u It-; 3 ? J>O

~

~t

~

~

~

->."!

1}~~~
·1·13

Y'ME OIRS OF
NUMERICAL.MATHEMATICSNUMBER

5

EDITORIAL BOARD

HIROSHI FUJITA (Chairman)
Department of Mathematics
Faculty of Science
University of Tokyo
SHIGERU FURUYA
Department of Mathematics
Faculty of Science and Engineering
Aoyama-gakuin University
SIN HITOTUMATU (Executive Editor)
Research Institute forMathematical Sciences
Kyoto University
MASAYA YAMAGUTI
Department of Mathematics
Faculty of Science
Kyoto University

The MEMOIRS OF NUMERICAL MATHEMATICS will publish
original papers concerned with numerical mathematics, andhenceforth will be published once a year.
All correspondence should be addressed to
Sin Hitotumatu
Research Instit.ute for Mathematical Sciences
Kyoto University, Sakyo-ku, Kyoto
606, Japan

CONTENTS

Kohei SATO

Equivalence Class and Invariant Figuresof Rational Iterations with Special
Reference to the Global Convergence
Properties of Newton's Method1

Mitsunobu KURIHARA

On Approximate Solutions of Nonlinear
Volterra Integrodifferentiai Equationsin Chebyshev Series
33

Kazuo ISHIHARA

The Buckling of Plates by the Mixed
Finite Element Method73

Equivalence Class and Invariant Figuresof·Rational Iterations
with special reference to
the Global Convergence Properties of Newton's Method

Kohei SatoA B S T R ACT
Basic concepts and theorems about global convergence featuresof rational iterations with one complex variable are summarized
here.

The concept of equivalence class of a rational function,

which is implicit in the works of

G. Julia

andreformulated in view of practical application.

oth~rs,is

Properties ofthe "domain of di·rect convergence" presented by Julia arederived more systematically from those of "invariant figures"for a given rational iteration.

Applying them to Newton's

method for solving polynomial equations, we have made a schemeof1.

sk~tching

the shapes of its convergence regions mechanically.

Introduction
The iteration by a rational function with one complex

variable is one of the pro~esses most frequently appearing
in numerical analysis.

The essential part of its global

convergence features was studied

by Gaston Julia in hiscomprehensive work [4J. His work, being concerned mainly
about the topological features of convergence areas, also
contains many hints on the numerical methods for1

determin~ngtheir boundaries.

On the other hand, in the field of numericalanalysis, we hardly find general treatments on global behaviorof this kind of iteration.
The aim of this paper is to sum up such global propertiesof rational iterations that are considered to be useful innumerical analysis and to establish a practical method for
determining the shape of its convergence areas.

Some ofthe concepts and theorems presented here, including theconcept of equivalence class and theorems about Newton's
iteration, are either implicit in Juia's work [4J orreadily derivable therefrom. However, they are reformulated
in view of numerical analysis and their implications in thepractically significant situations of applied mathematics
are clarified.

In particular, the idea of invariant figures

will turn out to be useful for investigating the propertiesof convergence areas.
1.1.

Definitions.

C we mean the closed complex

Byplane or the Riemann sphere.

A rational function

considered as a mapping from

C onto C.

by~,which

~l

=~,

~2

{~9,

-

= ~~,

~3

When two subsets

of

B

and

cedent ofis a sequence of

B

B

~2

,

, ... }

where

= ~~~ etc.

The set

~o(z)

-

Z,

I ~n(W) = z}{w~_n(z).

is represented byby the relation~1

The iteration

It[~J,

will be denoted by

It[~J

mappings:

is

~

A, B

~n(A)

=B

C are related to each otherof(n ~ 1), we call
It[~J.

Aan antecedenta consequent ofAin(0 =< m < n),Ais called an n-th ante-andB

the n-th2

cons~quentof

Especially, if

A.

When
a

fig~

teA)

= A. we callofIt[.].

C,A

0, a set of fixed points of

= x}
I lim
n07()() ·n(z){zis denoted by. , etc.

It[.].

are examples of the simplest (invariant) figures of
The setsi~plyan invariant figure or

(if

U.(x)

it by
• or x is evident from the context, we shall denoteior simply by

U(x)

~o

It is well known that,U).

x

1s a fixed point of

•.

of

U(x)

U(x), each of whose connected

is denoted by

~

U.(X)

0,

The interior, or the open kernpl,

component is called a convergence region (or a domain of convergence) ofIt[.]

toward

x.

U(x)

and

U(x)

are other

examples of figures ofIt[.], while a convergence region is

not always a figure ofIt[.].

1.2.

Equivalence of rational functions.

If

•

~

and

are rational functions and there exists a linear fractional
funct10nto•T

T.T_l = ~such thatand is denoted as

•

~

Q!

,~

or

•

is said to be~ ~.

equl~~lent

The following

propositions follow readily from this definition.
(1 )rfhe relation•(3) If
•
(2)

If

then

( 4)

If
then

(5 )

If

~,

Q!

T

deg

•=

and

A

is an n-th antecedent of

deg

~.

is an n-th antecedent ofT(A)~• 2-..

.--

then
~

----to-is an equivalence relation.

Q!

T(A)

and

A

T(B)

~,

then

T[U.(x)]in

It[.],

It[~J"

1n

is an invariant figure of

TtC·]s;It [~J .

is an invariant figure oforB= U~[T(x)Jand

T[U.(x)]x

be a fixed

U.(x)

or a

= U ~ [ 'I' ( x ) J .
1.3.

Classification of fixed points.

point of

•.

Then

x

point ofU.(x) - U.(x).

Let

is either a point ofIn the former case we callx

a stable fixed point and in the latter case an unstable fixed3

point.
point

Now we define the "dissipation factor" of the fixedxby(1.1 )

p(x)

={
,

~'(x)

if

x ri

00

l/~'(x)

if

x

=

00

,

which is invariant under an equivalence transformation, i.e.,

verified thatx

1 ~ Ip(x)1<if= pv[T(x)J.

p~(x)

then

O.~

is stable if

It can easily be

Ip(x)1 < 1

andunstable

Furthermore, it is known (see, e.g., [lJ,00.

[4], [5 J) that
(1 )It[~ J

the convergence oforder in a vicinity of
(2)

(3)

if
x

x

p(x) = 0,.

is of the first order in ao < I p (x) I < 1 ,.

if

I p (x) I = 1, eitherVex) J4 ~or every vic.inity ofcontains infinitely many non-convergennt sequences ofif,.

It[~ J

values generated by

( 4)

if

x

It[~J

the convergence ofvicinity of

is of the second or higher

I p (x) I > 1, then

Vex)

= ~.

We call the case(1)strongly stable, the case(2)

weaklystable, the case(3)

semistable .and the case(4)

proper-ly unstable.

It[~J

If

shows the convergence of the n-thorder in a vic-ini ty of a strongly' stable fixed point
may dallx

a stable fixed point of the n-th order, whichis the case when
or

x, we

x

(ri~)

is an

x = 00 is an n-ple pole of(n-l)-ple zero of~ (x).

It can elementarily be verified that there aredifferent fixed points of

~

p (x)

=1

deg ~ + 1

if and only if all of them have1.

dissipation factors other than
fixed point if~ '(x)

So we callx

a multiple

and a simple fixed point otherwise.

4

Table 1.

Classification of Fixed Points

stable fixed point

unstable fixed point

(lJ )
0

weaklystablestrongly
stable,'simple

multiple

Ip I =,1=0o < Ipl < 1

p =: 1second or
higher orderconvergence

first
order
convergencedomain ofconvergenceexistsp

1 <p ;'"1

The multiplicity ~f the" fixed point

domain ofconvergence
is absent

x

x

of

of the·equation~is the~(x)=xx ~ 00, 'whereas it is the multiplicity of the'root
the equation
1.4.

zt(l/z) =.1

Julia's theory.

Ip I

the fixed point is not containedin' any domain of convergence

the fixed point is containedin a domain of convergencemultiplicity of the root0

properly
unstablesemis table

simpl,e

(A)

when

0

when
of

x = 00.

,The results of Julia's study [-4]on the global convergence properties of rational iterations
may be summarized into the following four theorems.
[Theorem I]-

deg ~ > 2

If~

point, thenand~has no multiple fixed

has at least one properly unstable

fixed point".
[Theorem II]

Let

E

be the set of all antecedents of all(1) properly unstable fixed points and (2) semistab1e fixed
pOints whose dissipation factors are roots of
with

deg ~

~

2

andE'

be its derived set.

neighborhood of every point ofof all the points on
E'E'

I , of~Then every

contains antecedents

C except at most two points, andcQincides with the boundary ofarbitrary fixed point of~.

5

U(x), where

x

is an

[Theorem III]Letx

be a stable fixed point ofdeg ~ ~ 2, and the convergence region ofcontainsx

be dehoted by

B(x).
~,

at least one critical point of
of

U(x)

other thanif

U(x);l B(x).

[Theorem IV]
with

Let
~ ~

deg

which has

x

x

B(x)

Then

B(x)

which
contains

is an antecedent ofB(x)be a semistable fixed point of~2, and the convergence region of

It[~J

on its boundary be

is a

B(x),

If

x

p, there aresuch regions, each of them containing at least~.

one critical point ofprimitive root of
B(x)'s

It[~J

with

and every component

multiple fixed point with multiplicity
p - 1

~

1

If

with

is a multiple ofp(x)is an n-th

n > 2, then the number ofnand at least one of them

contains at least one critical point ofcases, every component ofU(x)In both~.

is an antecedent ofB( x) .
Theorem I

N + 1

is connected with the fact that, if

simpl~

fixed points

x

(i

i

~

has

= 1, ... , N + 1; N = deg ~)then we have
N+l

1

r
i=l 1 - P(xi)

(1.2)

= 1 ,

which will be used in the next section of this paper.
Theorem II
every point of

implies thatE'

E'

is a perfect set (i.e.,

is its accumulation point) and thata convergence region is either simply connected or infinitely
multiply connected.
Julia named the domain
Theorem IV

D(x)oin' Theorem III

the domain of d~rect conver6e~ce.

6

If

and
B(x)

= U(x)

(which is the case if
and

H(x)

containsx

N- 1

is a stable or a double fixed point~),

critical pOints of

H(x)

is

naturally called a complete convergence region.

2.

Special Classes of Rational Iterations
It[~Jand the global convergence features ofareknown~

those ofIt[']

can easily be derived therefrom.

Hence in investigating the global features of the iteration
by any given rational function, we may find a simplest
function in the equivalence class to which it belongs.

The

following theorems are useful for this purpose.
[Theorem 2.1]~

A rational functionof degreeNisequivalent to a fraction whose denominator is a polynomialof degree
Proof:

N- 1 .
~

If

holds true.

~,is a simple pole ofIf~

the theorem obviously

is not a simple pole of~,it can beverified elementarily that there exists at least one fixed
point

x

~

of

if we putsuch that

~'(x)

T~T_l

~Theorem

2.2J

degreeN

which is equivalent to

If

If

~

x

~

00.

Hence

00

00

~.

~

of

is equivalent to a polynomial is that it has
N~thorder.

is an N-th order stable fixed point ofis obviously a polynomial of degreexand

The n.a.s.c. that a rational function

a stable fixed point of the~0

T = 1/( z - x) , the point at infinity is a simple

pole of

Proof:~~,

N .and vice versa.

is a stable fixed point of the N-th order, thenthe point at infinity is an N-th order stable fixed point
of

T~T_l

where

T. = 1/ (z - x) , and vice versa.

7

[Theorem 2.3J

If a rational function~of degree

two stable fixed points of the N-th order, thenequivalent to

zN

N
~

has
is

and vice versa.

a be fixed pOints of the N-th order.
Then putting T = (z - a)/(z - a) if a F
and a Fand T = z - a if a' F
and a =
we have T~T_1 = zN
Proof:

Let

a

and

00

00

because

zN

00

00,

is the only polynomial that has two fixed pointsof the N-th order at

0

Now suppose that

f

and

00.

The converse is obvious.

is a rational function which is nota linear polynomial, and put~(z)(2.1 )= z - f(z)/f'(z).

It can easily be verified, by means of the Laurent expansions,that all the fixed points offactors being as shown in. fTable 2.

order

fixed point of

~

p(x)

.'

