Funkcialaj Ekvacioj, 12 (1969), 251-268

Determination of Stable Domains for BoundedSolutions of Simplified Equations

By Masahiro IWANO

(Tokyo Metropolitan University)

Dedicated to Professor Tokui Sat? on His sixty-third Birthday

Introduction

§1. Assumptions.Let there he given a system of two nonlinear ordinary differential equations

of the form

(A) $x^{¥sigma+1}y^{¥prime}=f(x, y, z)$ , $xz^{¥prime}=g(x, y, z)$ $(’=¥frac{d}{dx})$.

Here we assume that:i) a is a positive integer.

$¥mathrm{i}¥mathrm{i})$ $x$ is a complex independent variable.$¥mathrm{i}¥mathrm{i}¥mathrm{i})$

$y$ and $z$ are both scalars.$¥mathrm{i}¥mathrm{v})$ Both of $f(x, y, z)$ and $g(x, y, z)$ are scalar functions holomorphic in

$(x, y, z)$ for

(1. 1) $|x|¥leqq¥xi$ , $|y|¥leqq d$, $|z|¥leqq d$

and vanish at (0, 0, 0).We introduce the following assumption:Assumption I. The quantity $¥lambda$ given by

$¥lambda=f_{y}(0,0,0)¥equiv¥frac{¥partial}{¥partial y}f(x, y, z)|_{x=y=z=0}$

is different from zero.Then, as was already proved in M. Iwano [5] and as can be easily verified,

we can assume without loss of generality that:v) We have

$f_{x}(0, 0, 0)=0$ , $f_{z}(0,0, 0)=0$ , $g_{y}(0,0, 0)=0$.

We introduce a further assumption:Assumption $¥mathrm{I}¥mathrm{I}$ . The quantity $¥mu$ given by

$¥mu=g_{z}(0,0,0)=¥frac{¥partial}{¥partial z}g(x, y, z)|_{x=y=z=0}$

252 M. IWANo

has a positive real part.Under these assumptions, by the use of Hukuhara’s method [1] for formal

transformations of nonlinear ordinary differential equations (See also M. Iwano[3]$)$ , the following results are easily obtained:

Formal Solution. Equations (A) possess a formal solution of the form(F) $y¥sim¥sum_{p=0}^{¥infty}V(x)^{p}A_{p}(x)$ , $z¥sim¥sum_{p=0}^{¥infty}V(x)^{p}B_{p}(x)$ .

(1) $A_{p}(x)$ and $B_{p}(x)$ are holomorphic and bounded functions of $x$ for

(1. 2) $0<|x|<¥xi^{¥prime}$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$

and, moreover, admit asymptotic $e¥dot{x}pansi¥dot{o}ns$ in powers of $x$ as $x$ tends to the

origin through the sector $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ . Here the angles $¥underline{¥Theta}$ and $¥overline{¥Theta}$ are given by

either

(1. 3) $¥underline{¥Theta}=¥frac{1}{¥sigma}(¥arg¥lambda-¥frac{5¥pi}{2})+4¥epsilon$ , $¥overline{¥Theta}=¥frac{1}{¥sigma}(¥arg¥lambda+¥frac{¥pi}{2})-4¥epsilon$

or

(1. 4) $¥underline{¥Theta}=¥frac{1}{¥sigma}(¥arg¥lambda-¥frac{¥pi}{2})+4¥mathrm{e}$ , $¥overline{¥Theta}=¥frac{1}{¥sigma}(¥arg¥lambda+¥frac{5¥pi}{2})-4¥mathrm{e}$,

where $¥mathrm{e}$ is a sufficiently small positive constant.(2) $V(x)$ has the form, with an arbitrary constant $C$ ,

(1. 5) $V(x)=x^{¥mu}$ ($C+$blog $x$),

which is a holomorphic solution of an equation of the form(B) $xv^{¥prime}=¥mu v+bx^{¥mu}$ .

Here $b$ is a complex constant which may be zero. If $b¥neq 0$ , the quantity $¥mu$ isnecessarily a positive integer.

The equation (B) is called a simplified equation or a reduced equation.

§2. Review of Known Results.$1^{¥mathrm{o}}$ . The equations which were studied in Hukuhara [1] have a more general

form than (A) and M. Iwano [3, 4] investigated analytic meanings of the formalsolutions obtained by M. Hukuhara. If we apply directly the method of [4]

to equations (A) with formal solution (F), we can get the following result:Theorem A. The formal solution (F) is uniformly convergent whenever

the values of $x$ and $x^{¥mu}(C+b¥log x)$ satisfy inequalities of the form

(2. 1) $0<|x|<¥xi^{¥prime}$ , $¥underline{¥Theta}^{¥prime}<¥arg x<¥overline{¥Theta}^{¥prime}$ , $|x^{¥mu}(¥mathrm{C}+¥mathrm{b}¥log x)$ $|<¥delta^{¥prime}$ ,

where $¥underline{¥Theta}^{¥prime}$ and $¥overline{¥Theta}^{¥prime}$ are given by either

Stable Domains for Bounded Solutions of Simplified Equations 253

(2. 2) $¥underline{¥Theta}^{¥prime}=¥frac{1}{¥sigma}(¥arg¥lambda-¥arg¥mu)+¥epsilon^{¥prime¥prime}$, $¥overline{¥mathit{0}^{¥kappa^{¥prime}}}=¥frac{1}{¥sigma}(¥arg¥lambda-¥arg¥mu+2¥pi)-¥mathrm{e}^{¥prime¥prime}$

or

(2. 3) $¥underline{¥Theta}^{¥prime}=¥frac{1}{¥sigma}(¥arg¥lambda-¥arg¥mu-2¥pi)+¥mathrm{e}^{¥prime¥prime}$ , $¥overline{¥Theta}^{¥prime}=¥frac{1}{¥sigma}(¥arg¥lambda-¥arg¥mu)-¥epsilon^{¥prime¥prime}$

for a sufficientfy smdf positive constant $¥epsilon^{¥prime¥prime}$ .

It is natural to expect that, if $¥lambda>0$ and $¥mu>0$ , equations (A) have a generalsolution which tends to 0 as $x$ approaches the origin along the positive real axis.However, the above result does not give any information about the existence ofsuch a solution because the sectors $¥underline{¥Theta}^{¥prime}<¥arg x<¥overline{¥Theta}^{¥prime}$ with (2. 2) and (2. 3) can never

contain the positive red axis.$2^{¥mathrm{o}}$. Recently M. Iwano [5] devised a method to give analytic meanings to

formal solutions. As a special case of Theorem 1 in [5] we have the following:

Theorem B. Assume $b=0$ . Then the formal solution (F) is uniformlyconvergent for

(2. 4) $0<|x|<¥xi^{¥prime}$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ , $|x^{¥mu}C|<¥delta^{¥prime}$

and the sum is a solution of (A) which tends to 0 with the order of a certainpositive power of $x$ as $x$ approaches the origin along the positive red axis.

It is to be noted that the sectors $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ with (1. 3) and (1. 4) do

contain the positive real axis. The equations M. Iwano [5] studied have a moregeneral form than (A). Namely, in equations (A), $y$ and $z$ are resepctively$m-$ and $¥mathrm{n}$-column vectors; $f$ and $g$ are vectors; Assumption I must be replacedby the assumption that the Jacobian matrix $f_{y}(x, y,z)$ of $f(x, y, z)$ with respect

to $y$ is non-singular at $x=y=z=0$ ; after having applied suitable transformationswhich reduce all the components of the vector $f_{x}$ (0, 0, 0), the matrices $f_{z}(0,0,0)$

and $g_{y}(0,0,0)$ to 0, all the eigenvalues of the matrix $g_{z}(0,0,0)$ have positivereal parts and the equations corresponding to (B) have the form

$xv^{¥prime}=1_{n}(¥mu)v$,

where 1 $n(¥mu)$ is an n-hy-n diagonal matrix with diagonal elements $¥{¥mu_{k}¥}$ . Underfurther additional assumptions, he constructed a general solution which tends ta

0 as $x$ approaches the origin along the positive real axis. However, as he hasemphasized in [5], in the case when $b¥neq 0$ the improved method can not eitherbe applied in order to construct a bounded solution which tends to 0 with theorder of a positive power of $x$ as $x$ approaches the origin along the positivereal axis.

