu}W^{\xi}V^{\nu}= \prod_{|\alpha|\leq rn_{1} \{V_{\alpha}\}^{\nu_{\alpha}}.

Lemma 3.1. If the system (1.1) satisfies the condition q-B_{1},q-B_{2} and q-B_{3} , thenwe can reduct (1.1) into the system (3.1) with (3.2).

Proof. We multiply the both side of (1.1) by q-1 . Then we get \lambda(x)=1+(q-1)\rho(x)\squareand (3.1).
Remark. The assumption \rho(0)\neq(q^{i}-1)/(q-1) for i\in \mathbb{N}^{*} implies \lambda(0)\neq q^{\lambda} fori\in \mathbb{N}^{*}§4.

Proof of Theorem 1.4

In this section we will prove Theorem 1,4. This proof is divided into two steps; thefirst step is to construct a pair of formal power series solutions (û, \hat{v} ) of the system(1.1) and the other one is about the convergence of the pair of the formal power series
solutions (û, \hat{v} ).
§4.1.

Formal power series solutions

Let us show the system (1.1) has a pair of formal power series solutions of the form

(4.1)

û

=

\sum_{\dot{i}\geq 1}u_{i}(x)t^{i,}.\hat{v}=\sum_{i\geq 1}v_{i}(x)t^{i}.

Set v=v_{1}(x)t+v^{*} with v^{*}= \sum_{x\geq 2}v_{\lambda}(x)t^{i} . By substituting a pair of formal power series
solutions (û, \hat{v} ) into the system (3.1), then we have the following recurrence formulas:(4.2)\{\begin{ar ay}{l} qv_{1}(x)=c_{1}(x) (q-\lambda(x) u_{1}(x)=a(x)+b(x)v_{1}(x) \end{ar ay}

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JUN YAMAMOTO, HIROSHI YAMAZAWA

and for i\geq 2(4.3)\{q^i}(q^{i}-\lambda(x))u_{i}(x)=b(x)v_{i}(x)+\sum_{p+\mu+|\nu|+\xi=}\sum^{*}g_{p,\mu,\nu,\xi}(x)\prod_{i=1}^{\mu} q\iota_{t}u_{l_{i} (x)\prod_{|\alpha|\leqrn_{1} \prod_{j=1}^{\nu_{\alpha} \partial_{x}^{\alpha}u_{m_{\alphaj} (x)\prod_{k=1}^{\xi}v_{n_{k} (x)where\sum^{*}=

\sum

with

|l|_{\mu}=l_{1}+\cdots+l_{\mu}, |n|_{\xi}=n_{1}+\cdots+n_{\xi}and|m|_{rr\iota_{1},\nu}=|l|_{\mu}+|rn|_{77\iota_{1}\nu}+|n|_{\xi}+p=i\sum\sum m_{\alpha,j}\nu_{\alpha}.

|\alpha|\leq m_{1}J^{=1}

Therefore, by the assumption we find out that the system (3.1) has a pair of formal power
series solutions (û, \hat{v} ) whose coefficients are given by the above recurrence formulas.
§4.2.

Convergence of the formal power series solutions

We will show that the pair of the formal power series solutions (û, \hat{v} ) converges ina neighborhood of the origin

t=0 .

Since the fact that\hat{v}=\sum_{i\geq 1}v_{i}(x)t^{i} is written as

the form \hat{v}=v_{1}(x)t+v^{*} , we can rewrite the system (3.1) by the following system of
reduction equations:(4.4)\{ begin{ar y}{l (\sigma_{q}-\lambda(x)u=a(x)t+b(x)v^{*}+G_{2}(x)t,\sigma_{q}u,\{ partial_{x} u^{\alpha}\_{|\alpha|\leqm_{\imath} ,v^{*}) \sigma_{q}v^{*}=\sum_{i\geq2}c_{i}(x)t^{i}+\sum_{|\alpha|\leqm_{2}d_{\alpha} (x)t\partial_{x}^{\alpha}u. \end{ar y}

Remark. The function G_{2}(x)(t_{\backslash ,1}u, V, W^{*}) of (4.4) differs from (3.1) in the following
sense:

G_{2}(x)(t, u, V, W^{*})= \sum_{p+\mu+|\nu|+\xi\geq 2}g_{p\mu.\nu_{\backslash } \xi}^{*}(x)t^{p}u^{\mu}V^{\nu}W^{*\xi}
where

g_{p.\mu,\nu,\xi}^{*}(x)= \sum_{\xi_{1}+\xi_{2}=\xi}\frac{ _{ \imath} (x)^{\xi_ {1} (\xi_{1}+\xi_{2})!}{\xi_{1}!\xi_{2}! g_{p,\mu,\nu,\xi}(x).