1

strongly stable0m (~·2)

weakly stable1 - 11mm

properly unstable

1 + 11m

neither a zeronor a polestrongly stable

0

zero

m

weakly stablem/(m + 1)

polem (~ 2)

properly unstable

ml (m - 1)

zero
x:/-oopole

x· oo

are simple, their dissipation

Fixed points of ~(z) = z - f(z)/f'(z)where f is'a rational function of z

Table 2.

of

~

8

~

We callIt[~Jdefined bS (2.1) the Newton transform of

being Newton's method for solving the equation

= O.

fez)

Hereafter in this paper, the "Newton transform" will

be abbreviated as "N.T.".
[Theorem 2.4]

be the N.T. of another rational functionf

Proof:
let1

points ofdiffers~by the inverse of a natural nu~ber.

The necessity is evident from

Ai (i

to

is that thedissipation factor of every fixed point of
from~

The n.a.s.c. for a rational function= 1, ... , L) and~II j

Table 2.

= 1,

(jConversely,, M) be the fixed

with

(2.2)

and
~i

respectively, where

and

mj

are natural numbers.

Then

all these fixed points are simple so that from (1.2) we have
(2.3)

Hence, if we put
~(z)

(2.4)

= z - f(z)/f'(z)

where

(2.5)

if

~(oo)

llj){~k(zAi)

if

Ak = 00 ,j~k(z - llj)/¥(Z

Ai)

if

llk = 00 ,fez) =lJ(zJthen, referring to

for
~(z)

deg
and

values of

~

+ 1
~(z)

00,

Table 2, we have the relation

~(z)

(2.6)

f

n(z - llj)/~(Z - Ai)
j

=

~(z)

~ , ( z),=. ~, ( z )distinct points (fixed points of~).

must coincide with each other for all

z.

9

Hence

· The next theorem is derived directly from
[Theo~em ~.5]

Theorem 2.4.

A function equivalent to the N.T. of arational function is also the N.T. of a rational function.
On the other hand, from (2.3) we have
[Theorem 2.6]

The N.T. of a rational function has at least

one stable fixed point.
[Theorem 2.7]~

Ifis the N.T. of a rational function and~has only one unstable fixed point, thenis equivalent

to the N.T. of a polynomial.
Proof:~(z)If= z - f(z)/f'(z)unstable fixed point at·~o thatf~,thenis a polynomial.

and
00

~

has only one

is the only pole ofFrom this and the invarianceof' dissipation factors under equivalence transformation
follows the theorem.
It must be noted that the equivalence of N.T.'s of two
ratlonal functions does not mean the equivalence of therational functions themselves.
cf(z), wherec

is a constant other than

the same as that ofhand, if~

For instance, the N.T. off(z), whereas

is the N.T. offandcf ~ f.
S

0

or

1, is

On the otheris a linear

polynomial,

S~S-l

but that of

fS-l , which will be called a function

similar tof.

,~is (in general) not the N.T. of

..

10

SfS_l

f

Example 1.

If a quadratic function has no multiple fixed

point, it is equivalent to
~(z)

(2.7)
where

k

~

P ~ ±l

1,

~(z)

and

k(l =F p)

I p(oo) I > 1

(2.9)

are

2

p(±l) = 1 -we may suppose,. Since the dissipation factors of

the fixed pOints of the(2.8)

z2 - 1k(z - p)

- z -

=

k
k - 1

in

§ 1.1).

p(oo)

,

or

Re(k) > 1/2

without loss of generality (cf. Theorem I

Within the restriction of (2.9), the real parts of bothk(l + p)andk(l - p)may be less than1, which showsthat all fixe~ points of a rational function may beproperly unstable.
From

Theorem 2.2,

polynomial if either
From

Theorem 2.4,

function ifk(l + p)
and

k

~(z)

is equivalent to a quadratick(l - p)~(z)k(l + p)equals2.

is the N.T. of a rational

is a positive integer (greater thanis an even integer.

k(l + p)or1) andEspecially) if bothare positive even integers,

~(z)

k(l

~

is theN.T. of a polynomial.
If

k

=2

and

p

= 0, then~is the N.T. ofz2 - 1

which is similar to all quadratic polynomials with simple
zeros.

On the other hand, ~ ~ z2

if

k

=2

and

p

= 0.

The well-known properties of Newton's method applied toquadratic equations with simple roots can therefore be
deduced most naturally from the properties of11

It[z2J.

p)

'Note thatz/(z - 1).

z2

itself is the N.T. of the linear function

Furthermore,z2

is equivalent to2z - kz2(k ~ 0), the latter being the function iterated in the
widely used method for calculating the inverse of
Example 2.

k.

If a quadratic 'function has a multiple fixed

pOint, it is equivalent to eitherz + 1 + a/z(2.10)
The former,

ha~ing(a ~ 0)orz + liz .

a double fixed point at

00

and a simple

fixed point at-a, is equivalent to polynomials if and= 1.

The latter has a triple fixed point at

only ifa00

and is not equivalent to any polynomial.
Example 3.

Bailey's iteration for root extraction isdefined by(2.11)
wheret (z) -

n

~

2,

a

~

0

z[(n - l)zn + (n + l)a](n + l)zn + (n - l)a(cf. [7]).

,Th, dissipation factorsof the fixed points of this function are

.

Hen~e

odd.

,

p(a'lS) = 0(2.12,)
t,p(O) = p(oo) =

n + 1
n - 1

must be the N.T. of a rational function if't (z,) = z - f(z)/f'(z)In fact, we haven-1fez) = z -T(zn - a)(2.13)

nby putting.

It is easy to verify that all the stable fixed points ofare of the third order.
is equivalent toz3

Especially, if

from

Theorem 2.,.

12

is

t

n' = 2, the function

3.

Divergence Centers
If we cannot find a rational function~

class ofsuch that theE"

not an analytic curve [3],[4J.
the shape of the

E'

It[~J

of

segment~

circle or a closed line~.

in the equivalence

is known to be athen the

E'

It[~]

of

is

In this most general case,can be traced only numerically, forinstance, by plotting as many points ofE

as possible.

This method of numerical plotting has several theoreticaladvantages as will be shown'later.

For simplicity, the setof all unstable fixed points ofother than semistable ones~

with dissipation factors whose arguments are incommensurable
multiples ofwill be denoted by'/I'

antecedent of

A
0

A

A's

If the numerical values ofn-+co

A's E A,are known, those of

calculated by solvingknn

lim En = E.

then we haveo

If we write

A = {A }.

nand the n-th

0

0

V
A ,
En = k=O
k

(3.1)

€ A

by

A =- O.}

1

~(A)

k

1

A's E A etc.
2

2

= k-l
A successively.

ocan ,beHereafter

will be called a divergen6e center of the k-th order.

~xa~le4.

It can easily be verified that every, quadratic

polynomial is equivalent to4> (z) _ z2

(3.2)
where

p

is a constant.

case where
C1

and

co,

p

is real.

+ P ,Here we restrict ourselves to the
The fixe~ points of (3.2) are= (1 - II - 4p)/2,

rhe~r gi~§ipation

C2= (1 +'11

factors being

2Cl

4p)/2
) 2C2

and

we haveif(3.4)-3/4 < potherwise .

13

~

1/4 ,

0,

The divergence centers are calculated successively by meansof the formulae:(3.5)

A = -A
0

1

Rough sketches ofE'values of the points ofp

are shown in

,

A =
k

±.,t A - p

mechanically made from numericalES

for several real values ofFig. 1. (Cf. Example 5in§ 5. )

I' ....".

p ~ -2.500.

(k ~ 2)k-li' ·.....

!

'i-'-------"'1 "'--"-j"

.~

,,:

o

:

:

!

'!
! ,. • -2.000

I' ·j ,. • '1.0011

I

p. 0.000~.
········(r~~·
·
o

:

····
:

:
i:i.g.l..

.

1 ,. = 0.250-<>.".

~!
.i ,.\

P = 0.500

I'~

r

:

+-.r.

'\

i /'.~..

.......... ,. __ ..... _.. L~,_ ....... _.... ~. "" it
: ~'~

..

\0

....
:

:

:

"T'

"

t

Ju11~'1i 61niularitY fiet: lli' for It[z2 + p] when ptakes several real values (mechanically plotted fromnumerical values of the points of

14

Ea)

4.

Minimal Figures
If

A~ B

we calla r{gure

and both

A

and

B

are figures ofB

a subfigure ofA.

If every sub figureAsuch that

It[~].

Bof(4 .• 1 )bas the same closure as
It[~].

ofA, we callAa minimal figure

A minimal subfigure ofA

which is not containedin another minimal subflgure ofofAwill be called ~. kernelA.
For example, either ~'fixed point ofor a set of~

fixed points' of

~

deg ~

E, the set of all the divergence centers,~

2, thenIt[~].

is a'minimal figure ofis not a minimal figure of

It[~],.itikernel beingi.e., the set of all' unstable fixed points of
An n-cycle ofpointsXl, X2,

It[~],••• , xn

Ifi.e., the set of

n

~(=

EO)'

~.

distinctsatisfying(4.2)
is another example of minimal figure.
From the above definitions, the following proposit~ons
are easily derived.
(1)

If a minimal figure is a finite set, each of its elements
is either a fixed point or a member of some cycle.

(2)

If a minimal figure consists of a finite number of·connected components, each. component coincides with oneof its own consequents.

(3)

The kernel of a. figure which i, perfect (i.e., whichcoincides with its derived set) is perfect.

15

(4)

The kernel of a figure which is an open set is an open
set.

5.

Areas of Direct and Indirect Convergence
Hereafter we always suppose that
N :: deg(5.1)

The kernel of

U(x)

~

~

U(x)

and

2

will be denoted by

the area of direct convergence toward(4) of the preceding section,
component domain of

H(x)

H(x)

H(x)
x.

Especially, if

fixed point,H(x)As a naturalA

is nothing but Julia's "domain

x

R(x)

in thisis either a stable or a double

consequ~nceR(x).

of its definition,

It[~J.

a multiple fixed point,and called

is an open set.

coincides witha minimal figure of•

From the propertyof direct convergence" which is denoted by·paper.

yi ~

If

H(x)

x

H(x)

is

is either a stable oris also a minimal figure of

It[~J.

If

U(x)

yi

H(x), we callU(x) - H(x)indirect convergence towardxthe area ofand its connected component

a domain of indirect convergence towardx.

A domain of

indirect convergence that is an n-th antecedent ofwill be denoted byR(x).

denoted byThus we haveU( x )

(5.2)

5.1.
be~n

H(x).

co

co

n=on

n=O

H(x)

proved by Julia thata cyclotomic equation.

H(x)'s will be= V U ( x ) = UV H(x) •Properties ofcomponents if either

The union of all

H(x)

and

H(x)

I peA) I < 1

Hi&..

It has alreadyhas a finite number of
or

p(x)

is a root of

The following theorems will hold16

true even if

Ip(x)1.cyclotomic equation.
H(x)

and

H

for

=1

and

p(x)

is not a root of' a

For simplicity, we may write

H(x)

if

x

H

for

is either understood orindeterminate.
[Theorem 5.1J
onag(x).

Proof:

If

obvious.

There is at least one unstable fixed pointx
If

is a semistable fixed point,x

the theorem isis a stable fixed pOint, it is a simple

fixed point so that at least one fixed point
from

x

ag(x)

exists.

The fixed point~

or in a region

that is a subset of

y

and

~

a~

distinct

must lie either onbounded by a closed Jordan curveag(x).

In the former case,be unstable and the theorem holds.
both

y

y

must

In the latter case,being invariant figures, we can prove theexistence of at least one fixed point ona~by reductio

ad absurdum, through applying Caratheodory's theorem [9]td~.

This fixed point on[Theorem 5.2]

If

H(x)

g(x)'s, then every

is naturally unstable.

a~consists of a finite number ofH(x)coincides with one of its own

consequents, i.e., there exists
that
Proof:

H(x)is a figure ofnfor each

H(x)

such

It[~n].

This theorem is a special case of the property (2)of minimal figures mentioned in the foregoing section.
LTheorem 5.3J

If

H

is a figure ofIt[~n]'

it is multiply

connected if and only if it contains a pair of criticalpoints of~naround which the same branches ofexist.

17~-nProof:r

Let

fixed point ofIt[.n]·

be a connected component of•exists.

r

Then.-n(r)

one branch of

vicinity of a point onother thanE'

least LWo branches ofrcontains at leab\itself. (For, every

contains antecedents of all other

E' due to Theorem II.)

points onon which aomust be a figure of

B is multiply connected, aB

IfaD.-n(z) ~B must contain atHence

B divided by a closed curveon which a pair of critical points of.nexist.