254 M. $¥mathrm{I}_{¥mathrm{W}¥mathrm{A}¥mathrm{N}¥mathrm{O}}$

§3. Stable Domains.It is not so difficult to give some analytic meaning to formal solutions if

we do not attempt to find its maximal domain of validity. Namely, in thisdomain the formal solutions converge uniformly or they are asymptotic expan-

sions of analytic solutions. In the investigation of analytic meanings for formalsolutions, the estimation of the growth of a general solution for simplifiedequations near the singular point $x=0$ does play an important role.

$1^{¥mathrm{o}}$. In the proof of Theorem A the following lemma played an importantrole:

Lemma A. Let $(x_{1}, v^{1})$ be an arbitrary point in a domain of the form

(3. 1) $0<|x|<¥xi^{¥prime¥prime}¥omega(¥arg x)$ , $¥underline{¥Theta}^{r}<¥arg x<¥overline{¥mathit{0}¥sim}’$, $¥max$ $¥{|x^{¥mu}|, |v|¥}<¥delta^{¥prime¥prime}$ ,

where $¥omega(¥varphi)$ is a strictly positive valued and continuous function of $¥varphi$ for $¥underline{¥Theta}^{¥prime}¥leqq¥varphi¥leqq¥overline{¥Theta}^{¥prime}$.

Then there is a curve $¥Gamma_{x_{1}}$ , joining the origin and the point $x_{1}$ , such that:i) The curve $¥Gamma_{x_{1}}$ is entirely contained in the domain

$0<|x|<¥xi^{¥prime¥prime}¥omega(¥arg x)$ , $¥underline{¥Theta}^{¥prime}<¥arg x<¥overline{¥Theta}^{l}$

except for the origin.$¥mathrm{i}¥mathrm{i})$ As $x$ moves on the curve $¥Gamma_{x_{1}}$ , we have

(3. 2) $¥frac{d}{ds}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥geqq|¥lambda||x|^{-¥sigma-1}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥sin$$¥sigma ¥mathrm{e}$

with

$¥Lambda(x)¥equiv-¥frac{¥lambda}{¥sigma x^{¥sigma}}$ .

Here $s$ denotes the arc length of the curve $¥Gamma_{x_{1}}$ measured from the origin to thevariable point $x$ .

$¥mathrm{i}¥mathrm{i}¥mathrm{i})$ As $x$ moves on the same curve, we have

(3. 3) $¥frac{d}{ds}|x^{¥mu}|¥geqq|¥mu||x|^{-1}|x^{¥mu}|¥sin¥epsilon$.

Let two bounded domains $U$ and $V$ in the product space of the complex$x-$ and $¥mathrm{v}$-planes be given. We assume their boundaries depend continuously on tworeal positive parameters, say $U=U(¥xi, ¥delta)$ and $V=V(¥xi^{¥prime}, ¥delta^{¥prime})$ . Then we shall saythat $U(¥xi, ¥delta)$ and $V(¥xi^{¥prime}, ¥delta^{r})$ are equivalent if the following conditions are satisfied:

i) For any $V(¥xi^{¥prime}, ¥delta^{¥prime})$ there are suitable $¥xi$ , $¥delta$ such that $U(¥xi, ¥delta)¥subset V(¥xi^{¥prime}, ¥delta^{¥prime})$ .$¥mathrm{i}¥mathrm{i})$ Conversely, for any $U(¥xi, ¥delta)$ there are suitable $¥xi^{¥prime}$ , $¥delta^{¥prime}$ such that $V(¥xi^{¥prime}, ¥delta^{¥prime})$

$¥subset U(¥xi, ¥delta)$ .For example, a domain of the form (3.1) is equivalent to a domain of the

form

Stable Domains for Bounded Solutions of Simplified Equations 255

$0<|x|<¥xi^{J}$ , $¥underline{¥Theta}^{¥prime}<¥arg x<¥overline{¥Theta}^{¥prime}$ , $|y|<¥delta^{¥prime}$ .

The following facts are to be noticed: Since $x=0$ is an irregular type

singular point of equations (A), an inequality of the form (3.2) is essential inthe proof of the existence of asymptotic solutions for (A). Inequality (3.3)implies that the function $¥max$ $¥{|x^{¥mu}|, |V(x)|¥}$ is a monotone increasing functionof $s$ :

(3. 4) $¥frac{d}{ds}¥max$ $¥{|x^{¥mu}|, |V(x)|¥}¥geqq¥frac{1}{2}|¥mu|¥sin¥epsilon|x|^{-1}¥max$ $¥{|x^{¥mu}|, |V(x)|¥}$ .

$2^{¥mathrm{o}}$. In the proof of Theorem $¥mathrm{B}$ , we used the following lemma:Lemma B. Choose $(x_{1}, v^{1})$ in an arbitrary way from a domain of the form

(3. 5) $0<|x|<¥xi^{¥prime¥prime}¥omega(¥arg x)$ , $|v|<¥delta^{¥prime¥prime¥chi}(¥arg x)$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ ,

where $¥omega(¥varphi)$ and $¥chi(¥varphi)$ are strictly positive valued and continuous functions $¥varphi$ in$¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta}$.

Then there is a curve $¥Gamma_{x_{1}}$ such that:i) The curve $¥Gamma_{x_{1}}$ is contained in the domain

$0<|x|<¥xi^{¥prime¥prime}¥omega(¥arg x)$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$

except for the origin.$¥mathrm{i}¥mathrm{i})$ As $x$ is on the curve $¥Gamma_{x_{1}}$ , we have an inequdity of the form

(3. 2) $¥frac{d}{ds}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥geqq|¥lambda||x|^{-¥sigma-1}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥sin$$¥sigma¥epsilon$

with

$¥Lambda(x)¥equiv-¥frac{¥lambda}{¥sigma x^{¥sigma}}$ .

$¥mathrm{i}¥mathrm{i}¥mathrm{i})$ As $x$ moves $ori$ $¥Gamma_{x_{1}}$ , the value of the function $V(x)¥equiv x^{¥mu}C$ always stays

in the domain

$|v|<¥delta^{¥prime¥prime¥chi}(¥arg x)$ , $¥underline{}<¥arg x<¥overline{¥Theta}$.

It is to be noticed that: A domain of the form (3.5) is equivalent to $a$

domain of the form$0<|x|<¥xi^{¥prime}$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ , $|v|<¥delta^{¥prime}$ .

The function $¥max$ $¥{|x^{¥mu}|, |V(x)|¥}$ is equal to a constant multiple of $|V(x)|$ andis no longer an increasing function of $s$ . We have only, instead of (3.4), aninequality of the form

256 M. IWANO

$|¥frac{d}{ds}¥max$ $¥{|x^{¥mu}|, |V(x)|¥}|¥leqq|¥mu||x|^{-1}¥max$ $¥{|x^{¥mu}|, |V(x)|¥}$ .

The sector $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ is a maximal one such that we can determine acurve $¥Gamma_{x_{1}}$ along which an inequality of the form (3.2) is satisfied. A domainof the form (3.5) will be called a stable domain of the simplified equation (B)(with $b=0$) with respect to the monomial $¥Lambda(x)$ . “ Stable” means that the valuesof a general solution for (B) always remain in a small neighborhood of theorigin of the complex $¥mathrm{v}$-plane as $x$ moves on $¥Gamma_{x_{1}}$ .

