Ordinary and Partial Differential Equations: Proceedings of the Fifth Conference held at Dundee, Scotland, March 29-31, 1978 (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Edited by A. Dotd and B. Eckmann

827 Ordinary and Partial Differential Equations Proceedings of the

Fifth Conference Held at Dundee, Scotland, March 29 - 31, 1978

Edited by W. N. Everitt

Springer-Verlag Berlin Heidelberg New York 1980

Editor W. N. Everitt Department of Mathematics University of Dundee Dundee D1 4HN Scotland

AMS Subject Classifications (1980): 33 A10, 33 A35, 33 A40, 33 A45, 34Axx, 34 Bxx, 34C15, 34C25, 34D05, 34 D15, 34E05, 34 Kxx, 35B25, 35J05, 35 K15, 35K20, 41A60 ISBN 3-540-10252-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10252-3 Springer-Verlag NewYork Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin Heidelberg 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210

This volume is dedicated to the life, work and memory of ARTHUR

ERD~LYI

1908-1977

PREFACE

These Proceedings form a record of the plenary lectures delivered at the fifth Conference on Ordinary and Partial Differential Equations which was held at the University of Dundee, Scotland, UK during the period of three days Wednesday to Friday 29 to 31 March 1978. The Conference was originally conceived as a tribute to Professor Arthur Erd~lyi, FRSE, FRS, to mark his then impending retirement from the University of Edinburgh.

A number of his colleagues,

including David Colton, W N Everitt, R J Knops, A G Mackie, and G F Roach, met in Edinburgh early in 1977 in order to make provisional arrangements for the Conference programme.

At this meeting it was

agreed that Arthur Erd~lyi should be named as Honorary President of the Conference.

A formal invitation to attend the Conference was issued

to him in the autumn of 1977, and this invitation Arthur Erd~lyi gladly accepted, expressing his appreciation for the thought and consideration of his colleagues.

Alas, time, in the event, did not allow of these

arrangements to come about; Arthur Erd~lyi died suddenly and unexpectedly at his home in Edinburgh on 12 December 1977, at the age of 69. Nevertheless it was decided to proceed with the Conference; invitations had been issued to a number of former students, collaborators and friends of Arthur Erd~lyi to deliver plenary lectures.

The Conference

was held as a tribute to his memory and to the outstanding and distinguished contribution he had made to mathematical analysis and differential equations.

VI

These Proceedings form a permanent record of the plenary lectures, together with a list of all other lectures delivered to the Conference. This is not the time and place to discuss in any detail the mathematical work of Arthur Erd~lyi.

Obituary notices have now been

published by the London Mathematical Society and the Royal Society of London.

Those who conceived and organized this Conference are content

to dedicate this volume to his memory. The Conference was organized by the Dundee Committee; E R Dawson, W N Everitt and B D Sleeman. It was no longer possible to follow through the original proposal for naming an Honorary President.

Instead, following the tradition

set by earlier Dundee Conferences, those n~med as Honorary Presidents of the 1978 Conference were: Professor F V Atkinson (Canada) Professor H-W Knobloch (West Germany). All participants are thanked for their contribution to the work of the Conference; many travelled long distances to be in Dundee at the time of the meeting. The Committee thanks: the University of Dundee for generously supporting the Conference; the Warden and Staff of West Park Hall for their help in providing accommodation for participants; colleagues and research students in the Department of Mathematics for help during the week of the Conference; the Bursar of Residences and the Finance Office of the University of Dundee. As for the 1976 Conference the Committee records special appreciation of a grant from the European Research Office of the United

States Army; this grant made available travel support for participants from Europe and North America, and also helped to provide secretarial services for the Conference. Professor Sleeman and I wish to record special thanks to our colleague, Commander E R Dawson RN, who carried the main burden for the organization of the Conference.

Likewise, as in previous years, we

thank Mrs Norah Thompson, Secretary in the Department of Mathematics, for her invaluable contribution to the Conference.

W N Everitt

C O N T E N T S F. V. Atkinson Exponential behaviour of eigenfunctions and gaps in the essential spectrum ....

1

B. L. J. Braaksma Laplace integrals in singular differential and difference equations ...........

25

David Colton Continuation and reflection of solutions to parabolic partial difference equations .....................................................................

54

W. N. Everitt Legendre polynomials and singular differential operators ......................

83

Gaetano Fichera Singularities of 3-dimensional potential functions at the vertices and at the edges of the boundary .........................................................

]07

Patrick Habets Singular perturbations of elliptic boundary value problems ....................

I]5

F. A. Howes and R. E. O'Malley Jr. Singular perturbations of semilinear second order systems .....................

131

H. W. Knobloch Higher order necessary conditions in optimal control theory ...................

151

J. Mawhin and M. Willem Range of nonlinear perturbations of linear operators with an infinite dimensional kernel ............................................................

165

Erhard Meister Some classes of integral and integro-differential equations of convolutional type ............................................................

|82

B. D. Sleeman Multiparameter periodic differential equations ................................

229

Jet Wimp Uniform scale functions and the asymptotic expansion of integrals .............

251

Lectures ~iven at the Conference which are not represented by contributions to these Proceedings. N. I. AI-Amood Rate of decay in the critical cases of differential equations R. J. Amos On a Dirichlet and limit-circle criterion for second-order ordinary differential expressions G. Andrews An existence theorem for a nonlinear equation in one-dimensional viscoelasticity K. J. Brown Multiple solutions for a class of semilinear elliptic boundary value problems P. J. Browne Nonlinear multiparameter problems J. Carr Deterministic epidemic waves A. Davey An initial value method for eigenvalue problems using compound matrices P. C. Dunne Existence and multiplicity of solutions of a nonlinear system of elliptic equations M. S. P. Eastham and S. B. Hadid Estimates of Liouville-Green type for higher-order equations with applications to deficiency index theory H. GrabmUller Asymptotic behaviour of solutions of abstract integro-differential equations S. G. Halvorsen On absolute constants concerning 'flat' oscillators G. C. Hsiao and R. J. Weinacht A singularly perturbed Cauchy problem Hutson Differential - difference equations with both advanced and retarded arguments

XI

H. Kalf The Friedrichs Extension of semibounded Sturm-Liouville operators R. M. Kauffman The number of Dirichlet solutions to a class of linear ordinary differential equations R. J. Knops Continuous dependence in the Cauchy problem for a nonlinear 'elliptic'

system

I. W. Knowles Stability conditions for second-order linear differential equations M. KSni$ On C~ estimates for solutions of the radiation problem R. Kress On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies M. K. Kwon$ Interval-type perturbation of deficiency index M. K. Kwong and A. Zettl Remarks on Landau's inequality R. T. Lewis and D. B. Hinton Discrete spectra criteria for differential operators with a finite singularity Sons-sun Lin A bifurcation theorem arising from a selection migration model in population genetics M. Z. M. Malhardeen Stability of a linear nonconservative elastic system J. W. Mooney Picard and Newton methods for mildly nonlinear elliptic boundary-value problems R. B. Paris and A. D. Wood Asymptotics of a class of higher order ordinary differential equations H. Pecher and W. yon Wahl Time dependent nonlinear Schrodinger equations

XII

D. R a c e

On necessary and sufficient conditions for the existence of solutions of ordinary differential equations T. T. Read Limit-circle expressions with oscillatory coefficients R. A. Smith Existence of another periodic solutions of certain nonlinear ordinary differential equations M. A. Sneider On the existence of a steady state in a biological system D. C L Stocks and G. Pasan Oscillation criteria for initial value problems in second order linear hyperbolic equations in two independent variables C. J. van Duyn Regularity properties of solutions of an equation arising in the theory of turbulence W. H. yon Wahl Existence theorems for elliptic systems J. Walter Methodical remarks on Riccati's differential equation

Address list of authors and speakers N. AI-Amood:

Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH EH14 4AS, Scotland

R. J. Amos:

Department of Pure Mathematics, University of St Andrews, The North Haugh, ST ANDREWS, Fife, Scotland

G. Andrews:

Department of Mathematics, Heriot-Watt University, Ricearton, Currie, EDINBURGH EHI4 4AS, Scotland

F. V. Atkinson:

Department of Mathematics, University of Toronto, TORONTO 5, Canada

B. L. J. Braaksma:

Mathematisch Instituut, University of Groningen, PO Box 800, GRONLNGEN, The Netherlands

K. J. Brown:

Department of Mathematics

Heriot-Watt University,

Riccarton, Currie, EDINBURGH EHI4 4AS, Scotland P. J. Browne:

Department of Mathematics

University of Calgary,

CALGARY, Alberta T2N 1N4, Canada J. Carr:

Department of Mathematics

Heriot-Watt University,

Riccarton, Currie, EDINBURGH EH14 4AS, Scotland D. L. Colton:

Department of Mathematics

University of Delaware,

NEWARK, Delaware 19711, USA A. Davey:

Department of Mathematics

University of Newcastle-

upon-Tyne, NEWCASTLE-UPON-TYNE NEI 7RU, England P. C. Dunne:

Department of Mathematics

Heriot-Watt University,

Riccarton, Currie, EDINBURGH EHI4 4AS, Scotland M. S. P. Eastham:

Department of Mathematics

Chelsea College,

~ n r e s a Road, LONDON W. N. Everitt:

Department of Mathematics

The University, DUNDEE

DDI 4HN, Scotland G. Fichera:

Via Pietro Mascagni 7,00199 ROMA, Italy

H. Grabm~ller:

Fachbereich Mathematik, Technisehe Hochschule Darmstadt, D 6100 DARMSTADT, Sehlossgartenstrasse 7, West Germany

P. Habets:

Institut Mathematique, Universit~ Catholique de Louvain, Chemin du Cyclotron 2, 1348 LOUVAIN LANEUVE, Belgium

S. B. Hadid:

Department of Mathematics, Chelsea College, Manresa Road, LONDON

XIV

S. G. Halvorsen:

Institute of Mathematics, University of Trondheim, NTH, 7034 TRONDHEIM-NTH, Norway

D. B. Hinton:

Department of Mathematics, University of Tennessee, KNOXVILLE, Tennessee 37916, USA

F. A. Howes:

Department of Mathematics, University of Minnesota, MINNEAPOLIS, Minnesota 55455, USA

G. C. Hsiao:

Department of Mathematics, University of Delaware, NEWARK, Delaware 19711, USA

V. Hutson:

Department of Applied Mathematics, The University, SHEFFIELD Sl0 2TN, England

H. Kalf:

Fachbereich Mathematik, Technische Hochschule Darmstadt, D 6;00 DARMSTADT, Schlossgartenstrasse 7, West Germany

R. M. Kauffman:

Department of Mathematics, Western Wastington University, BELLINGHAM, WA 98225, USA

H. W. Knobloch:

Mathem. Institut der Universit~t, 87 WURZBURG, Am Hubland, West Germany

R. J. Knops:

Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH EHI4 4AS, Scotland

I. W. Knowles:

Department of Mathematics, University of the Witwatersrand, JOHANNESBURG, South Africa

M. K~nig:

Mathematisches Institut der Universit~t MUnchen, D 8 MI~NCHEN 2, West Germany

R. Kress:

Lehrstuhle Mathematik, Universit~t G~ttingen, Lotzestrasse 16.18, GOTTINGEN, West Germany

M. K. Kwong:

Department of Mathematics, Northern Illinois University, DEKALB, Illinois 60115, USA

R. T. Lewis:

Department of Mathematics, University of Alabama in Birmingham, BIRMINGHAM, Alabama 35294, USA

S. S. Lin:

Department of Mathematics, Heriot-Watt University, Riccarton, Currie, EDINBURGH EHI4 4AS, Scotland

M. Z. M. Malhardeen:

Department of Mathematics, Heriot-Watt University, Riecarton, Currie, EDINBURGH EHI4 4AS, Scotland

J. L. Mawhin:

Institut Mathematique, Universit~ Catholique de Louvain, Chemin du Cyclotron 2, 1348 LOUVAIN LANEUVE, Belgium

XV

E. Meister:

Fachbereich Mathematik, Technische Hochschule Darmstadt, 6100 DARMSTADT, Kantplatz I, West Germany

J. W. Mooney:

Department of Mathematics, Paisley College, High Street, PAISLEY, Scotland

R. E. O'Malley Jr:

Program in Applied Mathematics, Mathematics Building, University of Arizona, TUCSON, Arizona 85721, USA

G. Pagan:

Department of Mathematics, Royal Military College of Science, Shrivenham, SWINDON SN6 8LA, England

R. B. Paris:

Centre d'Studies Nuclearies, DP4PFC/STGI, Boite Postale No 6, 92260 FONTENAY-AUX-ROSES, Prance

H. Pecher:

Fachbereich Mathematik, Gesamthochschuie, Gauss-strasse 20, D 5600 WUPPERTAL I, West Germany

D. Rece:

Department of Mathematics, University of the Witwatersrand, JOHANNESBURG, South Africa

T. T. Read:

Department of Mathematics, Western Washington University, BELLINGHAM, Washington 98225, USA

B. D. Sleeman:

Department of Mathematics, The University, DUNDEE DDI 4HN, Scotland

R. A. Smith:

Department of Mathematics, University of Durham, Science Laboratories, South Road, DURHAM, England

M. A. Sneider:

Via A. Torlonia N.12, 00161ROMA,

Italy

D. C. Stocks:

Department of Mathematics, Royal Military College of Science, Shrivenham, SWlNDON SN6 8LA, England

C. J. van Duyn:

Ryksuniversiteit Leiden, Mathematisch Instituut Wassenaarseweg 80, LEIDEN, Holland

W. H. von Wahl:

Universitat Bayreuth, Lehrstuhl fur Angewandte Mathematik, Postfach 3008, D 8580 BAYREUTH, West Germany

J. Walter:

Institut f~r Mathematik, Universit~t Aachen, 51 AACHEN, Templergraben 55, West Germany

R. J. Weinacht:

Department of Mathematics, University of Delaware, NEWARK, Delaware 19711, USA

XVI

J. Wimp:

Department of Mathematics, Drexel University, PHILADELPHIA, PA 19104, U S A

A. D. Wood:

Department of Mathematics, Cranfield Institute of Technology, CRANFIELD Bedford MK43 OAL, England

A. Zettl:

Department of Mathematics, Northern Illinois University, DEKALB, Illinois 60115, USA

E X P O N E N T I A L B E H A V I O U R OF E I G E N F U N C T I O N S AND GAPS IN THE E S S E N T I A L S P E C T R U M F.V.Atkinson U n i v e r s i t y of Toronto I.

Introduction. In this paper we obtain c o n d i t i o n s on the c o e f f i c i e n t s in

c e r t a i n s e c o n d - o r d e r d i f f e r e n t i a l e q u a t i o n s w h i c h yield c o n c l u s i o n s r e g a r d i n g the spectra of a s s o c i a t e d d i f f e r e n t i a l operators.

Such results are p a r t i c u l a r l y w e l l - k n o w n for the case y" + ( I

- q)y = 0 ,

0 ~ t ~

;

(i.i)

we shall c o n s i d e r also the w e i g h t e d case y" +

(lw - q)y = 0 ,

(1.2)

and its v e c t o r - m a t r i x a n a l o g u e y" + again over

(IW

- Q)y = 0 ,

(0,o~), w h e r e

are square matrices. q, w, W

and

Q

positive,

and

W(t)

y

(1.3)

is a column-matrix,

and

W, Q

It will t h r o u g h o u t be assumed that

are continuous functions of

t , with

h e r m i t i a n and p o s i t i v e - d e f i n i t e .

c o n c l u s i o n s w i l l m o s t l y be of two kinds, spectrum contains the p o s i t i v e

w(t)

The

either that the

I - axis, or that certain

intervals n e c e s s a r i l y contain a point of the e s s e n t i a l spectrum. As a typical result in the first vein we cite that of ~nol

( 3, p.1562),

that for r e a l - v a l u e d

is a sequence of intervals bn - an ~

~

( a n'~ bn)

q(t),

, (bn - an )-I ~ q 2 ( t ) d t

the e s s e n t i a l s p e c t r u m a s s o c i a t e d w i t h p o s i t i v e semi-axis.

~

0 ,

(1.4-5)

(i.i) c o n t a i n s the

As to the second kind of result, we cite

the gap theorems for H a r t m a n and P u t n a m

for w h i c h there

with

(i.i) o b t a i n e d in the early paper of

(I0); c o n s i d e r a b l e extensions,

h i g h e r - o r d e r scalar equations, series of papers by E a s t h a m

c o v e r i n g also

have been given in a recent

(4,5,6).

We exploit here a l i t t l e - u s e d method,

in w h i c h we argue

s u c c e s s i v e l y between: (i) h y p o t h e s e s on the coefficients,

imposed in the m o s t general

cases over sequences of intervals, (ii) the e x p o n e n t i a l g r o w t h or decay, the h o m o g e n e o u s equation,

if any, of solutions of

again over s e q u e n c e s of intervals,

for the A -value in question,

and

(iii) a certain quantity the distance

of

~

@(A ), which in some sense measures

from the ~ essential

spectrum.

We make this last aspect precise, of

(1.3). We consider

measurable

operators

complex-valued

taking the general

in a hilbert

column-matrix

case

of locally

space

f(t)

functions

such

that " 2 d~f I f*(t)w(t)f(t) ilfll With

(1.6) we associate

dt

< co

a minimal

operator

(Tf) (t) = w-l(t) (Q(t) f(t) with domain f(t)

D(T)

support

in

T

- f"(t))

the set of continuously

with compact

p (~)

(1.6)

(0,~).

defined by ,

twice-differentiable

We then specify

is the largest number with the property lim inf

for every

ii(T -

sequence

{fJ

in D(T)

i fnl I = 1 , In practice

x

fn-

that

that

~ )'

(1.8)

such that,

as

n -~o~

0

we shall here bound

to have their support If

I )fn II ~ (

(1.7)

,

(1.9-10)

~(k

in intervals

)

by taking

the

(an, bn), where

~ ( k ) = 0, we have a standard

fn an ~

characterisation

of the

%

essential q

spectrum.

in (I.I) or

distance

of ~

If

(1.2)

~(~ ) >

Our approach from

different

to

hypotheses

6)

~ (~)

singular

integration.

The basic E(t)

of, roughly

y

of =

into two parts, and bring

the vast literature

(ii).

we see the

to light the connection seems due to (ii) come under

theory and that of asymptotic

idea is to assooiate,~ (1.3)

y*'y' +

speaking,

sequence

in that we do not argue

The idea of this argument

of stability

with a solution

is the

(6)).

(see the text of Glazman ( 8 , 181-183). Implications of the type leading from (i) to

the heading

or

this is the basis

(iii), but make a detour via

involved,

theory.

(see

from the direct

up the argument

with stability ~nol

differs

(i) immediately

is hermitian,

spectrum;

sequences"

as used by Eastham ( 4 -

In breaking

Q

is real, we have that

from the essential

of the method of "singular method

0, and

a function i y*Wy

Lyapunov

in various ways

such as

,

type. We do not attempt

on this topic.

