Points of Differential Perturbation Linear

Funkcialaj Ekvacioj, 11 (1968), 235-246

Perturbation of Linear Ordinary Differential

Equations at Irregular Singular Points

By Yasutaka SIBUYA*

(University of Minnesota)

Abstract

A block-diagonalization theorem for the equation $e^{¥sigma}dy/dx=x^{¥tau}A(x, ¥mu, ¥epsilon)y$ according tothe Jordan canonical form of $A(¥infty, 0,0)$ is proved. Whereas $A(x, ¥mu, ¥mathrm{e})$ is holomorphic in$¥mu$ , a transformation matrix, which is holomorphic in $¥mu$ as well as admits asymptotic $¥mathrm{e}¥mathrm{x}-$

pansions in both $x^{-1}$ and $¥epsilon$ , is constructed.

I. Introduction

0. An example.In order to explain our problem, we shall study a very special case as an

example. Let us consider a system of two equations:

(I) $dy/dx=x^{r}A(x)y$,

where $¥mathrm{y}$ is a two-dimensional vector, $A(x)$ is a two-by-two matrix whose com-

ponents are convergent power series in $x^{-1}$ , and $r$ is a nonnegative integer.

Put $A(x)=¥sum_{k=0}^{¥infty}A_{k}x^{-k}$ and assume that the matrix $A_{0}$ has two distinct eigenvalues

$¥lambda_{1}$ and $¥lambda_{2}$ . Then it is well known that, for any given direction $¥arg x=¥theta$ in the$¥mathrm{x}$-plane, there exist two linearly independent solutions $¥mathrm{y}_{1}(x)$ $¥mathrm{a}¥mathrm{n}¥mathrm{d}¥mathrm{y}_{2}(¥mathrm{x})$ suchthat

$y_{j}(x)¥equiv x^{¥sigma}j$ $(¥sum_{k=0}^{¥infty}p_{jk}x^{-k})¥exp Q_{j}(x)$

as $x$ tends to infinity in the direction $¥arg x=¥theta$ , where $Q_{j}(x)=¥lambda_{j}(r+1)^{-1}x^{r+1}+$

$¥sum_{k=1}^{r}q_{jk}x^{k}$ . The quantities $¥lambda_{j}$ , $¥sigma_{¥mathrm{j}}$ , $q_{jk}$ , and $p_{jk}$ can be found in a purely algebraic

manner, while the solutions $y_{j}(x)$ can be constructed essentially by the methodof successive approximations. Therefore, the following generalization is ratherobvious: Let us consider a system of the form

(II) $dy/dx=x^{r}A(x, ¥mu)y$ ,

where $¥mathrm{y}$ , $A$ and $r$ are the same as in (I), except that components of $A(x, ¥mu)$

are convergent power $¥mathrm{s}¥mathrm{e}¥mathrm{r}¥tilde{¥mathrm{i}}¥mathrm{e}¥mathrm{s}$ in $x^{-1}$ and parameters $¥mu=(¥mu_{1^{ }},¥cdots, ¥mu_{n})$ . Put $A(x,¥mu)$

$*)$ Sponsored by the Mathematics Research Center, United States Army, Madison,Wisconsin, under Contract No. : $¥mathrm{D}¥mathrm{A}-31-124-¥mathrm{A}¥mathrm{R}¥mathrm{O}$ -DA62 and $¥mathrm{N}¥mathrm{S}¥mathrm{F}$ $(¥mathrm{G}¥mathrm{P}-7041¥mathrm{X})$ .

236 Y. SIBUYA

$=¥sum_{k=0}^{¥infty}Ak(¥mu)x^{-k}$ and assume that the matrix $A_{0}(0)$ has two distinct eigenvalues.

Then the matrix $A_{0}(¥mu)$ has two eigenvalues $¥lambda_{1}(¥mu)$ and $¥lambda_{2}(¥mu)$ such that $¥lambda_{1}(0)¥neq$

$¥lambda_{2}(0)$ . Now it is easily proved that, for any given direction $¥arg x=¥theta$ in the$x-¥mathrm{p}¥mathrm{l}¥mathrm{a}¥mathrm{n}¥mathrm{e}$, there eixst two linearly independent solutions $y_{1}(x, ¥mu)$ and $y_{2}(x, ¥mu)$

such that

$y_{j}(x, ¥mu)¥equiv x^{¥sigma(¥mu)}j$ $(¥sum_{k=0}^{¥infty}p_{jk}(¥mu)x^{-k})¥exp Q_{j}(x, ¥mu)$

as $x$ tends to infinity in the direction $¥arg x=¥theta$ , where $Q_{j}(x, ¥mu)=¥lambda_{j}(¥mu)(r+1)^{-1}x^{r+1}$

$+¥sum_{k=1}^{¥gamma}qjk(¥mu)x^{n}$ . We can verify further that the quantities $y_{j}(x, ¥mu)$ , $¥lambda_{j}(¥mu)$ ,

$¥dot{¥sigma}j(¥mu)$ , $q_{jk}(¥mu)$ and $p_{jk}(¥mu)$ are convergent power series in $¥mu_{1}$ , $¥cdots$ , $¥mu_{m}$ . Let usnow relax the condition on the dependence $¥mathrm{o}¥mathrm{f}¥backslash ¥mathrm{A}$ on $¥mathrm{p}¥mathrm{a}¥mathrm{r}¥dot{¥mathrm{a}}$ met-ers. To do this,let us consider a system of the following form:

(III) $dy/dx=x^{r}A(x, ¥mathrm{e})y$,

where $y$ , $A$ and $r$ are again the same as in (I), except that components of$A(x, ¥mathrm{e})$ are convergent power series in $x^{-1}$ and that they $¥mathrm{a}¥mathrm{d}¥overline{¥mathrm{m}}¥mathrm{i}¥mathrm{t}$ asymptotic

expansions in powers of a parameter $¥epsilon$ as $¥mathrm{e}$ tends to zero. Put $A(x,¥mathrm{e})=¥sum_{k=0}^{¥infty}A_{k}(¥epsilon)x^{-k}$,

where matrices $A_{k}(¥mathrm{e})$ admit asymptotic expansions in powers of $¥epsilon$ as $¥mathrm{e}$ tendsto zero. Assume that $A_{0}(¥mathrm{e})$ has two eigenvalues $¥lambda_{1}(¥mathrm{e})$ and $¥lambda_{2}(¥mathrm{e})$ such that$¥lambda_{1}(¥mathrm{e})-¥lambda_{2}(¥epsilon)¥neq 0$ as $¥mathrm{e}$ tends to zero. Then it can be again proved that, for anygiven direction $¥arg x=¥theta$ in the $¥mathrm{x}$-plane, there exist two linearly independentsolutions $y_{1}(x, ¥mathrm{e})$ and $y_{2}(x, ¥mathrm{e})$ such that

$y_{j}(x, ¥epsilon)¥equiv x^{¥sigma_{¥mathrm{j}}(¥epsilon)}$ $(¥sum_{k=0}^{¥infty}p_{jk}(e)x^{-k})¥exp Q_{j}(x, ¥mathrm{e})$

as $x$ tends to infinity in the direction $¥arg x=¥theta$ , where $Q_{j}(x, ¥mathrm{e})=¥lambda_{j}(¥mathrm{e})(r+1)^{-1}x^{r+1}$

$+¥sum_{k=1}^{r}qjk(¥mathrm{e})x^{k}$. We can further verify that the quantities $¥lambda_{j}(¥mathrm{e})$ , $¥sigma_{j}(¥mathrm{e})$ , $qjk(e)$ and$p_{jk}(.¥epsilon)$ admit asymptotic expansions in powers of $¥mathrm{e}$ as $¥mathrm{e}$ tends to zero.However, the solutions $y_{j}(x, ¥mathrm{e})$ do not admit such asymptotic representationsunless we specify them more precisely. This difficulty is simply due to non-$¥mathrm{u}¥mathrm{n}¥mathrm{i}¥mathrm{q}¥mathrm{u}¥mathrm{e}¥mathrm{n}¥dot{¥mathrm{e}}¥mathrm{s}¥mathrm{s}$ of asymptotic solutions $y_{j}(x, ¥mathrm{e})$ . To see this, let us assume that$y_{j}(x, ¥mathrm{e})$ do have asymptotic representations in $¥mathrm{e}$ as $¥mathrm{e}$ tends to zero. Furtherassume that $¥mathrm{R}¥mathrm{e}$

$¥mathrm{Q}_{1}<¥mathrm{R}¥mathrm{e}¥mathrm{Q}_{2}$ along the direction $¥arg x=¥theta$ . Then$¥mathrm{y}(x, ¥mathrm{e})=y_{2}(x, ¥epsilon)+$ (elog e) $y_{1}(x, ¥mathrm{e})$

admits the same asymptotic representation in $x^{-1}$ as $y_{2}$ does. However, $y(x, ¥epsilon)$

does not admit an asymptotic representation in powers of $¥mathrm{e}$ as $¥mathrm{e}¥mathrm{t}¥dot{¥mathrm{e}}¥mathrm{n}¥mathrm{d}¥mathrm{s}$ to zero.Then the problem is to specify $y_{j}$ so that they admit also asymptoticrepresentations in $¥epsilon$ . To find an answer for this question, let us assume that

Perturbation of Linear Ordinary Differential Equations 237

-there exist two linearly independent solutions $¥tilde{y}_{1}(x, ¥mathrm{e})$ and $¥tilde{y}_{2}(x, ¥mathrm{e})$ such that

$x^{-¥sigma_{j}(¥epsilon)}¥exp¥{-Q_{j}(x, ¥mathrm{e})¥}¥tilde{y}_{j}(x, ¥mathrm{e})¥equiv¥sum_{/¥iota=0}^{¥infty}¥tilde{p}_{jh}(x)¥mathrm{e}^{h}$

as $¥mathrm{e}$ tends to zero uniformly with respect to $x$ in a small sector around thegiven direction $¥arg x=¥theta$ , where $¥tilde{p}_{¥mathrm{j}h}(x)$ admit asymptotic expansions in powersof $x^{-1}$ as $x$ tends to infinity in this small sector. Then we can prove that $¥tilde{y}_{j}$

also admit asymptotic representations in $x^{-1}$ as $x$ tends to infinity. To see this,assume that $¥mathrm{R}¥mathrm{e}$

$¥mathrm{Q}_{1}<¥mathrm{R}¥mathrm{e}¥mathrm{Q}_{2}$ in this small sector. Consider the relations between$¥{y_{1}, y_{2}¥}$ and $¥{¥tilde{y}_{1},¥tilde{y_{2}}¥}$ , and put

$¥tilde{y}_{j}(x, ¥mathrm{e})=¥gamma_{j1}(¥mathrm{e})y_{1}(x, ¥mathrm{e})+¥gamma_{j2}(¥mathrm{e})y_{2}(x, ¥mathrm{e})$ $(j=1,2)$

Then we get

$¥tilde{y}_{1}(x, ¥mathrm{e})=¥tau_{11}(¥mathrm{e})y_{1}(x, ¥mathrm{e})$ , $¥tilde{y}_{2}(x, ¥mathrm{e})¥equiv¥tau_{22}(¥epsilon)y_{2}(x, ¥mathrm{e})$

as $x$ tends to infinity.