§4. Main Result.The aim of this paper is to prove the following theorem:Main Theorem. The formd solution (F) is uniformly convergent whenever

the vdues of $x$ and $V(x)¥equiv x^{¥mu}(C+b¥log x)$ bdong to a domain of the form(4. 1) $0<|x|<¥xi_{1}$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$, $|v|<¥delta_{1}$ .

Here the angles $¥underline{¥Theta}$ and $¥overline{¥Theta}$ are the same as those appearing in (1.3). or (1.4).In the case when $b=0$, Theorem $¥mathrm{B}$ proves already Main Theorem. So, we

assume hereafter that the quantity $b$ is not zero and, consequently, the quantity$¥mu$ is equal to a positive integer.

The proof of Main Theorem is based on the lemma below which guaranteesthe existence of a stable domain of the simplified equation (B) with respect tothe monomial $¥Lambda(x)$ :

Fundamental Lemma. Let $f(¥varphi)$ and $¥omega(¥varphi)$ be functions which are strictly

positive valued, bounded and continuous in $¥varphi$ for $¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta}$. Consider a sub-domain, in the product space of the complex $x-$ and $v$-planes, whose points $(x, v)$

satisfy inequaIities of the form

(4. 2) $¥max$ $¥{|x|^{¥mu}, |v|¥}<¥delta^{¥prime¥prime}f(¥arg x)¥omega(¥arg x)^{¥mu}$ , $0<|x|$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ .

Let $(x_{1}, v^{1})$ be an arbitrary point belonging to domain (4.2) and determine thevalue of the arbitrary constant $C$ so that $V(x_{1})=v^{1}$ .

Then we can determine the functions $f(¥varphi)$ and $¥omega(¥varphi)$ , with the above specifiedproperties, in such a way that

i) There exists a curve $¥Gamma_{x_{1}}$ , joining the origin and the point $x_{1}$ , which isentirely contained in domain (4.2) except for the origin.

$¥mathrm{i}¥mathrm{i})$ As $x$ moves on the curve $¥Gamma_{x_{1}}$ , an inequality of the form

(4. 3) $¥frac{d}{ds}e^{-¥mathrm{R}¥mathrm{e}A(x)}¥geqq|¥lambda||x|^{-¥sigma-1}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥sin$ $ 2¥sigma¥epsilon$

is satisfied. Here $s$ denotes the arc length of the curve $¥Gamma_{x_{1}}$ measured from the

Stable Domains for Bounded Solutions of Simplified Equations 257

origin to the variable point $x$ .$¥mathrm{i}¥mathrm{i}¥mathrm{i})$ The values of the function $V(x)$ always remain in domain (4.2) as $x$

is on the curve $¥Gamma_{x_{1}}$ .$¥mathrm{i}¥mathrm{v})$ As $x$ is on the curve $¥Gamma_{x_{1}}$ we have an inequality of the form

(4. 4) $|¥frac{d}{ds}¥max$ $¥{|x|^{¥mu}, |V(x)|¥}|¥leqq¥frac{3¥mu}{2}|x|^{-1}¥max$ $¥{|x|^{¥mu}, |V(x)|¥}$

and, as $x$ moves on the segment joining the origin and the point $x_{1}$ , we have aninequality of the form

(4. 5) $¥frac{d}{d|x|}¥max$ $¥{|x|^{¥mu}, |V(x)|¥}¥geqq¥frac{¥mu}{2}|x|^{-1}¥max$ $¥{|x|^{¥mu}, |V(x)|¥}$ .

The functions $f(¥varphi)$ and $¥omega(¥varphi)$ will be determined in Section 5. It is clearthat a domain of the form (4.2) is equivalent to a domain of the form (4.1).The inequality (4.5) shows that the function $¥max$ $¥{|x|^{¥mu}, |V(x)|¥}$ is monotonouslyincreasing with respect to $|x|$ as $x$ moves away from the origin along a straightline.

I. Proof of Fundamental Lemma.

§5. Functions $¥omega(¥varphi)$ and $f(¥varphi)$ .We define $¥theta_{+}$ and $¥theta_{¥_}$ by

(5. 1) $¥theta_{+}=¥frac{1}{¥sigma}(¥arg(-¥lambda)-¥frac{¥pi}{2})$ , $¥theta_{-}=¥frac{1}{¥sigma}(¥arg(-¥lambda)+¥frac{¥pi}{2})$ .

Then the angles $¥underline{¥Theta}$ and $¥overline{¥Theta}$ are expressed as

(5. 2) $¥underline{¥Theta}=¥theta_{+}-¥frac{¥pi}{¥sigma}+4¥epsilon,¥overline{¥Theta}=¥theta_{-}+¥frac{¥pi}{¥sigma}-4¥epsilon$

which coincide with (1.3) or (1.4) according as we take $¥arg$ $(-¥lambda)=¥arg¥lambda-¥pi$ or$¥arg$ $(-¥lambda)=¥arg¥lambda+¥pi$ . Define the function $a(¥varphi)$ by

(5. 3) $a(¥varphi)=¥left¥{¥begin{array}{l}¥sigma(¥varphi-¥theta_{-}+2¥epsilon),¥theta_{-}+¥frac{¥pi}{2¥sigma}-2¥epsilon¥leqq¥varphi¥leqq¥overline{¥Theta},¥¥¥frac{¥pi}{2},¥theta_{+}-¥frac{¥pi}{2¥sigma}+2¥epsilon¥leqq¥varphi¥leqq¥theta_{-}+¥frac{¥pi}{2¥sigma}-2¥epsilon,¥¥¥sigma(¥varphi-¥theta_{+}-2¥epsilon)+¥pi,¥underline{¥Theta}¥leqq¥varphi¥leqq¥theta_{+}-¥frac{¥pi}{2¥sigma}+2¥epsilon.¥end{array}¥right.$

It is immediately seen that $a(¥varphi)$ is a continuous function of $¥varphi$ in $¥underline{¥mathit{0}^{n}}¥leqq¥varphi¥leqq¥overline{¥Theta}$ andsatisfies there the inequality

(5. 4) $ 2¥sigma ¥mathrm{e}¥leqq a(¥varphi)¥leqq¥pi-2¥sigma¥epsilon$.

258 M. IWANO

The function $¥omega(¥varphi)$ is to be defined as

(5. 5) $¥omega(¥varphi)=¥exp¥int_{¥theta^{*}}^{¥varphi}¥cot a(¥tau)d¥tau$,

where $¥theta^{*}$ is a fixed angle in $¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta}$. The function $f(¥varphi)$ is to be expressed

as

(5. 6) $f(¥varphi)=¥left¥{¥begin{array}{l}f_{-}(¥varphi)¥Delta¥leqq¥varphi¥leqq¥Theta,¥¥f_{+}(¥varphi)¥Theta¥leqq¥varphi¥leqq¥Delta¥end{array}¥right.--$

with

$¥Delta=¥frac{1}{2}(¥underline{¥Theta}+¥overline{¥Theta})$ ,

where

(5. 7) $¥left¥{¥begin{array}{l}f_{-}(¥varphi)=1+2|b|[¥int_{¥varphi}^{¥theta¥theta}|¥mathrm{c}¥mathrm{o}¥mathrm{t}a(¥tau)|d¥tau+¥overline{¥Theta}-¥varphi],¥¥f_{+}(¥varphi)=1+2|b|[¥int_{¥theta}^{¥varphi}|¥mathrm{c}¥mathrm{o}¥mathrm{t}a(¥tau)|d¥tau+¥varphi_{-}¥underline{¥Theta}].¥end{array}¥right.-$