(I.ii) to survey

Relevant

results linking

in a form going from w i t h real

q(t)

spectrum,

(iii)

(ii) and

to

(iii) are usually stated

(ii). Thus in the case of

b o u n d e d below,

and real

l

(i.i)

not in the e s s e n t i a l

it is known that there m u s t be a n o n - t r i v i a l solution

satisfying y(t) = for some

~ >

0(e-

~t)

0. This is due to ~nol

(1.12)

(see(8)

, p. 179), indeed

for the case of the m u l t i d i m e n s i o n a l Laplacian. case the conclusion is due to P u t n a m a s s u m p t i o n s that >

q(t)

lim sup q(t)

( i~

For the o r d i n a r y

w i t h the more special

is also b o u n d e d above, and that

. The s o l u t i o n a p p e a r i n g in (1.12) w i l l of

course be square-integrable,

and may be viewed as an eigen-

function a s s o c i a t e d w i t h some initial b o u n d a r y condition. We remark in passing that the c o n d i t i o n b o u n d e d b e l o w ensures that at

~

(i.I)

that

q(t)

be

is in the l i m i t - p o i n t c o n d i t i o n

. In w h a t follows, when linking

(ii) and

(iii) we shall

need similar more general c o n d i t i o n s w i t h the same effect, that the e s s e n t i a l spectrum will be non-vacuous, W e shall w e a k e n the p o i n t w i s e q

above to integral analogues,

Brinck

(2);

and complex That

so th

and p( I ) finite.

s e m i - b o u n d e d n e s s imposed on of the type i n t r o d u c e d by

further e x t e n s i o n s involve sequences of intervals, q

(1.12) m a y fail for real u n b o u n d e d

an e x a m p l e c o n s i d e r e d r e c e n t l y by H a l v o r s e n

q

may be seen from

( 9 ), who p r o v e s

also an interesting result in the converse sense, w i t h o u t b o u n d e d n e s s restrictions, of solutions of orders then

~

namely that if there exists a p a i r O(t-k-i/2),

0(t-k+i/2),

where

k~

0.

is not in the e s s e n t i a l spectrum.

We shall take up first the linking of depends on an integral inequality,

(ii) and

(iii). This

and simple i n e q u a l i t i e s

involving sequences. We shall then put this t o g e t h e r w i t h s t a b i l i t y - t y p e i n f o r m a t i o n so as to get results c o n c e r n i n g

~

( I ), and so on the e s s e n t i a l spectrum. We use the symbol *

t r a n s p o s e of a matrix,

to indicate the h e r m i t i a n c o n j u g a t e as in (1.6),

f , we take its n o r m as we take its n o r m as

max I Sfl

be d e n o t e d by

(i.ii).

Ifl = 4(f'f);

For a c o l u m n - m a t r i x

for a square m a t r i x

S

ISI in the o p e r a t o r sense, that is to say

subject to I ; we w r i t e

If~

= 1 . The identity m a t r i x will

Re Q = ½(Q + Q*)

In

~ 2

we prove the underlying

for w h i c h we need one-sided over an interval;,this of a result which, in

(I)

differential

restrictions

on the coefficients

forms the second-order m a t r i x analogue

in the

as a foundation

2n-th order scalar case, was used

for limit-point

criteria.

use this inequality

over a sequence of intervals,

with the hypothesis

that

exponential

behaviour

of intervals.

inequality,

~(k ) ~

we

combined

0, to yield a sort of

in an integral

We then specialise

In ~ 3

sense over sequences

the hypotheses,

for example

to make them ~hold over all intervals of a certain length,

so

as to obtain results of a known type on the exponential behaviour, (~)

including pointwise

>

behaviour,

0; our assumptions

usual pointwise bounds, hermitian

(or

q(t)

on

Q(t)

of solutions when

are weaker than the

and do not require that

real).

In

~5

we continue

Q(t)

be

this special

discussion

so as to complete

the argument of this paper in a

particular

case;

further assumptions,

additional

bounds on the coefficients,

by imposing

solutions

from behaving

positive,

and so force such

we go back to developing

exponentially

when

intervals,

intervals, abstracted

we develop more general criteria, spectrum to contain

~,o~),

the magnitude

of

together with

as to obtain order results as they overlap, provide

for

the results

~(~)

than those of

0,

In

~ 5, for the

over sequences of

and stability arguments )

as

k

-~

agree w i t h those of

(5, 10), but on the

hypotheses. discussions

It is a pleasure

with Prcfessors

to acknowledge

W. N. Everitt,

S. G. Halvorsen and A. Zettl,

helpful

W. D. Evans,

together with the o p p o r t u n i t y

to take part in the 1978 Dundee Comference on Differential Equations.

Acknowledgement

support of the National

so

; in so far

a different approach together w i t h variations

Acknowledgements:

7

we exploit the relation

behaviour, ~(~

~

over

behaviour over sequences

from the large ones.

C (~)'

to

on the coefficients

and in ~ 8

between the degree of exponential intervals,

is real and

the full force of the argument,

imply a certain degree of exponential of "small"

~

the

~ to be in the spectrum. In ~ 6

the effect that one-sided restrictions a sequence of "large"

in fact

we can prevent

is also made to the continued

Research Council of Canada,

through

2.

The basic inequality. The following result is very similar to Theorem 1 of

(1),

where the scalar

2n-th order case was dealt with. Subject

to the standing assumptions on Lemma i.

In the real interval

W , Q

we have

[a, b] let, for positive constants

A 1 and A 2 , and for a continuously differentiable matrix H(t) , there hold the inequalities (b-a)2W(t) ~/ AII, Let the column-matrix

Re Q(t)~/ z(t)

z" + ( ~ W

hermitian

H'(t), (b-a)IH(t) I ~ A 2, (2.1-3)

satisfy

- Q) z = 0 ,

a ~ t ~ b,

(2.4)

and write = max (Re I , 0) ,

(2.5)

v(t) = 6(b-a) -3 I(b - T)(T - a)dT .

(2.6)

Then

(zv +2z'v' ~ *w-l.zv"+2z,v, t %dt ~

C

z*Wz dt ,

(2.7)

where C = 2400! ~ A 1 1 + A1 2 + AI-2A22) .

(2.8)

In the proof which follows, all integrals will be over (a,b); the differential dt will be omitted. We note the estimates 0 ~ v ' ~ 3(b-a)-i/2,

Iv"l i 6(b-a) -2

(2.9)

We have first that the left of (2.7) does not exceed Al-l(b-a)2

~ (zv" + 2z'v')*(zv" + 2z'v')

2Al-l(b-a) 2 I (z*zv"2 + 4(z*'z'v'2)) ~_

72AI-2 I z*Wz

+ 8Al-l(b-a) 2 ~

z*'z'v '2

(2.10)

Here the first term on the right can be incorporated

in the

required bound (2.7-8). It remains to deal similarly with the second term on the right of (2.10). By an integration by parts, and use of (2.4), we have Iz*'z'v'2 = ~ z * ( ~ W

- Q) zv'2

Taking real parts, and using Iz*'z'v '2 ~_

-

(2.1-2),

(9/4) :(b-a)-2 Iz*Wz

+

2~z*z'v'v"

I z. z v 2

(2.9) we have - ~ z*H'zv '2

+ 4

z.zv

+

211)

Integrating by parts again, we have - '[z*H'zv 2 ,

=

2 Re~ z*Hz'v '2

+

2 ~z*Hzv'v"

(i/4)[ z*'z'v'2 + 4 I z*H2zv'2 + 18A2(b-a)-4~z*z (2.12) Turning to the last term in (2.11) we have, by (2.9), [z*zv"2 Combining

~

36(b-a)-4 Iz*z

<~ 36(b-a)-2Al-i I z*Wz .

(2.11-12) we obtain

(lj2) Iz. z v 2

{9 +j4 + A1 1(9A22 + 18A2 + •

This, together with

(b-a) -2 ~ z*Wz .

(2.13)

(2.10), yields a bound of the form (2.7-8).

The same result

holds, of course,

if

v

is replaced by l-v.

We supplement the above with a pointwise bound for Under the conditions of Lemma 1 we have z(t)v' (t) = ~(z'v' + zv")dT , v~ and so, by a vectorial version of the Cauchy-Schwarz z*(t) z(t)v'2(t)

~

(b-a)

I (z'v'+zv")*(z'v'+zv")dt

z .

inequality, •

Using the above estimations we get z*(t) z(t) ~

(b-a) 5(t-a)-2(b-t)-2(A3 ~ + A 4) ~ z*Wz at,

(2.14)

where A 3, A 4 depend only on AI, A 2 . Again, similar results were proved in ( 1 , p. 170) for the 2n-th order scalar case. 3.

Exponential behaviour. We now suppose that the hypotheses of Lemma 1 are satisfied

over a sequence of intervals 0 ~a

I ~

bI ~

a2 <

b2 ~

(ar, br), with . . . .

(3.1)

with the same constants AI, A 2 throughout. We will suppose also that 1 is such that ~( I ) ~ 0. We take some non-trivial solution y of (1.3), and write

Wr = Let

~i

function

£ u

(0, ~p ( l )), and, for

rs (t)

(3.2)

I y*Wy dt . ~T 0 ~ r <

s , let the

be defined by

Urs(t) = 0 ,

t ~

(ar, b s) ,

Urs(t) = y(t), t 6 [br, as],

and Urs(t) = in

Vr(t)y(t)

(ar, b r), (as , b s)

,

Urs(t) = (i - Vs(t))y(%),

respectively, where, as in (2.6),

%

P

vj(t)

= 6(bj-aj)-3J (bj - T)(T - aj)d~ .

By the definition of ~! i ) we have that, for given and sufficiently large r , say for. e ~ r O, II(T - I )Ursll 2 ~/

~

~12 ~ ~u*Wurs rs dt

~12 ~

y*Wy dt

>1 rl 2 ~

w.

(3.3)

On the other hand, by Lemma 1 and the fact that vanishes in

IIcT where

C

(T - t )Urs(t)

(br, a s ) , we have that

~)Ursll 2 .<

CCWr + w s)

is as in (2.8). Hence, using (3.3), we have Wr + ws

~

~12C -I ~

wj,

r~

rO .

(3.4)

Writing M1 = 1 +

(3.5)

FI2C -I ,

we then deduce from (3.4) that

wj

(3.6)

We distinguish

the two possibilities

In view of (3.4) the first implies that

wj

-.-->

co

.

(3.7)

while the second implies that

wj -~

0.

(3.8)

Thus either (3.7) 6r (3.8) is the case. We next note that it follows from (3.6) that

h__ wj and so, by (3.4),

>i

Mlk wm ,

m-k

ro

that

Wm_ k + win+k Suppose now that

~/

ko

(Ml-l)Mlk-i

wm •

(3.9)

is an integer such that

(MI-I)MI k°-I

~

2

It then follows from (3.9) that if

k ~

ko

the sequence

w m . win+k , Wm+2k . . . . is ultimately

convex,

or else ultimately

and so is either ultimately

decreasing,

Suppose for definiteness claim that there exist

m~

mI

mI and

according

or

w m , if

m ~/ m I , k ~

k~/

nlk o + n2(ko÷l)

ko2

k 1.

(3.10) is increasing

k = ko+l;

a finite number of such sequences, since any

to the cases (3.7-8).

such that

so that the sequence k = ko

increasing

that we have the case (3.7). We

m I , kI

Wm+ k > We choose

(3.10)

this involves

testing only

We can then take

can be represented

, with non-negative

if

k I = ko2 ,

in the form

integral

nl~ n 2 . It then

follows from (3.9) that Wm+k >/

½(M 1 - l)Mlk-i w m

(3.11)

subject to m ~ Similarly,

k+k I , 2k ~

kI .

in the case (3.8), we can choose integers

(3.12)

m I , kI

so that (3.10) is decreasing if

m~

m I, k ~

kI , and

can then deduce from (3.9) that, subject to (3.12), we have Wm_ k ~/

½(M 1 - l)Mlk-lwm .

We now introduce a quantity

~ = ~ (k), which measures

the exponential growth or decay of the \<~

(3.13)

w m | we set

=

lira sup k -1 lim sup !I in Wm+ k - in WmI.~ ~-9.~ ~ It follows from (3.12-13) that

(3.14)

in M 1 = in (1 +

(3.15)

~12C -1)

Our argument may be summed up, with a slight loss of precision, in Theorem 1.

Let, in the intervals [ar, br~

(br-ar)2W(t) ~ where

Hr(t)

and let wm

y(t)

AlI, Re Q(t) ~/ Hr'(t), l(br-ar)Hr(t)l~ A2,(3.16) is hermitian and continuously differentiable, be a non-trivial solution of (1.3). Then, with

given by (3.2) and ~ (~)

~/

by (3.14), we have

in (1 + ~2(~ )c-l).

This follows from (3.15), which holds for any We have discussed the case

~>

(3.17) ~l {

(0,~).

0 only| the result is

otherwise trivial. 4.

Corollaries regarding global exponential behaviour. We have now completed an argument to the effect that if

suitable conditions are imposed in sequences of intervals, and is not in the essential spectrum, then the solution exhibits exponential behaviour in an integral sense over this sequence of intervals. We shall later use this formulation to derive results about the essential spectrum. At this stage, however it is appropriate to give some more specialised and simpler formulations, including information on pointwise behaviour. We may work in terms of intervals of fixed length if

10 the coefficient

of

k

certainly

for (1.1).

us assume

that,

has a positive Taking

the more general

for some positive

W(t) >/ A l I ,

0 %

Re Q(t)~/ where in

H(t,

T)

Theorem

2.

essential

T)

(')

Under

denotes

we have for some

(4.1)

that we have,

and continuously d/dt

and 72 >

y 0

of unit length.

for any

A 2, T ~ t

~

T >w 0 , T+l,

(4.2)

differentiable

. We have then

the above assumptions,

spectrum

let

A1 ,

, IH(t, T ) I %

is hermitian

t , where

(1.3),

holds for any interval

we assume

H'(t,

case

and so

t < oo ,

so that the first of (3.16) For the rest of (3.16)

lower bound,

if

k

is a non-trivial

is not in the

solution

of (1.3),

either

e~ty(t)

--> 0 ,

(4.3)

or else e -2~t I Y2( -c ) d-~ t as

t -9

from Theorem e2~t

(4.3)

In the case certain

results

The equation

from

(4.5)

(1.1)

Q(t)

related

investigations statements

be seen by means

or else

0 .

q(t)

(4.5)

, the above

Snol and others,

W(t)

= I

q, or

and a pointwise by Rigler

are due to Kauffman

(4.3)

It would not seem that the corresponding

Q

i,

.

bound

(15);

y'(t)

(or (4.5))

ea~t

to in

other

(12,13).

may be made regarding that

extends

referred

bound on

been considered

of (2.13)

(4.4)

in view of (2.14).

a pointwise

with

has recently

Similar

->

, with real

of Putnam,

(1.3),

1 that we have either

I Y2(~)d~

since we do not assume

on

(4.4)

oO

It follows

We obtain

-~oo

. It may

imply that

o. pointwise

(4.6) statement

for

y'(t)

follows from this. A deduction of this kind may

indeed be made in the scalar case (1.1), with real real

q(t)

~

and

; however this will not be needed and we omit the

details. We have in any case, under the conditions of Theorem 2, that for some 72> 0

either

e27JtIly(t)12 + ly.(t)121 or else

-~9

0,

(4.7)

t+; e

-277tI Iy(~) 2 +

y'(-c

)t2

d~

-9~

(4.8)

.

t .

Simple conditions for the spectrum to contain

~0,Oo ).

Without at this point seeking the maximum generality,

we

note that some criteria for this situation are almost immediate. Theorem 3.

Let (4.1) hold and, with

differentiable,

T

--~ Oo

-9

0

(5.1)

. Let also ~ IQ(t)~ dt

as

continuously

let

T-11olW'(t)Idt as

W(t)

~

T --> oo . Then every real

0 0

(5.2)

is in the essential

spectrum. The hypothesis

(5.2) ensures (4.2), and so it is sufficSent

to show that no non-trivial

solution of (1.3) has the exponential

behaviour implied by (4.7) or (4.8). To this end one considers the growth or decay of E' =

~y*W'y

E(t) *

, as given by (1.11). We find that

Y*'QY + Y * Q Y

.

(5,3)

It follows that T-iIfE'E-lldt as

--~

0

(5.4)

T -->oo, which is inconsistent with (4.7) or (4.8)! this

proves the result. In particular,

we have the conclusion in the case (1,1) if

12 q(t)

(not necessarily real-valued)

some

p

with

1 ~ p

situation (with real complex 6.

q(t)

< oo q)

is in

LP(o, ~ )

for

. A recent discussion of this'

is due to Everitt ( 7 )3 spectra for

have been Cc,Bsidered recently by Zelenko (16).

Sequences of large intervals, We now revert to our main line of argument, in which we

consider the behaviour of the coefficients and of solutions over sequences of intervals, rather than over the whole semi-axis. We suppose that conditions are imposed on the coefficients which limit the exponential behaviour of solutions over a sequence of "large" intervals; within these large intervals we wish to be able to select "small" interval, satisfying the requirements of Theorem 1. The principle Theorem 4.

is embodied in

Let there be a sequence of intervals

~r' drD'

with 0 ~ cI

< d I G 02 <

....

(6.1)

and a positive, continuously differentiable function such that, on the

c r, d r , W(t) ~ ~2(t)l ~'(t)/~2(t)

and

?(t) /

I•

(t)dt

,

~

(6.2)

0 ,

--~ Oo

as t-~oo

as

,

(6.3)

r -->o~

(6.4)

G

Let

R(t), S(t)

be hermitian on the

continuously differentiable, S(t)

~c r, drY, with

R'(t)

continuous, and such that

on the ~cr, dr~ we have Re Q(t) ~/ R ' ( t ) + IR(t)l ~

and

K17(t)

(6.5)

,

(6.6)

,

I¢

~ I

S(t)

I

t) dt

(6.?)

<~- K2 ~ ~(t) dt e

--

'

13 where

K I, K 2

Let

~=

are positive constants. ~(~

)

be such that if

y(t)

is any non-trivial

solution of (1.3), and F(t) = then for any

(6.8)

y*'y' + y*Wy ,

~ >0

there is an

lln F(t") - in F(t')~ Then for any 6 - > 0

~

(~

r I such that, for

+ @)lTdt,

r >/r I ,

t', t" ~ ( C r , d r ) . ( 6 . 9 )

we have

In (i + ~2D-I)

~

2~-"

,

(6.10)

where

D -- lO 4 ~(~,÷ ÷ ~12)o---2 ÷ ~ Here

p

is as in (1.8), and

D

+ 2K22 t.

c6.n)

is a modification

of

the constant in (2.8); the numerical constants are of course not precise, and are inserted only to make plain their independence of the other parameters. similar in nature to ~

The quantity

is

in (3.14), or to Lyapunov exponents

or to the general indices of Bohl. If ~ ~--is arbitrary,

~

= 0 , the choice of

and (6.10) shows that

is in the essential spectrum.

If ~

as to obtain the best upper bound for

p = 0 , so that

0, we choose ~

~-- so

, at least so far

as order of magnitude is concerned. We use

G--to determine the

(am , bm). We may take it that

for some infinite sequence of k-values, f 2 k o - - ~ ~ 7 d t ~ 2(k+l)O ~- , c~ We determine 2k intervals (Ckr, Ck,r+l), Ckl = c k , such that

(6.12) starting with

~k~,~

7 d r - - ~ , r = 1 . . . . . 2k. (6.13) ckw It then follows from (6.7) that at least k of these intervals

must satisfy

14

ISl~-ldt We specify that the

(am , bm)

~< 2K2G--.

(6.14)

are to be those of the intervals

appearing in (6,13) which satisfy also (6.14) i k

runs through

a sequence of values satisfying (6.12). This process yields an infinite sequence of intervals

(am , bin), which are to be

numbered in ascending order. We shall have since

Cr--~ oo as

continuity of y

r -9 ~o

am -9 oo as

m -9 oo ,

: this follows from (6,4) and the

.

We next consider the lengths of the

(am, bin). Let

I

ly '(t)/72(t)l < ~ ' am % t Since

~< bm •

(6.15)

(6.16)

~ a ~ d t = O-- ,

we have, over

(am , bm), sup in 7 (t) - inf ln~ (t)

~o~

,

and so 7(am) exp (-o--~) ~< 7(t)

~ ~(am) exp(6--(~ ).