This system of reduction equations (4.4) has a pair of formal power series solutions(û, \hat{v}^{*} ) of the form
(4.5)

\hat{u}=\sum_{>i,1}u_{t}(x)t^{i},\hat{v}^{*}=\sum_{i\geq 2}v_{i}(x)t^{i}q ‐ANALOGUE OF A SYSTEM OF EQUATIONS FROM GEOMETRY

and the coefficients u_{i}(x) for i\geq 1 and v_{t}(x) for i\geq 2 satisfy the recurrence formulas(4.2) and (4.3).
Let us consider the following system of analytic equations:(4.6)\{begin{ar y}{l \sigmaX=\sigmaAt+\frac{l1I}(R-r)^{\gam \gam \iota^{*}\BY+\sum_{p+\mu+ |\nu+\xigeq2}\frac{t^p}G_{p,\mu n,\xi}^{* (R-r)^{7l *}(p+\mu+|\nu+ 2\xi-)}X^{\mu+|\nu}Y^{\xi} Y=\sum_{i\geq2}\frac{C_i}{(R-r)^{m *}(i-2)}t^{i+\sum_{|\alph|\leqm_{2} D_{\alph}tX \end{ar y}where M is in Lemma 2.2, A=\Vert a\Vert_{R}, B=\Vert b\Vert_{R}, G_{p\mu,\nu,\xi}^{*}=\Vert g_{p,\mu\nu\xi}^{*}\Vert_{R}, C_{i}=\Vert c_{i}\Vert_{R} for
any i\in \mathbb{N}^{*} and D_{\alpha}=\Vert d_{\alpha}\Vert_{R} for |\alpha|\leq m_{2}.

Lemma 4.1. The system of analytic equations (4.6) has the pair of holomorphic
solutions (X, Y) of the form

(4.7)

X(t, r)= \sum_{i\geq 1}X_{i}(r)t^{i}, Y(t_{\backslash }r)=\sum_{\dot{x}>2, \prime}Y_{i}(r)t^{i},

moreover the coefficients X_{i}(r), Y_{i}(r) have the following forms:

X_{i}= \frac{E_{i} {(R-r)^{m^{*}(i-1)}
Y_{i}= \frac{F_{i} {(R-r)^{\tau n^{*}(\dot{z}-2)}

for i\geq 1

fori\geq 2

with constants E_{1}=A and E_{i}, F_{i}\geq 0 for i\geq 2.

Proof. First we will show that the system of analytic equations (4.6) has a pair offormal power series solutions (X, Y) of the form (5.4). By substituting (X, Y) into thesystem (4.6), then we have the following recurrence formulas for X_{i}(r), Y_{i}(r) :X_{1}=Aand for i\geq 2\{beginary}{l\sigmaX_{z}=\fracM}{(R-r)^m{*}\BY_dot{i}+\sum_{p+\u|n+\xi=}sum^{*\fracG_{p\mu,n xi}^{*(R-r)m^{*}(p+\u |n+2\xi-)}prod_{i=1}^\mux_{iota\u}prod_{|\alph eqrn_{1}\prod_{j=1}^\nu_{alph}X_{m\alph}f rod_{k=1}^\xiY_{nk}\Y_{i=\fracC_{?}(R-r)^m{*}(i-2)+\sum_{|alph\eqrn_{2}D\alph}X_{z-1.\end{ary}This means that there exists a pair of unique formal power series solutions (X, Y) . Let

us show that this pair of formal power series solutions is holomorphic in a neighborhoodof the origin t=0 . Let consider the following functions:

171
171

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JUN YAMAMOTO, HIROSHI YAMAZAWA

F(t_{\dot{r} X, Y):= \sigma(X-At)-\frac{M}{(R-r)^{7 l^{*} \{BY+\sum_{p+\mu+ |\nu+\xi\geq 2}\frac{t^{p}G_{p\mu.\nu\xi}^{*}X^{\mu+|\nu|}Y^{\xi} {(R-r) ^{7 l^{*}(p+\mu+|\nu|+2\xi-2)} \}G(t, X, Y):=Y- \sum_{i\geq 2}\frac{C_{i} {(R-r)^{m^{*}(i-2)} t^{i}- \sum_{|\alpha|\leq m_{2} D_{\alpha}tX.