It is easily verified that the graph $(¥varphi, |¥cot a(¥varphi)|),¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta}$, has a straight

line $¥varphi=¥Delta$ as a symmetric axis. Hence we have $f_{¥_}(¥Delta)=f_{+}(¥Delta)$ . Since we have(5.4), we see that the functions $¥omega(¥varphi)$ and $f(¥varphi)$ thus defined are strictly positive

valued, bounded and continuous in $¥varphi$ for $¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta}$. From this it follows that a

domain of the form (4.2) is equivalent to a domain of the form (4.1).We see by inspection that $f_{¥_}(¥varphi)$ is a decreasing function of $¥varphi$ and $f_{+}(¥varphi)$

is an increasing function of $¥varphi$ . Hence the inequalities

(5. 8) $¥left¥{¥begin{array}{l}f_{-}(¥mathrm{a}¥mathrm{r}¥mathrm{g}x_{1})¥leqq f_{-}(¥mathrm{a}¥mathrm{r}¥mathrm{g}x)¥mathrm{i}¥mathrm{f}¥mathrm{a}¥mathrm{r}¥mathrm{g}x¥leqq ¥mathrm{a}¥mathrm{r}¥mathrm{g}x_{1},¥¥f_{+}(¥mathrm{a}¥mathrm{r}¥mathrm{g}x_{1})¥leqq f_{+}(¥mathrm{a}¥mathrm{r}¥mathrm{g}x)¥mathrm{i}¥mathrm{f}¥mathrm{a}¥mathrm{r}¥mathrm{g}x_{1}¥leqq ¥mathrm{a}¥mathrm{r}¥mathrm{g}x¥end{array}¥right.$

hold. Moreover we can assume without loss of generality that

(5. 9) $1¥leqq f(¥varphi)¥leqq 2$ $(¥underline{¥Theta}¥leqq¥varphi¥leqq¥overline{¥Theta})$ ,

(5. 10) $ 2|b|¥leqq¥mu$ .

Indeed, both of these inequalities will be satisfied only if the quantity $|b|$ issufficiently small. However, we can give to $|b|$ a value as small as we want by

applying, if it is necessary, a $¥mathrm{l}¥mathrm{i}¥mathrm{n}¥mathrm{e}¥mathrm{a}¥dot{¥mathrm{r}}$ transformation with constant coefficients.

§6. Determination of the Curve $¥Gamma_{x_{1}}$ .Denote by $(¥rho, ¥varphi)$ the polar coordinate of the variable point $x$ on $¥nabla x_{1}$ .

If $¥theta_{¥_}-2¥mathrm{e}<¥arg x_{1}<¥overline{¥Theta}$ or $¥underline{¥Theta}<¥arg x_{1}<¥theta_{+}+2¥epsilon$, the curve $¥Gamma_{x_{1}}$ consists of a cur-

Stable Domains for Bounded Solutions of Simplified Equations 259

vilinear part 1’:

(6. 1) $¥rho=|x_{1}|¥exp¥int_{¥arg x_{1}}^{¥varphi}¥cot a(¥tau)d¥tau$

for

$¥theta_{¥_}-2¥epsilon¥leqq¥varphi¥leqq¥arg x_{1}$ or $¥arg x_{1}¥leqq¥varphi¥leqq¥theta_{+}+2¥epsilon$

and of a rectilinear part $¥Gamma^{¥prime¥prime}$ :

(6. 2) $ 0¥leqq¥rho¥leqq|x_{1}|¥exp¥int_{¥arg x_{1}}^{¥varphi}¥cot a(¥tau)d¥tau$ , $¥varphi=¥theta_{¥_}-2¥mathrm{e}$ or $¥theta_{+}+2¥epsilon$ .

If $¥theta_{+}+2¥epsilon¥leqq¥arg x_{1}¥leqq¥theta_{¥_}-2¥mathrm{e}$, the curve $¥Gamma_{x_{1}}$ consists of a rectilinear part $¥Gamma^{¥prime¥prime}$

only:

(6. 3) $0¥leqq¥rho¥leqq|x_{1}|$ , $¥varphi=¥arg x_{1}$ .

Let $s$ be the arc length of the curve $¥Gamma_{x_{1}}$ measured from the origin to thevariable point $x$ . Since

$x=¥rho e^{i¥varphi}$, $i=¥sqrt{-1}$ and $ds=¥sqrt{(d¥rho)^{2}+(¥rho d¥varphi)^{2}}$,

an elementary calculation shows that, if $x$ is on the curvilinear part $¥Gamma^{r}$ , we have

$ ds=+¥frac{¥rho}{¥sin a(¥varphi)}d¥varphi$ or $ ds=-¥frac{¥rho}{¥sin a(¥varphi)}d¥varphi$

and hence,

(6. 4) $¥frac{dx}{ds}=+e^{i(a(¥varphi)+¥varphi)}$ or $¥frac{dx}{ds}=-e^{i(a(¥varphi)+¥varphi)}$

according as

$¥theta_{¥_}-2¥epsilon<¥varphi¥leqq¥arg x_{1}$ or $¥arg x_{1}¥leqq¥varphi<¥theta_{+}+2¥mathrm{e}$.

Since $¥Lambda(x)=-¥lambda/¥sigma x^{¥sigma}$ , it is easily found to be that

$-_{S}^{¥frac{d}{d}}¥Lambda(x)=¥mp¥lambda x^{-¥sigma-1}e^{i(a(¥varphi)+¥varphi)}$

and

(6. 5) $¥frac{d}{ds}(-¥mathrm{R}¥mathrm{e}¥Lambda(x))=¥mp|¥lambda|¥rho^{-¥sigma-1}¥cos(a(¥varphi)-¥sigma¥varphi+¥arg¥lambda)$

accordi.ng as $¥theta_{¥_}-2¥epsilon<¥varphi¥leqq¥arg x_{1}$ or $¥arg x_{1}¥leqq¥varphi<¥theta_{+}+2¥epsilon$ .

$¥backslash 7$ . Proof of Fundamental Lemma (Part I).Assertion (i). By definition, if $x¥in¥Gamma^{¥prime}$ , we have

260 M. IWANO

(7. 1) $x=x_{1}¥exp¥{¥int_{¥arg x_{1}}^{¥varphi}¥cot a(¥tau)d¥tau+i(¥varphi-¥arg x_{1})¥}$ ,

which immediately implies that

$|x|^{¥mu}=|x_{1}|^{¥mu}¥exp¥{¥mu¥int_{¥arg x_{¥mathrm{I}}}^{¥varphi}¥cot a(¥tau)d¥tau¥}$ .

On the other direction, since $x_{1}$ belongs to domain (4.2), $x_{1}$ satisfies

$0<|x_{1}|^{¥mu}<¥delta^{¥prime¥prime}f(¥arg x_{1})¥omega(¥arg x_{1})^{¥mu}$ .

By the definition of $¥omega(¥varphi)$ , we have at once$|x|^{¥mu}<¥delta^{¥prime¥prime}f(¥arg x_{1})¥omega(¥arg x)^{¥mu}$.

By virtue of (5.8), it is concluded that

(7. 2) $|x|^{¥mu}<¥delta^{¥prime¥prime}f(¥arg x)¥omega(¥arg x)^{¥mu}$, $x¥in¥Gamma^{¥prime}$ .

If $x¥in¥Gamma^{¥prime¥prime}$ , then $¥arg x$ is constant. Hence the inequality (7.2) continues hold-ing for $x¥in¥Gamma^{¥prime¥prime}$ if it holds at the point from which the rectilinear part starts.