(6.17)

Hence, by (6.16), (bin-am) ~ (am)eXp(-6-~) ~°-~<(bm-am ) V (am)eXp(~6)' We next calculate admissible values of For

AI

A I, A 2

in (3,16)..

we need, in view of (6.2), that A I <~_ (bm-am)2~2(t)

, am <~t ~ b m ,

and so, by (6.17-18), we may take A I = o-2exp( Passing to the remainder of (3.16), we take t Hm(t) = R(t) + ~ S(-~)d-C, so that, by (6.6), (6.14) and (6.16), ]Hm(t)) ~

R(t) + IO2 (~)I~-I(~)S(Y~)IdxZ ~(am)

exo(O--~) (KI + 2K2O-" ).

(6.18)

15 Hence (bm-a m) IHm(t)l~
(6.19)

We insert these values in (2.8), and obtain a value

Noting that

~

may have any positive value in (6.15) if

is suitably large, by (6.3), may replace

C

we see that for large

m

m we

by the value appearing in (6.11).

We now consider (3.14). We must first consider the variation of

(bm - am )

with

m . We claim that

I(bm÷k - am+k)/(b m - am)l 1/k as

m, k --~ o~ Ck+ 1 ~

am <

-9

1

(6.20)

, subject to bm <~ ...

< b m + k ~< dk+ I .

(6.21)

Write now sup I?'(t)/?2(t)l,

Ck+ l <

t~

dk+ 1 •

Using (6.17) with notational changes we have lln (bm+ k - am+k ) - in (bm - am)l~

linT(am+ k ) - in ~ ( a m ) l + 2 o ~

by (6.12) and (6.21). This proves (6,20), in view of (6.3)° Suppose now that

y

is a non-trivial solution of (1.3).

We have trivially that wm

~

(bm - am ) max F(t) ,

where the maximum is over ~am, b ~ . opposite sense we denote by (am, bin). We have

wm +

We now use ~2.13), which shows that y*'y' dt

~

ClW m

For an inequality in the

(o4 m, ~ m )

~ I F(t)dt g

,

(6.22)

the middle third of ¢~ y*'y'dt .

3

(6.23)

16 where

C1

does not depend on

m ; it is given, apart from a

numerical factor, by the expression in the braces in (2.13). Hence, by (6.23), 3-1(bm - am ) min F(t)

~

Wm(1 + C l) .

(6.24)

It follows that, subject to (6.21), lln Wm+ k - in Wml ~

sup in F(t) - inf in F(t)

+ in (3+3C l) +

+ lln (bm+k-am+k) - in (bm-am) ], where the "sup" and "inf" are over (Ck+ l, dk+l). In view of (6.20), it now follows from (3.14), (6.9) and (6.12) that

We now get (6.10) as a consequence of (3.17). This proves the result of Theorem 4. 7.

More on the case that ~0,o~)

is in the spectrum.

We go back to the topic of ~ 5, to generalise the approach there given. Whereas in

95 we needed that the solutions should

not grow or decay exponentially when considered over the half-axis, we can now exploit the situation that this is the case over a sequence of "large" intervals. As indicated after the statement of Theorem 4, it is a question of imposing extra hypotheses so as to ensure that

~=

0

in (6.9). The nature of

these extra hypotheses will depend on the stability argument being used. With the technique

indicated in (5.3) we have

Theorem 5. Let (6.1-4) hold and, with differen,tiable, let, as r ~

W(t)

oo ,

~-21W'1dt = o ~ ~ ~ dtl !

Ie~-ll = . Qldt o I!Jd ~. Then every real

~ ~/0

continuously

is in the spectrum.

(7.1)

(7.2)

17 Suppose that ~ is real and positive. equally use

E(t)

instead of

In (6.9) we may

F(t), and so we have (6.9) with

if

~'F.~-~l~t

= of~,dt

This follows from

I.

(7.3)

(7.1-2) and (5.3) since E(t)

as given by

(1.11) satisfies E(t) ~

ly'(t)l 2 + ~ ~ 21y(t)} 2

v criterion (1.4-5) of Snol we may

In particular, in the weaken (1.5) to

(7.4)

~

(bn - an)-l

!!q(t)Idt

--~ 0 ,

and dispense with the requirement that it was not required in Theorem 5 that

(7.5)

q(t) Q

be real-valuedl

be hermitian.

In a second example, we work in terms of an integral of Q(t) take

rather than its pointwise values. For simplicity, we now Q(t)

hermitian.

Theorem 6. Let (6.1-4) held, and also (7.1). Defining in the ~c r, dr~

R(t)

by

t R(t) = where

Q

t

,

Cr~

W-~(t)R(t)W~(t)

=

t ~

dr '

(7.6)

is hermitian, let R(t),

as

~~ Q(x= )d~

-~ oo

.

o ?(t)

Then the spectrum contains

(7.6-7)

£0,°0).

The proof is similar to that of the last theorem, using this time the Lyapunov or energy function El(t) =

yl*y I + ~ y*Wy ,

Yl = y' - Ry •

(7.8)

A simple calculation gives

E1 ' = 2 Re (~ y*RWy - Yl*RYl - Yl*R2y) + ~ y*W'y . Hence +

l,i

+ x

+

(7.9)

18 We then get an analogous result to (7.3). It is easily seen that E1

may replace

F

In particular, such that max O ~ h<_ 1

in (6.9) when

~\~ 0, in view of (7.6).

this covers the case of (1.1) with real

j iT+l, t q(t)dt

~

0

as

T

--~ ~o

.

q

(7,11)

T

For example, we can take

q(t) = t sin t 3 . More generally, we

can require that (7.11) hold as

T -~ oo through a sequence of

intervals whose length is unbounded. As a further example, we take the case of (1.2) with w

continuously twice differentiable,

for simplicity,

and

q

continuous and,

real. For a non-trivial solution we form the

energy-type function 2 _i 2 A E 2 ( t ) = y ' w 2 + ~ y W~ + ½ y y , w , w - 3 / 2 ,

(7.12)

for which

E2, = 2yy,qw-~I + ½y2qw 'w" 3 / 2 + ½ y y , ( w , w - 3 / 2 ) ' • Taking

(7.13)

~ real and positive, and assuming that

lw'w-3/ l <

(7.14)

we deduce that

o 0

X-½1qrw-½ ÷

l w.w- l +

By this means we obtain Theorem 7.

In a sequence of intervals Fcr , d ~

, with

L-

c ~ w ~ dt let

q w-1 -9

O,

w'w-3/2--~

, 0

and

w"w-2-9 0

(7.16) as

t

-9 =6 .

Then the spectrum includes ~ , c O ) . The proof follows the same lines, replacing by

F

in (6.9)

E 2 . The pointwise conditions imposed in the theorem are

easily generalised to integral conditions.

19 8.

Order of magnitude of gaps in the essential spectrum. In the hermitian case, to which we now confine attention,

t (~)

gives the distance of

and so, if -)Oo ~(~),

~

is real,

~

from the essential spectrum

the order of magnitude of

p(~ )

as

will be essentially the same as that of the function the length of a gap with centre /~

in the essential

spectrum~ it is the latter function which has been investigated by Eastham. In a series of papers (see (6)) he has extended earlier results on the relation between this order of magnitude and the continuity and similar properties of the coefficients, The present method, as we have developed it for the second-order case, seems to have similar power to the singular sequence method used in

( 6 ), at least in certain cases. It then

becomes a challenge to develop the stability techniques used here so as to cover the remaining cases obtained by the other method. We consider the matrix case (1.3), with hermitian and consider the order of natural choice of assume that

~"

for large real

in (6.10-11) is

~(~ ) ~

immaterial, and

t(X)

~

Q(t), . A

I / ~ ( ~ )! here we

O, since otherwise the choice of ~5- is

~ ( k ) = O.

We then have

~2 =

O(D)

,

and so, explicitly,

Suppose now that, as ~

"~,

{~) -- O{X½}, 1/~{~) -- O{X½). We c a n t h e n s i m p l i f y

e{X}

(8.1)

(8,2-3)

to

_-0'{~,~ 1j

{ x }?.

{8..)

Such a result will, of course, only be significant if

~ { X } = ocX½> . We proceed to examine some such cases.

{8.5}

20 As in the case of ~ 7, that in which the positive half-axis is in the spectrum,

one obtains a variety of results according

to the choice of "Lyapunov function".

Using E(t), as in (1.11),

we obtain, for the case of (1.3) with hermitian

Q(t),

Theorem 8. Let (6.1-4) hold. Let also (7.1°2) hold in the modifed form that "small o" °'big 0 ". Then, as

on the right is replaced by

~ -~oo ,

(~) = 0 ( ~ ~) . Using ( 5 . 3 )

(8.6)

we now find that, in the

~r' dr]'

~,=-l= 0(~-21.,i)÷ O(,k-~7-11Q1), where the last term is not significant,

(8.7)

The result now follows

from (8.4). If in (1.3)

W(t)

is constant,

say

W(t) = I, then the

first term on the right of (8.7) may be omitted, and we get

#(~) In particular, if and

w(t)

=

O(1)

(8.8)

(8.6) holds for ~he scalar case (1.2)

is bounded above and has a positive lower bound,

has a continuous bounded derivative,

and bounded. This is

and

q(t)

is real

a result of Eastham ( 5 ), who considers

the operator associated with (s(x))-l{ - (d/dx)(p(x)dy/dx)+

q(x)y~ = k y .

(8.9)

In this scalar case, one may transform to the situation (1.2) by means of the change of variable then given by

s(x)p(x)

dt = dx/p(x);

w(t)

is

. However we omit the details.

We next give results which fill in the range between (8.6) and the weakest significant assertion, namely that

p ( ~ ) = o(1).

(8.1o)

t

We now n e e d y e t a n o t h e r L y a p u n o v - t y p e f u n c t i o n ,

namely

21

E3(t ) = y*'y' + ~ Y * W l Y where

,

(8.11)

r t+ ~"~/~ Wl(t) :

k½

I W(% )d~.

(8.12)

t We now treat the case in which and in which

W(t)

~(t)

is a positive constant

is also bounded above. We write

Z (k ) =

lim sup t -~

max IW(t+s)-W(t)l. 0 < s < ~ -~

(8.13)

Simplifying the situation somewhat, we have Theorem 9. Q(t)

Let

W(t)

be continuous and bounded, and let

be hermitian and bounded. Let

positive-definite

W(t)

lower bound. Then, as

0(~):

O(x~(~)

+

also have a

~ ~ 00 ,

O.

(8.1.)

For the proof we note that E 3'

= 2 Re { y * ' ( ~ (WI-W) + Q)y~ + ~ Y*Wl'Y ,'

and so

We deduce that

'¢

_- 0

from which the result follows. We get the case (8.10) if and a range of intermediate

W(t)

is uniformly continuous,

cases are given by making

satisfy a HGlder condition with a suitable exponent. particular, condition,

we get (8.6) if

W(t)

W(t) In

satisfies a uniform Lipschitz

in addition to the hypotheses of Theorem 9. Of course

according to the present method we do not need to impose such conditions for all

t , but only on a sequence of intervals

of unbounded lengths. The case (8.10) is among those considered by Eastham ( 5 ), for the case (8.9). For results similar to (8.14) in the scalar

22 case reference may be made to Hartman and Putnam (i0), where the oscillation method is used. As is evident from Eastham's work, a reduction of the order of

~(~ )

depends on more drastic assumptions regarding the

smoothness of the coefficients and, if the present method is used, upon the use of a more developed Lyapunov function. We illustrate this here by obtaining (8.8), the case of bounded gaps, for the scalar case (1.2), with non-constant w(tl). We use now the function assume

w(t)

E2

given in (7.12), and

twice continuously differentiable, in a sequence

of intervals Kc r, dr~ satisfying (7.16). We have Theorem 10.

In addition to (7.16) let

w'w -3/2

be bounded on

the ~c r, dr~ , and let I~¢~lqlw-½ + lw°'lw-3/21 dt = 0 1 1 ~ w ~ £r

Then

I(~ ) = O (1)

dtl.

(8.15)

~" C~

as XA --) 00

In particular, if

w

.

is bounded from zero and bounded

above, it is sufficient that

q, w'

and

w"

be bounded on a

sequence of intervals of unbounded lengths. The result of the theorem follows from (7.15) together with the previous arguments. More elaborate Lyapunov functions than (7.12), involving higher derivatives, have often been used to get, stability theorems (13) . However it is not clear how to extend them to the matrix case (1,3), or how to form them in general.

23 REFERENCES I.

F. V. Atkinson, Limit-n criteria of integral type, Proc.Royal Soc. Edinb.(A),

2.

73(1975), 167-198.

I. Brinck, Self-adjointness and spectra of Sturm-Liouville operators, Math. Scand. 7(1959), 219-239.

3.

N. Dunford and J. T. Schwartz, Linear Operators. Part II, (Interscience, New York, 1963).

4.

M. S. P. Eastham0 Gaps in the essential spectrum associated with singular differential operators, Quart. Jour. Math. (Oxford)(2), 18(1967), 155-168.

5.

M. S. P. Eastham, Asymptotic estimates for the lengths of the gaps in the essential spectrum of self-adjoint differential operators, Proc. Yoral Soc. Edinb.(A), 74(1976), 239-252.

6.

M. S. P. Eastham, Gaps in the essential spectrum of evenorder self-adjoint differential operators, Proc. London Math. Soc.(3), 34(1977), 213-230.

7.

W. N. Everitt, On the spectrum of a second-order linear differential equation with a

p-integrable coefficient,

Applicable Analysis, 2(1972), 143-160. 8.

I. M. Glazman, Direct methods of qualitative spectral analysis of singular differential operators,

(Israel

Program for Scientific Translations, Jerusalem, 9.

1965).

S. G. Halvorsen, Counterexamples in the spectral theory of singular Sturm-Liouville operators, Mathematies no. 9/74, Matematisk Institutt, Universitet i Trondheim, Trondheim, Norway.

10. P. Hartman and C. R. Putnam, The gaps in the essential spectra of ~ave equations, Amer. Jour. Math. 72(1950), 849-862.

24 ll. R. M

Kauffman,

On the growth of solutions in the

oscillatory case, Proc. Amer. Math, Soc. 51(1975),49-54 12. R. M. Kauffman, Gaps in the essential spectrum for second order systems, Proc. Amer. Math. Soc. 51(1975),

55-61.

13. A. C. Lazer, A stability condition for the differential equation

y" + p(x)y = O, Michigan Math° Jour. 12(1965),

193-196. 14. C. R° Putnam, On isolated eigenfunctions bounded potentials,

associated with

Amer. Jour. Math. 82(1950),

135-147.

15. D. A. R. Rigler, On a strong limit-point condition and an integral inequality associated with a symmetric matrix differential 76(1976),

expression,

Proc. Royal Soc. Edinb.

(A),

155-159 •

16. L. B. Zelenko, Spectrum of SchrGdinger's complex pseudoperiodic Diff. Urav. 12(1976),

potential,

equation with a

Parts I and II,

806-814 and 1417-1426.

L A P L A C E INTEGRALS IN S I N G U L A R D I F F E R E N T I A L AND D I F F E R E N C E EQUATIONS by B.L.J. B r a a k s m a

0. I N T R O D U C T I O N

In this p a p e r we c o n s i d e r singular d i f f e r e n t i a l e q u a t i o n s

(0.i)

xl-P dy = f(x,y), dx

and d i f f e r e n c e e q u a t i o n s

y ( x p + i) = f(x,y(xP)).

(0.2)

Here p is a p o s i t i v e integer, y £ ~n and f(x,y) of x and y in a set S X £(0; p

6 ~n, f(x,y)

is an a n a l y t i c function

po ) w h e r e S = {x 6 ~ : Ixl > R, e < arg x < 8} and

> O. 0

Assume

(0.3)

f (x,y)

7 ~6I

b

(x) y~), w h e r e I = IN

and (0.4)

b

(x) N

We m a y t r a n s f o r m then

[ b k x k=0

(0.I) and

-k

as x + ~ in S.

dy (0.2) b y x p = ~ to e q u a t i o n s for ~

and y (~+i), but

(0.4) is an e x p a n s i o n in f r a c t i o n a l powers of ~. In general ~ will be a

singular p o i n t of the d i f f e r e n t i a l e q u a t i o n

(0.i) of rank at m o s t p. If D f(x,0) Y I + 0(x -2) as x + ~ then ~ is also a singular p o i n b of the d i f f e r e n c e e q u a t i o n (0.2). The c o n s t r u c t i o n of solutions of

(0. I) and

(0.2) near the s i n g u l a r p o i n t

o f t e n c o n s i s t s of two parts. I. The c o n s t r u c t i o n of a formal series w h i c h f o r m a l l y satisfies if the formal series for y and the a s y m p t o t i c series for b

(0. I) or

(0.2)

are s u b s t i t u t e d

in (0.3) and

(0.1) or

For example,

in several cases there exists a formal s o l u t i o n of the form

(0.2).

26

(0.5)

x

However,

X c x m o

-m

in g e n e r a l

t h i s formal

series d o e s not converge.

II.The p r o o f t h a t t h e r e e x i s t s a n a n a l y t i c as a s y m p t o t i c

expansion

this analytic

part.

First we consider + b(x).

the linear case of

W e a s s u m e t h a t the n x n - m a t r i x

as L a p l a c e

integrals.

w i l l be d e f i n e d a formal

s o l u t i o n of

w h i c h has

We c o n s i d e r

in sect.

(0.5)

s o l u t i o n w h i c h h a s t h e formal

as x ÷ ~ in a c e r t a i n

(0.1) or

A(x)

(0.2), w h e r e

and the n - v e c t o r of L a p l a c e

f(x,y)

b(x)

= A(x)y +

are r e p r e s e n t a b l e

integrals A 1 and A 2 which

to A. a n d (0.5) is 3 e x i s t s an a n a l y t i c s o l u t i o n y(x)

t h e n there

expansion

class A. w i t h the same h a l f p l a n e s 3 (cf. sect. 2 a n d 3).

solution

We shall c o n s i d e r m a i n l y

s h o w that if A and b b e l o n g

(0.2),

as a s y m p t o t i c

(0.I) a n d

two c l a s s e s

i. We w i l l

region.

and w h i c h

of c o n v e r g e n c e

is such that x-ly(x) as the L a p l a c e

is of

integrals

for A

and b

T h e class A 2 of L a p l a c e factorial solution

series of

expansions.

(0.i) or

(0.2)

since if the f a c t o r i a l the formal

series

In sect.

integrals

series

solution

there

to a formal

exists

which

exists

admit convergent

a factorial

solution

(0.5)

it m a y be c a l c u l a t e d

series

is i m p o r t a n t directly

from

(0.5).

a s s u m e t h a t f(x,y)

the n o n l i n e a r

is r e p r e s e n t a b l e

conditions

similar

exists a solution

in the form of a L a p l a c e

Instead of

ck x

-X k

(0.5) we m a y h a v e

,

case of

as a L a p l a c e to those

(0.I)

integral

and

(0.2).

formal integral

formal

Now we

o f a f u n c t i o n ~(t,y)

for the c l a s s e s A 1 and A2.

in this c a s e w e s h o w t h a t if there e x i s t s a

expansion.

of f u n c t i o n s

The problem whether

corresponding

4 and 5 we c o n s i d e r

which satisfies

consists

solution

(0.5)

which has

Also

then there

(0.5) as a s y m p t o t i c

solutions

X k ÷ ~ as k ÷

o of

(0.i)

or

(0.2).

For these

formal

solutions

a result

similar

to t h a t for

(0.5)

holds. Solutions

of

(0.i)

s t u d i e d b y Poincar~, others.

Following

or d i f f e r e n c e

Horn

= Yo +

Volterra

We show that a solution space o f a n a l y t i c

(0.2)

in the form of L a p l a c e

Horn,

Trjitzinski,

([8] - [12]) we t r a n s f o r m

equation

y(x)

into a s i n g u l a r

and

Birkhoff,

integrals

Turrittin,

have b e e n

Harris,

the d i f f e r e n t i a l

S i b u y a and

equation

(0.i)

(0.2) b y m e a n s of

S o

e -xpt w ( t ) d t

integral

equation

o f this i n t e g r a l

functions

for w

equation

with exponential

(here p = I in case of exists

bounds

in a s u i t a b l e

in a sector.