Then it follows

F(0,0,0)=G(0,0,0)=0, \det (\begin{ar ay}{l} \partial_{X}F(0,0,0)\partial_{X}G(0,0,0) \partial_{Y}F(0,0_{:}0)\partial_{Y}G(0,0,0) \end{ar ay})=\sigma\neq 0.

Therefore, by the implicit function:s theorem we get holomorphic solutions (X, Y) satis‐fying (X(0, r), Y(0, r))=(0,0) . We will prove the latter in the statements by induction
on i . Since X_{1}=A holds from the recurrence formulas, it is clear for i=1 . For thegeneral case i\geq 2 we haveY_{i}= \frac{C_{i} {(R-r)^{m^{*}(i-2)} +\sum_{|\alpha|\leq r \iota_{2} D_{\alpha}X_{i-1}= \frac{C_{i} {(R-r)^{rn^{*}(i-2)} +\sum_{|\alpha|\leq m_{2} D_{\alpha}\frac{E_ {i-1} {(R-r)^{7n^{*}(i-2)}=\frac{C_{i}+\sum_{|\alpha|\leqm_{2} D_{\alpha}E_{i-1} {(R-r)^{m^{*} (\dot{\tau}-2)}

and

X_{i}

= \frac{\Lambda I}{\sigma(R-r)^{m^{*} (B\frac{F_{i} {(R-r)^{m^{*}(i-2)} +\sum_ {p+\mu+|\nu|+\xi=i}\sum^{*}\frac{G_{p,\mu,\nu,\xi}^{*} {(R-r)^{m^{*}(p+\mu+|\nu| +2\xi-2)}\cros \prod_{i=1}^{\mu}\frac{E_{l_{t} {(R-r)^{7n^{*}(l_{i}-1)} \prod_{|\alpha|\leqm_{1} \prod_{j=1}^{U_{\alpha} \frac{E_{rn_{\alpha}J }{(R-r)^ {rn^{*}(m_{\mathfrak{a}g -1)} \prod_{k=1}^{\xi}\frac{F_{n_{k} {(R-r)^{7n^{*}(n_ {k}-2)}= \frac{M}{\sigma(R-r)^{m^{*} }(\frac{BF_{i} {(R-r)^{r \iota^{*}(i-2)}+ \sum_{p+\mu+|\nu|+\xi=i}\sum^{*}\frac{G_{p_{:}\mu_{U_{)} \xi}^{*} {(R-r)^{7n^ {*}(p+\mu+|\nu|+2\xi-2)} \prod_{R-}^\{p\rodm_{+u,(r})^i{=\ta1uEr\_iotal^_{*}(|zl_}{\mu}|_{m_{1}\nu}+|n_{\xi}-(\mu+|\nu|+2\xi) }17\gam at}|\alpha|\leqr\iota_{1}\prod_{j=1}^{\nu_{\alpha} E_{r\iota_{\alphaj}\prod_{k=1}^{\xi}F_{n k})= \frac{\lambda I}{\sigma(R-r)^{m^{*} (\frac{BF_{i} {(R-r)^{\tau n^{*}(\dot{i} -2)}+\sum_{p+\mu+|\nu|+\xi=}\sum^{*}\frac{G_{p.\mu\nu.\xi}^{*}\prod_{i=1}^{\mu}E_ {l_\tau}\prod_{|=1^{E_{m_{\alpha}\prod_{k=1}^{\xi}F_{n_{k} \alpha\leq7. n_{1}^{\prod_{jJ}^{\nu_{\alpha} {(R-r)^{7n^{*}(i-p)+7n^{*}(p-2)}=\frac{\LambdaI/\sigma(BF_{i}+\sum_{p+\mu+|\nu|i}+\xi=\sum^{*}G_{p\mu_{:}\nu. \xi}^{*}\prod_{i=1}^{\mu}E\iota_{i}\prod_{|\alpha|\leq\gam a\gam a\iota_{1} \prod_{j^{=1}j ^{\nu_{\alpha} E_{\taur\iota_{\alpha} \prod_{k=1}^{\xi}F_{71}k)} {(R-r)^{7n^{*}(i-1)}

173

q ‐ANALOGUE OF A SYSTEM OF EQUATIONS FROM GEOMETRY\squareWe give the following proposition in order to show the convergence of the pair of theformal power series solutions (û, \hat{v}^{*} ).
Proposition 4.2. For any

0<r<R we have

(4.8)

\Vert q^{i}u_{i}\Vert_{\Gamma}, \Vert\partial_{x}^{\alpha}u_{i}\Vert_{r}\leq X_{i}(4.9)

1v_{i}\Vert_{t}\leq Y_{i}for i\geq 1_{:}|\alpha|\leq m^{*}for i\geq 2.