Conversely,

if

B contains such branches of .-n(z)

(l

B,

dB

has at least one branch other thanritself so that

As was already painted out by JuJla, a

B that is notthen.-n(f) ~B cannot be simply connected.
simply connp.cted is infinitely multiply connected.
The next theorem can readily be derived from
[Theorem 5.4J

(1)

B is a figure ofIf

It[.n J , it issimply connected if there is only one unstable fixed
point of

.n

on

dB;

simply connected if there is only one critical point

(2)

of

(3)

·n in D . ando'multiply connected if all the critical points ofare contained in

5.2.
fronl

Theorem 5.3.

B.

Divergence centers on

dR __'On the other hand,

Theorem 5.1

and the definition ofR, we haveaR

contains at least one

A
and non[Theorem 5.5J
that

.n

A suchmm < n.

Furthermore, sincebranches ifx~s.-l(z) ~B(x)has two or moreeither a stable 6r a multiple fixed point,18

there must be at least one}onag(x).

Hence thefollowing theorem is verified by mathematical induction.
[Theorem 5.6J

Ifx

is either a stable or a multipleaR(x~fixed point, there exists onof any order not less thana divergence cente

v '

n.

The following theorem can be proved in a similar way.
[Theorem 5.7J

equation~

If

p(x)

is a root of a cyclotomic

there exist onag(x)divergence centers of all orders.

These theorems are useful for numerically tracing theshapes of convergence regions by the method described in[Theorem 5.8]

If it is known thatIt[~nJ

region which is a figure ofhas

M branches, then

B

aB

§ 3.

is a simply connected~-n(Z)~ B

and thatis a closed Jordan curve or

not according as the number of fixed points of

on

~n

ag

equalsM- 1

Proof:

If

is a simply connected domain, there exists aBor not.

univalent analytic functiongwhich maps

interior of the unit circle

c

[9J.

Since

the
B ontodef'I'= g~ng-1

maps the interior ofConto itself and its inverse

has

C,

'I'

M branches on

points on

C.

In order thatcurve, these fixed points of

has

M- 1

aB
'¥

aD
o

contains

M- 1

proved that the mappingaB

different fixed

may be a closed Jordanon

one-to-one. to the fixed points of
if

'1'-1

C
~n

must correspondon . aD., Conversely,odifferent fixed points, it can beaB ~ C

must be a closed JorQan curve.

19

is a bijection and hence

5.3.
N = deg

Polynomial case.
~~of order~(z)If

2, the point at infinity is a stable fixed pointNso that we have~-l(z)

and all branches of~.

critical point ofandTheorem 5.8

(1)

B(oo)

to

are around00

which is a

Therefore, applying

Theorem 5.3

8(00), we haveis simply connected if and only if it contains

no finite critical point of(2)

E'

is a polynomial with,.

~is a closed Jordan curve if and only if no finite

critical point is in

8(00)

~

and

has

N - 1

different

unstable fixed points.
Example 5.

In caseconstant,

B(ooJ= z2 + P~(z)p

is a realis simply connected if and only if

8(00), which is equivalent to~-2elementarily.

On the other hand,

fixed point if-3/4 < potherwise (cf.

Example 4).

if and only if-3/4 < p

Example 6.

N ~ 3,

If

where

~

~

1/4

P ~ 1/40 ¢as is verified

has only one unstable~

and two unstable fixed points

So' E'
1/4.

is a closed Jordan curve(See

Fig. 1.)

it sometimes happens that a multiply

connected convergence region coexists with a simply connectedone.

For instance,

B(oo)It[z3 - 3z + 3Jofis multiply

connected because it contains a finite critical point
while another'finite critical point
stable fixed point and hence
connected,

Moreoyer::

~B(l)

8(1)

1

-1,

coincides with a= H(l)

is simply

is a closed Jordan curve

because it contains only one single critical point of20~.

Fixed points other than
Therefore

E'

1

~

and

are properly unstable .

crete closed Jordan curves which separate

6.

U(l)

from

U(~).

Newton's Method Applied to Polynomial Equations
Let

and

fez)

4>(z)

N (~ 2)be a polynomial withbe its Newton transform.

(6.1)

N*

d~f

"-.,

Zeros ofa~.

consists of a countably infinite number of dis-

f

~

deg f

different zeros

We have
N = deg 4>

are stable fixed points of.

4>, while the point

infinity is the only unstable fixed point of4>,with thedissipation factor(6.2)
Hence, the following properties ofIt[4>]

are readily

derived from the theorems in the preceding section.
(1)

Every

B is a simply connected infinite domain, while

D (n ~ 1) is a simply connected finite domain.
n

(2)

ag

B contains

is a closed Jordan curve if and·only if

only one critical point of

4>

which is a simple criticalpoint.

aD(n ~ 0) contains divergence centers of all ordersnequal to or greater thanof orders less thann,

and never contains thosen.

B which is bounded by a closed Jordan
curve will be called a normal B' Since there are N
For aimpllg1ty. a

different stable fixed points and no more thandifferent critical points of

2N - 24>, the following two

propositions are derived from this definition and (2).

21

(4)

At least two(5)

If oneB

B's

are normal.

is complete, all the otherB's

are normal.

On the other hand, it can be proved from (2) and~-l(z)

Caratheodory's theorem thatboundary of a normal

(6)

If

on

0

A's (nncase of~,mi~contains only oneaB2), the arrangement of

A

and

I

o ~ k < n}{AIkbeing the same as that on the unit circle in the

aB

6.1.

which implies the following fact.

is normal, thenD

2n - lB,

has two branches on theIt[z2].

Center of gravity of roots.
andG

Now we define symbols

as follows.

a stable fixed point of ~ (i.e., a root of f = 0);o. (another index i
is attached as the subscript if~necessary).
~(k ~ 1).

: a root of.

mt : the multiplicity of

as a root of

G : the center of gravity of the roots of

f

= o.

f = 0, i.e.,N

G = l m·u·/N* .
i=l ~'O ~(6.3)
Note that, if

(6.4)

~(z)

This implies that,Iz - GI=G +is sufficiently large, we have

z - G
p(oo)

if a figure ofin every neighborhood ofIt[~]contains a point

00, it is a repe~ition of similar

configurations which are arranged asymptotically like ageometrical progression withFor example, if

G as the center of similitude.

B is normal and. we define

exterior of the circle with center

22

G

X(R)

and radius

as theR, thenag r"\ X(R)is approximated by(6.5)

U'{p(oo)n(z - G)n-O

+ G I z e.. A}

,

where

(6.6)

A =

ag" X(R) - aB t\X[p(oo)R]ag".

usually consists of two half-closed finite arcs ofag

Hence it is inferred thatis an analytic curve only

when it coincides with a straight line which passes

aDo-In the other (general) case,must be a curve somewhat

like a branch of hyperbola --- more exactly,
has enumerable common

t~ngents

G.

X(R) r"\aB

with a hyperbola-likeanalytic curve whose asymptotes are two straight linesaB

G, and all the divergence centers on

passingaBonodes of

6.2.

are

N = 2, both

B's are normal

and complete (irrespective of the values ofN*) and hence

Quadratic case.

(6.7)

If

(i = 1, 2)an(lll.') = aU(ll') = E'
o O.
ol.

is reduced to an infinite closed Jordan curve.
be verified tnat
ml > m2 ·
ml

in

§ 6.1

. 6.3.

while
if

if

G = A
1

Therefore,

= m2

if

It can easily

E'

i~~s

is a straight line passingGa non-analytic curve as was described

ml F m2

Main part of

aB~

If

M > 3, every neighborhoodof an arbitrary point onaBof closed Jordan curves.

We shall call each of these closed

contains an infinite number

Jordan curves a Jordan component ofTheorem II,aBo

the exterior (with respect to

Jordan component contains at least one

23

From Julia's

B) of every

conv~rgenceregiontoward another stable fixed point.
It[~J+s an invariant figure ofHence the closed curve
if and only if its exterior

contains another domain of direct convergence, i.e., if andonly if it is an infinite Jordan curve.

dB

component of

Such a Jordan

and the union of all the main components ofdB

Both a main component and the main part ofminimal figures of

It[~J,

aB

dB

are

the main part being the kernelof the union of all the Jordan components ofNote thatwill beaBocalled the main part of

(7)

dB

will be called a main component ofdB.

is a tree-like chain of infinitely manyclosed curves and the tops of its infinitely many "twigs"
do not belong to any Jordan component.
On the other hand, the following propositions are provedby the same

reasoning~s.wasused in the proof of

Theorem

5.8.
If

(9)

Each of the main components of
A
1

and

M ~ 3,

aB

(8~

2n-l

has

M- I

main components.

aB

contains only one

A's (n ~ 2), their arrangement being thensame as that on the boundary of a normalD.
oFurthermore, the foregoing description on the shape of the

B applies also to the shape of each
aB in this case. Hence a normal B canboundary of a normal
main component of

be considered a~ a special

B whose boundary consists of

only one main component.

6.4.

Mechanical sketching.

Based upon the above proper-

ties of Newton's method for polynomial equations, we have24schemed aproces~of making a sketch of the convergenceregions mechanically.
the zeros ofFirst, we obtain numerically all

f, all the critical points of~and allthe divergence centers of a few lowest orders.

Then, wedraw smooth curves each of which contains divergenceaBcenters of lowest orders on a main component ofandis asymptotic to a pair of half-straight lines starting
from

ro

G.
and

Let the union of all such curves be denoted by~-n(ro)

by

Then

rn·

rn

gives a sketch ofI 0 ~ k $ n} and, among the parts of rn approximating
aD's , those for lower values of k will approach their
{aD
k

k

exact form asnincreases

theoretically, at least.

In practice, we cannot draw too microscopic details of theconvergence regions.

We have omitted them in the process ofnumerical calculation throughneglecting such divergencecenters that are too close to those of lower orders.
If we calculate all~'sparallel tork

and plotrk )their locations (except those which are too close towith appropriate symbols, most remarkable convergence regions
toward each zero offwill be discriminated from thosetoward other zeros.

(See Figs.

2--5~

where the convergenceregions toward a' fixed point identified by a numeral are
distinguished by the same small numerals at

6.5.

~'s

in them.)Equations whose roots are all simple.

the critical points of~are zeros offf"

N = N*,

If
and

A's

are

1

zeros oftof'.

On the other hand, ~(z)[f(z)/(z - ~)J'
o

=0

if

~
0

25

= ~o ~ zis equivalentis a simple zero off.

It[~J.

Hence follows the following property of
If all the zeros of(10)

are simple, thenfis normal if no zero offll

is in it,

1°

B

2°

B is complete if all the zeros of3°

the gravity center of all the critical points ofisfll

are in it,
t

G,

G,

the gravity center of all the

A's

is

the gravity center of all the

lJ'S

contained in1andU(ll.)oJ.

1is

I llj/(N - 1) def= Gi •

(6.7)

jii 0

G =

We may restrict ourselves to the cases where~ithout°

loss in generality, since the other cases are

obtained simply by parallel displacements therefrom.
Examole 7.

In casefez)= z3 + pz + q ,where~ = 4p 3 + 27q2 ~are real constants satisfying
origin coincides withG

and the only zero of

p

and

q

° , thef".

So

B

is complete or normal according as it contains° or not.

Since all the coefficients ofmust be real

B(~)

if

~

t

is real,0, which happens only whenof the complete

B

is

E'

p~ >

lJ

o

0.

The boundary

so that each of its two main

components must coincide with the boundary of a normalp~If< 0, then we have

p <

B.

° so that there are two

real

A's, one of which must be a common boundary point of

two

B's
If

toward conjugate complex zeros ofp~= 0, the origin is the onlyA1common to the boundaries of all the three26

f.
which must be
B's.

Thus in this case, there are four types of global
convergence features ofIt[~J

as shown in• 3 _ 20z _ 25 • 0

Fig. 2 .

.3 + Z _ 2 • 0

III

•

,,
,,
(31
---~--------~------~---+--------.-----------,,
,,
,

il)
(21

•

~3 _ a • 0(a ~ 0)• 3 _ 2. - , • 0

131

•

(2)

•

(3)

•

iii)
(:n

•

,,

iv)

,•
••

"

I

••
•
I
I0
0

'F1lZ •."

2.