(0.2)). Banach

T h i s leads to a

27

solution of Volterra

(0.i) or

(0.2) with the d e s i r e d properties.

integral equations

In sect.

6 we give applications

when formal solutions of solutions

in asymptotics

(0.i) and

in the sense m e n t i o n e d

for linear equations

of the results

Malmquist

above.

Also an application

[18],

2-5. Here we show analytic

to a reduction

The differential

[7]

and Iwano

[14],

(0.i) has been investigated

in [2], where also functional

in y has been considered

[8] - [12], T r j i t z i n s k y

Harris and Sibuya

[24, ch. ll]. The linear case of

type are considered.

[4].

theorem

is given.

[16], T u r r i t t i n

ris Jr. and myself

in sect.

(0.2) exist to w h i c h correspond

Our results are related to the w o r k of Horn

aleo W a s o w

The r61e of singular

has been explained by Erd61yi

equation

differential

equations

(0.I) where f(x,y)

[17], [15], cf.

by W.A. Harof a certain

is a polynomial

in [i].

I. LAPLACE INTEGRALS A N D F A C T O R I A L SERIES We shall consider cases w h e r e

the differential

grals. We use two classes of Laplace DEFINITION

and difference

(0.3) holds and the c o e f f i c i e n t s

I. Let p be a positive

@i ~ arg t ~ 82 }

inclusive

b

integrals.

integer,

equations

ii) ~

of

(0.2) in

They are d e f i n e d as follows:

@I ~ @2' ~ ~ 0. Let S 1 = {t 6 ~ :

the p o i n t 0. Then a I (@i' @2' g' P) is the set of

functions ~ such that I i) t i - p ~ 6 C (SI, ~n) and, if @i < @2' then ~ is analytic o S1

(0.I) and

b e l o n g to a class of Laplace inte-

in the interior

S 1.

(t) = O(1) exp

(~lltl)

as t ÷ ~ o n S 1 for all ~i > ~"

m

iii) ~

(t) ~

X tOm m=l

t ~ -i as t + 0 on SI, where ~ m 6 ~n, m = I, 2 . . . .

Let (i.i)

G1~

Gi(~)

= {x 6 ~ : B @ £ [@1,@2]

Then AI(01,

@2' ~' p) is the set of analytic

such that Re

functions

(xPe i@) > g}.

f : G1 + ~n such that

i0 (1.2)

~(x) = fo +

s~e o

e -xpt ~ ( t ) d t ~

f

o

+ L ~(x), p

x£Gi(~) ,

where fo £ ~n and ~ 6 a I (@i' 02' U, P)We now define

subsets a 2 (m, ~) and A 2 (~, U) of a I (@, 8, g, i) and

A I (@, 8, g, i) where m 6 ~, ~ + 0, ~ ~ 0 and @ = - a r g

~. Let us agree that a

function is analytic on a closed set if it is continuous lytic in its interior.

on the set and ana-

28

D E F I N I T I O N 2. Let ~ 6 ~, m ~ 0, O = - arg m, H > 0. Let S 2 = $2(~) be the compon e n t of {t E ¢ : Ii - e-~t I < I}

t h a t c o n t a i n s the ray arg t = @. T h e n a2(~,~)

is

the set o f f u n c t i o n s 02: $2(~0) ÷ Cn such that: i) 02 is a n a l y t i c on S 2(~). ii) 02(t) = 0 (i) exp

(HiItl)as t + ~ on s2(w)

for all ~I > ~"

Let G 2 = G2(~I) = {x E ¢: Re(xe i8) > ~}. T h e n A2(m,~) functions f: G2(U) + Cn w i t h the r e p r e s e n t a t i o n

is the set of analytic

(1.2) w h e r e p = 1 such that

02 6 a 2 (~,~) and fo 6 ~n. For short we will o f t e n d e n o t e the classes of Laplace integrals A d e f i n e d a b o v e b y AI,

A 2 or A 1 (H), A2(H)

if we o n l y w a n t to stress the v a l u e of the

p a r a m e t e r ~. Moreover, we w i l l use a similar d e f i n i t i o n for m a t r i x functions. It is w e l l k n o w n (1.3)

(cf. D o e t s c h

[3, p.45,

174] that f 6 AI(@I,

@2' ~' p) implies

f(x) ~ fo + I F (m) q)mX-m as x ~ m= i

on any c l o s e d subsector of G 1 of the form: -

½7 - 8 2 + £ ~ arg x p ~ ½7 - 8 1 - E, E > 0.

For short we shall say in this case that Conversely,

(1.3) holds o n closed subsectors of G I.

if f is a n a l y t i c on a closed sector G such that G I C G ° and

(1.3) h o l d s

o n G, then f E A 1 (@I' 82' D' P) for some ~ ~ 0. If f E A 2 (~, ~) then f is r e p r e s e n t a b l e by a factorial series (1.4)

f(x) = f

+ o

m~fm+ I

E m=0

~

(~ + I) ...

, x E G2(~) , (~ + m)

where f

6 ~n if m 6 ~ (cf. D o e t s c h [3, p.221]. m Conversely, if (1.4) holds, then f has a L a p l a c e integral r e p r e s e n t a t i o n

(1.2) w i t h p = I, @ = -arg ~ u n d e r s o m e w h a t weaker c o n d i t i o n s on ~ than in d e f i n i t i o n 2: 02(t) = 0(i)

exp

(~lltl) o

o < e < ~ and 02 is a n a l y t i c in S 2 If f E A2(~, of G2:

~), then

IIm ~t I ~ ~ - 6 for all HI > D~"

(1.3) w i t h p = 1 h o l d s as x ÷ ~ on any closed subsector

larg x - O I ~ ½7 - ~ (0 < e < ½z). Conversely,

w i t h p = i holds as x + ~ on factorial

as t ÷ ~ o n (~)

series

if f 6 A2(~, ~) and

(1.3)

larg x - 6 I < ½n - g, then we m a y c o n s t r u c t the

(1.4) from the a s y m p t o t i c series: we may e x p a n d each term in

(1.4) in an a s y m p t o t i c p o w e r series, c o m p a r i s o n w i t h

(1.3) n o w gives a r e c u r s i o n

formula for the fm+1" A l t e r n a t i v e l y we may w r i t e x -m as a factorial series; s u b s t i t u t i o n in (1.3) and c o m p a r i s o n w i t h

(1.4) g i v e s also a r e c u r s i o n f o r m u l a for

fm .For the explicit form of this formula c f . W a s o w [23,p.330] In this w a y w e sum a s y m p t o t i c series for functions in A2(~,~) b y factorial series. This is a useful p r o p e r t y since factorial series converge u n i f o r m l y in half planes.

This p r o p e r t y

will be u s e d in the following sections w h e r e we encounter formal p o w e r series solu-

29

tions w h i c h u n d e r c e r t a i n c o n d i t i o n s are a s y m p t o t i c e x p a n s i o n s of solutions in A2(~,~)

and c o n s e q u e n t l y m a y be summed to any d e g r e e of a p p r o x i m a t i o n

rial series.

If m > I, then S 2 ( m w ) C S 2 (~) and so A2(~,~)

ly factorial series

c A2

(m~,~). Consequent--

(1.4) also are r e p r e s e n t a b l e b y factorial series

with parameter me instead of

by facto-

(1.4) on G 2

~ if m > I.

If fl' f2 6 Aj then also fl f2 6 Aj since

q91 ~ %92 6 aj

if

~9I, q02 6 aj

.

2. THE L I N E A R D I F F E R E N T I A L E Q U A T I O N

We n o w c o n s i d e r the d i f f e r e n t i a l e q u a t i o n

(0. I) in the case that it is

linear and that it is a c o u p l e d s y s t e m of a s y s t e m w i t h a s i n g u l a r i t y of the first k i n d and a s y s t e m w i t h a s i n g u l a r i t y of the second kind. To formulate this we p a r t i t i o n n x n - m a t r i c e s along the n I - th row and column w h e r e 0 < n I < n:

where

Mj h

is

vectors f =

an

n.

x n h matrix,

n2 = n

-

n 1.

A corresponding

2

after the n I - th component will be used.

partitioning

of

N o w consider the system

(2.1)

xl-P

d__yy= A ( x ) y + b(x) dx

w h e r e p is a p o s i t i v e integer, and c o n c e r n i n g A and b we assume either case I : A, b 6 A I (81' @2' ~' p) or case 2: p = i and A, b 6 A 2 ( ~ , ~). T h e n we have r e p r e s e n t a t i o n s

(2.2)

A(x) = A

+ Lp~(X)

o

'

b(x) = b

o

+ L 8(x) p

and a s y m p t o t i c e x p a n s i o n s

(2.3)

A(x)

~

Z m=O

A x -m, b(x) ~ [ b x -m as x ÷ m m=0 m

in closed subsectors o f G 1 in case I and G 2 in case 2. We assume 21 Allm = 0, A 12m = 0, b Im = 0 if m = 0,1,..., p-l; A 0 = 0, (2.4) A 22 + ptI is n o n s i n g u l a r in S in case j o n2 j "

30

Then

THEOREM

we have

i. Suppose

Xo

c

m

x - m ~s a formal

solution of (2.1). Then there exists

an analytic solution y of (2.1) which belongs

to A I ( 0 I, 0 2 , v, p)in case i and

to A 2 ( ~ , ~) in case 2 such that -m (2.5)

y(x)

~

X 0

c x m

as x ~ ~ on any closed subsector of G 1 in case i and of G 2 in case 2. The solution y with these properties

is unique.

REMARK.

formal

In c a s e

factorial

PROOF.

2 we m a y

series

Let u =

which

sum the

satisfies

(2.1)

N-I -m Z c x , a partial m 0

xl-P

d_uu = A ( x ) u dx

solution o n Re

sum of the

+ b(x)

~ c x -m to a c o n v e r g e n t

(xe l@

> ~

formal

(cf.

sect.

solution.

i).

Then

- c(x)

where

c 6 A. a n d c 1(x) = 0 ( x - p - N ), c 2 (x) = 0(x -N) as x + ~ on c l o s e d s u b s e c t o r s 3 of G.. H e n c e w i t h y - u = v w e g e t x I-p d v = A(x) v + c(x) as e q u a t i o n e q u i v a l e n t 3 dx t o (2.1). So it is s u f f i c i e n t

to p r o v e

p + N - i, b 2 = 0, h = 0,I, assume

this

the theorem

..., N-I

latter

condition

B(t) ~

Z m=N

in c a s e b hI = 0, h = 0,

for a s u f f i c i e n t l y

large

f r o m n o w o n or e q u i v a l e n t l y

by

.. .,

integer

(2.2)

(cf.

N. W e (1.3)

m__ 1 (2.6)

We seek a solution A.. 3

If y = L

(2.7)

8m t p

y of

as t ~ 0 in Sj,

(2.1) w h i c h

is 0 ( x -N)

~h1 = 0 if N < h < N + p - i.

as x ~ ~,

and which

belongs

to

w is of c k a s s A. t h e n P 3

xl-P

dxd--YY= L p

(-ptw) , A ( x ) y

= Lp(A0w

Hence

+ ~ * w).

(2.1) h a s a s o l u t i o n y = L w o f c l a s s A. iff - p t w = A w + ~ * w + B an~[ P 3 o w 6 a . ( ~ ) . T h i s e q u a t i o n f o r w is a s i n g u l a r V o l t e r r a i n t e g r a l e q u a t i o n . 3 1 l-If t P v 6 C ( S , %n) we d e f i n e 3 (2.8)

Tv = -

(A + p t I) -I o

With

(2.9)

~ = -

(A

+ p t o

I)-is

(~ * v).

31

the equation

(2.10)

for w

is equivalent to

w = Tw + ~.

The assumptions

(2.11)

on A 0 imply that

(A° + p t I) -i = diag {p-lt-iInl,

(A~ 2 + p t In2)-l}

and that (2.12)

(A 22 + p t I )-i and t(A + p t I) -I 0 n2 o

are uniformly bounded on S.. So if n I > 0 then T is singular in t = 0. 3 We solve (2.10) in a Banach space V N of functions v : S. ÷ ~n such that N 3 -t

P v is analytic

(2.13)

II

in S. and 3 N i-vl~ = sup It p v(t)] t6S. J

Here ~i is a fixed number,

_l/lit] e

<

Pl > l/ where p is the parameter

or A 2 ( ~ , l/). It is clear that V N is a Banach space with definition will be used for m a t r i x - v a l u e d Since b £ Aj , it follows from Using

(2.9),

(2.6),

(2.11) and

l/ in AI(81,

norm

(2.14)

tk-i

(2.2) that 8(t) = 0(e l/lltl) as t + ~

Moreover,

in Sj.

(2.12) we deduce ~ 6 V N-

6 Vp,

2h

A 6 Aj,

(2.2) and

6 Vl, h = 1,2. Since

, tm-i = B(k,m) tk+m-I if Re k > 0, Re m > 0,

-~ we see that t p

If.IfN. A similar

functions.

Next we show that T maps V N into V N. From the assumption (2.4) we deduce that l h

82, p, p)

1- N+--L (~ * v) I and t

P

(~ * v)

are analytic on S. if v 6 V N. J

if t 6 S. then 3 _N_ I

I t-l((~ * v) l(t)l < (ll°~ll]lp + ]] ~1211 p/ e!/ll t ] Ilvl! N I t - 1 (1 , t p Hence, by

(2.15)

(2.11)

]]{ (Ao + p t i) - 1 ( ~ *

v)}lll N

t~ 1 (tl £1 IIp+ll £2 IIp) tlvtlN

Similarly

l/llti I(~ * v) 2(t) l < II all i e and therefore

i

-- - I

IIv II N Itp

-- - I

* tp

1

)l-

82

(2.16)

I[{(A° + p t I) -I

p p < ll~lq1 livllN Bc!,~)

(a • v ) } 2 1 1 N

_

i It p

sup t6S ] Hence exists

a

with

(2.8)

constant

and

(2.12)

(A 22 o

we see

K independent

of

+ p t I

) - 1 ln2

that

N such

T maps

VN i n t o

VN a n d

that

there

that

[ITII
p

Choosing

N sufficiently l a r g e we see t h a t T is a c o n t r a c t i o n on V N if o . Consequently t h e r e e x i s t s a u n i q u e s o l u t i o n of (2.10) in V N if N > N

N > N --

o

--

Now going

backwards

y = 0 ( x -N)

we easily

verify

as x + ~ in c l o s e d

Hence,

if N > N --

o f the o r i g i n a l

and

Pl

. o

that

subsectors > Z there

y = L w satisfies (2.1) a n d P . ] exists a unique solution y = co + L w of G

o

p

equation

(2.1),

without

assuming

(2.6),

such

that

1-! t

P w is a n a l y t i c

w(t)

on S. a n d 3 N-I X m= 1

=

m

c

N

-- -i -- -I tp + 0(tP ) as t ÷ 0 in S. , 3

m F (m/p) Plltl

W (t)

= O(e

Now the uniqueness

implies

solution

y 6 Aj (~i)

solution

y belongs

COROLLARY.

) as

such

that

that

t

÷ ~ in

w does

(2.5)

to t h e c l a s s

S.. 3

not depend

holds.

o n N. H e n c e

By v a r i a t i o n

we h a v e

of ~i w e

a unique

see t h a t

this

A. (~).~3

We make the same assumptions as in theorem i except that the cases

i and 2 are modified as follows: Assume

(2.17)

where ~ ,

A(x)

bh,

=

p-i Z h=0

x - h ~(xP),--

h = 0,1 . . . . .

p-l,

are

b(x):

p-i X h=0

x - h b~(xP),-"

of class A I ( 8 1 , 8 2 ,

~,

I) in case i and of

class A 2 (~,~) in case 2. Then, if 5-~ c m x -m is a formal solution of (2. i), t h e r e exists an analytic solution y(x)

= X ~ - ~" 0 x -h ~ h ( X p) where ~h 6 Aj (p) and

holds as x + ~ in

(2.18)

-

--2 - 82 + e _< p a r g x _< ~ - 81 - e(e

where 81 = 82 = - a r g ~ in case 2.

> 0),

(2.5)

33 This

may

be s h o w n

using

a rank

reduction

scheme

of T U R R I T T I N

[21]:

substitute

~T

x = ~i/p, u(~) = (~o(~)

'

"'''

(~))T

Yp-I

v(~) = (~o(~)..... ~p-i (~))T. Then

(2.1)

is e q u i v a l e n t du --= d~

(2.19)

M(~)u

where

From

(2.4)

M(~)

to

+ v(~),

~ Z M ~-m, 0 m

we may deduce

of Mo w i t h

nlp.

So we m a y

apply

In c a s e p = 0 in m a y be t r a n s f o r m e d

o

(2.1)

Hence

Mo

theorem we have

to t h e c a s e

AIo © 1

I

1 p

that M ° has nlP

multiplicity

nonsingular.

M

rows

(2.19)

a regular

"'-

A op-I

..... " .A I o

"-

of z e r o s

is s i m i l a r I to

A1 o •

to d i a g

both

0 is e i g e n v a l u e

{ 0 , M 22}O w h e r e

and the result

singular

p = I by dividing

and that

"A o o

point sides

M 22o is

follows.

in ~. T h i s o f the

[] case

equation

b y x: d y _- x-i A ( x ) y d-~

(2.20)

If ~(x)

= x -I A(x),

involved: now give

Here

we have ~ = 0 a n d so w e n e e d n o t p a r t i t i o n the m a t r i c e s o n I = n, p = I a n d (2.4) is s a t i s f i e d for (2.20). O u r r e s u l t s

we take Laplace

3. T H E L I N E A R

transforms

DIFFERENCE

we consider

(3.1)

By means

+ x-lb(x)

related

to f o r m a l

EQUATION

the equation

z ( x p + i) = A(x) z(x p) + b(x)

of the substitution

(3.2)

we transform

z(x)

= y ( x I/p)

(3.1)

into

.

power

series

solutions.

34

y ( ( x p + i) I/p) = A ( x ) y ( x )

(3.3)

We distinguish

t w o cases.

c a s e 2 w e assume:

i we assume:

A, b 6 AI(81,

p = i, A, b 6 A 2 ( w , p). We a s s u m e

Ao = diag

in c a s e

82, p, p) a n d in j:

{In I' A ~ 2}' blo = 0, A Ibm = 0, b Im = 0 if h = 1,2; m=l, .... i~-i

if k 6 ~ k

(3.4)

In c a s e

+ b(x).

{0} t h e n 2 k ~ i ~ S j ; ~

A22 -t - e I is n o n s i n g u l a r o n2 H e r e S. a n d G. are d e f i n e d ] ]

~

Icos 8 I if 81 ~ @ < ~2;

on 8.. 3

in d e f i n i t i o n

j o f sect.

i. T h e n we h a v e

2. Suppose ~ c x -m is a formal solution of (3.3) a n d Re t is bounded o m above on sj. Then there exists a function y E A I ( 8 I, 8 2 , u, P) in case i,

THEOREM

Y 6 A 2 ( ~ , p) in case 2 which satisfies

(3.3) if (xP+l) I/p E Gj_ and such that

with (3.2) is satisfied, and which satisfies of G

]

(3.1)

(2.5) as x ÷ ~ on closed subsectors

in case j. The function with these properties is unique.

PROOF: instead

The proof of

is q u i t e

similar

to t h a t o f t h e o r e m

y ( ( x p + i) I/p) = L

(e -t w(t)) (x) , if P

Hence the integral

equation

(3.5)

(e -t I - A )-i o

w(t)

Instead of

i. T h e d i f f e r e n c e

is t h a t

(2.7) we h a v e

(2.11)

(3.6)

=

and

(2.10) w i t h

(xP+l) I/p 6 G.. ]

(2.8) a n d

(2.9) n o w r e a d s

(~ * w + 8)(t) .