Proof. We prove the evaluations by induction on i . For the case i=1 it follows by
the definition of A . For the case i=2 by recurrence formulas and induction hypothesis
we have\Vertq^{2}v_{2}\Vert_{r}=\Vertc_{2}+\sum_{|\alpha|\leqm_{2} d_{\alpha} \partial_{x}^{\alpha}u_{1}\Vert_{r}\leq C_{2}+\sum_{|\alpha|\leq\tau n_{2} D_{\alpha}X_{1}=Y_{2}
therefore we have

\Vert v_{2}\Vert_{r}\leq X_{2}.
For? \geq 3 we have in the same manner\Vertq^{i}v_{i}\Vert_{\gam a}=\Vertc_{i}+\sum_{\alpha1\leqm_{2} d_{Q} \partial_{x}^{\alpha}u_{i-1}\Vert_{r}\leq C_{i}+\sum_{|\alpha|\leq 7n_{2} D_{\alpha}X_{i-1}\leq\frac{C_{\dot{\tau} {(R-r)^{rn^{*}(?-2)} +\sum_{|\alpha|\leq m_{2} D_{\alpha}X_{i-1}=Y_{i}

thus we get\Vert v_{z}\Vert_{\tau}\leq Y_{i}.

Hence on v_{i}(x) for i\geq 2 we get the inequalitiy (4.9) in Proposition 4.2. On the other174

JUN YAMAMOTO. HIROSHI YAMAZAW’A

hand, for i\geq 2 we have

\Vert(q^{i}-\lambda)\iota\iota_{i}\Vert_{r}=\Vertbv_{i}+\sum_{p+\mu+|\nu|+\xi=}\sum^{*}g_{p_{\dot{4}\mu,\nu,\xi}^{*} \prod_{i=1}^{\mu}q_{l t}u_{l t}\prod_{|\alpha|\leqm_{1}\prod_{j=1} ^{\nu_{\alpha}\partial_{x}^{\alpha}u_{rn_{\alphaj}\prod_{k=1}^{\xi}v_{n_{k} \Vert_{r}\leqBY_{i}+\sum_{p+\mu+|\nu|+\xi=}\sum^{*}G_{p,\mu_{:}\nu_{:}\xi}^{*} \prod_{i=1}^{\mu}X_{l i}\prod_{|\alpha|\leq\gam ar\iota_{1}\prod_{j=1}^{\nu_ {\alpha}X_{m_{\alphag}\prod_{k=1}^{\xi}Y_{n_{k}\leqBY_{i}+\sum_{p+\mu+|\nu|+\xi=}\sum^{*}\frac{G_{p,\mu,\nu,\xi}^{*} {(R-r) ^{r \iota^{*}(p+\mu+|\nu|+2\xi-2)} \prod_{i=1}^{\mu}X_{l_{t} \prod_{|\alpha|\leq rn_{1} \prod_{j=1}^{\nu_{\alpha} X_{m_{\alphag} \prod_{k=1}^{\xi}Y_{n_{k}= \frac{(R-r)^{rn^{*} {l\downar ow I}\sigma X_{i}therefore by Lemma 2.3 we have

\Vert u_{i}\Vert_{r}\leq\frac{1}{\sigma q^{i} \frac{(R-r)^{7t1^{*} {M}\sigma X_{i}\leq\frac{X_{\dot{i} {q^{i} .

(4.10)

Let give estimates of the derivative of u_{i}(x) for i\geq 2 . By applying Lemma 2.1 to (4.10)
with m_{1} ‐times, then we have for |\alpha|\leq m_{1}\Vert\partial_{x}^{\alpha}u_{i}\Vert_{r}\leq\frac{(m^{*}(\dot{i}-2)+1)+\cdots+ (m^{*}(i-2)+m^{*})}{\Lambda Iq^{i} e^{m^{*} \frac{E_{i} {(R-r)^{7n^{*}(i-2)+71L^ {*} }\leq\frac{(m^{*}(\dot{i}-2)+2m^{*})+\cdots+(m^{*}(\dot{i}-2)+2m^{*})}{Mq^{i} e^{m^{*} \frac{E_{i} {(R-r)^{7n^{*}(?-1)}(m^{*}ie)^{m^{*}}=\overline{j1Iq^{l}}X_{i}\leq X_{i}.