Four types of global convergence featurea of Newton's method
applied to the equation z 3 + pz + q = 0
where p and q
are real and satisfy b. - 4p3 + 27q2 :f 0i) p < 0, b. < 0ii) P > 0,b. > 0,.

iii) p - 0,

27b. > 0., iv) p < 0, b. > 0 .

Example 8.

In casefez)= zN - a(a ~ 0, N ~ 3), theorigin is the Qnly zero offlland the only

same time.

B's

are normal and, from

Hence all theAat the1

Theorem 5.5, the origin-is
common to the boundaries
of all theg's and all

the

D's.

Every

one

D

1
1

N - 2

if
~'s

U

has

N = 3, andN ~ 4.

if

Every finite divergencecenter is therefore common
to the boundaries of
N + N(N - 2)

= N(N - 1)

convergence regions.
Fig. 2 iii)

Fig. 3(See

Convergence regions of Newton's method
applied toFig. 3.)andz4 - a = 0,

where

a

a positive real constant.

Example 9.

In casefez)= zN - azorigin is the only zero ofsame time.

Therefore,N - 1

at theis complete and the other

~'s

being

B because

ag(O)

aB(O)

= E'.

B's

consists

The total(N - l)(N - 2), every

to the boundaries of two

and

B's

A
1

N - 2- D's.
1every finite divergence center is common tocomponents ofthis case,f = 0

components, each of which must coincide with theboundary of anothernumber ofB(O)and a root of

From (8), the main part ofare all normal.
offll(a ~ 0, N ~ 3), the

~

aD(O)
o

= E'.

(See

Fig. 4.)

Therefore,

N - I

JordanNote that, inis equivalent to a polynomial.

28

is common

is

131

•

,,
,,
,,
131
-----0----,,,
,
,,,
,,
,
,,
,
,,,
,

,

"

------------0---121

,
,,,

.

!~,

~l'- -----:.~---------------------- -------

I

'.

121

,

i)

Fig. 4.

Convergence regions of Newton's method applied toi)Example 10.
zeros

ii)

z3 - az - ~

and

ii)

Let us take up the case where(i= 1, " ' , N)

For simplicity,

B(~i)

f

N ~ 3

and

will be meant by

Bi.

where

(a > 0).

z4 - az - 0

N real

has
~i

o

~i+1

<

0

.

It is known

contains only one zero of f"
if 2 < i ~ N - 1.
8i
(Cf. [7],[8J.) Therefore, M = 2 inand 8N' whereas
81

that

M= 3

in the other

B's.

In other words,

Hl

and

normal, whereas the boundary of each of the otherBN

D's ,has0

two main components.

Each of the main components crosses

the real axis atwhich is a simple zero ofA1

it can be proved thata8 + 1 .

ag il' v ,

arewherefromshare::; a maln, cumponenl withagi (2 ~ i ~ N - 1) iscornman to a pair of Jordan components of agio
(See Fig. 2
1

i) ,

Fig

Every divergence center on

0

4 i)

and i Fig. 5.)29

-~-~-

Fig. S.

Convergence regions of Newton's method applied toz4 - lSz 2 - 10z + 24 • 0 whose roots are all real.

A Q K NOW LED GEM E N T S
This study started from preliminary experimental researct
about the convergence regions of Newton's method applied to
cubic equations, which was performed by
Mr. Masato takeichi

Mr. Toshiyuki Iida,

and the author as a generalization of

RaIl's "convergence map" [12J and was reported at the meetingof the Union of the Japanese Scientists and Engineers (Nikkagiren) two years ago.
sincere thanks toIri

The author would like to express his

Prof. Sigeiti Moriguti

andProf. Masao

for many theoretical suggestions and .continuing encour-

agement.

He is also indebted toDr. Tsutomu Date

andProf. Toshio Uno,

Mr. Takahide Ogawa

whose comments on

several theoretical issues were inspiring to the author .
. Numerical calculations of the ex.amples contained in this30

pa~erwere conducted by the

HITAC 8800-8700

computer systemin the Computer Centre of the University of Tokyo, and themechanical plotting of convergence regions was made with theflat-bed type X-Y plotter connected to the system.

REF ERE N C E S~:JG. Konigs,

int~grals

Sur lesfonctionelles.

de certaines

Ann. de l'Ecole Normale~quations

Sup~rieure,Suppl. nO 2, nOs 24, 25 (1884), pp. 3 - 41.
[2]

M. Fatou,~quations

15 Oct.

[3J

Sur les solutions uniformes de certaines
fonctionelles.

(1906), pp. 546 - 548.

M. Fatou,

Sur les substitutions rationelles.

Acad. Sci. Paris,[4J

C. R. Acad. Sci; Paris,

G. Julia,

C. R.

21 Mai (1917), pp. 806 - 808.

Memoires sur l'iteration des fonctions

rationelles.

J. deMath~matique,7 e serie, Tome IV,

nO 1 (1918), pp. 2 - 245.
[5]

J. F. Ritt,

On the Iteration of Rational Functions.

Trans. Amer. Math. Soc.,[6J

N. Bourbaki,

21 (1920), pp. 348 - 356.

Les structures fondamentales de l'analyse

III, Topologie generale.

Actualite Sci. Ind.,

Hermann.& Cie, Paris (1948).

[7]

A. S. Householder,Principles of Numerical Analysis .

. McGraw-Hill, New'York (1953).

[8]

S. Gorn,

Maximal Converge~ce Interval and a Gibbs-Type

Phenomenon for Newton's Approximation Procedure.
Ann. of Math.,59-3 (1954), pp. 463 - 491.

31

[9J

J. A. Jenkins,
Mappings.

[lOJ

Univalent Functions and Conformal

Springer, New York (1958).

Y. Yoshida,

Tenshfigoron Nyfimon (Introduction to

the Theory of Point-Sets).
[llJ

[12J

Y. Yoshida,

BaifOkan, Tokyo (1960).

Kansfiron (Functions of a Complex

Variable).

Iwanami Shoten, Tokyo (1965).

B. RaIl,

Computational Solution of Nonlinear

Operational Equations.

John Wiley & Sons, New York

(1969).

Kohei Sato
Department of Mathematical Engineering
and Instrumentation Physics
Faculty of Engineering
University of Tokyo
Tokyo, Japan32

On Approximate Solutions of Nonlinear Volterra
Integrodifferential Equations in Chebyshev Series

Mitsunobu Kurihara( University of Yamanashi, Japan )O.

Introduction
The present paper is concerned with the multi-point boundary

value problem for nonlinear Volterra integrodifferential equations.
Let(0. 1)ddtx(t)=f(t,x(t),Jt g(t,s,x(s»ds)-1

be a given system of nonlinear integrodifferential equations
subject to a given multi-point boundary condition(.0. 2)

Here x(t) is the unknown vector valued function of t defined on
the interval J={tl-l<t~l}·. Denote that

S={(t,s)l-l~s<t<l}.

Assumethat f(t,x,y) and g(t,s,x) are vector valued twice continuously
differentiable functions of (t,x,y) and (t,s,x) on the domains
JxDxE and SxD respectively,where D and E are bounded open domains
in the Euclidean space with the Euclidean nor~» " which satisfy

the relation(0. 3){xiII xll<2max{\\g(t,s,x)1I1 (t,s,x)e-SxD}}CE.

Moreover assume that Li(i=O,l, ••••• ,N) are square matrices,

~

is a

-vector and ti(;t=O,l, ••••• ,N) are given points belonging to theinterval J such that-l=t o<t 1 <o····<t. N=1.
It is clear that our boundary value problem'includes as the
special cases Cauchy problem, two-point boundary value problem
and Hukuhara's problem.

33

The system of linear integrodifferential equations whichcorresponds to the system (0. 1) is written in the form
(0. 4)~tX(t)=A(t)X(t)+Jt B(t,s)x(s)ds+h(t),-1where A(t) and B(t,s) are square matrix valued continuous functionsof t on the ihterval J and of (t,s) on the domain S respectively
and h(t) is a vector valued continuous function of t on theinterval J.
In the present paper we shall construct the Green function of
the multi-point boundary value problem (0. 4) and (0. 2). The
Green function plays an impotant role in studying approximate
solutions of the nonlinear integrodifferential equations (0. 1)
and the boundary'conditi~n (0. 2). We shall prove three main
theorems on approximate solutions of the boundary value problem(0. 1) and (0. 2). The first theorem, Theorem 4.1, says that for
any isolated solution there exists an approximate solution
accurately as it is desired by computing finite Chebyshev
polinomial series. The second theorem, Theorem 4.2, says that theobtained Chebyshev approximate solution corresponds one to one tothe isolated solution. The third theorem, Theorem 4.3, says that
one can always assure the existence of an exact solution bychecking several conditions on the obtained Chebyshev approximate
solution and further it gives a method to obtain an error boundof the obtained approximate solution.
The analogou~ theorems were originally proved by M. Urabe [5J,

[6] Qopcerning the multi-point boundary value problem for nonlinear
differential equations. Later concerning nonlinear integral
equations, the theorem analogous to Theorem 4.3 was proved by
M. Shimasaki and K. Kiyono [2] and the theorems analogous to
Theorem 4.1 and Theorem 4.2 were proved by the author [7J especiallyfor nonlinear integral equations of Fredholm type. On the other
hand K. Tsuruta and K. Ohmori [3] proved the theorem analogous to
Theorem 4.3 concerning the Cauchy problem for nonlinear integrodifferential equations and gave some examples of the numericalsolutions with their aposteori error bounds.
Throughout the present paper we denote Euclidean norms forvectors and matrices by the symbol II

II. Moreover for any vector

valued function x=x(t) continuous of t on the interval J we use
two kinds of norms IIxllc and IIxlIQ which are defined as follows
(0 ..

5)

IIxllc=ma,x{lIx(t)JlltEJ}

and(0. 6)In order to prove Theorem 4.1, we use the following lemma
proved by M. Urabe [4] based on Newto~~Raphson's jrocedure fornonlinear algebraic equations.
Lemma 0.1
(0. 7)

Let

F(Cl)=O

be a given real system of equations, where Cl and F(Cl) are voctors

35

of the same dimension and F(a) is a continuously differentiable
function of a defined in some region n of the a-space. Assume
that the system (0. 7) has an approximate solution a=a for which
the determinant of the Jacobian matrix J(a) of F(a) with respect
to a does not vanish and that there are a positive constant a anda nonnegative constant K<l such that(1)
(0. 8)

na={allla-aU~8}Cn,(2) . \\J«(l)-J(a)n~K/M(3)

for any aen a ,Mr/(l-K)<a,where rand M are numbers such that
( 0 . 9)I\F (a )1\ <r

andIIJ- l (a)1I <M.

-

Then the system (0. 7) has one and only one solution a=a in Qa
and for a=ait holds that
(0.10)

1.

detJ(a);iO

and

1\ a-al\~Mr/ (l-K') .

Some Properties of Chebyshev Series
Denote by Tn(t) Chebyshev polynomial of degree n, that is,(1. 1)

Tn(t)=cosne

as t=cosefor n=0,1,2,· •• ••. Then it is well known that for any continuous
funotion x(t) of t on the interval J we have Chebyshev polynomial
series expansion of the form00

(1. 2)x(t)~Lea T (t),n=O n n n

where
(1.

3)

e =1

e n =12 for n=1,2,·····

and

o

and
(1. 4)For the expansion (1. 2) we obtain the Parseval's equality of theform(1. 5)

2~

2

IIxl\ Q= l lian II •n=O

In particular for finite Chebyshev series of the formm

(1. 6)

x (t)= Lea T (t)mn=O n n n

we haveand(1. 7)

where a=(aO,al,·····,am). In fact the inequalities (1. 7) areproved making use of the Parseva1's equality (1. 5) and Schwarz's
inequality.
Suppose that x(t) is a continuously dlfferetiab1e functionof t on theinte~val

J. Let the Chebyshev polynomial series of thederivative of x(t) be00

ddtx(t)~ L e a'T (t).
n=O n n n

Then it is easily proved that(1.