(2.12) we n o w h a v e

(e -t I - A )-I = d i a g o

{ (e -t - I ) - i i

(e-tI n I'

n2

- A22) -I} o

and (3.7)

(e-tI n2

- A22) -I and t ( e - t I o

- A )-i o

are uniformly

b o u n d e d on S.. H e r e w e u s e the f a c t t h a t le-tl ÷ ~ as t ÷ ~ o n S.. ] ] w i t h t h e s e a l t e r a t i o n s we m a y s h o w t h a t a l l s t e p s in the p r o o f o f t h e o r e m i

with

slight modifications If R e t is b o u n d e d

valid.

remain valid,

and t h e o r e m

b e l o w o n S. t h e a s s e r t i o n

2 follows.

of theorem

[]

2 does not remain

In t h i s c a s e Re t ÷ ~ and e !t ÷ 0 as t + ~ o n Sj. H e n c e t ( e - t I

is n o t b o u n d e d

o n S, (cf. ]

(3.6)), and t h e p r o o f o f t h e o r e m

- Ao)-i

2 does not go through

in t h i s case. If A -I e x i s t s w e m a y m o d i f y o

that proof

for t h i s case.

First we may solve

35

(3.5) in a n e i g h b o u r h o o d

of 0 in S.. Then the solution may be extended to a 3 in S. (cf. sect. 4.3). We may estimate this solution by ] the right hand side of (3.5) since (e-tI - Ao)-I is bounded in a

global solution majorizing

neighbourhood

of ~ in S.. Applying Gronwall's lemma we get an exponential bound ] for the solution. In this way we get a solution of (3.3) in Aj(p') for some

~' > p. We do not present details of the proof sketched is a special

case of theorem 6 in sect.

However,

a result corresponding

above,

to theorem 2 in the case that Re t is

bounded below on S. also may be o b t a i n e d by transformation 3 ~(x) = z(l-x). Then ~(x p + i) = A-I(x e ~i/p) ~ w h i c h is of the same type as A2(~,

-i

'

(x e~i/P),

Hence we deduce

(xp) - A-l(x e ~i/p)

of

b

(3.1). Let

ix e ~i/p)

,

(3.1). We now assume A -I, b 6 AI(8 I, 82' p' p) or

p). Then it is easily seen using

A

since this result

5.

b(x e ~I/p)

(1.2) that

6 A 1 (e I + ~, 82 + z, p, p) or A2(-~,

1.1).

from theorem 2 :

T H E O R E M 3. Suppose A -I , b 6 AI(81 , 82, ~, p) in case i and 6 A2(~, p) in case 2.

Assume

(2.3) as x ÷ ~ on G. and (3.4) holds in case j, j/= 1,2. Assume R e t i S ] and ~ c x -m is a formal solution of (3.3). Then there exists ] o m a function v E A (6 g~, ~, p) in case I, v 6 A~(,~, ~) in case 2 such that

bounded below on s

y(x)

= v((xP-l)l}p)Isat~sfies

(3.3) if

(xP-l)I/PZ6

Gj and

(2.5) as x ÷ - on Gj.

This solution is uniquely determined. Corresponding COROLLARY:

to the corollary of theorem

We make the same assumptions

and 2 are modified as follows: Assume belong to Al(el,

1 in sect.

2 we now have

as in theorem 2 except that the cases 1 (2.17) where ~ ,

g2, u, i) in case 1 and to A2(~,

hh, h = 0, 1 . . . . .

~) in case 2. Then,

p - I,

if

X~ c x

ts a formal solution of (3.3), there exists an analytic solution m p-i y(x) = h~0 "x-h ~ h (xp) where ~h 6 Aj and (2.5) holds as x ÷ ~ in (2.18) where 81 = 82 = - arg ~ in case 2.

4. THE N O N L I N E A R D I F F E R E N T I A L We consider

(4.1)

EQUATION.

the differential

equation

xl-P d_yy = f(x, y) dx

in the case that f(x, y) satisfies theorem I. We again consider

conditions

similar to those in sect.

2,

two cases j = i or 2 and use the same notation

S. and 3

36

G

as in d e f i n i t i o n

j of sect.

I. A s s u m e

3 H. (j = 1 o r 2). L e t Po > 0, p > 0, p a p o s i t i v e ] c a s e j = 2. W e h a v e

HYPOTHESIS

(4.2)

f(x, Y) I= f°(Y)

where

fo(y)

(4.3)

and t

+ {Lp~(y,

t)} (x), if

P qg(y, t) a r e a n a l y t i c m

qD(y, t) ~

uniformly

on A

Moreover,

if ~I > Z

Z m=l

(0; Do).

%01n (y)t p

in ~

integer,

(x, y) 6 G.] x A

p = I

in

(0; po ) ,

(0; p O ) x Sj, fo(0)

= 0 and

as t + 0 in S 3

Here ~m(y)

is a n a l y t i c

then there exists

in A

a constant

(0; po ), m = I, 2,

K depending

....

o n Pl s u c h t h a t

1 (4.4)

I~(Y,

t) l ! K

In t h e f o l l o w i n g =

(\~i' "'''

~n ) 6

~%0

It p

I exp

(~lltl)

on ~

we assume either hypothesis

1 = ~n

we d e n o t e

(0; DO) x S O . O H I or h y p o t h e s i s

H 2. If

I~I = ~i + "'" + ~n and

(y, t) = ~Ivl%0(Y' t)

and similarly

for ~

f

o

(y).

~Yl "'" ~nyn Df w i l l b e the d e r i v a t i v e of f • W e use the p a r t i t i o n i n g o o as in sect. 2. O u r m a i n r e s u l t is: THEOREM

4. Suppose A

o

of m a t r i c e s

and v e c t o r s

j = I or 2 and in case j:

= Dfo(0)

= diag

22 {O n , A ° }

'

22 A ° + p t I is n o n s i n g u l a r

o n S. 3

i II,~1 (4.5)

Suppose

{~

%0 (0, t)} 1 = 0 ( t

{~

fo(0)} I : 0 if l~l ! P

that

p

(4.1) possesses

) a s t + 0 in S. if ]

IvI < p ,

a formal solution I c x -m. Then there exists a m

number ~' > ~ and an analytic solution y o f (4.1) suchlthat y E A 1 (81 , 8 2 , p' , p) in case i and y E A 2 (~, ~') in case 2 and such that

(2.5) holds as x + ~

on

closed subsectors of G.. This solution is unique. 3 The proof will be given with estimates, derived,

in sect.

a neighbourhood solution solution

in sect.

in s e v e r a l

4.2 an i n t e g r a l

steps:

in sect.

equation

4.1 w e g i v e s o m e l e m m a s

equivalent

to

4.3 w e s h o w t h a t a s o l u t i o n o f t h i s i n t e g r a l

o f t = 0, in sect.

in S. a n d in sect. ] o f (4.1).

4.4 t h i s s o l u t i o n

4.5 we estimate

(4.1) w i l l be equation

will be extended

the solution

and we obtain

exists

to a the

in

37

In section 4.6. we c o n s i d e r some g e n e r a l i z a t i o n s o f t h e o r e m 4.

4.1.

SOME

ESTIMATES

LEMMA 1. Let P, h and l be positive numbers. T h e n t h e r e K(n,l) independent of p such

(4.6)

~ m=0

PROOF. The

ml+h-I P F(ml+h)

exists a positive constant

that

< K(h,l) m a x (0h-l , e p) . --

e s t i m a t e is e v i d e n t for p < i. If p > i, we use the H a n k e l - i n t e g r a l

for the g a m m a f u n c t i o n and w e get for the l e f t h a n d side of

I

2~i

~

(0 +)

m~O

I

es

(~)ml+h-I

s

ds

(0 +)

I

s

(4.6):

I

2~i

e~S ( l - s - 1 ) - i

s - h as .

In the last integral w e choose as p a t h o f i n t e g r a t i o n a loop e n c l o s i n g the n e g a t i v e axis and the p o i n t s s = exp

(2gni/1), g 6 ~. The r e s i d u e in s = i gives

the main c o n t r i b u t i o n to the integral as P + + ~- In this w a y we obtain

L E M M A 2. Let p be a positive number or p = ~ and sj(p) = s 5 ~ n~

(4.6)°

(O;p),where

j = i or 2. Suppose z 6 c ( s . ( p ) , ~n)- a n d z is analytic in s.(p). Assume that 3 3

(4.7) where

I Z ( t ) i~ M I t l 1-1 exp ( ~ l l t l ) M > O, 1 > O, ~i ~

(4.8)

IZ*~

(t)

I

<

0 are constants. Then

MI~I FIUl (1)

-

~ O, where z * v i s n

if t 6 S.(p), ]

Itl I~11-1 e ~lltl

if

t £ S.(p),u 6 I,

r(l~ll)

J

the convolution o f

~lfactors

zl,

v2 f a c t o r s z 2 . . . . . . .

and

factors z . n

pROOF: T h e p r o o f e a s i l y follows b y

i n d u c t i o n using

(2.14).

L E M M A 3. Let z satisfy the conditions of lemma 2. Suppose dr6 %n if v 6 I and

there exist positive constants K I and Pl such that

I%I <_ Ki~ 1- I ~ I , ~

(4.9) Then

d z *~

~, ,~+ o.

is uniformly convergent on compact subsets of

v6I

w~O

in S~ (p) a n d 3

Sj (po) , analytic

38

(4.10)

X I d~ z*~(t) vEI

I ~ KoKIk I sup {Ikltl I-I, e kl Itl} eUl Itl i

i f t 6 s j ( p ) , where k 1 = (p~l NnF(1))l/1 and Ko i s a c o n s t a n t only depending on 1. Moreover, if p = + ~, then

(4.11)

L P

{ X 9EI

d z*V(t) w

} =

X ~El

d w

(L z) v P

on {x E ~: Re (xp e i8) > Pl + kl} where the path of integration in the Laplace integral is arg t = 8. PROOF:

The proof follows using lemma 2 in combination with (4.9), and lemma i.

LEMMA 4. Let h be a positive number and S ] (p) i f ~ E I. (4.12)

t l-h ~ (t): sj (p) ÷ ~n be analytic in

Assume

[qw(t) l ~ K 2 p~ 191 Ith-ll e p21tl

, if t E S, (p), ~ 6 I,

]

where K 2 and P2 are positive numbers. Then

X

q

* z *~ (t)

is analytic in S (p), ]

(4.13)

[ I~9 * Z*~ (t) I -< K'O K2 k21-h max ([k2tl h-l, elk2tl) e71tl ~EI

if t 6 Sj(p), where

~ = max (PI' P-)' k. = (Mn r(i)) I/i and K' is a constant dec z z Pl

pending only on h and

(4.14)

Lp(~EiX ~

i. In case p = ~ we have

* z *~) (x) = ~EIX (Lpq) (x) (LpZ)V(x)

,

on {x 6 ~: Re (xPe i@) > ~ + k2} , where the path of integration in the Laplace integral is arg t = e.

PROOF.We use lemma 2. From (4. 12) and (4.8) we deduce (4.15)

K2

ID9 * z*W (t) l < -

r(l~ll)

e PIITI dT I -< -F(I~ll) -~

(Mr (i)) IVl ft ~

(MF(1)) I~l jltl ~

I (t-T) h-I e

u21t-TITI~II-I

o

h+]~ll-1 Itl

B(h,I~ll).

89

With lemma i the result easily follows.

4.2. REDUCTION OF THE DIFFERENTIAL EQUATION TO AN INTEGRAL EQUATION From hypothesis H. we deduce ] (4.16)

fo(y) =

[ b ~6I

y , <0 (y, t) =

[ By(t) Y ~6I

,

v+o if y 6 A ( 0 ; P ) , o (4.17)

b

-

t 6 S~ and ] 1

SVfo (0) , ~ ( t )

= m--~ 1 ~v q) ( o , t ) i f

~ 6 I,

t 6 Sj

.

From Cauchy's formula for derivatives of analytic functions and hypothesis H. we 3 deduce Ib I ~ KoPo[~l , l~(t) l~ KoP °-l~i

(4.18)

1 -i ~lltl It~ 1 e ,

where Ko is the maximum of K (cf. (4.4)) and maxl fo(Y) lon ~ (0;Po)" Let N 6 IN, where N will be chosen later on sufficiently Define UN(X)

=

m N-I N-I -- -i X c m x -m ' zN(t ) = X cm {F(m-)} -I t p p i 1

Then u N = LpZ N and uN6J''3

(4.19)

x I-p

large.

Moreover

fUN(X) I < Po if Ixl is sufficiently large.Let

d d-x U N - f(x, UN(X))

From the assumption that

Z 1CmX

-m

= gN(x)-

is a formal solution of

(4.1) and hypothesis

H. we may deduce that J (4.20) From

gN(x) = 0(x-P-N),

(4.19) and

(4.21)

gN =

g~(x) = 0(x -N) as x + ~ in G.. ]

(4.17) we may infer with lemmas 3 and 4 Lp YN' - YN (t) = PtZN(t)

+

~6ZI { ~ *

z*VN + bY zN*m} (t).

These lemmas also imply YN 6 aj(~ 2) for some ~2 > ~ " From N

(4.22)

~

(t)

= 0(tP),YN(t)

N 2

=

O.t p{

Substituting y = u N(X) + v in (4.1) we get

) as

t +0

in

S.. ]

(4.20) we deduce

40

(4.23)

xl-P

We show that

dxdV = f(x, uN(x)

+ v) - f(x, UN(X))

(4.23) has a solution v = L

w 6 J(v') 3

P w

E a.(~') ]

- gN(x).

for some U' > ~2 iff

and

(4.24)

-ptw(t)

= AoW(t)

+ H(ZN, w) (t) - XN(t)

Here (4.25)

H(z, w) =

7 ~EI

b

{(z + w) *~ - z*~ }

+

I~I>2

E ~6I

B * { (z + w) *~ -z .9}

~+0

First we remark that H(z, w) exists

in S~ if z, w 6 aj(~') and that H(z, w) Ea=3 ~'') ] > z' on account of lemmas 3 and 4. Moreover, these lemmas imply that

for some ~"

L p ( A o W + H(ZN, w)) (4.23) and

(x) = f(x, ~N(X)

+ v(x))

- f(x, U N(X))

on Gj( ~" ). Hence

(4.24) are equivalent.

Next we (4.26)

rearrange

the terms

H(z, w) = ~(z,

.) * w +

I vEI

{B

(Z,.) * W *~ + b w'W}, ~

I~I>2 where

e(z,.)

= DqO(0,.)

+

{ ~! ID3m

r

f O (0)Z*9 + 9--~ i D ~ 9 ~ ( % ")* z *~ } ,

vEI (4.27) ~

(z,t)

=

8 (t) +

E

(v+O) O {b +O z *O +

We prove this for z = z N and w 6 aj(p').

M, and the estimate

~+o

( ~ ) ~ n

(4.27) converge uniformly

J~I+Iol .

8 +g ~ z*O} (t) .

To this end we use the estimate

•

for Z = z N and for z = w wlth ~, replaced by ~

in

.

,

>

p,

(4.7)

1

, 1 = - - a n d a suitable

P

Then lemmas 3 and 4 imply that the series

and absolutely on compact sets in S. and that ] -i

l~(z N, t) l < MIItP

eP31t I I

(4.28)

1 -i 189(Z N, t) I ~ M 1 ( ~ ) I W I I t P

for some ~3 > ~

e~31t I I

and a constant M 1 independent of ~. A second application

lemma 4 shows that the series in compact ~ets in Sj, and so

(4.26)

(4.26) are u n i f o r m l y follows from

absolutely

of

convergent on

(4.25) if z = ZN, w 6 aj(pl).

41

4.3. LOCAL SOLUTION

OF THE INTEGRAL

We first solve where S.(£) 3

(4.24)

EQUATION

in a Banach

= S. n ~ (0,£) 3

sp~ce VN(a)

of functions

such that tl-p w is analytic

w: Sj(e)

in S.(c) 3

÷ ~n

and

N (4.29)

I IWlIN = sup {It

Here g will be chosen We rewrite

(4.24)

P w(t) I: t 6 S

3

later on, a > 0. as w = Tw, where

(4.30)

Tw = -(A ° + ptI) -I H(ZN, w) + ~N

(4,31)

~N = (Ao + ptI)

Using

(4.22)

and

formal

solution

-I

YN "

(4.5) we deduce

Choose M 2 > ICll of

(£)}.

~N 6 V N, so in particular

~N 6 VN(¢).

{F(p-l)} -I +i, where c I is the first coefficient

in the

(4.1). Then i

(4.32)

IzN(t)I < (M2-1)

itp

i o n S <6 )

--

for some sufficiently for w 6 V N(£)

3

small positive

£

o

O

depending

on N. We consider

H(z, w)

and z 6 V(6) where i

(4.33)

V(£) = {s 6 Vl(e) : Iz(t) I

In particular of

z N + w ° 6V(a)

<_ M 2 Itp

if w ° 6 VN(£)

I on Sj(¢)].

and £ is sufficiently

small,

because

(4.32). We first estimate

a and B

0 < £ <__ £i" The estimates

defined by

(4.27)

(4.18) , the assumption

imply that there exists a constant

M 3 independent

on S (£), if z 6 V(E) and 3 and lemmas 3 and 4

(4.5)

of z, t,N and ~J such that

i Is(z, t) l<_ M31

tP

(4.34)

!, I~lh(z, t) l < M 3 if h = 1,2 i -i

18~(Z, t) I <__ M 3 (Q~) I~I

if z 6 V(£),

t 6 Sj(a)

Now choose

Itp

[, "~6I,

and 0 < £ ~ e I.

N > 16M3, N > p. Then we have analogous

to

(2.15)

N --

I{(Ao + ptI)-I

e(z,.)

*

w(t)}ll

if w 6 VN(g) , 0 < e <_ el, t 6 Sj(6), the same conditions

_< 81 [[WI[N

z 6 V(g).

Analogous

Itp to

-i

I (2.16) we have under

42

N

* w(t)}21

I{(Ao+ ptI) -I ~(z,.) where K' is a constant 0 < s2

independent

< K'

I I wl IN [tl p

of z, w and t. Hence there

exists

E2,

(cf. (4.31)),

and

~ E1 such that i

(4.35)

If (AO + P tI)-[ ~(z,.)

if w ~ VN(£), Let L =

<

~

l lwl IN ,

0 < ~ _< E 2 .

211~NI IN , where the norm is the norm in VN(E 2)

(4.36) From

z 6 V(s),

* W 1 IN

Hi(z , w) = H(Z, w) - ~(z,.) (4.26),

(4.34)

(4.37)

. w.

and lamina 4 we deduce 2N

Jill(z, w)J < M 4 ltJ p

if t 6 S . ( s ) , z 6 V(e) 3 Here M 4 is a constant

,

w 6 VN(E), independent

I JWllN

,

IlW[IN _<

2L,

0

< e

<

e 2.

of z, w, t , g. Choose

E so small that 0 < e < ~2'

2N sup It t£S. (g) ]

P

and that z N + w 6 V(E)

M4 < i 4

(Ao + ptI)-ll

if w 6 VN(E) , j lwl JN < 2L. Let D(S)

J lwJ 'iN < L}. Then we deduce

from

(4.35)

- (4.37) _

(4.38)

II (Ao + ptI)-I

if wl, w 2 6

D(E).

Wl, w 2 6

D(e),

then H(ZN, (4.25)

quently

there exists a unique

(4.38)

OF THE SOLUTION

the solution

that T maps D(e)

WllIN, into D(£).

w of

imply that T is a contraction

solution

of w = Tw, so of

OF THE INTEGRAL (4.24)

on D(e).

(4.24),

o

denotes

EQUATION

is known on S.(p)

the set S translated

Conse-

in D(E).

for some p > 0 (cf.4.2):.