Therefore we get the desired results.

\squareThus, by summing up from Proposition 4.2 we have|\^{u}|\leq X, |\hat{v}|\leq Y.
Therefore the formal power series solutions (û, \hat{v} ) of (3.1) converges in a neighborhoodof the origin

t=0.

§5.

Formal solutions for the particular case of (1.1)

In this section we will construct formal solutions as singular solutions for the partic‐
ular case of the system (1.1). Let us consider the following system:(5.l)\{ begin{ar ay}{l (\sigma_{q}-\lambda(x)u=a(x)t+v u\partial_{x}u \sigma_{q}v=b(x)t+\partial_{x}u \end{ar ay}

q ‐ANALOGUE OF A SYSTEM OF EQUATIONS FROM GEOMETRY

where(t, x)\in \mathbb{C}_{t}\cross \mathbb{C}_{x}, \lambda(x), a(x), b(x)\in \mathcal{O}(D_{R_{0}}) .

Definition 5.1. Let us denote by\bullet\bullet\bullet\mathcal{R}(\mathbb{C}\backslash \{0\}) the universal covering space of \mathbb{C}\backslash \{0\},S_{\theta} the sector in \mathcal{R}(\mathbb{C}\backslash \{0\});\{t\in \mathbb{C}\backslash \{0\};|\arg t|<\theta\},
S(\epsilon(s))=\{t\in \mathbb{C}\backslash \{0\}:0\prime<|t|<\epsilon(\arg t)\} for some positive‐valued function \epsilon(s)defined and continuous on \mathbb{R}.

Definition 5.2. We define the set \overline{\mathcal{O} + of all functions u(t, x) satisfying the following
conditions:

1. u(t, x) is holomorphic in S(\epsilon(s))\cross D_{R} for some \epsilon and R>0,

2. there is ana>0 such that for any \theta>0 and compact subset K of D_{R}\max_{x\in K}|u(t, x)|=O(|t|^{a})as tarrow 0 in

S_{\theta}.

Theorem 5.3. Set \lambda_{q}(x)=\log\lambda(x)/\log q . If the holomorphic function \lambda(x) satisfies\lambda(0)\neq q^{i} for i\in \mathbb{N}^{*} , then there exists a pair of formal solutions of (5.1) having an
expansion of the following forms:(5.2)\{begin{ar y}{l U(\varphi)=\sum_{>i,1}u_{i(x)t^{i}+\sum_{k\leqi+j-1,\geq1}\varphi_{ j},k (x)t^{i+j\lambda_{q}(x) \logt)^{k} V(\varphi)=\sum_{1}v_{i(x)t^{i}+\sum_{\geq?\geqk\leqi+j-1_{j}1,\psi_{,jk} (x)t^{i+j\lambda_{q}(x) \logt)^{k} \end{ar y}where \varphi=\varphi_{01,0}(x) is any holomorphic function on D_{R_{0}}, u_{i}(x), v_{i}(x)\in \mathcal{O}(D_{R_{0}}) for
i\geq 1 and \varphi_{i,j,k}(x), \psi_{i,j,k}(x)\in \mathcal{O}(D_{R_{0}}) for j\geq 1, k\leq i+j-1.

Proof. Set for I\geq 1(5.3)

u_{I}(t, x)=u_{I}(x)t^{I}, \varphi_{I}(t, x)= \sum \sum \varphi_{i_{\mathcal{J}},k}(x)t^{i+j\lambda_{q}(x)}(\log t)^{k}i +j=I,j \geq ı k\leq i+j-1

and for I\geq 2(5.4)

v_{I}(t.x)=v_{I}(x)t^{I} \psi_{I(t,x)=\sum_{i+j=I,?,j\geq 1}\sum_{k\leq i+j-1} j}\psi_{i,j,k(x)t^{i+\lambda_{q}(x)}(\log t)^{k}Then we can rewrite the form (5.2) as follows:(5.5)

U( \varphi)=\sum_{I\geq 1}(U_{I}+\varphi_{I}), V(\varphi)=v_{1}+\sum_{I\geq 2} (v_{I}+\psi_{I}) .