This proves Assertion (i).Assertion (ii) $¥sim$ If $x¥in¥Gamma^{¥prime}$ , formula (6.5) gives

(7. 3) $¥frac{d}{ds}e^{-¥mathrm{R}¥mathrm{e}A(x)}=¥mp|¥lambda|¥rho^{-¥sigma-1}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥cos(a(¥varphi)-¥sigma¥varphi+¥arg¥lambda)$

according as $¥theta_{¥_}-2¥epsilon<¥varphi¥leqq¥arg x_{1}$ or $¥arg x_{1}¥leqq¥varphi<¥theta_{+}+2¥epsilon$ .To simplify the description we choose the argument of ?1 so that, for

example, $¥arg$ $(-¥lambda)=¥arg¥lambda+¥pi$. Then, by using (5.3), a simple calculation showsthat

$-¥frac{3¥pi}{2}+2¥sigma¥epsilon¥leqq a(¥varphi)-¥sigma¥varphi+¥arg¥lambda¥leqq-¥pi+2¥sigma¥epsilon$ , $¥theta_{¥_}-2¥epsilon¥leqq¥varphi¥leqq¥overline{¥Theta}$

and

$-2¥sigma¥epsilon¥leqq a(¥varphi)-¥sigma¥varphi+¥arg¥lambda¥leqq¥frac{¥pi}{2}-2¥sigma¥epsilon,¥underline{¥Theta}¥leqq¥varphi¥leqq¥theta_{+}+2¥epsilon$ .

Hence from (7.3) we can derive the inequality (4.3) for $x¥in¥Gamma^{¥prime}$ .

If $x¥in¥Gamma^{¥prime¥prime}$ , we have $s=¥rho=|x|$ and $¥arg x$ is constant and satisfies$¥theta_{+}+2¥epsilon¥leqq¥arg x¥leqq¥theta_{-}-2¥epsilon$.

An easy consideration shows that

$|¥arg¥lambda-¥sigma¥varphi+¥pi|¥leqq¥frac{¥pi}{2}-2¥sigma¥epsilon$, $¥theta_{+}+2¥epsilon¥leqq¥varphi¥leqq¥theta_{-}-2¥epsilon$.

Since

Stable Dornains for Bounded Solutions of Simplified Equations 261

$-¥mathrm{R}¥mathrm{e}¥Lambda(x)=|¥lambda|(¥sigma¥rho^{¥sigma})^{-1}¥cos(¥arg¥lambda-¥sigma¥varphi)$ ,

we have therefore

$¥frac{d}{ds}(-¥mathrm{R}¥mathrm{e}¥Lambda(x))=-|¥lambda|¥rho^{-¥sigma-1}¥cos(¥arg¥lambda-¥sigma¥varphi)¥geqq|¥lambda|¥rho^{-¥sigma-1}$ sin2 $¥sigma ¥mathrm{e}$ ,

which proves the inequality (4.3) for $x¥in¥Gamma^{¥prime¥prime}$ .

Thus Assertion (ii) has been proved.

§8. Proof of Fundamental Lemma (Part II).Assertion (iii). A direct calculation yields

$V(x)=(¥frac{x}{x_{1}})^{¥mu}v^{1}+bx^{¥mu}¥log¥frac{x}{x_{1}}$ .

Hence

$|V(x)|¥leqq|¥frac{x}{x_{1}}|^{¥mu}|v^{1}|+|b||x|^{¥mu}|¥log¥frac{x}{x_{1}}|$ .

If $x¥in¥Gamma^{¥prime}$ , we have by (7.1)

$|V(x)|¥leqq¥exp¥{¥mu¥int_{¥arg x_{1}}^{¥varphi}¥cot a(¥tau)d¥tau¥}¥times$

$¥chi[|v^{1}|+|b||x_{1}|^{¥mu}|¥int_{¥arg x_{1}}^{¥varphi}¥cot a(¥tau)d¥tau+i(¥varphi-¥arg x_{1})|]$.

However, since $(x_{1}, v^{1})$ is in domain (4.2), $|x_{1}|^{¥mu}$ and $|v^{1}|$ satisfy the same ine-quality:

$|x_{1}|^{¥mu}$ , $|v^{1}|<¥delta^{¥prime¥prime}f(¥arg x_{1})¥exp¥{¥mu¥int_{¥theta^{*}}^{¥arg x_{1}}¥cot a(¥tau)d¥tau¥}$ .

From this it follows by the definition of $¥omega(¥varphi)$ that

$|V(x)|<¥delta^{¥prime¥prime}¥omega(¥varphi)^{¥mu}[f(¥arg x_{1})+|b|f(¥arg x_{1})¥{|¥int_{¥arg x_{1}}^{¥varphi}|¥cot a(¥tau)|d¥tau|+|¥varphi-¥arg x_{1}|¥}]$ .

By using (5.8) and (5.9) we see that the expression appearing in the bracketdoes not exceed the expression

$f(¥arg x_{1})+2|b|¥{|¥int_{¥arg x_{1}}^{¥varphi}|¥cot a(¥tau)|d¥tau|+|¥varphi-¥arg x_{1}|¥}$

where $¥varphi$ varies in the interval $¥theta_{¥_}-2¥epsilon¥leqq¥varphi¥leqq¥arg x_{1}$ or $¥arg x_{1}¥leqq¥varphi¥leqq¥theta_{+}+2¥epsilon$ . Henceby using (5.7) with $¥varphi=¥arg x_{1}$ it is found to be that the above expression isequal to $f(¥varphi)$ . Thus we have

$|V(x)|<¥delta^{¥prime¥prime}f(¥arg x)¥omega(¥arg x)^{¥mu}$ for $x¥in¥Gamma^{¥prime}$ ,

which proves that the values of $V(x)$ always remain in domain (4.2) for $x¥in¥Gamma^{¥prime}$ .

262 M. IWANO

On the other hand, it is easily seen that: the assertion that $V(x)$ stays indomain (4.2) for $x¥in¥Gamma^{¥prime¥prime}$ is an immediate consequence of inequality (4.5) whichwill be proved later.

Thus Assertion (iii) has been proved.Assertion (iv). Let $L_{x_{1}}$ be a curve joining the origin and the point $x_{1}$

and $t$ be its length measured from the origin to the variable point $x$ . Then itis easy to verify that

(8. 1) $¥frac{d|x|^{¥mu}}{dt}=¥mu|x|^{¥mu-1}¥frac{d|x|}{dt}=¥mu|x|^{¥mu}¥mathrm{R}¥mathrm{e}(x^{-1}¥frac{dx}{dt})$

and

(8. 2) $|V(x)|^{-1}¥frac{d|V(x)|}{dt}=¥mathrm{R}¥mathrm{e}(V^{-1}¥frac{dV}{dt})=¥mu ¥mathrm{R}¥mathrm{e}¥{(1+¥frac{b}{¥mu}¥frac{x^{¥mu}}{V(x)})x^{-1}¥frac{dx}{dt}¥}$.

Proof of Inequality (4. 4). Let $x^{*}$ be an arbitrary point on the curve $¥Gamma_{x_{1}}$ .

Then we have either

(8. 3) $¥max$ $¥{|x|^{¥mu}, |V(x)|¥}=|x|^{¥mu}$ at $x=x^{*}$

or

(8. 4) $¥max$ $¥{|x|^{¥mu}, |V(x)|¥}=|V(x)|$ at $x=x^{*}$ .

Observe that relations (6.4) imply that

$|¥frac{dx}{ds}|=1$ for $x¥in¥Gamma_{x_{1}}$ .

Then it follows from equations (8.1) with $t=s$ that we have inequalities of theform

(8. 5) $|¥frac{d|x|^{¥mu}}{ds}|¥leqq¥mu|x|^{¥mu-1}=¥mu|x|^{-1}¥max$ $¥{|x|^{¥mu}, |V(x)|¥}$

at a point $x^{*}$ such that (8.3) holds.On the other hand, at a point $x^{*}$ for which (8.4) is satisfied, we have by

(5.10)

(8. 6) $¥frac{1}{2}¥leqq ¥mathrm{R}¥mathrm{e}(1+¥frac{b}{¥mu}¥frac{x^{¥mu}}{V(x)})¥leqq¥frac{3}{2}$ .