This means that $¥tilde{y}_{j}$ also admit asymptotic representations in $x^{-1}$ . In case when$¥mathrm{R}¥mathrm{e}$

$¥mathrm{Q}_{1}=¥mathrm{R}¥mathrm{e}¥mathrm{Q}_{2}$ along the given direction $¥arg x=¥theta$ , we must modify our methodto get the same conclusion. In this paper, we shall discuss this problem in avery general case.

1. Main theorem.Consider a system of linear ordinary differential equations

(1. 1) $e^{¥sigma}dy/dx=x^{¥gamma}A(x, ¥mu, ¥mathrm{e})y$ ,

where $r$ and $¥sigma$ are nonnegative integers, $x$ is a complex independent variable,$¥mathrm{s}$ is a complex parameter, $¥mu$ is an $¥mathrm{w}¥mathrm{z}$-dimensional vector whose components arealso parameters, $y$ is an $¥mathrm{w}$-dimensional vector whose components are unknownfunctions of $(x, ¥mu, ¥mathrm{e})$ , and $A(x, ¥mu, ¥mathrm{e})$ is an n-hy-n matrix whose componentsare single-valued, bounded and holomorphic functions of $(x^{-1}, ¥mu, ¥mathrm{e})$ for

(1. 2) $|x|¥geqq R_{0}$ , $|¥mu|¥leqq¥mu_{0},0<|¥mathrm{e}|¥leqq ¥mathrm{e}_{0}$, $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ ,

the quantities $R_{0}$ , $¥mu_{0}$ , $¥mathrm{e}_{0}$ and $¥rho_{0}$ being positive constants. Assume further that$A$ admits an asymptotic expansion

(1. 3) $A(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{h=0}^{¥infty}A_{h}(x, ¥mu)¥mathrm{e}^{h}$

in powers of $¥mathrm{e}$ as $¥mathrm{e}¥sim 0$ in the sector: $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ uniformly for $|x|¥geqq R_{0}$ and$|¥mu|¥leqq¥mu_{0}$ , where the coefficients $A_{h}(x, ¥mu)$ are holomophic in $(x^{-1}, ¥mu)$ for $|x|¥geqq R_{0}$

and $|¥mu|¥leqq¥mu_{0}$.

Let $¥lambda_{1}$ , $¥cdots$ , $¥lambda_{S}$ be the distinct eigenvalues of $A_{0}(¥infty, 0)$ and let $n_{1}$ , $¥cdots$ , $n_{S}$ denotetheir respective multiplicities. Assume that

238 Y. $¥mathrm{s}_{¥mathrm{I}¥mathrm{B}¥mathrm{U}¥mathrm{Y}¥mathrm{A}}$

(1. 4) $A_{0}(¥infty, 0)=[_{0}^{A_{1}}.0¥circ..$ $A_{2}.00¥circ.$

.$.000.$

.$...........$

.

$.000.$

.$A_{s}.00¥circ..]$

where

(1. 5) $A_{j}¥circ=¥left¥{¥begin{array}{llllllll}¥lambda_{j} & d_{j1} & 0 & 0 & ¥cdots & 0 & 0 & 0¥¥0 & ¥lambda_{j} & d_{j2} & 0 & ¥cdots & 0 & 0 & 0¥¥¥cdots & ¥cdots & ¥cdots & ¥cdots & ¥cdots & ¥cdots & ¥cdots & ¥cdots¥¥ 0 & 0 & 0 & 0 & ¥cdots & 0 & ¥lambda_{j} & d_{jn_{j}-1}¥¥0 & 0 & 0 & 0 & ¥cdots & 0 & 0 & ¥lambda_{j}¥end{array}¥right¥}$

the quantities $d_{jk}$ being 0 or 1.Put

(1. 6) $¥omega_{jk}=¥arg$ $(¥lambda_{j}-¥lambda_{k})$ $(j ¥neq k)$

and(1. 7)

$¥omega_{+}=¥max_{(j.k)}¥omega_{jk}$, $¥omega_{-}=¥min_{(j.k)}¥omega_{jk}$ .

Let us denote by $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ the sector in the $¥mathrm{x}$-plane which is defined by

(1. 8) $¥rho_{1}¥leqq¥arg x¥leqq¥rho_{2}$ .

In this paper we shall prove the following theorem.

Theorem. Assume that the sector $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ is contained in the sector

(1. 9) $-¥frac{3¥pi}{2}-¥omega_{¥_}+¥sigma¥rho_{0}<(r+1)¥arg x<¥frac{3¥pi}{2}-¥omega_{+}-¥sigma¥rho_{0}$

for a suitable choice of $¥omega_{jk}$ $(j ¥neq k)$ . Then there exist a sufficientiy targe positivenumber $N$, sufficientfy small positive numbers $¥mathrm{e}_{1}$ and $¥mu_{1}$ , and an n-by-n matrix$P(x, ¥mu, ¥mathrm{e})$ satisfying the following conditions:

(i) for every nonnegative integer $h_{0}$ , the matrix $P$ can be written in $a$

form:

(1. 10) $P(x, ¥mu, ¥mathrm{e})=¥sum_{h=0}^{h_{0}}P_{h}(x, ¥mu)¥mathrm{e}^{h}+¥mathrm{e}^{h_{0}¥dagger 1}Qh_{0}(x, ¥mu, ¥mathrm{e})$,

where $P_{h}(x, ¥mu)$ are holomorphic for $|x|¥geqq N$, $x¥in ¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , $|¥mu|¥leqq¥mu_{1}$ and admit

for $|¥mu|¥leqq¥mu_{1}$ uniform asymptotic expansions

(1. 11) $P_{h}(x, ¥mu)¥equiv¥sum_{k=0}^{¥infty}P_{hk}(¥mu)x^{-k}$

as $x$ tends to infinity through the sector $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , and $Q_{h_{0}}$ is holomorphic for$|x|¥geqq N$, $x¥in ¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , $|¥mu|¥leqq¥mu_{1}$ , $0<|¥mathrm{e}|¥leqq ¥mathrm{e}_{1}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ and admits for $|¥mu|<$

$¥mu_{1},0<|¥epsilon|¥leqq ¥mathrm{e}_{1}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ a uniform asymptotic expansion

(1. 12) $Q_{h_{0}}(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{k=0}^{¥infty}Q_{h_{0}k}(¥mu, ¥mathrm{e})x^{-k}$

Perturbation of Linear Ordinary Differential Equations 239

.as $x$ tends to infinity through the sector $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ ;

(ii) the quantities $P_{hk}(¥mu)$ are holomorphic for $|¥mu|¥geqq¥mu_{1}$ , and the quantities$,Q_{h_{0}k}(¥mu, ¥mathrm{e})$ are holomorphic for $|¥mu|¥leqq¥mu_{1},0<|¥mathrm{e}|¥leqq ¥mathrm{e}_{1}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ and admit for$|¥mu|¥leqq¥mu_{1}$ uniform asymptotic expansions

(1. 13) $Qh_{0}h(¥mu, ¥mathrm{e})¥equiv¥sum_{h=0}^{¥infty}Qh_{0}kh(¥mu)¥mathrm{e}^{h}$

as $¥mathrm{e}$ tends to zero in the sector $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ , where the coefficients are holomor-

phic in $¥mu$ for $|¥mu|¥leqq¥mu_{1}$ ;

(iii) the matrix $P_{00}(0)$ is nonsingular;

(iv) the transformation(1. 14) $y=P(x, ¥mu, ¥mathrm{e})z$

reduces System (1. 1) to

(1. 15) $e^{¥sigma}dz/dx=x^{¥gamma}B(x, ¥mu, ¥mathrm{e})z$

where(1. 16) $B(x, ¥mu,¥epsilon)=$ diag $(B_{1}(x, ¥mu, ¥mathrm{e}), ¥cdots, B_{s}(x, ¥mu, ¥mathrm{e}))$,

and $B_{¥dot{f}}(x, ¥mu, ¥mathrm{e})$ is an $n_{j}-by-n_{j}$ matrix such that

(1. 17) $B_{j}(x, ¥mu, ¥mathrm{e})=A_{j}¥circ+O(|x^{-1}|+|¥mu|+|¥epsilon|)$ .

Remarks. (1) Previously we studied System (1. 1) in case when $A$ doesnot depend on $¥mu$ . [2] By using the same method, we can construct $P$ which.satisfies all conditions of our theorem for a fixed $h_{0}$ . Our main concern is to

.construct $P$ in such a manner that $P$ is independent of $h_{0}$.

(2) The theorem of P. F. Hsieh and Y. Sibuya [1] is a special case of ourtheorem in which $¥sigma=0$ and $A$ does not depend on $¥mathrm{e}$.

2. Lemmas on nonlinear systems.

In the same way as in our previous paper [2], we can reduce our mainproblem to the study of a nonlinear system

(2. 1) $e^{¥sigma}dy/dx=x^{r}f(x, y, ¥mu, ¥mathrm{e})$ ,

where $r$ and $¥sigma$ are nonnegative integers, $¥mathrm{e}$ is a complex parameter, $x$ is a com-

-plex independent variable, $¥mu$ is an $¥mathrm{m}$-dimensional vector whose components are,also parameters, $y$ is an $¥mathrm{n}$ -dimensional vector, and $f(x, y, ¥mu, ¥epsilon)$ is an w-dimen-sional vector whose components are single-valued, bounded and holomorphicfunctions of $(x^{-1}, y, ¥mu, ¥mathrm{e})$ for

(2. 2) $|x|¥geqq R_{0}$ , $|y|¥leqq¥delta_{0}$, $|¥mu|¥leqq¥mu_{0},0<|¥mathrm{e}|¥leqq ¥mathrm{e}_{0}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ ,

the quantities $R_{0}$ , $¥delta_{0}$ , $¥mu_{0}$ , $¥mathrm{e}_{0}$ and $¥rho_{0}$ being positive constants. In addition weshall suppose that $f$ admits for

(2. 3) $|x|¥geqq R_{0}$ , $|y|¥leqq¥delta_{0}$, $|¥mu|¥leqq¥mu_{0}$

. a uniform asymptotic expansion

240 Y. SIBUYA

(2. 4) $f(x, ¥mathrm{y}, ¥mu,¥epsilon)¥equiv¥sum_{h=0}^{¥infty}f_{h}(x, y, ¥mu)¥mathrm{e}^{h}$

as $¥mathrm{e}$ tends to zero through the sector

(2. 5) $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ ,

where the coefficients $f_{h}$ are assumed to be single-valued, bounded and holo-morphic in the domain (2. 3).