8)(ale n-l. )a'n-1 -a'n+1 =2na n37(n=1,2,·····),

where a n (n=1,2,·····) are the coefficients of the expansion(1. 2) of x(t). Moreover making use of the relation (1. 8), we
obtcrin00(1. 9)

a'=l:2e I (n+2p-l)a +2 1·
n
np=l
n p-

We define the operator P which expresses the truncation ofmthe Chebyshev polynomial series (1. 2) of the operand discarding
the terms of the order higher that m, thatis~for any continuousfunction x(t) of t on the interval J expanded in the form (1. 2)(i.lO)
If x=x(t) is a continuously differentiable function of t on theinterval J, it is proved that for m=O,l,·····(1.11 )

H( I-Pm) xU C~a (m) 1\( I-Pm-l) ~~\IQ <a (m) II~~ llQ'

(1.12)\I(I-Pm)x!\Q<a l (m) n(I-Pm_l)~~IIQ<al (m)II~~UQ'(1.13)(\~t ( I-Pm) xII c~(m+2) II (I-Pm-l) ~~ I~ + \Ie I-Pm+l ) ~~ IIQ

and(1.14)
Here and hereafter I ii the identity operat~r,

P~l=O

and a(m) andal(m) are monotone decreasing functions of m satisfying(1.15)

1:2/(m+l)<a(m)</2/m and 0l(m)=l/(m+l).

These properties of Chebyshev polynomial series
detail in the paper by M. Urabe [6J.

are proved in2.

Linear Integro-Differential Equations
In the present section we study a system of linear integro-

differential equations (0. 4). We put(2. 1)

Q(t,S)=A(t)+JtB(t,S)dSs

and consider the matrix equation(2. 2)

R(t,Sl=I+J:R(t,UlQ(u,SldU

on the domain S. It is well known that there exists uniquely thecontinuous function R(t;s) satisfying the equation (2. 2). R(t,s)
is also the unique solution of the adjoint equation(2. 3)~SR(t,Sl=-R(t,slA(Sl-J:R(t,ulB(U'SldU

satisfying the condition(2. 4)

R(t,t)=I (identity matrix)

on the domain S. The function R(t,s) is called Resolvent matrix
with respect to the matrices A(t) and B(t,s).
We introduce the following lemma proved by Tsuruta K. and
K. Ohmori [3].
Lemma 2.1

The resolvent matrix R(t,s), which is theunique solution of the equation (2. 3) satisfying the condition(2. 4) on the domain S, 1s differen~iable with respect to t andsatisfies the equat~on(2. 5)~tR{t,~)=A(t)R(t,S)+JtB(tJU)R(U,S)dU.
s

39

Let us consider the Cauchy condition(2.6)·
Then we shall prove the following lemma.
Lemma 2.2

For any vector Xo and any continuousfuncti~nh(t) the system (0. 4) subject to the condition (2. 6) isequivalent to the system(2. 7)

X(t)=R(t,-l)Xo+f t R(t,s)h(s)ds-1

on the interval J.
In fact, similarly to the proof by Grossman S. I. and R. K.
Miller [lJ, for any solution x=x(t) of the system (0. 4) satisfying~hecondition (2. 6) we haveda
Jt_l{R(t,s)dSx(s)+[asR(t,s)JX(S)}dS

Then it follows from the equation (2. 3) that
X(t)-R(t,-l)Xo-J t R(t,s)h(s)ds-1=Jt R(t,S)[A(S)X(S)+J s B(s,u)x(u)du+h(s)Jds
-1

-1

a
Jt R(t,s)h(s)ds+ J t [asR(t,s)Jx(s)dS-1-1a= ft_l[R(t,S)A(S)+ ftsR(t,U)B(U,S)dU+asR(t,S)]X(S)dS=0.

40

Conversely if x(t) solves the system (2. 7), it follows from. theequation (2. 5) and the condition (2. 4) thatd
Jt B(t,u)x(u)du-h(t)dtx(t)-A(t)x(t)--1=~tR(t,-l)Xo+ft ~tR(t,S)h(S)dS+h(t)-1-A(t)[R(t,-l)X o +f t R(t,s)h(s)ds]
-1

-f

u

t

-1

B(t,U)[R(U,-l)X O+J

R(u,s)h(s)ds]du-h(t)-1d R(t,-l)-A(t)R(t,-l)- Jt_lB(t,U)R(u,-l)dU]X
=[at
o

a
Jt B(t,u)R(u,s)du]h(s)ds+ Jt [atR(t,S)-A(t)R(t,s)-1s=0.

It is clear that x(t) satisfies the condition (2. 6). This
completes the proof of Lemma 2.2.
We shall construct the Green function H(t,s) for the system(0. 4) and the multi-point boundary value condition (0. 2).
Lemma 2.3

Let R(t,s) be the resolvent matrix withrespect to the matrices A(t) and B(t,s). If the matrix(2. 8)

is nonsingu1ar, then for any constant vector ~ and any continuousfunction h(t) tbe system (0. 4) subject_to the condition (0. 2)
is equjval~lIt to tile system
41

(2. 9)X(t)=R(t,-1)G- 1 i+J 1 H(t,s)h(s)ds,-1

where for t k _ 1 <t<t k (k=1,2,·····,N)-R(t,-l)G-

1 N

L LiR(ti,s)+R(t,s);t _l~s<t_

i=p

P

P

(p=1,2,·····,k-1)
1 N-R(t,-l)G- i~kLiR(ti,s)+R(t,S);tk_1<s<t(2.10)

H(t,s)=-R(t,-l)G-

1 N

L LiR(ti,s)i=k;t p. l<s<t=p(p=k+1,·····,N).
In fact, by Lemma 2.2, for any constant vector Xo any solutionof the system (0. 4) satisfying the condition (2. 6) is expressedin the form (2.

7). The solution (2. 7) satisfies the boundary

condition (0. 2) if and only ifi=N
L Lix(ti)=Gx O+ NL Li Jti R(ti,s)h(s)dS.
i=O
i=O
-1

(2.11)
Substituting (2.11) into (2. 7), we have th~ desired equality(2.9) with the function (2.10). This completes the proof of
Lemma 2.3.
The function H(t,s) in (2.10) is called Green function forthe multi-point boundary value problem (0. 4) and (0. 2) orGreen function with respect to the matrices A(t), B(t,s) and Li(i=O,l,·····,N). If we put(2.12)

X(t)=J l H(t,s)h(s)ds,-1

then x(t) satisfies the equations (0. 4) and the homogeneous
boundary condition

by Lemma 2.3. Therefore the expression (2.12) defines a boundedlinear mapping in the normed space C(J) which is defined toconsist of all continuous vector valued functions of t on theinterval J. For brevity we express (2.12) in the form
(2.13)

x=Hh,where H;C(J)~C(J).

Hereafter the mapping H defined in (2.12) or (2.13) is called
H-mapping with respect to the matrices A(t), B(t,s) and L.J. (i=O,l,• •••• ,N). It it noted by Lemma 2.3 that the H-mapping can bealways defined so far as the matrix (2. 8) is nonsingular for the
resolvent matrix R(t,s) with respect to A(t) and B(t,s) and thematrices Li (i=O,l,·····,N). The norms of the H-mapping are definedcorresponding to the norms(0. 5) and (0. 6) of vector valued

functiQns belQng1ng to the space.C(J). Hence we have two kinds ofnorms \\Hllc and II HI\Q' which are defined in the usual ways in normed
spaces.

43

3.

Isolated Solutions
We return to our multi-point boundary value problem (0. 1)

and (0. 2) under all assumptions written in the section O. Denoteby C(J;D) the space of all continuous vector valued functions oft on the interval J lying in the domain D for any teJ. Let

. 1)

(3.

fx(t,x,y), f y (t,x,y) and g x (t,s,x)

be Jacobian matrices of the functions f(t,x,y) and g(t,s,x) withrespect to the variables x) y and x respectively. We define forany function x=x(t)€C(J;D) the functions~(t;x)=f (t,X(t),I t g(t,s,x(s»ds),x-1~(t,s;x)=f y (t,X(t),f t-1 g(t,s,x(s»ds)g x (t,s,x(s»,where we note that for any function x=x(t)EC(J;D) and for any tEJ
the vector

It g(t,s,x(s)dSEE by the relation ·(0. 3).
-1

Let

x=~(t)

be any solution of the system (0. 1) satisfying

the condition (0. 2) lying in the domain D for any tEJ. The systemof linear integrodifferential equations
ddty(t)=~(t;~)y(t)+It ~(t,s;~)y(s)ds-1

is called the first variation equations of (0. 1) with respect to6

the solution x=~(t). When we denote by R(t,s) the resolvent matrix
with respect to the matrices
x=~(t)~(t;~)

and ~(t,s;~), the solution

is called isolated solution of the boundary value problem

44(0. 1) and (0. 2) if the matrix(3. 4)~G=N

L Lift(ti,-l)

i=O

is nonsingular. The word "isolated" comes from the following fact.
Lemma 3.1

If the matrix (3. 4) is nonsingular, then,besides the solution x=~(t), there is no other solution of theboundary value problem (0. 1) and (0. 2) in a sufficiently smallneighborhood of x=~(t).
A

The above lemma can be proved in the following way. Since G
is nonsingular, by Lemma 2.3, there is the H-mapping "H with respect
to the matrices ~(t;x),~(t,s;~)and Li(i=O,l, ••••• ,N). Let us

take a positive number E so that
(3. 5)

E<1/3I1HIlC·

For such E, by the ulliform continuities of the derivatives of thefunctions f(t,x,y) and g(t,s,x) and by the definitions of ~(t;2)
and ~(t,s;~) in (3. 2), there exists a positive constant 0 suchthat
(3.6 )

u={xlll x-'X'(t)II~c5 for some tE-JKD

and(3. 7)

and(3. 8)

for any z=z(t)E.C(J;D) satisfying II zU C
Suppose that, besides x=~(t), there is a solution x=x(t) of
the boundary value problem (0. 1) and (0. 2) satisfying II x-'SC'I\ c~c5 ..

45

Then if we put(3. 9)

y(t)=x(t)-~(t),we have(3.10)andd
Jt g(t,s,x(s»ds)
dty(t)=f(t,x(t),-1-f(t,~(t),ft g(t,s,~(s»ds).
-1

Using a mean value theorem, we rewrite (3.12) in the form~tY (tl- J~ [~( t ;i+ey)y (t)+C~1

(t ,5 ;i+ey)y (5 )d5 ]de.t(t;~)y(t)+ft ~(t,s;~)y(s)ds+h(t),
-1

where
h( t )--

J~ [~(t ;2)-Ht ;It+ey)]y (t )de .

_flft [~(t,s;~)-~(t,s;~+ey)Jy(s)dsde.
o -1

Not+n~

that for '~nye~[03l] ~+ey=~+e(x-~)eC(JiP)~

that is,lIeyl\c~o, we have by (3. 7) and (3. 8)

II hUc~e:IlYllc+2e:/Iy IIc=3e:/lyllc·
On the other hand, by Lemma 2.3, for the equations (3.13) andthe boundary condition (3.11) we havey(t)= [ 1 "H(t,s)h(s)ds.
-1

Thus it follows that

By the inequality (3. 5) this implies that\I YI\C=O,
that is,y(t)=O

for any tEJ.

This completes the proof of Lemma 3.1.

4.

Main Theorems
In order to obtain an approximate solution of the multi-

point boundary value problem (0. 1) and (0. 2), we consider finite
Chebyshev polynomial series with unkn6wn coefficients aO,a l ,·····,am' that are(4. 1)

Here Tn(t)(n~Qal,·····)are Chebyshev polynomials defined in(1. 1) and en (n=O,l,·····) are constants defined in (1. 3). It
is reasonable to determine the m+l coefficients aO,al,····,am in
the finite Chebyshev polynomial series (4. 1) so as to satisfy
the conditions that xm(t) lies in D for any t€J and that
(4. 2)d X (t)=p l[f(t,x (t),
dt
m
~
m

47

Jt g(t,s,x. m(s»ds)]
-1

and

N

I Lix (t i )=1,

(4. 3)
wh~re

i=O

m

Pm is the truncation operator defined in (1.10). In what

follows the finite Chebyshev polynomial series (4. 1) satisfying(4. 2) and (4. 3) is called Chebyshev approximation of order m.
In the present paper we shall prove the following three main
theorems.
Theorem 4.1

Suppose that there exists an isolated

solution x=i(t) of the boundary value problem (0. 1) and (0. 2)
lying in D for any tEJ. Then for some sufficiently large rna there
exists a Chebyshev approximation x=xm(t) of any order m~mO suchthat

uniformly on the interval J as m~.
Theorem 4.2

The Chebyshev approximation x=im(t) statedin Theorem 4.1 is determined uniquely in a sufficiently smallneighborhood of the solution x=i(t) provided that the order m of
the Chebyshev approximation x=im(t) is sufficiently high.
Theorem 4.3

Assume that the boundary value problem (0. 1)

and (0. 2) has an approximate solution x=i(t), for which there area positive constant 6, a nonnegat~ve

constan~K<l and the matrices

A(t) and B(t,s) continuous of t on the +nterval J and of (t,s) on
the domain S respectively such that

N

(1)

G= ~ LiR(ti,-l) is nonsingular
i=O

(2)

u={xll\x-x(t)l\~o

for some tE JKn

U~(t;x)-A(t)II~K/2Ml for any tEJ and any XEC(J ,U)(4. 4)(4)

I\qr(t,S;X)-B(t,S)I\~K/4Ml for any (t,s)E-S andfor any XEC(J,U)

Here R(t,s) is the resolvent matrix with respect to the matrices
A(t) and B(t,s).