Then from (8.2) with $t=s$ we can derive inequalities

(8. 7) $|¥frac{d|V(x)|}{ds}|¥leqq¥frac{3¥mu}{2}|x|^{-1}|V(x)|=¥frac{3¥mu}{2}|x|^{-1}¥max¥{|x|^{¥mu}, |V(x)|¥}$

for a point $x=x^{*}$ at which (8.4) holds.From (8.5) and (8.7) it is easily seen that the inequality (4.4) is satisfied

Stable Domains for Bounded Solutions of Simplified Equations 263

for $x$ on the curve $¥Gamma_{x_{1}}$ .

Proof of Inequality (4. 5). Let $x^{*}$ be an arbitrary point on the segmentjoining the origin and the point $x_{1}$ . In this case we have

$t=|x|$ , $x^{-1}¥frac{dx}{dt}=|x|^{-1}$ .

If equation (8.3) holds at $x=x^{*}$ on the segment, we have, by using equalities(8.1) with $t=|x|$ , the relations

(8. 8) $¥frac{d|x|^{¥mu}}{d|x|}=¥mu|x|^{¥mu-1}=¥mu|x|^{-1}¥max$ $¥{|x|^{¥mu}, |V(¥dot{x})|¥}$ .

If we have equality (8.4) at $x=x^{*}$ , inequalities (8.6) are satisfied at $x=x^{*}$ .Hence, from equations (8.2) with $t=|x|$ it follows that

(8. 9) $¥frac{d|V(x)|}{d|x|}¥geqq¥frac{¥mu}{2}|x|^{-1}|V(x)|=¥frac{¥mu}{2}|x|^{-1}¥max¥{|x|^{¥mu}, |V(x)|¥}$

at a point $x=x^{*}$ such that we have (8. 4).We see by virtue of (8.8) and (8.9) that inequality (4.5) is satisfied on a

segment starting from the origin. This proves Assertion (iv).

§9. Fundamental $¥bm{¥mathrm{I}}¥bm{¥mathrm{n}}¥bm{¥mathrm{e}}¥bm{¥mathrm{q}}¥bm{¥mathrm{u}}¥bm{¥mathrm{a}}¥bm{1}¥bm{¥mathrm{i}}¥bm{¥mathrm{t}}¥bm{¥mathrm{i}}¥bm{¥mathrm{e}}¥bm{¥mathrm{s}}_{¥wedge}$

By virture of (4.3) and (4.4) we have at once

$¥frac{d}{ds}[¥max ¥{|x|^{N¥mu}, |V(x)|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}]¥geqq$

$¥geqq|x|^{-¥sigma-1}¥max$ $¥{|X|^{N¥mu}, |V(X)|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)(|¥lambda|}$ $¥sin 2¥sigma ¥mathrm{e}-¥frac{3N¥mu}{2}|x|^{¥sigma})$

for $x$ on the curve $¥Gamma_{x_{1}}$ . If $|x|$ satisfies the inequality

(9. 1) $3N¥mu|x|^{¥sigma}¥leqq|¥lambda|$ sin2 $¥sigma ¥mathrm{e}$ ,

then we have

(9. 2) $¥frac{|¥lambda|¥sin 2¥sigma¥epsilon}{2}|x|^{-¥sigma-1}¥max¥{|x|^{N¥mu}, |V(x)|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥leqq$

$¥leqq¥frac{d}{ds}[¥max ¥{|x|^{N¥mu}, |V(x)|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}]$ for $x¥in¥Gamma_{x_{1}}$ .

Inequality (4.5) implies that we have the inequality

(9. 3) $¥frac{N¥mu}{2}|x|^{-1}¥max¥{|x|^{N¥mu}, |V(x)|^{N}¥}¥leqq¥frac{d}{d|x|}¥max¥{|x|^{N¥mu}, |V(x)|^{N}¥}$

for $x$ on a segment starting from the origin.

264 M. $¥mathrm{I}_{¥mathrm{W}¥mathrm{A}¥mathrm{N}¥mathrm{O}}$

II. Proof of Main Theorem.

§10. Preliminary Transformation.We put

(10. 1) $P_{N}(x, v)=¥sum_{p=0}^{N-1}v^{p}A_{p}(x)$ , $Q_{N}(x, v)=¥sum_{p=0}^{N-1}v^{p}B_{p}(x)$ ,

so that $¥{P_{N}(x, V(x)), Q_{N}(x, V(x))¥}$ is the sum of the first $¥mathrm{N}$-terms of theformal solution (F).

If we apply the change of variables

(10. 2) $y=¥eta+P_{N}(x, V(x))$ , $z=¥zeta+Q_{N}(x, V(x))$

to equations (A), the transformed equations can be written as

$(10. 3)_{N}$ $x^{¥sigma+1}¥eta^{¥prime}=¥lambda¥eta+F(x, V(x), ¥eta, ¥zeta)$ , $x¥zeta^{¥prime}=G(x, V(x),¥eta, ¥zeta)$ .

Then it is easy to verify that there are positive constants $¥xi^{¥prime}N$ , $¥delta^{¥prime}N$ and $d^{¥prime}N$ suchthat $F(x, v, ¥eta, ¥zeta)$ and $G(x, v, ¥eta,¥zeta)$ are holomorphic and bounded functions of$(x, v, ¥eta, ¥zeta)$ for

$(10. 4)_{N}$ $0<|x|<¥xi^{¥prime}N$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ , $|v|<¥delta^{¥prime}N$ , $|¥eta|<d^{¥prime}N$ , $|¥zeta|<d^{¥prime}N$ .

Since equations $(10.3)_{N}$ admit a formal solution of the form

$¥zeta¥sim¥sum_{p=N}^{¥infty}V(x)^{p}A_{p}(x)$ , $¥zeta¥sum_{¥tilde{p}=N}^{¥infty}V(x)^{p}B_{p}(x)$ ,

we see that $F$ and $G$ satisfy inequah.ties of the form

(10. 5) $¥left¥{¥begin{array}{l}|F(x,v,¥eta,¥zeta)|¥leqq A^{¥prime}(|¥eta|+|¥zeta|)+B¥max¥{|x|^{N¥mu},|v|^{N}¥},¥¥|G(x,v,¥eta,¥zeta)|¥leqq A,,(|¥eta|+|¥zeta|)+B¥max¥{|x|^{N¥mu},|v|^{N}¥}¥end{array}¥right.$

and $F$, $G$ satisfy Lipschitz’s conditions with respect to $¥eta$ and $¥zeta$ :

(10. 6) $¥left¥{¥begin{array}{l}|F(x,v,¥eta_{1},¥zeta_{1})-F(x,v,¥eta_{2},¥zeta_{2})|¥leqq A^{¥prime}(|¥eta_{1}-¥eta_{2}|+|¥zeta_{1}-¥zeta_{2}|),¥¥|G(x,v,¥eta_{1},¥zeta_{1})-G(x,v,¥eta_{2},¥zeta_{2})|¥leqq A,,(|¥eta_{1}-¥eta_{2}|+|¥zeta_{1}-¥zeta_{2}|)¥end{array}¥right.$

provided the arguments belong to domain (10.4) $N$ . Here $A^{¥prime}$ and $A^{¥prime¥prime}$ are positiveconstants independent of $N$. In particular, owing to the assumption (v) inSection 1, we can give the quantity $A^{¥prime}$ a value as small as we want if theconstants appearing in (10.4) $N$ are sufficiently small. Hence we assume that$A^{¥prime}$ satisfies the inequality

(10. 7) $8A^{¥prime}<|¥lambda|$ sin2 $¥sigma¥epsilon$ .

$B$ may depend on $N$.

Remark. $F$ and $G$ satisfy actually inequalities of the form

$|F|¥leqq A^{¥prime}(|¥eta|+|¥zeta|)+B|v|^{N}$ , $|G|¥leqq A^{¥prime¥prime}(|¥eta|+|¥zeta|)+B|v|^{N}$.