Choose to 6 S.(p)3 with 21P < jtol < P" Let sT3 = s.(p)] N (s.-t3o ) and+ Here S + t

If

w 2) - H(ZN, w I) = H(z N + w I, w 2 - w I) in view of

Hence

Suppose

and

_

Wl)IIN < : ii w 2

With w I = 0 this implies

(4.25).

4.3. EXTENSION

1

H(ZN + W, W2

= {w 6 VN(E):

that

over t . Then S.(p) o 3

3 = S~+t3o"

and S~ are convex ]

sets. We transform decomposition analytic

the integral

equation

of u . v for scalar

in its interior:

+ on S. using the following ] u, v continuous on S.(p)G S~ and ] 3

(4.24)

functions

43

(4.39)

(u ~v)

(to+tl)

= {U(to+

.)~ v}

(tl) +

{V(to+

.)•

u}

(t I) + R(u,v)(tl),

where t R(u,v) (tl) = S o v(t + t I - T)U(T) t 1 o

(4.40)

[0, t l ]

and the paths of integration

and

[tl,

R(U, v) = R(V, u) and R(u, v) only depends Using induction

v

~k

+

if k >

k-I E { v *(k-l-j) j=l

for an n-vector

function we have

Ikl > 2

~(z,

+

to

Sj(p).

Here

on the values of u and v on [to, ti].

(to + tl) = k {V(to + ")~ v~(k-l))"

k 6 I,

=

t o ] belong

we may show

* R(v, v~J)}

2 and v ~° • R = R, v ~I ~ R = v ~R.

(4.41)

d~, t I E Sj,

tl)m

(tl),

From this formula we may deduce

z whose components

zmk(t

o

+ tl) -

n I

~=~

satisfy

k

] I

]

nE kl-i z~(k_(j+l)el ) m R(Zl' E i=i j=l n ~k E Zn n . . . . . l=l

(tl) +

the assumptions

{z.(to+ ]

that

above and

.)~ z~(k-ej ) } (ti) =

Zl~J ) (tl) +

~kl+ 1 *k I ,kl_ 1 *k 1 Zl+l ~ R(Zl , Zl_l ~-.. • Zl ) (tl)

.

Here e. is the n-vector whose components are zero except for the j-th component ] which is equal to one. From the definition of R and ~ it follows that ~ ( z , t I) is determined

on S? by the values of z on S.(p). ] ] If z satisfies (4.7) with ~I = o and 1 = 1 on Sj(p)

and

then we have from

(4.8)

(4.40) IR(z l, Z~ j)

(tl) I _< 2 M j+l (j_l)-------T PJ if j ~ i, t I E Sj ,

~k 1 *kl_ [ *k 1 IR(z I , Zl_ 1 *--- * z I ) (tl)l

ki+

....

k1

2 2(Mp) -i

{P(kl-l):

(kl+ ... + k]_l-l):}

if t I 6 S-. ] Combining these formulas with (4.41) and (4.8) we see that there exists a constant Mo independent of t I and z but depending on p and M such that

44

(4.42)

]~(z,

tl)II _< ~]-~°

, t I C Sj .

Now we transform of

the operator T into an operator ~ on $7. Let w be the solution ]n (4.24)of w = Tw on S.(P)3 (cf.(4.30)). If z 6 C (S~, (~) then we define

(4.43)

(~z)

(tl) = -(A ° + p(t ° + tl)

I) -I {~ ~ z(tl)

+ ~

(tl)}

if t I 6 S~. and n

* z(t I) = ~(ZN,.)*

z(t I) +

[ V61

, ('~-e,)

[ j:l

~j {8 (ZN,.) * w v

3

I~I>_2 * (~-e)

+ b w

] } • zj (t I) ,

~ ( t I) : ~(z N, to+ -) % w(t I) ~ R(~(ZN,

+

E v6I

{Sv(ZN,

.) , Rv

(w,.) +

~

.) ,w)(t I) - YN(to + t I) +

(zN, to+ .) * w *v +

I~I>_2 + R(Sv(z N, Using lemmas 2 and 4,

-), W *v) + b v R (w, .) }(tl). (4.28),

(4.42) we may deduce

that ~ and ~ exist and are

continuous on $7 and analytic in ($7) °. These functions only depend on the values 3 ] of w in S. (p). 3 The definitions of T and ~ in (4.30) and (4.43) and (4.39)imply (~)

(to+t I) = {~ W(to + ")}

(tl) if t16 $73 n (sj(p) -to).

Hence w

solution of ~z = z on this set. Since the linear Volterra

(to + .) is a

integral

equation

z = ~ z has a unique solution in S7 w h i c h is analytic in (sT) O and continuous 3 3 on S~, this solution is a continuation of w(t + .). Denoting this continuation o J also by w(t + .) we see that w = T w on S.(O) U S < b e c a u s e of the relation of T o 3 3 and ~. Varying t o we get a unique solution of w = T w on S_3 (3),z hence on S.. 3 Thus we have shown that (4.24) has a unique solution w in S. w h z c h is continuous o ] on S. and analytic in S.. 3 ]

4.5. E X P O N E N T I A L

ESTIMATE

N O W we estimate (4.44) We rewrite

FOR THE SOLUTION

the solution w of

g(p) : sup {lw(t) l: t 6 S., 3 (4.24)

in the form

(4.24) on S,. Let 3 ItI: p}, if p > 0 .

(4.30) and use estimates

for

(A + ptI) o

-i

from

45

(2.12)

for H(ZN,

w)

from

(4.26),

(4.28)

and

the f a c t t h a t YN 6 Gj(~2) ( c f . s e c t . 4 . 2 ) . constants

M and d such that

(4.45)

g(p) <

(Tlg)

for

(4.18), and for ~N f r o m

(4.31)

and

T h e n w e see t h a t t h e r e e x i s t s p o s i t i v e

@ > 1 we have

(p) ,

where 1

= M {eP3 p +

( ' r i g ) (p)

~

2

d m g *m (p)

~

m:2 By c h o o s i n g

M > sup {g(p):

Following

Walter

m=1

p £ [0,1]}we

[23,

p.17]

d m (p~- - 1 e P B P ) :~ g~m( p } }

X

+

get

we f i r s t

(4.45)

solve

for p > 0.

v = Ttv.

If

u = Llv,

then

1 u(x)

= M(x-P3)-I

+ t,~ 5-

dmum(x) + M I" (1) ( x - p 3) P

m=2

This equation

has a unique

in x I/p, p o s i t i v e u(x) Let V contain

solution

= M x -I + 0 ( x

dmum(x)

V of ~ w h i c h

is a n a l y t i c

1

P) as x p ~

Re x ~ P4' w h e r e

1 (Lllu) (p) = M + 2 ~

=

X m:l

u in a n e i g h b o u r h o o d

for x > 0, x 6 V and 1

the halfplane

v(p)

P

P4 > ~3" T h e n

;4~ i~

ePX

{u (x) -Mx-l }dx'

P4-i ~

if p > 0. It f o l l o w s 0(exp

p4p)

t h a t v is r e a l - v a l u e d ,

as p + +~.

Suppose

In p a r t i c u l a r

there exists

g(po ) = V(Po).

Then

v(0)

we h a v e g(0)

= M

Po > 0 s u c h t h a t 0 < g(p)

(4.45)

gives

(4.46)

a contradiction.

lw(t) I ~ K O e x p

for some c o n s t a n t

Consequently

and v(p)

=

< v(p)

if 0 < p < Po a n d

implies

g ( p o ) < (Tlg) (po) < (TlV) (po) = V(Po) which

(cf.[3, p.174])

< v(0).

,

H e n c e g < v on ]R + and so (P41tl),

if t 6 Sj

K . o

LpW e x i s t s

o n Gj a n d i s

a solution

of

(4.23).

Therefore

y = UN+

+ L w is a s o l u t i o n of (4.1). In the s a m e w a y as in the p r o o f o f t h e o r e m i we m a y P s h o w y 6 J j ( P 4 ) a n d (2.5) as x ÷ ~ o n G . T h i s c o m p l e t e s the p r o o f of t h e o r e m 4. ]

46

4.6 A G E N E R A L I Z A T I O N In s e v e r a l

(4.47)

cases

y(x)

=

(4.1) h a s f o r m a l

Z

dk x

solutions

of the form

-k.K

Ikl=1 where k =

..., k ) 6 ~ g + 1 (g a p o s i t i v e i n t e g e r ) , K = (i, 0 if j = i , ..., g and k . < = k o + klKl + ... + k g < g (cf. sect. 6, (ko,

application

V).

In t h e s e

5. A s s u m e

THEOREM

(4.48)

c a s e s we h a v e the f o l l o w i n g

hypothesis

A m = F(~)

generalization

of t h e o r e m

4.

H I . Let

Dqgm(0) , b v o = by' b v m _- l__u! F(~)agp q)m(0), m = I, 2 . . . .

and assume A ll

= 0,

A 12 = 0 ,

m

b1

m

= 0 if

m = 0 .....

p-1

and

w 6

I,

I~1

+ 1,

~m

(4.49) A 21 = 0, A 22 + ptI o o

If

(4.1)

has a formal

solution

an analytic solution y = L w P oo (4.50)

y(x) N

y

is n o n s i n g u l a r (4.47) of(4.1)

on S I.

then there exists a real number" ~' > ~ and on GI(~')

(cf.

(l.l))such

that

-k. K

dk x

Ikl=* as x + ~ on closed subsectors PROOF. with

The proof

the s e q u e n c e Then

is s i m i l a r

Ikl> i be a r r a n g e d 11, 12,

of GI(V').

This solution is unique.

to t h a t of t h e o r e m

in o r d e r o f i n c r e a s i n g

.... H e n c e 0 < Re I i < R e 12 <

(4.47) m a y be r e w r i t t e n

as y(x)

= Z c 1

(4.51)

UN(X)

w h e r e N is c h o s e n

4. L e t the set of n u m b e r s magnitude

k. K

of t h e i r r e a l p a r t s to

... and Re Am ÷ ~ as m + ~.

x -Am . Let m

N-I -i = Y c x m, U N = L p Z N , i m

t

N-I 1 m -i zN = Z c {F(--m)} -I t p i m p

in s u c h a w a y t h a t Re IN_ I < Re A N.

In g e n e r a l

u N ~ A I. W i t h

(4.19) w e d e d u c e giN(X) = O ( x - p - I N ) ,

and

(4.21).

Instead

of

2 gN(x)

-I = 0(x

N)

(4.22) w e n o w h a v e

h

h

l

TiN(t)= 0 ( t p N) , T N2 ( t ) = 0 ( t P N w i t h y = u N + v, v = L p W w e d e d u c e

(4.23)

) as t + 0 in S I

-

(4.27).

Because

of

(4.51) w e m a y

47

estimate e and 8~ in

(4.27) w i t h lemmas 3 and 4 w i t h 1 = Ii/p. The r e s u l t is

(4.28) w i t h I/p r e p l a c e d b y Ii/p. In sect. 4.3 we a d a p t the d e f i n i t i o n of VN(e) b y r e p l a c i n g N / p b y IN/p (cf.

(4.29)). In

ii/p and in

(4.32) - (4.34) and

(4.45) we n o w have to r e p l a c e i/p b y

(4.37) N/p b y IN/p. W i t h s u c h m o d i f i c a t i o n s the r e a s o n i n g in sect.

4.2 - 4.5 r e m a i n s v a l i d and the p r o o d of t h e o r e m 5 is completed. REMARK. Cases where a formal s o l u t i o n of

(4.1) also c o n t a i n s l o g a r i t h m i c terms like

in Iwano [15] m a y be t r e a t e d u s i n g s o l u t i o n s y = L w w h e r e the e x p a n s i o n of w n e a r P the origin c o n t a i n s l o g a r i t h m i c terms.

5. THE N O N L I N E A R D I F F E R E N C E E Q U A T I O N W e n o w c o n s i d e r the d i f f e r e n c e e q u a t i o n (5.1)

z(~+l)

= f( I/p, z(~))

or e q u i v a l e n t l y w i t h

(3.2) :

y((xP+l) I/p) = f(x, y(x)) ,

(5.2)

w h e r e f s a t i s f i e s h y p o t h e s i s H. (j=l o r 2) as in sect. 4. W e also a s s u m e 3 (5.3)

A

(5.4)

A 22 - e-tI is n o n s i n g u l a r on S., o n2 3

(5,5)

2k~i ~ S. if k 6 ~ \ { 0 } p + cos 8 > 0 if 81 < @ < e2 3 , _

(5.6)

A

o

= Df

22 o

o

(0) = d i a g {Inl, A 22} o '

is n o n s i n g u l a r if Re t is b o u n d e d b e l o w on S.. 3

T h e n we h a v e T H E O R E M 6. Assume hypothesis H.,

(5.3) -

3

(5.6), with j=l or2. Assume

1-19t ~0(O,t)} 1 = 0 ( t

P

(5'.7) ~

Suppose

fo(O) = 0 if

) as t ÷ 0 in S. if 191 < p 3

l~I < p

(5.2) has a formal solution I c x -m. Then there exists a number

i m ~' > ~ and an analytic solution y of (5.2) such that y 6 A 1 (e l, e2,v', p) in case I a n d y 6 A2(~o, ~') in case 2 and such that (2.5) holds on closed subsectors of G . 3

This solution is unique.

PROOF. T h e p r o o f is a m o d i f i c a t i o n of that of t h e o r e m 4. T h i s m o d i f i c a t i o n is the same as u s e d in the p r o o f of t h e o r e m 2: the l e f t h a n d side o f

(4.24) n o w

48

reads e-twit)

and in

(4.30),

(4.31)

etc.

we r e p l a c e

(A

+ ptl) -I by

(A -e-tI) -I

91 In v i e w of

Theorem

Analogous

(5.4)

(5.6)

the m a t r i x

6 is an e x t e n s i o n to t h e o r e m

7. Assume

THEOREM

and

o

(A -e-tI) is b o u n d e d on S . ~ A(0;I) o 3 and the m a t r i c e s in (3.7) a r e b o u n d e d on S. N A(0;I) and e v e n on S~ if 3 3 Re t is b o u n d e d a b o v e on S.. Using these m o d i f i c a t i o n s the p r o o f of t h e o r e m 4 3 goes through. D REMARK.

(3.6),

of a r e s u l t of Harris

5 we have the following

hypothesis

HI,

(5.3)

generalization

(5.6) with

-

a~id S i b u y a

[7].

of t h e o r e m

6.

j=l and in the notation

of

(4.48): A 11 = 0, A 12 = 0,if m=l,..., m m

(5.8)

If (5.2) has a formal solution

p-l;

(4.47)

b uI m :

o if

m=o . . . . .

p-1

and ~ ~ ~ , b t + I~

then there exists a real number

~' > p and an analytic solution y = L w of (5.2) on GI(p') (ef.(l.l))such that P (4.50) holds as x + ~ closed subsectors of GI(P'). This solution is unique. PROOF.

The p r o o f

is a m o d i f i c a t i o n

tions as in the p r o o f of t h e o r e m

of that of t h e o r e m

5, w i t h a n a l o g o u s

modifica-

6.

6. A P P L I C A T I O N S In this s e c t i o n we first g i v e s u f f i c i e n t series

solutions

applied.

Finally

of

(0.I)

and

we d e d u c e

(0.2)

exist,

a reduction

conditions

in o r d e r

so t h a t the p r e v i o u s

theorem

for linear

t h a t formal

theorems

differential

m a y be

equations.

I. A f o r m a l n o n - t r i v i a l

s o l u t i o n ~ c x -m of (2.1) w i t h b ~ 0 e x i s t s if (2.4) is o m A 12 = 0, A II is s i n g u l a r and A 11 + mI is n o n s i n g u l a r for m = I, 2 .... P P i P nl N o w we m a y choose for c an e i g e n v e c t o r c o r r e s p o n d i n g to the e i g e n v a l u e 0 of A II satisfied,

o

and e 2 = 0. o S u c h a formal is satisfied, m = i, 2,

p

solution

exists

for

(3.3),and

A 12 = 0, A 11 is s i n g u l a r P P

so for

(3.1), w i t h b -= 0 if

and A II + m I is n o n s i n g u l a r P P n1

(3.4)

for

...

The difference

in the c o n d i t i o n s

for

(2.1)

and

(3.3)

stems

from the formal

relations: x l-P

dy d~

= X - mc x - m - p , m 1

(6.1) y((xP+l)

I/p)

- y(x)

= X i

Formal equations

solutions

(2.1)

and

c x -m-p m

of the type c o n s i d e r e d

(3.3)

under

~ ~ + P

X g=l

x-Pg}. g+l

above exist for the nonhomogeneous

the same c o n d i t i o n s

except

t h a t n o w A II a l s o h a s P

49

to be n o n s i n g u l a r ; treatment

then c

d i f f e r s from c in the p r e v i o u s case. A d e t a i l e d o o s o l u t i o n s of (2. i) a n d (3.3) has b e e n g i v e n b y ~ i r r i t t i n [19 ].

of formal

II. R e s u l t s

for formal

s o l u t i o n s ~ c x %-m a n a l o g o u s to t h e o r e m s 1-3 m a y be o b t a i n e d o m of the e q u a t i o n s (2.1) and (3.3) v i a the s u b s t i t u t i o n

by a t r a n s f o r m a t i o n y(x)

= x%~(x).

formal

The e q u a t i o n

for ~

is of the same type as that for y and has the

solution

~- c x -m o m The a p p l i c a t i o n s I and II of t h e o r e m

concerning

factorial

series

as s o l u t i o n s

solutions

h a v e the same h a l f p l a n e

Turrittin

g e t s a smaller

represents

the s o l u t i o n

III. A m a t r i x

solution

i contain of

of c o n v e r g e n c e

halfplane

the r e s u l t s

of T u r r i t t i n

(2.1) w i t h b -= 0. However, as the c o e f f i c i e n t s ,

of c o n v e r g e n c e

for the f a c t o r i a l

[18]

here the whereas

series which

(cf.[2]). of

(2.1) a n d

(3.3) w i t h b ~ 0 m a y be o b t a i n e d

as follows.

Assume

A 12 = 0 and there is no p a i r of e i g e n v a l u e s of A 11 w h i c h d i f f e r b y a P P p o s i t i v e i n t e g e r in the case of (2.1), w h e r e a s in the case of (3.3) we assume the same

for p Ap11 . If the a s s u m p t i o n s

satisfied

and b ~ 0 we m a y c o n s t r u c t

and U ( x ) x

of

subsectors

(3.3)

concerning

s u c h that U £ J. and U(x) ÷ 3 for the d i f f e r e n c e

o f G.. We g i v e the p r o o f 3 pA~l Y(x)

= U(x)x

in

(3.3).

1,2 41 or 3 are

A in t h e o r e m s

an n x n l - m a t r i x

solution

U(x)x ~ of

(2.1)

as x ÷ ~ in c l o s e d equation

(3.3).

Substitute

T h e n we g e t _A 11

U((xP+I) I/p)

The r i g h t h a n d matrices

= A(x)

side d e f i n e s

U. P a r t i t i o n i n g

U(x)

a linear

(l+x -p)

P

transformation

T in the space V of n x n l-

these m a t r i c e s

(TU)I(x)

= {All(x)

(TU)l(x)

= ul(x)

Ul(x)

after the n l t h r o w we get ii -A + Al2(x) U2(x)} (l+x -p) P ,

hence + x-P(A~IuI(x)

- U I ( X ) A ll)p + 0(x-P-l).

ii ii t U1 b N U. - U , N has as e i g e n v a l u e s he Ii p i i p d i f f e r e n c e s of the e i g e n v a l u e s of A . H e n c e p times this t r a n s f o r m a t i o n has P eigenvalue 0 with eigenvector I and no o t h e r i n t e g e r is eigenvalue. A l s o n1 (TU)2(x) ~ A 22 U 2 (x) as x + ~ in G . H e n c e all c o n d i t i o n s of a p p l i c a t i o n I are o 3 s a t i s f i e d and the r e s u l t for U follows. Similarly, t h e d i f f e r e n t i a l e q u a t i o n Here the linear

transformation

(2.1) m a y be t r e a t e d

(cf.[2]).