Let

(2. 6) $f(x, y, ¥mu, ¥mathrm{e})=¥hat{f}_{0}(x, ¥mu, ¥mathrm{e})+A(x, ¥mu, ¥mathrm{e})y+O(|y|^{2})$ ,

where $¥hat{f}_{0}$ is an $¥mathrm{w}$-dimensional vector, and $A$ is an n-hy-n matrix. It is easilyseen that $A$ admits for

(2. 7) $|x|¥geqq R_{0}$ , $|¥mu|¥leqq¥mu_{0}$

a uniform asymptotic expansion

(2. 8) $A(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{h=0}^{¥infty}A_{h}(x, ¥mu)¥mathrm{e}^{h}$

as $¥mathrm{e}$ tends to zero through the sector (2. 5), where the coefficients $A_{h}(x, ¥mu)$ aresingle-valued, bounded and holomorphic in the domain (2. 7).

Assume that the matrix $A_{0}(¥infty,0)$ has the Jordan canonical form with non-zero eigenvalues $¥lambda_{¥mathrm{I}}$ , $¥cdots$ , $¥lambda_{n}$ , and that

(2. 9) $¥hat{f}_{0}(0,¥mu, ¥mathrm{e})¥equiv 0$.

Put

(2. 10) $¥omega_{j}=¥arg¥lambda_{j}$ $(j=1,2, ¥cdots,n)$

and(2. 11)

$¥omega_{+}=¥max_{j}¥omega_{j}$ , $¥omega_{-}=¥min_{j}¥omega_{j}$ .

Let us denote by $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ the sector in the $¥mathrm{x}$-plane which is defined by

(2. 12) $¥rho_{1}¥leqq¥arg x¥leqq¥rho_{2}$ .

Now we make the following assumption:Assumption (A). The sector $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ is contained in the sector

(2. 13) $-¥frac{3¥pi}{2}-¥omega_{¥_}+¥sigma¥rho_{0}<(r+1)¥arg x<¥frac{3¥pi}{2}-¥omega_{+}-¥sigma¥rho_{0}$

for a suitable choice of $¥omega_{j}$ .We shall state four lemmas under Assumption (A).Lemma 1. There exists a solution of System (2. 1):

(2. 14) $y=¥varphi(x, ¥mu, ¥mathrm{e})$

such that(i) $¥varphi$ is single-valued, bounded and holomorphic for

(2. 15) $|x|¥geqq N$, $x¥in ¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , $|¥mu|¥leqq¥mu_{1},0<|¥epsilon|¥leqq ¥mathrm{e}_{1}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{¥mathrm{J}}$

Perturbation of Linear Ordinary Differential Equations 241

if $1/N$, $¥mu_{1}$ , $¥epsilon_{1}$ are sufficiently small;

(ii) $¥varphi$ admits for(2. 16) $0<|¥mathrm{e}|¥leqq ¥mathrm{e}_{1}$ , $|¥arg¥epsilon|¥leqq¥rho_{0}$, $|¥mu|¥leqq¥mu_{1}$

a uniform asymptotic expansion

(2. 17) $¥varphi(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{k=1}^{¥infty}¥varphi_{k}(¥mu, ¥mathrm{e})x^{-k}$

as $x$ tends to infinity in $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , where the coefficients $¥varphi_{h}(¥mu, ¥mathrm{e})$ are single-

valued, bounded and holomorphic for (2. 16) and they admit for $|¥mu|¥leqq¥mu_{1}$ uni-

form asymptotic expansions

(2. 18) $¥varphi_{k}(¥mu, ¥epsilon)¥equiv¥sum_{h=0}^{¥infty}¥varphi_{kh}(¥mu)¥mathrm{e}^{h}$

as $¥epsilon$ tends to zero in the $sect¥dot{o}r$ $|¥arg¥epsilon|¥leqq¥rho_{0}$ , the coefficients $¥varphi_{kh}(¥mu)$ being holo-morphic $f¥dot{o}r$ $|¥mu|¥leqq¥mu_{1}$ .

Lemma 2. Suppose that there exists a solution of System (2. 1):

(2. 19) $¥mathrm{y}=¥psi(x, ¥mu, ¥mathrm{e})$

such that(i) $¥phi$ is single-valued, bounded and holomorphic for (2. 15);

(ii) $¥phi$ admits for(2. 20) $|x|¥geqq N$, $x¥in ¥mathrm{S}(¥rho_{1},p_{2})$ , $|¥mu|¥leqq¥mu_{1}$

a uniform asymptotic expansion

(2. 21) $¥phi(x, ¥mu,e)¥equiv¥sum_{h=0}^{¥infty}¥phi_{k}(x,¥mu)¥epsilon^{k}$

as $¥epsilon$ tends to zero in the sector $|¥arg¥epsilon|¥leqq¥rho_{0}$, where the coefficients $¥phi_{h}(x, ¥mu)$ aresing&-valued, bounded and holomorphic for (2. 20) and they admit for $|¥mu|¥leqq¥mu_{1}$

uniform asymptotic expansions

(2. 22) $¥psi_{h}(x, ¥mu)¥equiv¥sum_{k=1}^{¥infty}¥phi_{hk}(¥mu)x^{-k}$

as $x$ tends to infinity through $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , the coefficients $¥psi_{hk}(¥mu)$ being holomorphic

for $|¥mu|¥leqq¥mu_{1}$ . Then

(2. 23) $¥phi(x, ¥mu, ¥mathrm{e})-¥varphi(x, ¥mu, ¥epsilon)¥equiv 0$

as $x$ tends to infinity through $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ uniformly for $(¥mu, ¥mathrm{e})$ in the domain(2. 16).