~(t;x)and qr(t,Sjx) are the functions defined in(3. 2) by use of the Jacobian matrices of the functions f(t,x,y)and g(t,s,x). Ml and M2 are constants satisfying(4. 5)
respectively, where H is the H-mapping with respect to A(t),B(t,s) and L1 (i=0,1, ••••• ,N). rand E are constants satisfying theinequalities(4. 6)\I~tX(t)-f(t'X(t),Jt g(t,s,x(s»ds)ll~r for any tEJ-1

and(4. 7)
Then there exists uniquely an exact solution x=~(t) of theboundary value problem (0. 1) and (0. 2) lying in U for any t~J.
Moreover this is'an isolated solution and it holds that
(4. 8)The proof of these theorems are gi·ven later in the present
paper.
49

The coefficients a=(a O,a1 ,·····,am) of our desired Chebyshev
approximations x=x (t) in (4. 1) of order m are determined by themequations (4. 2) and (4. 3) if x (t) lies in D for any tEJ. Themequations (4. 2) and (4. 3) are equivalent to the system ofnonlinear algebraic equations(4. 9)
whereN(4.10)

FO(a)=

L Lix (t i )-1

i=O

m

and
(4.11)m-l=

L e F +l(a)Tn (t),n=O n nwhich implies by the expressions (1. 4) and (1. 9)(4.12)1 JI

Fon (a)=~
1
1T n-

-1

[f(t,xm(t),Jt-1g(t,s,xm(s»ds)Tn l(t)-

00

x(1_t 2 )-1/2]dt_l2e -1n

L (n+2P-2)an+2p_2

p=l

for n=1,2,·····,m. The system (4. 9) is called determining equationsof Chebyshev approximations. For a solution ;=(i o ,i 1 ,·····,im) ofth~system (4. 9) the finite Chebyshev polyn9mial seriesmxm(t)= n=O
L e i T (t)n n n
is a Chebyshev approximation of order m~

50

5.

Inequalities for Determining Equations
Suppose that

x=~(t)

is an isolated solution of the boundary

value problem (0. 1) and (0. 2) lying in D for any tEJ. Thereexists a positive constant 0 such that
u={xIJlx-~(t)l\<o(5.1 )

Denote that~for some t~J}CD.

=p ~. It follows from the inequalities
m m(l.ll)-(l.l~)

that
2)

1\ ~m-iItC<Kl a (m) /m(m-l) ,(5. 3)

Iljtm-iYQ~Kl/(m+l)m(m-l)(5. 4)

II ~tim-~tinc<Kl [(m+2)/m(m-l)+a(m+l)/(m+l) ](5. 5)\I ~t~m- ~t jtllQ<Kl [{m+2/ I2m(m-l) +1/ (m+2) (m+l) ],(5.

where Kl is a constant satisfying

ft

d3
d2
tl-3xI\Q=II~f(t,i(t),
get ,s ,i(s) )ds )I\Q~Kl'
dt
dt
-1

(5. 6)

Kl may depend only on the structure of the given system (0. 1).
Hereafter we denote by K's the constants depending only on thestr~ctureof the given system (0. 1).

In order to determine a domain where the function F(m)(a)of the system (4. 9) of determining equations is well defined, wechoose the number ml(5. 7)

su~ficiently

Kla(m)/m(m-l)<olarge such thatfor any m>ml'

This is possible by the properties (1.15) of the constant a(m).
Then it~ollowsimfrom the inequalities (5. 2) and (5. 7) that(t )EUeD

for any teJ.
51

Let us put
m(5. 8)~(t)=(P ~)(t)= I eaT (t)mmn=O n n n

and(5. 9)&=(A§. l ' ••••• , §.m) •<:1 0 ,

Define the domain(5.10)The domain nm is the desired domain where the function F(m)(a) is
well defined. In fact, if we put for any vector a=(aO,al,·····,am)belouging to the domain nm

then we obtainx (t )EUCDm

for any teJ

since for any tEJ\l

II x m(t )-~ (t)1l < xm(t )-~m (t)\I + lI~m (t) -i (t)1I<12m+ 1 Ua-& \\ + lI~m -~llc</2m+l[o-K l o(m)/m(rn-l)]112m+l+K l o(m)/m(m-l)=O
by the inequa11ttes (5, 2) and (1.

7). Therefore it follows fromthe expression (4.10) and (4.12) that the function F(m)(a) is
continuously differentiable of a on the domain nm.
)
Let Jm(a) be the Jacobian matrix of the function F (m).. (a.
Tc

investigate the properties of the matrix Jm(a), we consider asystem of linear equations of the form52

(5.11)

Jm(a)~+y=O,

where

a=(a

a

0' l '••••• a )EQ'mm'

Let us put thatmm-1y(t)= Leu T (t) and h(t)=L e c +IT (t).
n=O n n nn=O n nn
Then we can prove that the system (5.11) is equivalent to theboundary value problem(5.12)
andddty(t)=P m~(t;x)

wherel[~(t;x m)y(t)+f t V(t,s;xm)y(s)ds]+h(t),~1and V(t,Sjx) are the functions defined in (3. 1)

a.nd (3. 2).

Substitutinu~ (t) into x (t) in the equations (5.13),Pmm
equivalently doing a into a in the system (5.11), we obtain(5.14)d.

dty(t)=~(tj~)y(t)+ft~(t,s;~)y(s)ds+h(t)+p(t),

-]

where
(5.15)

P(t)=-(I-Pm_1)[~(t;~)y(t)+Jt V(t,s;~)y(s)ds]
-1

-P

m-

l{[~(t;~)-~(t;~m)]y(t)+f t-1 [V(t,s;~)-~(t,s;tmJ]y(s)ds}.
,
53

Let ~(t,s) be the resolvent matrix with respect to the~(t;~)

matricesand ~(t,s;~). The matrix

is nonsingular from the isolatedness of the solution x=~(t).
Then, applying Lemma 2.3 for the boundary value problem (5.12)
and (5.14), we havel'"
A-I c +J A
y(t)=-R(t,-l)G
H(t,s)[h(s)+p(s)]ds,o-1A

where H(t,s) is the Green function with respect to the matrices~(t;~),~(t,s;~)and Li (i=O,l,····· ,N),. Denote by H the H-mapping

eith respect to the above matrices. This implies that

(5.16)
where

On the other hand for the function (5.15) we have

+I\[~(t;~)-~(t;im)]y(t)+f:1['¥(t,S;l!:)-'¥(t,S;ll:m)]Y(S)ds Il Q
by the inequality (1.12) and the Parseval' s, equality (1. 5).
Moreover us1ng Schwartz's inequality and a mean value theorem,
we obtain

54

· dt
d [4> (t ;~)y(t)+ Jt '1' (t, s ;i)y(s )ds J1\ Qn-1~1I~t4>(t;~)y(t)IIQ+"4>(t;~){Pm_1[~(t;~m)y(t)+Jt '1'(t,s;~m)Y(S)dSJ+h(t)}UQ
-1

Jt a

+\1 '1' (t ,t ; X') y (t ) t a t 'l' (t ,s ; i. ) y (s) dsll Q-1

andII [4> (t; ~) -4> (t ; ~m) Jy (t) +Jt ['1' (t ,s; ~) -'1' (t ,s; ~m) Jy <s) ds{\ Q-1Then for the inequality (5.16) we have

Il y\\ ~M1\1 coli + IU-' ItQ (1 +0 1 (m-1) K 3 ) II h\l Q+URIIQ[K201(m-1)+K4o(m)/m(m-1)]UY"Q
using the inequality (5. 2). If we choose a number m2~ml
sufficiently large, then we have for any m>m 2(5.17)

where M 1s a gonstant independent of m. The inequality (5.17) isequivalent to the inequality55(5.18)by the Parseval's eqiality (1. 7) for the finite Chebyshev
polynomial series yet) and h(t). From the inequality (5.18) andthe relation (5.11) it readily follows that for anym~m2(5.19)
and(5.20)The inequality (5.20) will play an impotantrole in the proof o~Theorem 4.1.
Let
a. ' = ( a'0' a'l ' ••••• , a')and a" = ( a"0' a"l ' ••••• , a")mm

be arbitrary vectors be J 0nging to the domain Om. For any vector~=(UOJU'J·····Jum)

we consider the systems of linear equations(5.2~)

where

Let us put
m

ITI

r

r

x'(t)=
e a'T (t), x"(t)=e a"T (t)mn=O n n nmn=O n n n "mmh' (t) =e c' T (t) and h" ( t ) =e c" T ( t ) .
n=O n n+l n
n=O n n+l n

r

r

56

Then, corresponding to the systems (5.21), we haveN

L L yet i )=-c'0i=O i(5.22)~ty(t)=Pm_1[~(t;X~)y(t)+Jt ~(t,s;x~)Y(S)dS]+h'(t)
-1

and
N

L L yet i )=-c"0i=O i(5.23)ddty(t)=Pm_1[~(t;x~)y(t)+Jt ~(t,s;x~)Y(S)dsJ+h"(t)-1respectively by the definition of Jm(a). From (5.22) and (5.23 1
it read~ly follows that
(5.24)

c'=c"
o 0

and
(5.25)

h'(t)-h"(t)=-P m- l{[~(t;x')-~(t;x")]y(t)
m
m
t

+J

-1

[~(t,s;x')-~(t,s;x")]y(s)ds}.
m.
m

The relation above implies that

II h' -h"\1 Q~K511 x~-x~1I c lIy \I Q.
Then by the relation (1. 7) and (5.24) we obtain

II y' -y"l! ~K4/2m+1I1a' -a"lln E;; \I,which implies ,from the systems (5,21) that for an~ m~~ and any
a' , a"en m(5.26)

57

The inequality (5.26) will also play an impotant role in the
proof of Theorem 4.1.

6.

Proof of Theorem 4.1
Suppose that there exists an isolated solution x=i(t) of theboundary value problem (0. 1) and (0. 2) lying in D for any t~J.
It is concluded that in the previous section 5 that there exists
a positive constant 0 satisfying the relation (5. 1) and a number
m2 sufficiently large such that for any m>m 2 the inequalities(5. 2)-(5. 5) hold and such that the function F(m)(a) of thedetermining equations (4. 9) is continuously differentiable of ain the domain ~m and its Jacobian matrix Jm(a) has the inverse

J~l(a) at a=& satisfying the inequality (5.20) and satisfies theinequality (5.26), where i m, ~ and ~ m are defined in the formula(5. 8), (5. 9) and (5.10) respectively.
Let us put(6. 1)

and(6. 2)dl[f(t,~ (t),dt ~ m(t)-P m-·
m

It
-1

g(t,s,~m(s»ds]=h(t).

Using the inequality (5. 2), we have for the relation (6. 1)(6. 3)58

We rewrite the equation (6. 2) in the formddh(t)={--i(t)---i(t)}dt mdt+(I-Pm_l)[f(t,~(t),J:lg(t,s,2(S»dS)]-pl[f(t,i (t),f t g(t,s,i (s»ds)
mm-1m-f(t,i(t),J t g(t,s,i(s»ds)]-1

using the fact that x=i(t) is a solution of the equations (0. 1).
Then by the inequalities (5. 5), (1.12), (1. 5) and (5. 3) and amean value theorem used in (3.12) we obtain(6. 4)

II hll~Kl [/m+2/12m(m-l )+1/ (m+2) (m+l)]+K 1 /m(m-l)+K 6 Kl /(m+l)m(m-l).