By substituting the pair of formal solutions (5.5) into the system (5.1), then we havethe following recurrence formulas for u_{I},v_{I}and \varphi_{I}, \psi_{I} :\{ begin{ar ay}{l} qv_{1}(x)=b(x) (q-\lambda(x) u_{1}(x)=a(x)+v_{1}(x) \end{ar ay}

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and for I\geq 2\{ begin{ar y}{l q^{I}v_{I}(x)=\partial_{x}u_{I-1}(x) (q^{I}-\lambda(x)u_{I}(x)=v_{I}(x)+\sum_{I 1}+I_{2}=Iu_{I {\imath} (x) \partial_{x}u_{I 2}(x) \end{ar y}and:(\sigma_{q}-\lambda(x))\varphi_{1}(t_{:}x)=0
and for I\geq 2(5.6)\{beginary}{l \sigma_{q}\psiI(t.x)=\partil_{x}\varphi_{I-1}(t,x) \sigma_{q}-\lmbda(x)\vrphi_{I}(t,x)=\psi_{I}(t,x)+\sum_{I1}+2=I}(u_{1}(t,x)+\varphi_{I1}(t,x) \partil_{x}(uI2}(t,x)+\varphi_{I2}(t,x) \sum_{I1}+2=I}u_{1(t,x)\paril_{x}uI2(t,x).\end{ary}By assumption, it is clear that u_{I}(x) and v_{I}(x) are determined for I\in \mathbb{N}^{*} . We take any
holomorphic function \varphi(x)\in \mathcal{O}(D_{R_{0}}) and put \varphi(x)=\varphi_{0,1,0}(x) . For I\geq 2 we will showthat \varphi_{I} and \psi_{I} are determined by induction. By the above recurrence formulas, it isobvious that \psi_{I} is determined for I\geq 2 if \varphi_{I} is determined for I\geq 1 . Therefore, forverifying that it is sufficient to see that \varphi_{I} is determined for I\geq 2 . We substitute thefunction \varphi_{I} of the form (5.3) into the second equation of (5.6), then we have(q^{i+j\lambda_{q}(x)}- \lambda(x) \varphi_{i}j,k(X)+\sum^{i+j-1-k}q^{i+ j\lambda_{q}(x)}(k+k')!k!k'!(\log q)^{k^{f} \varphi_{i,j,k+k'}(x)k'=1=\psi_{i_{j},k}(x)+i_{1}\sum+i_{2}=i [j\partial_{x}\lambda_{q}(x)u_{i_{1} (x)\varphi_{i_{2},jz_{2}+j- 1}(x)k\leq i_\sum{2}+j-1\{\partial_{\mathcal{I} (u_{i_{1} (x)\varphi_{i_{2} ,j_{:}k(x) + j\partial_{x}\lambda_{q}(X)u_{i_{1} (x)\varphi_{i_{1j},k-1}(x)\}]i_{1}\geq 1,i_{2}\geq 0++ \sum \{\varphi_{i_{1j1},k_{1} (x)(\partial_{X}\varphi_{i_{2} 02,k_{2}(x)+ 12xq(x)\varphi_{i_{2} j_{2},k2-{\imath}(x) \}i_{1}+i_{2}=ij_{1}+j_{2}=j_{j1_{\dot{i}}}j_{2}\geq 1k_{1}+k_{2}=kwhere

j_{1}+j_{2^{=}j . \geq2\sum_{i_{1}+i_{2}=i},\varphi_{i_{1j_{1:} k_{1} (x) \partial_{x}\varphi_{i_{2_{:} j_{2},k_{2} (x)\equiv0.

k_{1}+k_{2}=i+j-1

The definition of \lambda_{q}(x) and this recurrence formulas tell us that the system (5.1) has\squarethe pair of formal solutions in the form (5.2).

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References[1] R. Bielawski, Ricci‐flat Kähler metrics on canonical bundles. Math. Proc. Cambridge
philos. Soc. 132 (2002)_{\backslash } no. 3: 471‐479.
[2] R. Gérard and H. Tahara, Singular Nonlinear Partial DiffeTential Equations, Vieweg, 1996.
[3] H. Yamazawa, Holomorphic and Singular Solutions of q‐Difference‐Differential Equationsof Briot‐Bouquet Type, Fuckcialaj Ekvacioj, 59 (2016), 185‐197.

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