Stable Doma ins for Bounded Solutions of Simplified Equations 265

As we have already seen in the proof of Fundamental Lemma, the function$|V(x)|$ is not a monotone function of $s=|x|$ but the function $¥max$ $¥{|x|^{¥mu}, |V(x)|¥}$

is an increasing function of $s=|x|$ as $x$ moves away from the origin along astraight line. By this reason, for the proof of Main Theorem, it is convenientto use the inequalities (10.5) instead of the above inequalities.

§11. Main Problem and Proof of Main Theorem.

We make a further transformation of the form

(11. 1) $¥eta=e^{¥Lambda(x)}Y$, $¥zeta=Z$ $(¥Lambda(x)¥equiv-¥frac{¥lambda}{¥sigma x^{¥sigma}})$,

so that $(10.3)_{N}$ are reduced to the equations

$(11. 2)_{N}$ $Y^{¥prime}=x^{-¥sigma-1}e^{-¥Lambda(x)}F(x, V(x), e^{A(x)}Y, Z)$ , $Z^{¥prime}=x^{-1}G(x, V(x), e^{¥Lambda(x)}¥mathrm{Y}, Z)$ .

We shall prove that uniform convergence of the formal solution (F) resultsfrom the solution of the following problem:

Main Problem. Equations (11.2) $N$ have a solution $¥{¥Phi_{N}(x, V(x))$ , $¥Psi_{N}(x$ ,$V(x))¥}$ whenever the values of $x$ and $V(x)$ belong to a domain of the form

$(11. 3)_{N}$ $0<|x|<¥xi^{¥prime¥prime}N$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ , $|v|<¥delta^{¥prime¥prime}N$ .

Here $¥xi_{N}^{¥prime¥prime}$ and $¥delta^{¥prime¥prime}N$ are suitably chosen positive constants and $¥Phi_{N}(x, v)$ and $¥Psi_{N}(x, v)$

are scalar functions holomorphic and bounded in $(x, v)$ for $(11.3)_{N}$ .Moreover a solution of equations (11.2) $N$ satisfying the conditions

$(11. 4)_{N}$ $Y=O(e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥max ¥{|x|^{N¥mu}, |V(x)]^{N}¥})$ , $Z=O(¥max ¥{|x|^{N¥mu}, |V(x)|^{N}¥})$

is unique.Assume Main Problem has been solved. The expressions $¥{¥varphi_{N}(x, V(x))$ , $Q_{N}$

$(x, V(x))¥}$ , where$¥varphi_{N}(x, v)=P_{N}(x, v)+e^{¥Lambda(x)}¥Phi_{N}(x, v)$ , $Q_{N}(x, v)=Q_{N}(x, v)+¥Psi_{N}(x, v)$ ,

are a solution of equations (A) provided the values of $(x, V(x))$ belong todomain (11.3) $N$ . We can assert that this solution is independent of $N$. Indeed,let $N^{¥prime}>N$ be any integer. We observe that the expressions

$e^{-¥Lambda(x)}(¥varphi¥wedge’’(x, V(x))-P_{N}(x, V(x)))$ , $QN^{r}(x, V(x))-Q_{N}(x, V(x))$

are a solution of equations (11.2) $N$ and satisfy conditions (11.4) $N$ if $(x, V(x))$

belongs to the common part of domains $(11.3)N^{J}$ and (11.1) $N$ . However, sincesuch a solution is unique, this solution must coincide with the solution $¥{¥Phi_{¥mathrm{A}^{¥Gamma}}$

$(x, V(x))$ , $¥Psi_{N}(x, V(x))¥}$ , which implies that $¥mathit{9}N^{l}(x, V(x))¥equiv¥varphi_{N}(x, V(x))$ and$QN^{¥prime}(x, V(x))¥equiv Q_{N}(x, V(x))$ for $(x, V(x))$ in the common part mentioned above.This proves our assertion.

266 M. $¥mathrm{I}_{¥mathrm{W}¥mathrm{A}¥mathrm{N}¥mathrm{O}}$

We denote by $¥{¥varphi(x, V(x)), Q(x, V(x))¥}$ the solution thus $¥mathrm{o}¥mathrm{b}¥mathrm{t}¥mathrm{a}¥mathrm{i}¥mathrm{n}¥dot{¥mathrm{e}}¥mathrm{d}$. Then,by analytic continuation, the functions $ff(x, v)$ and $Q(x, v)$ are defined in adomain $¥mathrm{o}¥dot{¥mathrm{f}}$ the form (4.1). Clearly the functions $Q^{)}(x, V(x))$ and $Q(x, V(x))$

admit the asymptotic expansions (F) as $|V(x)|$ tends to 0. On the other hand,$v=0$ is an interior point of domain (4.1) in which the functions $E^{)}(x, v)$ and$Q(x, v)$ are defined. Hence, by virtue of Cauchy’s theorem, $ff(x, V(x))$ and$Q(x, V(x))$ admit uniformly convergent expansions in powers of $V(x)$ . By theuniqueness of asymptotic expansions, these convergent expansions must coincidewith the asymptotic expansions (F). This proves that the formal solution (F)

is uniformly convergent as the values of $x$ and $V(x)$ stay in domain (4.1).Thus Main Theorem has been proved.

§12. Solution of Main Problem.

Consider, instead of domain $(11.3)_{N}$ , a set of points $x$ and $v$ satisfying

inequalities of the form$(12. 1)_{N}$ $¥max$ $¥{|x|^{¥mu}, |v|¥}<¥delta_{N}f(¥arg x)¥omega(¥arg x)^{¥mu}$, $0<|x|$ , $¥underline{¥Theta}<¥arg x<¥overline{¥Theta}$ ,

which is a subdomain in the product space of the complex $x-$ and $¥mathrm{v}$-planes.

Here $¥gamma(¥varphi)$ and $¥omega(¥varphi)$ are the same functions as those that appeared in Funda-mental Lemma in Section 4.

Let $¥Phi$ be the family of pairs $¥{¥varphi(x, v), ¥psi(x, v)¥}$ , where $¥varphi(x, v)$ and $¥psi(x, v)$ areholomorphic and bounded functions of $(x,v)$ in domain $(12.1)_{N}$ and satisfy thereinequalities of the form

$(12. 2)_{N}$ $|¥varphi(x,¥mathrm{v})|¥leqq ¥mathrm{K}¥max$ $¥{|x|^{N¥mu}, |v|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}A(x)}$ , $|¥psi(x, ¥mathrm{v})|¥leqq ¥mathrm{K}¥max$ $¥{|x|^{N¥mu}, |v|^{N}¥}$ .

Here $¥mathrm{K}$ is a positive constant and satisfies

(12. 3) $¥mathrm{K}(¥delta_{N}f(¥theta)¥omega(¥theta)^{¥mu})^{N}<d^{¥prime}N$ for $¥underline{¥Theta}¥leqq¥theta¥leqq¥overline{¥Theta}$ .

Let $(x_{1}, v^{1})$ be an arbitrary point in domain $(12.1)_{N}$ and determine thevalue of the arbitrary constant $C$ so that $V(x_{1})=v^{1}$ . We define the functions$¥Phi(x, v)$ and $¥Psi(x, v)$ by the integrals

(12. 4) $¥Phi(x_{1}, v^{1})=¥int_{0}^{x_{1}}ff(x, V(x))dx$ , $¥Psi(x_{1}, v^{1})=¥int_{0}^{x_{1}}ffi(x, V(x))dx$ ,

(12. 5) $¥left¥{¥begin{array}{l}¥ovalbox{¥tt¥small REJECT}(x,v)=x^{-¥sigma-1}e^{-¥Lambda(x)}F(x,v,e^{¥Lambda(x)}¥varphi(x,v),¥psi(x,v)),¥¥ffl(x,v)=x^{-1}G(x,v,e^{¥Lambda(x)}¥varphi(x,v),¥psi(x,v)).¥end{array}¥right.$

The integration of the first formula of (12.4) is along the curve $¥Gamma_{x_{1}}$ thatappeared in Section 6 and the second one is along the segment $¥overline{¥mathrm{O}x}_{1}$ joining theorigin and the point $x_{1}$ .