It follows from the inequalities (6. 3) and (6. 4) that
(6. 5)

By the definition of the function F(m)(a) of the determining
equations (4. 9) the boundary value problem (6. 1) and (6. 2) isequivalent to a system
F(m) (Il)=p (m) .
This implies that

Then there exists s number m~m2 such that for any m~m3

59

(6. 6)for some constant K7 by the asymptotic behavior (6. 5).
We shall apply Lemma 0.1 in the section 0 to the determining
equations (4. 9) to' complete the proof of Theorem 4.1. In orderto check the conditions (0. 8) in Lemma 0.1 we choose an arbitrary
nonnegative constant K<l and put

where constants K5 , M and Kl are defined in (5.26), (5.20) and(5. ,6) respectively. There exists a number m4~m3 so that[M/(1-K)JK 7m- 3/ 2 <ol/12m+l
for any m>m 4 sincem- 3/ 2 /2m+l=O(m- l / 2 ) as m~.
If we choose a number om such that
(6. 7)

then we obtain(6. 8)° }(n.ru ={a 11\ a-lilt =< m n m.
rn

In fact, for any aEn o and m~m4~m3
mt!a-lil\<om<olI12m+l<[o-K 1 a(m 3 )/m 3 (m 3-1)JI12m+l~[o-Kla(m)/m(m-l)J/12m+l,

which implies afO rn

60

Moreover it follows from the inequality (5.26) that
(6. 9)

HJm(a) -J m(&)\1 <K5 /2m +l l\a-&"<K5/2m+lo~K5°l~K5(K/K5M)=K/M

for any aEQ oand any m~m4. Finally by the inequalities (6. 6) andm(6. 7) we haveThus the inequalities (6. 8), (6. 9) and (6.10) show that theconditions (0. 8) in Lemma 0.1 are fulfilled. Hence by Lemma 0.1
we see that the determining equations (4. 9) has one and only one
solution a=; in the domain nosatisfyingmand(6.11)
If. we put

andmxm(t)= ncO
1 eaT
(t),
n n n
then x=x (t) is i Chebyshev approximation and satisfies.
m(6.12)

61

for any m~m4 fr.9m.the inequalities (1. 7) and (6.11). The
inequality (6.12) implies the uniform convergence of the Chebyshev
approximation x (t) to the solution i(t).
m. In order to prove the uniform convergence of the derivativesof the Chebyshev approximations xm(t), we have
(6.13)~tXm(t)-~t~(t)=-(I-Pm_l)[f(t,~(t),Jt g(t,s,~(s»ds)]-1-f(t,~(t),Jt g(t,s,i(s»ds)].
-1

On the other hand~ by the inequalities (1.11) and (1.12), we haveII(I-Pm_l)[f(t,~(t),Itg(t,s,~(s»ds)]\lc-1~K1C1(m-1)/(m-1)

and by the inequalities (1. 5), (1. 7), (5. 3) and (6.11) we havet g(t,s,x (s»ds)
lipm- l[f(t,xm(t),I -1m-f(t,~Ct),It g(t,s,~(S»dS)]"c-1<';2(m-l)+lK6nxm-~IlQ<K6';2m-l{lIxm-~mUQ+lI~ni-~IIQ}<K612m-l{[M/(1-K)]K7m-3/2+Kl/(m+1)mCm-1)}.
Then for the relation (6.13) we have62

ll~tXm-~tjtllc=o(m-1) as m-+oo,
which implies the uniform convergence of the derivatives ~tXm(t)dto dtx(t).
This completes the proof of Theorem 4.1.

7.

Proof of Theorem 4.2
Let x=i(t) be an isolated solution of the boundary value

problem (0. 1) and (0. 2) lying in D for any tEJ. There exists a
positive constant 0 satisfying (5. 1). We choose an arbitrary
constant E satisfying O<E~O. Then there exists a number mO suel!
that for any m>mO

(7.

1)

°l(m-l)=l/m<E.

Suppose that for any m~mO there are two Chebyshev
approximations

satisfying(.7. 2)
We denote thatuE={xlllx-~(t)l\~E for some tE:J}CU:::D.

Then it follows that for any m>mo xm(t) and x'(t)be:ong to themdomain UE for any tEJ. Let us puty(t)=x (t)-x'(t).
m

m

63

By the definition 'of Chebyshev'lappro.~.1l'nati.ons· .:we.'_'.haveand(7. 4)t-f(t,x~(t),I g(t,s,x~(S»ds)J.
-1

We rewrite the above equations (7. 4) in the form
(7. 5)~ty(t)=~(t;~)y(t)+It ~(t,s;~)y(s)ds+h(t),
-1

where

h(t)=-(I-Pm_1)[~(t;~)y(t)+Jt ~(t,s;~)y(s)ds]-1+

Jt-1 [~(t,s;xm6 )-~(t,s;~)Jy(s)ds}d6,and
6 (t)=X m
Xm'(t)+6[Xm(t)-x'(t)].
mNoting that -6
xm(t)~Ue for any tEJ and 66 [0,1), we obtain, by thesame-argument as that used in proceeding from (5.16) to (5.17),(7. 6)

from (7. 1) and (7. 2).

64

On the other hand, applying Lemma 2.3 to the bounda:ry value
problem (7. 5) and (7. 3), we obtainly(t)= JA

H(t,s~~(s)ds,-1"'-

where H(t,s) is the Green function with :respect to the matrices~(t;f),~(t,s;~~and Li(i=O,l, ••••• ,N). Then it follows from

(7. 6) that
(7. 7)

Since E is arbitrary, the above inequality (7. 7) implies thatthat isy(t)=o for any teJ
by the Parseval's equality for the finite Chebyshev polynomial
series yet).
This proves the uniqueness of Chebyshev approximations andhence completes the proof of Theorem 4.2.

8.

Proof of Theorem 4.3
For the given approximate solution x=x(t) of the boundary

value problem (0. 1) and (0. 2) we put

(8.

1)

-

~ t x (t ) =f (t , x (t ) ,Jt g ( t , s , X( s ) ) ds ) +q '( t )
-1

and

65

(8. 2)Introducing the matrices A(t) and B(t~s), we rewrite the equations(8 .1) in the form
(8. ,3)~tX(t)=A(t)X(t)+It B(t,s)x(s)ds+h(t;x)+q(t).
-1Here we denote thath(t;X)=f(t,X(t),I t g(t,s,x(s»ds)-1-A(t)X(t)-f t B(t,s)x(s)ds-1

for any x=x(t)EC(J;D). Applying Lemma 2.3 to the boundary value
problem (8. 3) and (8. 2), we have
(8. 4)

X(t)=R(t,-1)G- 1 1 f +J l H(t,s)[h(s;x)+q(s)]ds,-1

where H(t,s) is the Green function with respect to the matrices
A(t), B(t,s) and Lt(i=O,l,.··.·,N).
To seek an exact solution of the system (0. 1) satisfying
the boundary condition (0. 2), we consider the iterative process
x +l(t)=R(t,-l)G- 1 1+ II H(t,s)h(s;x )dsn(8. 5)-1n(n=O,l,·····)For the iterative process (8. 5) we shall prove that it can becontinued infinitely in the space C(J;D) and that66

(8. 6)and(8. 7)
for n=O,l,·····. In fact, for n=O the inequality (8. 6) isev1dent.
Since
Xl(t)-XO(t)=R(t,-l)G-l(t-t')-f l H(t,s)q(s)ds,-1

then by the assumptions of the theorem we obtain\lxl-xOllc<M2€:+Mlr,
which implies by the condition (5) in (4. 4)\~l-xO"C«l-K)B<B.

This proves (8. 7) for n=O. To prove our statement by induction,
let us suppose that the iterative process (8. 5) has been
continued up to n-l and we have obtained (8. 6) and (8. 7) up ton-l. Then by the inequality (8. 7) for n-l we can make xn+l(t)and from (8. 5) we have
(8. 8)
whereh(t·x, n )-h(t·x' n-l )

-f(t,x n- l(t),J t g(t,s,x n- l(s»dS)-1-A(t)[Xn(t)-Xn_l(t)]-f t B(t,s)[xn(s)-Xn_l(s)]ds.
-167

Moreover, by a mean value theorem we obtainh(t;xn )-h(t;xn- 1)

It

I

8
= I {~(t;x n8 )[x n (t)-x n- l(t)]+~(t,s;x )[x (s)-xnnn- 1(S)ds}d8
O
-1

I

t

-A(t)[x (t)-xnn-l(t)]--1.

B(t,s)[x (s)-xnn-l(s)]ds,where
xn8 (t)=x n- 1(t)+8[Xn (t)-x n- l(t)]~C(J;D).
It follows thatI\h(t;xn )-h(t;xn- 1)11

I II

+ t
1I~(t,s;x 8 )-B(t,s)llllx (s)-x 1(s)lld8dS.
_IOn
nnHence for the relation (8. 8) we have

II xn -xn _ l Uc< IIH lIcllh (t; xn ) -h (t ; xn_l)\l C~Ml (K/2M l +2K/4M l ) IIxn -xn_lU C=K IIxn-xn_ll\ Cby the assumption (3) and (4) in (4. 4) of the theorem. Thisimplies (8. 6) by the assumption of the indu.c.tion and(8. 9)

IIxn+l-XoUC~(Kn+Kn-l+ ••••• +K+l)lIxl-Xolic< (M 2 E+M l r) / (l-K )~O.

This completes the induction and hence we see that the iterative
process (8. 5) can be continued infinitely and satisfies
inequalities (8. 6) and (8. 7) for every n.
68By the inequalities (8. 6) and (8. 7) we see that the sequence{Xn(t)}~C(J;D)

obtained by the iterative process (8. 5) converges

uniformly to a function i(t)EC(JjD). It readily follows from (8. 9)and (8. 5) that\\ i-x I\c~(M2E+Mlr) / (l-K )~oand(8.10)

i(t)=R(t,-l)G-l~+fl H(t,s)h(sji)ds.
-1

The equation (8.10) implies by Lemma 2.3 thatand~ti(t)=A(t)i(t)+ft B(t,s)i(s)ds+h(tji)-1=f(t,i(t),f t g(t,s,i(s»ds).
-1

Therefore the function i(t) is a solution of the boundary value
problem (0. 1) and (0. 2) belonging to the space C(JjU).
In order to prove the uniqueness of the solution of ourboundary vR1ue problem, we consider another solution x=i'(t) of
the problem (0. 1) and (0. 2) belonging to the space C(JjU). ThenN

(8.11)

L L it(t )=~

i=O i

i

and

69

(8.12)~t~'(t)=f(t,i'(t),Jt g(t,s,~'(s»ds)-1=A(t)~'(t)+Jt B(t,s)~'(s)ds+h(t;i').
-1

Equivalently the relations (8.11) and(8.13)(8.1~)

imply that~'(t)=R(t,-l)G-l!+Jl H(t,s)h(s;i')ds.
.

-1

Subtracting (8.13) from (8.10), we obtain
IIx-x' Ilc~K\lX-X" l\c'which implh~s
1\ x-x' Uc=O.

This proves the uniqueness of the solution of the problem (0. 1)
and (0. 2) lying in U for any tEJ.
In order to prove the isolatedness of the solution x=i(t),it is enough to see that the matrix

L'>.

is nonsingular, where R(t,s) is the resolvent matrix with respect
to the matrices ~(t;i) and ~(t,s;~). Suppose that ~ is singular.
Then there exists a nontrivial vector c such that
r:-

Gc=O.

For such c let us put"'-y(t)=R(t,-l)c.
Then y=y(t) is a nontrivial solution of the boundary value problem

70

of the form

C8.14)
and(8.15)ddty(t)=~(t;~)yCt)+

ft ~(t,s;~)y(s)ds-1=A(t)y(t)+f t BCt,s)y(s)ds-1+[~(t;~)-A(t)]YCt)+Jt [~(t,s;~)-B(t,s)]y(s)ds.
-1

Applying Lemma 2.3 to the problem (8.14) and (8.15), we obtainy(t)=f 1 H(t,s){[~(s;i)-A(s)]y(s)-1+Js[~(s,u;i)-BCs,u)]y(u)du}ds.

-1

Then, by the assumption", ('3) and (4) in (4. 4), we have

"YUC~"HUC(K/2Ml+2K/Ml)~Y"C=K"Y"~,whicn implies thatl\y\!c=O.
D

This is contradiction. Hence the matrix G is nonsingular, thatls, the solution x=i(t) is isolated. This completes the proof of
Theorem 4.3.