IV. We n o w g i v e s u f f i c i e n t

conditions

such t h a t there e x i s t s

c x -m of (4.1) in case h y p o t h e s i s H and 1 m 3 (4.48) w e d e d u c e the formal e x p a n s i o n

(4.5)

a formal

are satisfied.

From

solution (4.2)

and

50

f(x, y) : (2 0

A x-m)y + [ m ~EI

( [ b x -m) y~ + [ m=0 vm m=l

b

x -m om

I~I>S F r o m this we m a y d e d u c e that there exists a formal solution of

(4.1) if n 1 = 0

o r if n. ~ I and i)

Alllhas P

no eigenvalue

(6.2)

which

is

a negative

integer

and

b I = 0 if 2 < I~I < p + l - m %TH ---

or ii) A II has e i g e n v a l u e -i, but no other n e g a t i v e integer is e i g e n v a l u e of A II , P P (6.2) h o l d s and A 12 p

(A22)-Ib 2 = b 1 o ol op+l

In these cases we m a y a p p l y t h e o r e m 4. An a n a l o g o u s result

holds for the d i f f e r e n c e e q u a t i o n

(4.5) we n o w assume ii replaced by p A P

(5.3)-(5.6)

V. A formal solution

(4.47) of

H~ and ]

o

(5.2): instead of Ii

i) or ii) above w i t h A

P

(4.1) a s s u m e d in t h e o r e m 5 exists if h y p o t h e s i s

(4.5) are s a t i s f i e d and

a) ~ i' "''' < g b) k

and the c o n d i t i o n s

are e i g e n v a l u e s of -A II w h e r e R e < • > 0, j= I, ..., g; P ]

+ kl
k1 +

.

..

+ kg_

>

2

g

<

or

g k

is n o t e i g e n v a l u e of -A II if ko, p I + ... + k g = i ' k o > 0;

..., k

g

c) there exists at m o s t one p o s i t i v e integer w h i c h is e i g e n v a l u e of 1 is such an e i g e n v a l u e

'

then 1 6 {~i

'

"'''


op+h

=

0,

b2 oh

E IN and

Ail ; if P = 0 for -

h = i, ..., 1 - I and

AI2 p

(A22)-i o

b2 = b I . ol op+l

The n u m b e r of formal solutions of A II + K.I , j=l, P 3 n1 solutions of systems

(4.47) d e p e n d s on the d i m e n s i o n s of the n u l l s p a c e s

..., g. We refer to H u k u h a r a

[13] who c o m p u t e d all formal

(0.i). T h e o r e m 5 is r e l a t e d to results of Iwano

A formal solution

(4.47) of

c o n d i t i o n s as above except that

(5.2) a s s u m e d in t h e o r e m 7 (4.5) is r e p l a c e d b y

[14].

exists u n d e r the same

(5.3)-(5.6)

and A

ii

P r e p l a c e d b y p A II P VI. We m a y a p p l y IV to obtain a b l o c k d i a g o n a l i z a t i o n for linear systems (6.3)

x l - P d_yy = C ( x ) y dx

a n a l o g o u s to W a s o w

[24, t h e o r e m 12.2] and M a l m q u i s t

[16, sect. 3 ].

is

51

T h e o r e m 8. Suppose C 6 A ( U ) , ]

Let X I

.. '

"

~ '

C(x) ~ E C x -m, C = d i a g {C 11 C 22 ) o Q m 1 o o C 1 are h . ~ those of C 22

be the eigenvalues of r

o

r+l'

""

'

n

o

"

Assume Z !p (lg - I h) ~ S 9 if g _< r < h. Then there exists a transformation y : T ( x ) z which takes (6.3) into

(6.4)

x l - p dz _ ~(x)z, ~(x) = diag {~ll(x) dx

C~92(x)}

such that T and ~ 6 A (~' ) for some ~ ' > ~ and ~Ii = cll , ~22 = C22 3 o o o 0 If CI2(x) , C 21 (x) = 0(x -N) , then T(x) = I + 0(x -N). P R O O F . F i r s t we substitute y = Q(x)y, w h e r e

ox/to fix t Then

(6.3) is t r a n s f o r m e d into

(6.5)

x l - P d w = D(x)w, dx

w h e r e Dl2(x) ~ 0 iff (6.6)

x l - p ~xx d Q 12 = c ii (x) QI2 _ Q12C22 (x) _ Q12c21 (x)Q 12 + cl2(x).

N o w C II 12 12 22 o Q -Q Co d e f i n e s a linear t r a n s f o r m a t i o n in the linear space of 12 r x (n-r)-matrices Q w i t h e i g e n v a l u e s hg- hh, g = I, .... r; h = r + I, ..., n. Hence we may use a p p l i c a t i o n IV w i t h n I = 0 and a solution of A.(~') ]

for some ~i > Z" In a similar way we m a y t r a n s f o r m

(6.6)

(6.6) to

exists in (6.4). []

A special case of t h e o r e m 8 w i t h j = 2 has b e e n g i v e n b y Turrittin[22]. T h e o r e m 8 w i t h j = i c o r r e s p o n d s to t h e o r e m 12.2 in W a s o w

[24] w i t h a d i f f e r e n t

sector. R e s u l t s of this type m a y be u s e d to reduce linear d i f f e r e n t i a l and d i f f e r e n c e e q u a t i o n s to canonical forms, cf. M a l m q u i s t

[16], T u r r i t t i n

[18] ,[20].

REFERENCES [i]

BRAAKSMA,

B.L.J., Laplace integrals, factorial series and singular diffe-

rential equations, Proc. B i c e n t e n n i a l c o n g r e s s of the W i s k u n d i g Genootschap, A m s t e r d a m 1978. [2]

BRAAKSMA, B.L.J.

& W.A. Harris, Jr., Laplace integrals a n d f a c t o r i a l

series

in singular differential systems. To appear in A p p l i c a b l e Mathematics. [3]

DOETSCH, G., Hendbuch der Laplace Transfor~ationj Basel, 1955.

Band II. B i r k h ~ u s e r Verlag,

52

[4]

ERDELYI, A.,

The integral equations of asymptotic theory, in

Asymptotic

Solutions of Differential Equations and their Applications, edited by C.H. Wilcox, John Wiley, New York, [5]

HARRIS, Jr., W.A. & Y. SIBUYA, Amer. Math. Soc., 70

[6]

1964, 211-229.

Note on linear difference equations, Bull.

(1964)

123-127.

Asymptotic solutions of systems of nonlinear

HARRIS, Jr. W.A. & Y. SIBUYA,

difference equations, Arch. Rat. Mech. Anal., 15 (1964) 377-395. [7]

HARRIS, Jr., W.A. & Y. SIBUYA,

On asymptotic solutions of systems of non-

linear difference equations, J.reine angew. Math., 222 [8]

HORN, J.,

(1966)

120-135.

Integration linearer Differentialgleichungen durch Laplacesche

Integrale und Fakultdtenreihen. Jahresber. Deutsch. Math.Ver., 24 (1915) 309-329; 25 [9]

HORN, J.,

(1917) 74-83.

Laplacesche Integrale als L~sungen yon Funktionalgleichungen,

J. reine angew. Math., [i0]

HORN, J.,

146 (1916) 95-115.

Verallgemeinerte Laplacesche Integrale als L~sungen linearer

und nichtlinearer Differentialgleichungen. Jahresber. Deutsch. Math. Vet., 25 (1917) 301-325. [11]

HORN, J.,

Uber eine nichtlinea~e Differenzengleichung, Jahresber. Deutsch.

Math. Ver., 26 (1918) 230t251. [12]

HORN, J.,

Laplacesche Integrale, Binomialkoeffizientenreihen und Gamma-

quotientenreihen in der Theorie der linearen Differentialgleichungen. Math. Zeitschr., 21 [13]

HUKUHARA, M.,

(1924) 85-95.

Integration formelle d'un syst~me des ~quations di~rentiel -

les non lin~aires dans le voisinage d'un point singulier. Ann. Mat. Pura Appl., [14]

IWANO, M.,

(4) 19 (1940) 35-44.

Analytic expressions for bounded solutions of non-linear

ordinary differential equations with an irregular type singular point. Ann. Mat. Pura Appl., (4) 82 (1969) 189-256. [15]

IWANO, M.,

Analytic integration of a system on nonlinear ordinary

differential equations with an irregular type singularity. Ann. Mat. Pura Appl., [16]

MALMQUIST,

(4) 94 (1972)

109-160.

J., Sur l'~tude analytique des solutions d'un syst~me d'¢quations

diff~rentielles dans le voisinage d'un point sin~lier d'ind~termination, II. Acta Math., 74 (1941) 1-64. [17]

TRJITZINSKY, W.J.,

Laplace integrals and factorial series in the theory

of linear differential and difference equations, Trans. Amer. Math. Soc., 37 (1934) 80-146.

53

[18]

TURRITTIN,

H.L.,

Convergent solutions of ordinary lineal~ homogeneous

differential equations in the neighbourhood of an irregular singular point. Acta Math., 93 (1955) 27-66. [19]

H.L., The formal theory of systems of irregular homogeneous linear difference and differential equations, Bol. Soc.Math. Mexicana

TURRITTIN,

(1960)

255-264.

H.L., A canonical form for a system of linear difference equations, Ann. Mat. Pura Appl., 58 (1962) 335-357.

[20]

TURRITTIN,

[21]

TURRITTIN,

H.L.,

Reducing the rank of ordinary differential equations.

Duke Math. J., 30 (1963) 271-274. [22]

TURRITTIN,

H.L.,

Solvable related equations pertaining to turning point

problems, in Asymptotic Solutions of Differential Equations and their Applications. Edited by C.H. wilcox, John Wiley, New York, 1964, 27-52. [23]

WALTER, W.,

Differential- and Integral Inequalities. Springer Verlag,

Berlin, [24]

WASOW, W.,

1970.

Asymptotic expansions for ordinary differential equations.

Interscience

Publishers,

New York,

1965.

CONTINUATION AND REFLECTION OF SOLUTIONS TO PARABOLIC PARTIAL DIFFERENTIAL E~UATIONS

David Colton * Dedicated to the memory of my teacher and friend Professor Arthur Erd~lyi

I. Introduction. As is well known, a solution of an ordinary differential equation can be continued as a solution of the given differential equation as long as its graph stays in the domain in which the equation is regular. On the other hand the situation for solutions of partial differential equations is quite different since a solution of a partial differential equation can have a natural boundary interior to the domain of regularity of the equation (c.f.~7]). exceptional circumstances

In fact it is only in very

that one can prove that every sufficiently

regular solution of a partial differential equation in a given domain can be extended to a solution defined in a larger domain.

In the

general case continuibility into a larger domain depends on the solution of the partial differential equation satisfying certain appropriate boundary data on the boundary of its original domain of definition, the classical example of this being the Schwarz reflection principle for harmonic functions.

In the past twenty-five years there has been

a considerable amount of research undertaken to determine criteria for continuing solutions of partial differential equations into larger domains and in these investigations two major directions stand out:

* This research was supported in part by NSF Grant MCS 77-02056 and AFOSR Grant 76-2879.

55

i) reflection principles,

and 2) location of singularities

of locally defined integral representations.

by means

Until quite recently

both of these approaches have been confined to the case of elliptic equations. The generalization harmonic

functions

pendent variables

of the Schwarz reflection principle

to the case of elliptic equations

for

in two inde-

satisfying a first order boundary condition along

a plane boundary was established by Lewy in his seminal address to the American Mathematical Lewy considered

Society in 1954 ([20]).

In this address

the elliptic equation

Uxx + u yy + a(x,y)u x + b(x,y)Uy + c(x,y)u = 0

(I.i)

defined in a domain D adjacent on the side y < 0 to a segment o of the x axis.

On o, u(x,y) was assumed to satisfy the first order

boundary condition ~(X)Ux(X,O ) + ~(X)Uy(X,O)

+ y(x)u(x,O)

Then under the assumption that u(x,y)

= f(x) •

e C2(D) A ~ ( D

(1.2)

L/ o), ~(z),

~(z), y(z) and f(z) are analytic in D u o ~ D* (where D* denotes the mirror image of D reflected across o), ~(z) # 0 and ~(z) # 0 throughout D ~ o ~ D * ,

and the coefficients

of (i.i) expressed in terms of

the variables z = x + iy

( i . 3) z* = x - iy are analytic functions of the two independent and z* for z e D V

o~

complex variables

z

D*, z* E D v o ~ D*, Lewy showed that u(x,y)

could be continued into the domain D ~ o ~ D* as a solution of (i.I). In particular Lewy showed that the domain of dependence associated

with a point in y > 0 is a one dimensional y < 0.

line segment lying in

Lewy also gave an example to show that an analogous result

was not valid in higher dimensions, even for the case of Laplace's equation in three variables satisfying a linear first order boundary condition with constant coefficients along a plane.

This

problem of the reflection of solutions to higher dimensional elliptic equations across analytic boundaries was taken up by Garabedian in 1960 ( ~ 2 ~ )

who showed that the breakdown of the

reflection property is due to the fact that the domain of dependence associated with a solution of an n dimensional elliptic equation at a point on one side of an analytic surface is a whole n dimensional ball on the other side.

Only in exceptional circumstances

does some kind of degeneracy occur which causes the domain of dependence to collapse onto a lower dimensional subset, thus allowing a continuation into a larger region than that afforded in general.

Such is the situation for example in the case of the

Schwarz reflection principle for harmonic functions across a plane or a sphere (where the domain of dependence degenerates to a point) and the reflection principle for solutions of the Helmholtz equation across a sphere (where the domain of dependence degenerates to a one dimensional

line segment - c.f.

[41).

Such a degeneration can

be viewed as a form of H u y g e n ' s p r i n c i p l e

for reflectio ~ analogous

to the classical Huygen's principle for hyperbolic equations, and in recent years there have been a number of intriguing examples of when such a degeneracy can occur (c.f. [9], [21]). The second major approach to the analytic continuation of solutions to elliptic partial differential equations is

through

57

the method of locally defined integral representations.

This approach

is based on the use of integral operators for partial differential equations relating solutions of elliptic equations to analytic functions of one or several complex variables and has been extensively developed by Bergman ([i]), Vekua ([242) , and Gilbert ([12). The main idea is to develop a local representation of the solution to the elliptic equation in the form of an integral operator with an analytic function with known singularities as its kernel and with the domain of the operator being the space of analytic functions. The problem of the (global) continuation of the solution to the elliptic equation can then be thrown back onto the well investigated problem of the continuation of analytic functions of one or several complex variables.

A simple example typical of this approach, and

one which exerted a major influence on much of the subsequent investi/

gations, was that obtained by Erdelyi in 1956 on solutions of the generalized axially symmetric potential equation u

+u xx

yy

where k is a real parameter ([8~).

+k -y

u

y

=

0

(1.4)

Erd~lyi's result was to show that

if u(x,y) is a regular solution of (1.4) in a region containing the singular line y = 0 and u(x,O) is a real valued analytic function in a y-convex domain D (i.e. if (x,y) e D then so is (x,ty) for -l
u(x,y) is a regular solution of (1.4) /

in D.

For generalizations of this result of Erdelyl the reader is

referred to Gilbert's book ( ~ ) .

The method of integral operators

outlined above has a variety of applications, among them being the

58 analytic continuation of the solution to Cauchy's problem for elliptic equations which arises in connection with certain inverse problems in fluid mechanics

(c.f. [iI], [14]).

More recently the author used such

an approach to establish a relationship between the domain of regularity of axially symmetric solutions of the Helmholtz equation defined in exterior domains and the indicator diagram of the far field pattern

([2]). In contrast to the rather extensive investigations

into the pro-

blem of continuing solutions of elliptic equations, until recently relatively little work has been done in connection with the corresponding problems for parabolic equations.

One reason for this is of course the

fact that solutions of parabolic equations do not enjoy the same regularity properties as solutions of elliptic ~quations,

i.e., in general

solutions of parabolic equations with analytic coefficients are not analytic functions of their independent variables.

Until about ten

years ago there were (to the author's knowledge) only two rather isolated results on the global continuation of solutions to parabolic equations, both of them concerned with the heat equation in one space variable. The first of these was the fact that solutions of the one dimensional heat equation defined in a rectangle and satisfying homogeneous Dirichlet or Neumann data on a vertical side of the rectange could be reflected as a solution of the heat equation into the mirror image of the rectangle ([25]).

On the other hand it was shown by Widder in [26] that

if h(x,t) is a solution of the one dimensional heat equation h

xx

= h

(1.5)

t

which is analytic in x and t for Ix[ < x

o'

Itl < t , then h(x,t) can o

be continued as a solution of the heat equation into the entire strip

59

Ixl < =, Itl < to, and expressed there as a uniformly convergent series of heat polynomials h(x,t) =

Z a h (x,t) nn n=O

(1.6)

where the heat polynomials are defined by /~ h (x,t) = ~]: E n k=O

x n-2ktk (1.7) (n-2k) ~k:

This result is noteworthy since it is one of the few cases where it can be stated that eyery sufficiently regular solution of a partial differential equation defined in a given (real) domain has an automatic continuation into a larger (real) domain regardless of what the boundary data is.

Note that such behaviour is only true for

analytic solutions of the heat equation, and not in general for classical solutions which are infinitely differentiable, but not analytic, in the time variable. Hence in the development of the reflection and continuation properties of solutions to parabolic equations with analytic coefficients one can expect a different behaviour depending on whether or not the solutions are analytic in the time variable.

In this connection it is worthwhile to note that if u(~,t)

is a solution of a linear parabolic equation with analytic coefficients defined in a cylinder~.x(O,T)

in ~Rn+l, and if u(~,t) assumes analytic

Dirichlet data on the analytic boundary ~.C~x(O,T), then u(~,t) is an analytic function of its independent variables i n - / ~ x(O,T)(c.f. ~0]). In the following sections we shall survey, with an outline of proofs, some recent results on the reflection and continuation of solutions to linear parabolic equations with analytic coefficients.

We

shall restrict ourselves to parabolic equations of second order, although

60 corresponding results should also be valid for certain higher order equations.

The theory we shall present is far from complete.

In

particular we have omitted questions concerning removable singularities as well as the problem of backward continuation of solutions to parabolic equations.

Some aspects of this last topic will be

discussed in a survey paper to be presented at the conference on "Inverse and Improperly Posed Problems in Differential Equations" to be held next year in Halle.

II. Parabolic ERuations in One Space Variable. The basic idea behind Lewy's reflection principle for elliptic equations was to develop an integral operator which allowed the general reflection problem to be reduced to one in which the Schwarz reflection principle for analytic functions could be applied.

This

is also the main idea behind the method for developing reflection principles for parabolic equations, where in this case the basic theorem employed is the reflection principle for solutions of the heat equation.

The operators employed in this analysis are no longer

based on the idea of a complex Riemann function as in Lewy's work but instead on a generalization of the transformation operators for ordinary differential equations developed by Gelfand, Levitan and Marcenko in their investigations of the inverse scattering problem in quantum mechanics (c.f. [7]).

To introduce these generalized trans-

formation operators we first consider solutions of the parabolic equation Uxx + a(x,t) Ux + b(x,t)u = u t

(2.1)

defined in a domain D of the form D = {(x,t): Sl(t)<x<s2(t)10
61 Eunctions of x and t in D u ~

~ D* where o is the arc x = Sl(t) and

3* is the reflection of D across ~ defined by D* = {(x,t): 2Sl(t) s2(t) < x < Sl(t) , 0 < t < to }"

Assuming that x = Sl(t) is analytic

and making the change of variables = x - sl(t) (2.2)

T = t changes

(2.1) into an equation of the same form but now defined

in a domain D to the right of the t axis with lateral boundary x = O.