Stable Domains for Bounded Solutions of $Sin¥iota plified$ Equations 267

By virtue of Fundamental Lemma in Section 4, as $x$ moves on the curve $¥Gamma_{x_{1}}$

the values of $x$ and $V(x)$ always stay in domain (12.1) $N$ . Hence $¥ovalbox{¥tt¥small REJECT}(x, V(x))$

is a holomorphic function of $x¥in¥Gamma_{x_{1}}$ except for the origin. By the same lemmawe see that $¥max^{¥prime}$ $¥{|x|^{¥mu}, |V(x)|¥}$ is a monotone increasing function of $s=|x|$ as $x$

moves on the segment $¥overline{¥mathrm{O}x}_{1}$ and does not exceed the value $¥max$ $¥{|x_{1}|^{¥mu}, |v^{1}|¥}$ for $x$ onthis segment. Hence $ff(x, V(x))$ is a holomorphic function of $x$ on this segmentexcept for the origin. On the other hand, inequalities (10.5) and (12.2) $N$ implythat

(12. 6) $¥left¥{¥begin{array}{l}|¥ovalbox{¥tt¥small REJECT}(x,V(x))|¥leqq(2A^{¥prime}¥mathrm{K}+B)|x|^{-¥sigma-1}¥max¥{|x|^{N¥mu},|V(x)|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x)}¥¥|_{¥mathrm{c}}¥mathfrak{X}(x,V(x))|¥leqq(2A,,¥mathrm{K}+B)|x|^{-1}¥max¥{|x|^{N¥mu},|V(x)|^{N}¥}.¥end{array}¥right.$

Since the rectilinear part of the curve $¥Gamma_{x_{1}}$ is in the sector $|¥arg$ $(-¥lambda)-¥sigma¥arg x|$

$¥leqq¥frac{¥pi}{2}-2¥sigma¥epsilon$ for $x$ of which we have $¥mathrm{R}¥mathrm{e}$ $¥Lambda(x)>0$ , the integrand of the first formula

of (12.4) tends to 0 exponentially as $x$ approaches the origin along this curve.Hence the first integral of (12.4) is convergent. If $N¥mu>1$ , the integrand of thesecond formula of (12.4) tends to 0 as $x$ approaches the origin along $¥overline{¥mathrm{O}x}_{1}$

and the second integral of (12.4) is convergent.Thus we see that the mapping $f$ :

$¥{¥varphi(x, v), ¥psi(x, v)¥}¥rightarrow$ $¥{¥Phi(x, v), ¥Psi(x, v)¥}$

has a well defined meaning.We shall prove that the mapping $x$ has a fixed point.Since {0, 0} $¥in¥Phi$ , the family $q$ is a non-empq set. It is clear that:I. $q$ is a convex, closed and normal set.We assume the following propositions have been established:$¥mathrm{I}¥mathrm{I}$ . $x$ mapps $q$ into itself. Namely,

1) We have the inequalities

(12. 7) $¥left¥{¥begin{array}{l}|¥Phi(x_{1},v^{1})|¥leqq ¥mathrm{K}¥max¥{|x_{1}|^{N¥mu},|v^{1}|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}¥Lambda(x_{1})},¥¥|¥Psi(x_{1},v^{1})|¥leqq ¥mathrm{K}¥max¥{|x_{1}|^{N¥mu},|v^{1}|^{N}¥}.¥end{array}¥right.$

2) $¥Phi(x, v)$ and $¥Psi(x, v)$ are holomorphic functions of $(x, v)$ in $(12.1)_{N}$ .$¥mathrm{I}¥mathrm{I}¥mathrm{I}$ . $x$ is a continuous mapping with respect to the topology of uniform

convergence on compact subsets.Then, by virtue of a fixed point theorem (See for example M. Hukuhara

[2]$)$ , it is concluded that:There is a member $¥{¥Phi_{N}(x, v), ¥Psi_{N}(x, v)¥}$ of $g$ which corresponds to a fixed

point of the mapping $x$ .In order to get the solution of Main Problem, we assert that:$¥mathrm{I}¥mathrm{V}$ . The expression $¥{¥Phi N(x, V(x)), ¥Psi_{N}(x, V(x))¥}$ is a solution of equations

268 M. IWANO

$(11.2)_{N}$ provided $(x, V(x))$ belongs to $(12.1)_{N}$ .

V. A solution of equations (11.2) $N$ such that we have (11.4) $N$ is unique.

The proof of these propositions can be carried out in almost exactly thesame way as in the proof of Theorem $¥mathrm{B}$ in Section 11 of M. Iwano [5] (Seepp. 215-226). But, in order to show how to determine the constants $N$, $¥mathrm{K}$, $¥delta_{N}$ ,

we shall give the proof of inequalities (12.7).By virtue of (12.6) and (9.2) we have readily an inequality of the form

(12. 8) $|¥Phi(x_{1}, v^{1})|¥leqq¥frac{2(2A}{|¥lambda|¥mathrm{s}}¥frac{¥prime ¥mathrm{K}+B)}{¥mathrm{i}¥mathrm{n}2¥sigma¥epsilon}¥max$ $¥{|x_{1}|^{N¥mu}, |v^{1}|^{N}¥}e^{-¥mathrm{R}¥mathrm{e}A(x_{1})}$

if the quantity $¥delta_{N}$ is chosen so small that

$3N¥mu[¥delta_{N}¥max_{¥varphi}¥{f(¥varphi)¥omega(¥varphi)^{¥mu}¥}]^{¥sigma/¥mu}¥leqq|¥lambda|¥sin 2¥sigma ¥mathrm{e}$.

Since we have (10.7), we see that, by taking the quantity $¥mathrm{K}$ sufficiently large,the first inequality of (12.7) results from (12.8).

By using (12.6) and (9.3) we see that

$|¥Psi(x_{1}, v^{1})|¥leqq¥frac{2(2A^{¥prime¥prime}¥mathrm{K}+B)}{N¥mu}¥max$ $¥{|x_{1}|^{N¥mu}, |¥iota¥prime^{1}|^{N}¥}$ .

We first take the quantity $N$ large enough to have$ 8A^{¥prime¥prime}<N¥mu$ .

Next, by taking, if it is necessary, $¥mathrm{K}$ much larger, we have the second inequalityof (12.7). Finally, for the constant $¥mathrm{K}$ thus determined, we have to take thequantity $¥delta_{N}$ so small that inequality (12.3) is satisfied.

References

[1] M. Hukuhara, Integration formelle d’un systeme d’equations differentielles nonlineaires dans le voisinage d’un point singulier. Annali di Matematica Pura edApplicata, Serie 4, 19 (1940), 35-44.

[2] M. Hukuhara, Renzokuna Kansu no Zoku to Syazo. Mem. Fac. Sci. Kyushu Univ.Ser. A. 5 (1950), 61-63.

[3] M. Iwano, Integration analytique d’un systeme d’equations differentielles non line-aires dans le voisinage d’un point singulier, I. Annali di Matematica Pura edApplicata, Serie 4, 44 (1957), 261-292.

[4] M. Iwano, Integration analytique d’un systeme d’equations differentielles non line-aires dans le voisinage d’un point singulier, II. ibid. 47 (1959), 91-150.

$¥mathrm{L}^{¥ulcorner}$ 5] M. Iwano, Analytic expressions for bounded solutions of non-linear ordinarydifferential equations with an irregular type singular point, ibid. 82 (1969), 189-256.

(Ricevita la 20-an de augusto, 1969)