Lemma 3. Suppose that a solution (2. 19) satisfies Conditions (i) and(ii) of Lemma 2. Then, for each $h_{0}$ the solution $¥psi$ can be written in a form

(2. 24) $¥phi(x, ¥mu, ¥epsilon)=¥sum_{h=0}^{h_{¥dot{0}}}¥psi_{h}(x, ¥mu)¥mathrm{e}^{h}+¥epsilon^{h_{0}+1}¥varphi_{h_{0}}(x, ¥mu, ¥mathrm{e})$

where $¥varphi_{h_{0}}(x, ¥mu, ¥epsilon)$ satisfies $Con$-ditions (i) and (ii) of Lemma 1.Lemma 4. There exists a $solutio¥dot{n}(2.19)$ of System (2. 1) which satisfies

242 Y. SIBUYA

Conditions (i) and (ii) of Lemma 2.Remarks. (1) Lemma 1 can be proved in the same manner as we proved

a similar lemma in our previous paper. [2; page 42, Lemma 2], Therefore,we shall omit the proof of Lemma 1.

(2) Lemma 3 can be derived from Lemma 2. In fact, put

$y=¥sum_{h=0}^{h_{0}}¥psi_{h}(x, ¥mu)¥mathrm{e}^{h}+¥mathrm{e}^{h_{0}+1}z$

and derive a system of differential equations for $z$ . Then

$z=¥mathrm{e}^{-h_{0^{-1[¥phi(x,¥mu,¥mathrm{e})-¥sum_{h=0}^{h_{0}}¥psi_{h}(x,¥mu)¥mathrm{e}^{h]}}}}$

is a solution which satisfies Conditions (i) and (ii) of Lemma 2. Therefore,use Lemma 2 to complete the proof of Lemma 3.

(3) Since the proof of our main theorem can be reduced to the proofs ofthese four lemmas, it is only necessary to prove Lemmas 2 and 4. Lemma 4will be proved in Chapter $¥mathrm{I}¥mathrm{I}$ , and Lemma 2 will be proved in Chapter $¥mathrm{I}¥mathrm{I}¥mathrm{I}$ .

II. Proof of Lemma 4

3. A formal solution.We shall construct a formal solution of System (2. 1)

(3. 1) $y=¥sum_{h=0}^{¥infty}¥phi_{h}(x,¥mu)¥mathrm{e}^{h}$

whose coefficients $¥psi_{h}(x,¥mu)$ satisfy the conditions of Lemma 4. We shall studyonly the case $¥sigma=0$ . The case $¥sigma>0$ can be treated in a similar manner.

The quantity $¥psi_{0}(x, ¥mu)$ must satisfy the system

(3. 2) $dz/dx=x^{¥gamma}f_{0}(x,z, ¥mu)$ ,

where(3. 3) $f_{0}(x,z, ¥mu)=f_{0}(x, 0, ¥mu)+A_{0}(x, ¥mu)z+O(|z|^{2})$ .

By using Lemma 1, we can construct $¥phi_{0}(x, ¥mu)$ . Then let us put

(3. 4) $y=u+¥phi_{0}(x, ¥mu)$

to derive(3. 5) $du/dx=x^{¥gamma}¥{f(x, u+¥psi_{0}(x, ¥mu), ¥mu, ¥mathrm{e})-f_{0}(x, ¥psi_{0}(x, ¥mu), ¥mu)¥}$

$=x^{¥gamma}¥{g_{0}(x, ¥mu, ¥mathrm{e})+B(x, ¥mu, e)u+O (|u|^{2})¥}$ ,

where(3. 6) $g_{0}(x, ¥mu, ¥mathrm{e})=f(x, ¥psi_{0}(x, ¥mu),¥mu, ¥mathrm{e})-f_{0}(x, ¥psi_{0}(x, ¥mu),¥mu)$

$¥equiv¥sum_{h=1}^{¥infty}f_{h}(x, ¥psi_{0}(x, ¥mu),¥mu)¥mathrm{e}^{h}$,

and $B(x, ¥mu, ¥epsilon)$ is an w-by-n matrix such that

(3. 7) $B(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{h=0}^{¥infty}B_{h}(x, ¥mu)¥mathrm{e}^{h}$

Perturbation of Linear Ordinary Differential Equations 243

and(3. 8) $B_{0}(x, 0)=A_{0}(¥infty, 0)+O(x^{-1})$ .

Now $¥psi_{h}(x, ¥mu)(h ¥geqq 1)$ must satisfy a linear system

(3. 9) $d¥phi_{h}/dx=x^{¥gamma}¥{B_{0}(x, ¥mu)¥psi_{h}+¥varphi_{h}(x, ¥mu)¥}$ ,

where $q_{h}$) is a polynomial in $¥psi_{k}(k <h)$ and their derivatives. Thus we canconstruct a formal solution (3. 1) whose coefficients $¥psi_{h}$ satisfy the conditions ofLemma 4.