71

References[lJ

Grossman S. I. and R. K. Miller; Perturbation theory for
Volterra integro-differential systems. J. Diff. Eqs., 8,457-474 (1970).

[2J

Shimasaki M. and T. Kiyono; Numerical solution of integral
equations in Chebyshev series. Numer. Math., 21, 373-380(1973).

[3]

Tsuruta K. and K. Ohmori; A ·posteriorLerrorc..estimation for
Volterra integro-differential equations. Memoirs of Numer.
Math., No.3, 33-47 (1976).

[4]

Urabe M.; Galerkin's procedure for nonlinear periodic systems.
Arch. Rat. Mech. Anal., 20, 120-152 (1965).

[5]

Urabe M.; An existence theorem for multipoint boundary value
problems. Funk. Ekvac., 9, 43-60 (1966).

[6)

Urabe M.; Numerical solution of multipoint boundary value
problems in Chebyshev series Theory of the method. Numer.
Math., 9, 341-366 (1967).

[7J

Kurihara M.; On the existence of an approximate solution in
Chebyshev series of a nonlinear integral equation of Fredholm
type. Memoirs of Numer. Math., No.3, 49-67 (1976).

72

The Buckling of Plates by the Mixed Finite Element Method
By
Kazuo ISHIHARA *1. Introduction
We shall consider the mixed finite element approximation applied
to the buckling problem of the thin plate with the clamped boundary
condition:
I::.l::.u

L ,. ij axiax j = °
i,j=l

+ A

(1)

2

a2u

u

Here

n

= au/av =

n,

in

°

on an.

is a bounded convex domain in the x l x 2 -plane with boundary

an consisting of a finite number of smooth arcs,
operator,a/av

I::.

is the Laplaceis the outward normal derivative along an, and Tij(i,j=1,2) are given smooth functions such that++

The buckling of the plate is possible only for certain definite values
ofA. The minimum of these values determines the critical bucklingload. The associated buckling configuration is the function
ponding to the buckling loaducorres-A. A simple case is buckling under pure

*Department of Mathematics, Faculity of Science, Ehime University,
Matsuyama 790, Japan.

73

compression
6.6.U+ A6.U = O.

The aim of this paper is to give the rates of convergence forthe finite element approximate solutions of the eigenvalues and theeigenfunctions,by applying the mixed method with piecewise linear

polynomials proposed by Miyoshi [5J,[6J.

2.
Let

Notations and weak formulation

L?(n) be the real space of square integrable functions on

n, whose inner product and norm are denoted by ( , ) and U.11,
spectively. Let

Hm(n)

re-

be the real m-th order Sobolev space(m=1,2,

3,···) provided with the' norm
2 1/2

luflm = (Here

a=(S,y)lal~mis a two-component index with non-negative integers,

lal= S + Y'llhe spac es

L I/ Dau ll)

H1Un
0

a
D

and

=

a~x~ dx~.

'1andH'-(n)
0

are given by

Hi(n) = {f; f~ Hl (n) ,0

H~(n) = {f; fEo H2 (n),

on

dn},

f = df/dv = 0

on

f = 0

()n}.

Tn orner to deal with the buckling problem (1) in a weak form,
let us define bilinear forms~u,v> =

L

< ,>

and

[, J

(Dau,Dav)}lal =22[u,yJ =

Li ,j=l

(T ij ()u/()x i ,3v/()x j ).

74

by

We assume that[u,uJ:2=O
and that the norm 1/1'111

8'll lininduced by[,]is equivalent to the norm

H~(g). The standard weak form of (1) consists of finding areal eigenvalue A and a non-zero eigenfunction<u,<P> = A[u,<PJ
tha~

It is well known

for each

all the eigenvalues

0'< Al ~ A2~ •••u E: H~ (n)defined by<pE-H6(n).
{Ai}

are arranged as<00.

The multiplicity of each eigenvalue is always finite. The corresponding eigenfunctions are denoted by{u i }

with the normalization con-ditionwhere 0ij is Kronecker's delta. It is also well known that the eigen00

functions{ui}i=lbelong to the space

sufficiently smooth and that

H~(n)(\H4(n)belong to

when

an

H~(n)nH3(n)

is
when

n 1s a convex polygon! From the Rayleigh principle the eigenvalues
are characterized by
Ai =

<u,u>
min
[u,u]
2
u Eo HO (n)u:\:O[u,uj]=O
j=l, ... ,i-l

and the minimum is attained by

i=1,2,···,ui •In order to construct the mixed finite element scheme, we introduce another formulation: Findsuch that

75

2

L

i,j=l

[(au/axi,a~ij/axj) + (Uij'~i~)J = 0.J~ijE H1 (n) (~12=~21)'for each

(2)

2

L

2

i,j=l

(aUij/aXj,a~/axi) - A(
L TijUij'~) = 0
..
i,j=l

for each1~E- Ho(n).

We assume that· u~H~(n)"H3(s'n. The solution (u,Ull,U12'U21,U22) of(2) is related by
1~i,j:!:2.

V by

We define a spaceV ={(u,U11'U12'U21,U22)~H~(n)XH1(n)XH1(n) XH 1 (n)XH 1 (n);U12 =U 21 ,i~j[(au/aXi,a~ij/aXj) + (Uij'~ij)J = 0
1~ij E-H (n)for each
and a bilinear form
E(U,W)
where

E(U,W)(~12=~21)}'by= (U 11 ,W 11 ) + (U 12 ,W12 ) + (U 21 ,W 21 ) + (U 22 ,W 22 )U = (U11,U12,U21,U22)'

W ~ (W11,W12,W21,W2?)' The eigenvalues

are characterized byAi=

min E(U,U){u,U}4:V III ull' 2
u+O[u,u.jJ=O
j=l,··,i-1

i=1,2, ... ,3. Convergence of the finite element scheme
For simplicity, we assume that the domain
Then the eigenfunctionubelongs to76n

is a convex polygon.

~~(n)('\H3(n). The domain is'decomposed into disjoint triangular elements in the usual manner. Let
Pi,l~i~n

tion

Th

(or

Pi,n+l~i~n+J)

which belong tobe the nodal points of the triangula-n (or an). Herelength of all the triangular elements.
triangulationThhis the largest sideis uniform in the interior of

Miyoshi ([5J,[6J). Let

assume~It is also{~i}(i=1,2, ••• ,n+J)

nthat thein the sense of

be the piecewise linear

functions such thatA<l>i(P,i) = 0i,1'

Let

~h

be the subspace ofbe the subspace ofHl(n)1 ~ i,j ~ n+J .
r,.

and

H~(n)

We now define the mixed finite element solution
'"
U

r..}

spanned by {<I>l,···,<I>n+J'" ~ YO'h '"1\
Uij E- Yh, ' "U12 =U22 )(u21 )/,,,,1'I'

A

CA,u,Ull,U12,U21'of the consistent mass scheme for the

problem (2) by

We define a space vh by
h
;to
A
A
/\
/\
h
h
h
h
h
V = {(u,Ull'U12'U21'U22) ,"Yo xy J( Y )( Y )( Y .

11By (3) and (4), we can obtain a set of the matrix equations
tv

Kx '+ My: 0,

K and

where

~

Ky + XGx = 0,

r are elastic stiffness matrices,

stiffness matrix, andM

G

is the geometricis the consistent mass matrix. The eigeh-{1} n of (3) are arranged a8. i i=lvaluesO<~1~1?~ ... ~1n·~~i~
Ai Ai Ai Ai
The corresponding eigenfunctions (ui,U )=(ui,Ull'U12'U21,U22) (i=1,2,... ,n)can be normalized by"1\[ui,ujJ="[ui,uiJ~O.

<5 ij ,

Then the eigenvalues are characterized by

l'i =

,

"min h(~,U)E- V
u:\=O[G,o.jJ=O

i=1,2,···,n,j=l,···,i-l""i ).
and the minimum is attained by (ui,UConsider the static plate bending problem with the clamped boundary
condition:

66w = l'w = dWjdV = 0
Here

f

in

St,

011

aQ.

is a given function belonging to

1 2 (Q).

Miyoshi proved thefollowing proposition ([5J,[6J).
Proposition 1.

Letw

be the solution of (5) and

A

/\

'"

"
A
W ) • yh X. yh X. yh X yh X yl1 (W12 =W 21 ) be the finite element solution
22
0

defined by

78t\(w,Wll,W12,W2l'

=0
(6)

2

A

l

i,j=l

A

A

(aWij/aXj,a$/ax i ) + (f,$) = 0
for each~ Eo yh .

Thenrl w - ~lIl ~ C UfU h l / 2 ,
where

C

is a constant independent ofh.

We can now obtain the following results using Proposition 1.

The

proof will be published elsewhere.
Theorem 1.

Let

1 i · be the approximate eigenvalue offor sufficiently small
dependent ofhh, there exists a constant

Ai.

Then

C1 which is in-such that

I Ai - ~i I ~ Cl h l / 2 .
Theorem 2.

Let

Ai+p<Ai+p+l)' andLet

Ai... ,n).

p+l (p%O, Ai _ l < Ai ="'=

ui,···,ui+p'be the corresponding eigenfunctions.

Auk

be of multiplicity'

be the approximate eigenfunction corresponding toThen fur sufficiently small

which is independent ofhA

Ak (k=1,2,h, there exists a constantsuch thatj =i, ... , i+p,where
dist{u, B} = inf Illu - bOl.
bEB

As a corollary to Theorem 2, we have

79

C2

Corollary 1.
small

h

If

Ai

there exists a

is a simple eigenvalue, then for sufficiently
c~nstant

C3

which is independent ofh suchtbat

Remark.

Furthermore, we can propose the generalized mixed mass

scheme with a parameter0~a~1([2J)

a,Kx + {aM + (l-a)Ml}y
where

Ml

= 0,

,..,

"'"

Ky + AGx= 0,

is the diagonal lumped mass matrix. The error estimateofthe above scheme is similar to the one of the consistent mass scheme.

4. Numerical example
To show the validity of the theoretical results, we deal with thefollowing example of the square plate, which is the same as the one
given by Weinstein and Stenger ([7J, p193).

H:

Example ..

=0

in

Q,

u = aU/dv _. 0

on

an.

tllm + >'tlu

Although the exact first eigenva:ue (buckling ioad) Al

is not known,

they obtained the inequality
5.30362:S Al~ 5.31173.
We divide

n

into

nXn

choices for the parameter

uniform mesh as shown in Figure 1.

e

are

80

0, 0.5, 1.

Our

Table 1 shows thenumerical results, from which we can see that the approximate first
eigenvalues converge ashtends to zero. All numerical computations

were performed on the FACOM 230-28 computer at Ehime University, andthe FACOM 230-75

mesh(n X n)
h

e

0
0.5
1

comp~terat Kyushu University.

Figure 1.

Mesh pattern (3 X 3)

Table 1.

The results for Example

7x7

8X8

9X 9

10)< 10J2rr/6[2rr/7[2rr/8[27T/94.820075.45497
6.31138

4.941375.42827
6.036425.022235.405595.86073

5.07883
5.38766
5.74163

Acknowledgements
The author would like to express his thanks to Professor a Fujita
of the University of Tokyo and Dr. M. Tabata of Kyoto University
for their advices.

91

References[lJ Brezzi, F.

: On the existence, uniqueness and approximation of

saddle-point problem arising from Lagrangian multipliers. R. A.
I. R. 0., 8-R2 (1974) 129-151.
[2J Ishihara, K. : Convergence of the finite~:ement

method applied

to the eigenvalue problem 6U+AU=0. Publ. RIMS, Kyoto Univ., 13(1977) 47-60.
[3J Kikuchi, F. : Convergence of the ACM finite element scheme forplate bending problems. Publ. RIMS, Kyoto Univ., 11 (1975) 247265.
[4J Kikuchi, F.

: On a finite element scheme based on the discrete

Kirchhoff assumption. Numer. Math., 24 (1975) 211-231.
[5J Miyoshi, T. : A finite element method for the solutions of fourthorder partial differential equations. Kumamoto J. Sci. (Math.),
9 (1972) 87-116.
[6J Miyoshi, T. : A mixed finite element method for the solution of
the von Karman equations. Numer. Math., 26 (1976) 255-269.
[7J Weinstein, A. and Stenger, W.

: Methods of intermediate problems

for eigenvalue problems, theory and ramifications. Academic Press
1972.

82