4. Existence of an actual solution.By virtue of Borel-Ritt theorem [3; page 43], we can construct a vector-

valued function $G(x, ¥mu, ¥mathrm{e})$ such that(i) $G$ is single-valued, bounded and holomorphic for (2. 15);(ii) $G$ and dGjdx admit for (2. 20) uniform asymptotic expansions

$G(x, ¥mu, ¥mathrm{e})¥equiv¥sum_{h=0}^{¥infty}¥psi_{h}(x, ¥mu)¥epsilon^{h}$ ,

(4. 1)$dG(x, ¥mu, t)jdx$ $¥equiv¥sum_{h=0}^{¥infty}¥phi_{h}^{¥prime}(x, ¥mu)¥mathrm{e}^{h}$

$ias$ $¥mathrm{e}$ tends to zero in the sector $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ . Put

(4. 2) $y=u+G(x, ¥mu, ¥mathrm{e})$

to derive

(4. 3) $e^{¥sigma}du/dx=x^{¥gamma}[c(x, ¥mu, ¥mathrm{e})+C(x, ¥mu, ¥mathrm{e})u+O(|u|^{2})]$ ,

where(4. 4) $c(x, ¥mu, ¥epsilon)=f(x, G(x, ¥mu, ¥mathrm{e}),¥mu, ¥epsilon)-x^{-¥gamma}¥epsilon^{¥sigma}dG(x, ¥mu, t)jdx$

and $C(x, ¥mu, ¥mathrm{e})$ is an w-by-w matrix such that

(4. 5) $C(x, ¥mu, ¥epsilon)¥equiv¥sum_{h=0}^{¥infty}C_{h}(x, ¥mu)¥mathrm{e}^{h}$

and(4. 6) $C_{0}(x, 0)=A_{0}(¥infty,0)+O(x^{-1})$ .

Since (3. 1) is a formal solution of System (2. 1), we have

(4. 7) $c(x, ¥mu, ¥epsilon)¥equiv 0$

uniformly for (2. 20) as $¥epsilon$ tends to zero in the sector $|¥mathrm{a}¥mathrm{I}¥mathrm{g}$ $¥mathrm{e}|¥leqq¥rho_{0}$.Then in the same way as we did in our previous paper [2], we can con-

struct a solution of System (4. 3):

(4. 8) $u=¥hat{G}(x, ¥mu, ¥mathrm{e})$

such that $¥hat{G}$ is single-valued, bounded and holomorphic for (2. 15) and $¥hat{G}¥equiv 0$

uniformly for (2. 20) as $¥epsilon$ tends to zero in the sector $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$. This com-

pletes the proof of Lemma 4.

244 Y. SIBUYA

III. Proof of Lemma 2

5. A linear system.

Let (2. 14) be a solution of System (2. 1) which satisfied Conditions (i)and (ii) of Lemma 1. Put

(5. 1) $y=u+¥varphi(x, ¥mu, ¥mathrm{e})$

to derive(5. 2) $e^{¥sigma}du/dx=x^{¥gamma}g(x, u, ¥mu, ¥mathrm{e})$

where(5. 3) $g(x, u, ¥mu, ¥mathrm{e})=f(x, u+¥varphi, ¥mu, ¥mathrm{e})-x^{-r}¥mathrm{e}^{¥sigma}d¥varphi/dx$ .

Since (2. 14) is a solution of System (2. 1), we have(5. 4) $g(x,0, ¥mu, ¥mathrm{e})¥equiv 0$ .

Hence we can write $g$ in the form(5. 5) $g(x, u, ¥mu, ¥epsilon)=E(x,u, ¥mu, ¥mathrm{e})u$ ,

where $E(x, u, ¥mu, ¥epsilon)$ is an n-by-n matrix whose components are single-valued,bounded and holomorphic for $(x, ¥mu, ¥mathrm{e})$ in the domain (2. 15) and $|u|¥leqq¥delta$ , $¥delta$

being a small positive constant. Furthermore(5. 6) $E(^{-}-x,0,¥mu,¥epsilon)=A_{0}(¥infty,0)+O(|x|^{-1}+|¥mu|+|¥mathrm{e}|)$.

Now let (2. 19) be a solution of System (2. 1) which satisfied Conditions(i) and (ii) of Lemma 2, and put

(5. 7) $u_{0}(x,¥mu, ¥epsilon)=¥psi(x, ¥mu, ¥mathrm{e})-¥varphi(x, ¥mu, ¥mathrm{e})$ .Then $u=u_{0}(x,¥mu, ¥mathrm{e})$ is a solution of System (5. 2), which is $¥mathrm{s}¥mathrm{i}¥mathrm{n}¥mathrm{g}¥mathrm{l}¥mathrm{e}-¥mathrm{v}¥dot{¥mathrm{a}}¥mathrm{l}¥mathrm{u}¥mathrm{e}¥mathrm{d}$

bounded and holomorphic for (2. 15), and(5. 8) $u_{0}(x,¥mu, ¥mathrm{e})=O(|x|^{-1}+|¥mathrm{e}|)$ .

Put(5. 9) $F(x, ¥mu, ¥mathrm{e})=E(x, u_{0}(x, ¥mu, ¥mathrm{e}),¥mu,¥epsilon)$ .

Then $u=u_{0}(x, ¥mu, ¥epsilon)$ is a solution of the linear system(5. 10) $¥mathrm{e}^{¥sigma}du/dx=x^{¥gamma}F(x,¥mu, ¥mathrm{e})u$ .

Notice that(5. 11) $F(x, ¥mu, ¥mathrm{e})=A_{0}(¥infty,0)+O(|x|^{-1}+|¥mu|+|¥mathrm{e}|)$ .

6. A block-diagonalization theorem.Let us assume that

(6. 1) $A_{0}(¥infty,0)=$ diag $(A_{1^{ }}¥mathrm{o},¥cdots,¥mathrm{o}A_{s})$ ,where

(6. 2) $A_{j}¥circ=¥nu_{j}I_{j}+N_{j}$

a$¥dot{¥mathrm{n}}¥mathrm{d}$ the quantities $¥nu_{j}$ are distinct eigenvalues of $A_{0}(¥infty, 0)$ , $I_{j}$ is the n-by-nidentity matrix, and $N_{j}$ is an n-by-n nilpotent matrix. Let us fix a direction

Perturbation of Linear Ordinary $Differef/tial$ Equations 245

(6. 3) $¥arg x=¥theta$

so that(6. 4) $¥rho_{1}<¥theta<¥rho_{2}$ .

Let us elassify 1, 2, $¥cdots$ , $s$ into three classes $J_{1}$ , $J_{2}$ and $J_{3}$ so that

(6. 5) $¥mathrm{R}¥mathrm{e}$$[¥nu_{¥dot{f}}x^{¥gamma+1}¥epsilon^{-¥sigma}]¥left¥{¥begin{array}{l}>0j¥in J_{1},¥¥=0j¥in J_{2},¥¥<0j¥in J_{3}¥end{array}¥right.$

for $¥mathrm{a}¥mathrm{r}¥mathrm{g}¥mathrm{x}=¥theta$ and $¥epsilon>0$ . Then choose a small sector $¥mathrm{S}(¥theta)$

(6. 6) $|¥arg x-¥theta|¥leqq¥rho$

so that

(6. 7) $¥mathrm{R}¥mathrm{e}$

$[¥nu_{j}x^{r+1}¥mathrm{e}^{-¥sigma}]¥left¥{¥begin{array}{l}>0j¥in J_{1}¥¥<0j¥in J_{3}¥end{array}¥right.$

for $x¥in ¥mathrm{S}(¥theta)$ and $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$, where we assume that $¥rho_{0}$ be a small positive con-stant. Furthermore, we can assume that, if $j¥in J_{2}$, we have

(6. 8) $¥mathrm{R}¥mathrm{e}$

$[¥nu_{j}x^{r+1}¥epsilon^{-¥sigma}]>0$

for some direction in the sector $¥mathrm{S}(¥theta)$ .Now by choosing $¥rho$ and $¥beta ¥mathrm{o}$ sufficiently small and by using the same method

as in our previous paper [2], we can construct an n-by-n matrix $Q(x,¥mu, ¥epsilon)$

such that(6. 9) $Q(x,¥mu, ¥epsilon)=I+O(|x|^{-1}+|¥mu|+|¥epsilon|)$

uniformly for $|¥mu|¥leqq¥mu_{1}$ , $|x|¥geqq N$, $x¥in ¥mathrm{S}(¥theta)$ , $0<|¥epsilon|¥leqq¥epsilon_{1}$ , $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ , where I isthe n-by-n identity matrix, and that the transformation

(6. 10) $v=Q(x, ¥mu, ¥epsilon)u$

reduces System (5. 10) to

(6. 11) $e^{¥sigma}dv/dx=x^{¥gamma}G(x, ¥mu, ¥epsilon)v$,

where(6. 12) $G(_{¥backslash }x, ¥mu,¥epsilon)=$diag $(G_{1}(x, ¥mu, ¥epsilon), ¥cdots,G_{s}(x,¥mu,¥epsilon))$

and $G_{¥mathrm{j}}(x, ¥mu, ¥mathrm{e})$ is an $¥mathrm{n}_{¥mathrm{j}}-¥mathrm{b}¥mathrm{y}-¥mathrm{n}_{j}$ matrix such that

(6. 13) $G_{j}(x, ¥mu, ¥mathrm{e})=A_{j}¥circ+O(|x|^{-1}+|¥mu|+|¥mathrm{e}|)$ .

It is evident that(6. 14) $v=Q(x, ¥mu, ¥mathrm{e})u_{0}(x, ¥mu,¥epsilon)$

is a bounded solution of System (6. 11). Hence the components of this solutionmust be identically equal to zero except for $j¥in J_{3}$ . This implies that

(6. 15) $Q(x, ¥mu, ¥epsilon)u_{0}(x, ¥mu, ¥mathrm{e})¥equiv 0$

uniformly for $(¥mu, ¥epsilon)$ as $x$ tends to infinity in $¥mathrm{S}(¥theta)$ . Since $¥theta$ is an arbitrarydirection in $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ , we get

,246 Y. SIBUYA

(6. 16) $u_{0}(x, ¥mu, ¥mathrm{e})¥equiv 0$

uniformly for $(¥mu, ¥mathrm{e})$ as $x$ tends to infinity in $¥mathrm{S}(¥rho_{1}, ¥rho_{2})$ . This completes the proofof Lemma 2.

Remark. In this proof of Lemma 2, we assumed that $¥rho_{0}$ is small. If $¥rho_{0}$

is not sufficiently small, then cover the sector $|¥arg$ $¥mathrm{e}|¥leqq¥rho_{0}$ by a finite number ofsmall sectors, and use the same proof for each sector.

References

[1] P. F. Hsieh and Y. Sibuya, Note on regular perturbations on linear ordinary

differential equations at irregular singular points, Funkcialaj Ekvacioj, 8 (1966),99-108.

[2] Y. Sibuya, Simplification of a system of linear ordinary differential equationsabout a singular point, Funkcialaj Ekvacioj, 4 (1962), 29-56.

[[3] W. Wasow, Asymptotic expansions for ordinary differential equations, Inter-science, New York, 1965.

(Ricevita la 2-an de septembro, 1968)