A further change of variables of the form fx u(x,t) = v(x,t) exp {- 12 I a(s,t)ds} J0

reduces

(2.3)

(2.1) to an equation of the same form except that a(x,t)

is now equal to zero.

Hence without

loss of generality we can

consider equations of the form u

xx

+ q(x,t)u = u

(2.4)

t

where q(x,t) is analytic in D u ~ ~ D* and D = {(x,t):O<x<s2(t), O
is a classical

solution of (2.4) defined

Then u(x,t) can be represented

in the form ([3])

u(x,t) = (I + T)h (2.5) = h(x,t) +

(s,x,t)h(s,t)ds

where h(x,t) is a solution of the heat equation hxx = h t in ~ problem

~

~* and E(s,x,t)

(2.6)

is a solution of the initial value

62 E

- E

xx

ss

E(x,x,t)

+ q(x,t)E = E

fx

= - ~

t

q(s,t)ds

(2.7)

O

E(-x,x,t)

= O .

A solution of (2.7) can be constructed by iteration and is analytic for -s2(t)<x<s2(t )

-s2(t)<s<s2(t )

O
The fact that

O

every solution of (2.4) defined in D u o ~ D *

can be represented

in the form (2.5) follows from the fact that ~ is a Volterra operator and hence I + T is invertible. for parabolic equations various modifications

The reflection principle

is obtained by a judicious application of

of the operator

(2.5).

This will be illus-

trated in the proof of the following theorem: Theorem 2.1: continuously

Let u(x,t) be a classical solution of (2.1) in D, differentiable

~(t)u(sl(t),t)

in D k7 ~, and satisfying

+ 8(t)Ux(Sl(t),t)

+ y(t)ut(sl(t),t)

= f(t)

(2.8) on o, where Sl(t), ~(t), B(t), y(t) and f(t) are real analytic on (O,to).

If, for t E (O,to), the non zero vector ](t) =

(~(t), y(t)) is never tangent to x = Sl(t) x = Sl(t)) and never parallel to the x axis), then u(x,t) ution of (2.1) into D ~ Proof:

(or always tangent to

to the x axis (or always parallel

can be uniquely continued as a sol-

o~)D*.

By the above discussion we can reduce the problem to the

case when a(x,t) = O(i.e. equation

(2.4)), Sl(t) = O, and assume

that B(t) = 0 or B(t) # O for t ~ (O,to). an appropriate non-characteristic

Furthermore,

by solving

Cauchy problem for (2.4)

(to be

discussed at the end of this section) we can assume that f(t) = O.

63 Hence without loss of generality we can consider the problem of continuing solutions of (2.4) subject to (2.8) with Sl(t ) = 0 and f(t) = O, where u(x,t) is defined in D and is continuously differentiable in D ~

o.

We shall only consider the special case when

6(t) # 0 and y(t) # 0 for t c (O,t o) and refer the reader to [5] for full details.

In this special case by making a preliminary

change of variables of the form U(X,t) = v(x,t) exp {-

~(~)dT}

(2.9)

0 we can assume without loss of generality that ~(t) = 0 and the boundary condition on o is Ux(O,t ) + q(t)ut(O,t) = 0 where q(t) =

Y(t)16(t).

(2.10)

We can now show that in D, u(x,t) can be

represented in the form XK(1)

u(x,t) = h(1)(x,t) + h(2)(x,t) +

(s,x,t)h

(i)

(s,t)ds

~0

(2.11) + {XK(2)(s,x,t)h(2)(s,t)ds JO where K(1)(s,x,t) is the solution of K (I) _ K (I) + q(x,t)K (I) = K (I) xx ss t 'x

K(1)(x,x,t) = - ~

i

(2.12)

q(s,t)ds 0

K(1)(O,x,t) = O, s K(2)(s,x,t) is the solution of K (2) - K (2) [q(x,t) xx ss +

q (t)] K(2)

q~)'J

K(2) (x,x,t) = _ ½ ix Eq(s,t ) J0 K(2)(O,x,t) = O,

= K (2)

t q (t) ~ds q(t)

(2.13)

and h(2)(x,t) = -n(t)h~l)(x,t) where h(1)(x,t) is a solution of (2.6) in D satisfying h(1)(O,t) = O.

K(1)(s,x,t) and K(2)(s,x,t) can be

X

constructed by iteration and are analytic for -s2(t)<x<s2(t), -s2(t)<s<s2(t), O
The reflection principle now follows from

(2.11) and the fact that h(1)(x,t) can be continued into D ~ ~ by the reflection principle for the heat equation.

~*

The uniqueness

of the continuation follows from Holmgren's uniqueness theorem. From the above analysis it is see that for the original problem the domain of dependence associated with a point in X<Sl(t) is a one dimensional line segment lying in X>Sl(t). It would be desirable to know if the restriction on the direction of the vector ~(t) can be removed. As a corollary to Theorem (2.1) we have the following version of Runge's theorem for solutions of (2.1) defined in a domain D = {(x,t): Sl(t)<x<s2(t), O
ytic for O
Let u(x,t) e C2(D) ~ C°(~) be a solution of (2.1) in

D and assume that the coefficients of (2.1) are analytic for -~<x<=, O~t@t o.

Then for every c>O there exists a solution Ul(X,t) of (2.1)

in -~<x<~, O
There exists a solution Ul(X,t) s C 2 ( D ) ~ CI(~) satisfying

analytic boundary data on x = Sl(t ) and x = s2(t) such that the above inequality is valid.

By reflecting Ul(X,t) repeatedly across

the arcs x = Sl(t) and x = s2(t ) it is seen that Ul(X,t) can be continued as a solution of (2.1) into the infinite strip -~<x<~, O~t~t o •

65 We now turn to the case when the solution of (2.1) is known to be analytic in some neighbourhood

of the origin and make the

assumption

that the coefficients

of (2.1) are analytic for -=<x< ~,

-t
In this case we have that u(0,t) and Ux(O,t)

are anal-

ytic functions of t for -t
are analytic functions of t for -t
Cauchy-Kowalewski

theorem and Holmgren's uniqueness

is analytic in a neighbourhood Widder's

Hence by the theorem h(x,t)

of the t axis for -t
theorem referred to in the Introduction h(x,t)

By

is analytic

in the strip -=<x<=, -to
and hence from (2.5) and the anal-

yticity of the kernel E(s,x,t)

so is u(x,t).

Theorem 2.2:

Let the coefficients

strip -~<x<=, -to
and let u(x,t) be any solution of (2.1)

that is analytic for -Xo<X<Xo, stant x .

of (2.1) be analytic in the

-to
for some positive con-

Then u(x,t) can be analytically

continued into the strip

O

-~<x<~

-to
We close this section be giving a simple derivation of Widder' result on the analytic continuation of analytic solutions of the heat equation which was used to prove Theorem 2.2

Suppose h(x,t)

is an analytic solution of the heat equation for -Xo<X<Xo, -to
=

i - 2hi

J

E X (X,t-T)h(O,T)dT

It-el

- 2~ii

~

in the form (~5])

=

E(x,t_T)hx(O,r)d ~

Jt-T I = 6

(2.14)

66 where oo

E

E(x,t) =

x2j+l (-I) J ~'

(2.15)

j=O (2j+l) : tj+l Since E(x,t) is an entire function of x we immediately have the result that h(x,t) is analytic in the strip -~<x<~, -t
If

O

we further assume that h(O,t) and h (O,t) are analytic for Itl
and r e p r e s e n t

t h e s e J u n c t i o n s by t h e i r

Taylor series

oo

h(O,t) =

E b tn n= O n

,

Itl < t o (2.16)

oo

hx(O,t ) =

)2 Cn tn n=O

Itl < t O

we have from (2.14) and termwise integration h(x,t)

where hn(X,t)

=

that for -~<x< ~, -to
E a h (x,t) n= 0 n n

are the heat polynomials

(2.17)

defined in (1.7) and a2m=bm ,

a2m+l=Cm for m = O,1,2, .... Returning to the proof of Theorem 2.1 we note that from the above results a special solution of (2.4) satisfying

(2.8)

(for

Sl(t ) = O) can be obtained in the form u(x,t) = (~ + ~)h(x,t) where h(x,t)

is given by (2.14) with h(O,t) and hx(O,t) chosen appropriately.

Ill.Parabolic

Equations

in Two SEace Variable @.

In contrast to the case of parabolic equations variable,

in one space

the results on the global continuation of solutions

parabolic equations

in two space variables

are rather limited.

in the case of analytic solutions of parabolic equations space variables with analytic coefficients

to Even

in two

it is not yet known

whether or not such solutions can be reflected across a plane boundary on which they assume homogeneous

Dirichlet data.

However

67 it has been shown by Hill that the domain of dependence associated with an analytic solution of a parabolic equation in two space variables at a point on one side of a plane on which the solution vanishes is a one dimensional line segment on the other side (El6]). This suggests the possibility of establishing a reflection principle for parabolic equations in two space variables, in the case of analytic solutions.

at least in

However this has yet to be done.

The main problem seems to be to establish the domain of regularity in the complex domain of the normal derivative of the solution evaluated along the plane on which the solution vanishes.

This

suggests the problem of determining the domain of regularity in the complex domain of analytic solutions of parabolic equations, given either their domain of real analyticity or the domain of regularity of the Cauchy data in the complex domain.

As will be seen,

modest results such as these can be applied to derive a Runge approximation property for parabolic equations as well as a global representation of the solution to certain non-characteristic problems arising in the theory of heat conduction.

Cauchy

Although we

shall derive these basic results in this section, we shall postpone their application to problems associated with the heat equation until Section IV. We consider analytic solutions of parabolic equations of the form + c(x,y,t)u - d(x,y,t)u t L [u~ = Uxx + u YY + a(x,y,t)u x + b(x,y,t)Uy

(3.1) --0 and for the sake of simplicity make the assumption that the coefficients of (3.1) are entire functions of their independent complex

68 variables.

By making the nonsingular change of variables mapping

the space of two complex variables into itself defined by z = x + iy

(3.2) z* = x - iy we can rewrite (3.1) in its complex form as

(3.3) i i I i where A = [(a + ib), B = ~(a - ib), C = [ c, D = ~ d.

The basic

tool used in the investigation of analytic solutions of (3.1)

(or

(3.3)) is the Riemann function for (3.3) first introduced by Hill in [16].

This is defined to be the unique solution R(z,z*,t;~,~*,T)

of the (complex) adjoint equation

-- V-zz ,

My]

3(B~/) + C _ ~ r + _ ~ ~z*

a(AV) 3z

(D'V') = 0 (3.4)

satisfying the initial data R(z,~*,t;~,~*,T)

= --!-I exp {

B(q,~*,t)d~}

t-~

(3.5) R(¢,z*,t;C,¢*,T)

= ~

exp {

do}

along the planes z -- ¢ = ~ + iQ, z* = ~* = $ - in. function can be constructed by iterative methods

The Riemann

([3],

[16]) and

(under the assumption that the coefficients of (3.1) are entire) is an entire function of its six independent complex variables except for an essential singularity at t = T.

In the case of the heat

equation (3.6)

u xx + uyy = u t the Riemann function is given by R(z z*,t;¢,¢*,T) •

= ~

I

exp {

(z-C) ( z * - ¢ * ) 4 (t-T)

} .

(3.7)

69 Now let u(x,y,t) be an analytic solution of (3.1) defined in a cylinder Dx(O,T) where D is a bounded simply connected domain. Assume that D contains the origin and that d(x,y,t) > 0 in Dx(O,T). By standard compactness arguments we can conclude that if ~

C

D,

O

6o > O, then u(x,y~t) is analytic in some thin neighbourhood in C 3 of the product domain box ~o,T-~o3.

We now use Stokes theorem

applied to u and R log r (where R is the Riemann function and r = /(x-~) 2 + (y-n) 2 to represent u(x,y,t) in terms of the Riemann function, where the domain of integration is the torus D

x O

with ~ L =

{t:It-T [ = ~}.

This yields the result that

4~ 2 ~D x~ho where~

D x ~L o

(3.8)

is the adjoint operator to (3.1) and

H[u,v3 = {(VUx - UVx + auv)dydt - (VUy

UVy + bul)dXdtv

(duv)dxdy}. (3.9)

Since ~

[R log r] is an entire function of its independent complex

variables except for an essential singularity at t = r and H[u, R log r~ is analytic for (x,y,t) ¢ ~Do x ~ ¢ ~o' T - 6 ~ ,

and 6 O

in D x (O,T).

~ Do,

we have the following theorem (Note that subject

to the above restrictions D Theorem 3.1:

, ~ ¢ Do, ~

are arbitrary): O

Let u(x,y,t) be an analytic solution of (3.1) defined Then'U-

(z,z*,t) = u(x,y,t) is analytic in D x D* x (O,T)

where D = {z:z ~ D}, D* = {z*: z* e D}. Theorem 3.1 is analogous to the result that every regular solution of a linear ordinary differential equation with entire coefficients defined on a finite interval can be extended to an entire function

70 of its independent singularities entire.

complex variable.

For partial d~ffe~emtial

can of course occur even though the coefficients

are

However Theorem 3.1 shows that in the case of analytic solutions

of parabolic equations gularities

in two space variables

the location of the sin-

in the complex domain is determined

location of the singularities

in two independent variables

in a simple way by the

on the boundary of the domain of def-

inition of the solution in the real domain.

Bergman

e~fnations

For elliptic equations

analogous results have been found by

([I]), Lewy (E20]) and Vekua

([24]).

Using Theorem 3.1 we can now represent?~(z,z*,t)

in D x D* x (O,T)

in terms of the Riemann function and the Goursat data on the characteristic hyperplanes

z = O and z* = O in a manner almost identical

to the

use of the classical Riemann function to solve the Goursat problem for hyperbolic

equations

in two independent variables.

Since this Goursat

data is defined in a product domain we can apply Runge's theorem for several complex variables sets by polynomials

to approximate

and hence, since the Riemann function is entire

except at t = T, approximate-~f(z,z*,t) entire solution of (3.1). classical

this data on c o m p a c t s u b -

on compact subsets by an

If we now note that if d(x,y,t)

> O every

solution of a parabolic equation with analytic coefficients

defined in a cylindrical

domain with analytic boundary can be approx-

imated by a solution having analytic boundary data, and hence from the Introduction

is an analytic solution of the parabolic equation,

we can now deduce the following version of Runge's theorem ([3]): Corollary:

Let u(x,y,t) be a classical

where d(x,y,t)

> O in D x (O,T).

solution of (3.1) in D x (O,T)

Then for every compact subset D cD o

and positive constants ~ and e there exists an entire solution Ul(X,y,t) of (.3.1) such that

71

max

lu(x,y,t) -Ul(X,y,t)I

< e .

o We now turn our attention

to the problem of determining

domain of regularity of solutions to non-characteristic

the

Cauchy pro-

blems for parabolic equations with data prescribed along an analytic surface.

Such problems arise if inverse methods are used to study

free boundary problems in the theory of heat conduction

(c.f. Section

IV) and a local solution can always be found by appealing to the CauchyKowalewskl

theorem.

However in addition to being impractical

for com-

putational purposes such an approach does not provide us with the required global solution to the Cauchy problem under investigation, and hence we are lead to the problem of the analytic continuation solutions to non-characteristic We shall accomplish

of

Cauchy problems for parabolic equations.

this through the use of the Riemann function in

connection with contour integration techniques and the calculus of residues fn the space of several complex variables. We assume that u(x,y,t)

is an analytic solution of (3.1) in a

domain containing a portion of the non-characteristic S along which u(x,y,t) the intersection one dimensional F(x,y,t)

= O.

assumes prescribed

analytic

analytic Cauchy data.

surface Let

of the plane t = constant with the surface S be a curve ~(t).

Suppose ~(t) is described by the equation

Then since S is analytic we can write F.Z+Z* £ 2

'

z-z* t) = O 2i '

(3.10)

....

where z and z* are defined 5y (3.2) and this is the equation for ~(t) in (z,z*) space.

We now choose C(t) to be an analytic curve lying on

this complex extension of ~(t) and for (~n,r)

not on S let G(t) be a

72 cell whose boundary consists of C(t) and line segments lying on the characteristic planes z = ~ = ~+i~ and z* = ~ = t-in respectively which join the point (~,~) to C(t) at the points P and Q respectively. If we now use Stokes theorem to integrate RL[I~r]--~rM[R]

(where R is

the Riemann function and L and M are defined by (3.3) and (3.4) respectively) over the torus {(z,z*,t):(z,z*) -~r(~,~,T) =

~.... ~ 4~i J

e G(t),It-T I = 5} we have

[ R ( P , t ) ~ (P,t) + R(Q,t)7-F(Q,t)]dt

It-~1 =~

(3.II)

2~i

[t-T I=6 C(t) - (BR~-+

½ R T_)-z - ½ Rz-O-)dzdg

where we have suppressed the dependence of the Riemann function on the point (~,~,T) and an expression of the form 7jr(P,t) is a function of three independent variables, i.e.7_~(P,t) =~r(~l,~2,t) where ( 1, 2) are the Cartesian co-ordinates of the point P in C 2. For (~, ~,~) sufficiently near the initial surface S and for sufficiently small, (3.11) gives an integral representation of the solution to the non-characteristic Cauchy problem for (3.1) with analytic data prescribed on S.

Equation (3.11) can now he used to

obtain a global solution by deforming the region of integration, provided a knowledge of the domain of regularity of the Cauchy data and analytic function F(x,y,t) is known.

In particular such a pro-

cedure yields results on the analytic continuation of solutions to parabolic equations along characteristic planes in terms of the domain of regularity of the Cauchy data and the domain of regularity of the function describing the non-characteristic surface along which

73 the Cauchy data is prescribed.

For example suppose for each fixed

t, t e [O,TJ, z = $(~,t) maps the unit disc conformally onto a domain D t and let the analytic surface S be described by im ~-l(z,t) = 0

(3.12)

Assume that ¢(z,t) and ¢-l(z,t) depend analytically on the parametec t.

Then setting ~(~,t) = ¢(~,t) we have that the equation

for C(t) is given by @-l(z,t) = ~-l(z*,t).

Hence if we assume

that the Cauchy data is analytic in D t for each t we have from (3.11) the following result (c.f.~]): Theorem 3.2:

Let u(x,y,t) be an analytic solution of (3.1) which

assumes analytic Cauchy data on the surface (3.12) where @(z,t) conformally maps the unit disc onto the domain D t.

If the Cauchy

data for u(x,y,t) is analytic in D t then, for each fixed t, 7J(z,z*,t) = u(x,y,t) is an analytic function of z and z* in O t x Dr* where O t" = {ze:z* e Ot). Theorem 3.2 is a generalization of the corresponding result obtained by Henrici for elliptic equations in two independent variables ([lhJ).

For partial progress on extending the results

of this section to parabolic equations in three space variables we refer the reader to [23S.

IV. Applications to Problems in Heat Conduction. In this last section we shall apply the results of the previous two sections to derive an explicit analytic representation of the solution to the inverse Stefan problem and to construct some complete families of solutions to the heat equation which are suitable for constructing approximate solutions to the standard initial-boundary value problems arising in the theory of heat

74

conduction (c.f.[3]).

We first consider the Stefan problem.

This

is a free boundary problem for the heat equation and we are interested in the inverse problem where the free boundary is assumed to be known a priori.

Such an approach allows one to construct a

variety of special solutions from which a qualitative idea can be obtained concerning the shape of the free boundary as a function of the initial-boundary data.

In certain physical situations, e.g.,

the growing of crystals~ the inverse problem is in fact the actual problem that needs to be solved.

The simplest example of the type

of problem we have in mind is the single phase Stefan problem for the heat equation in one space variable, which can be mathematically formulated as follows:

to find u(x,t) and s(t) such that u

xx

= ut;

O<x<s(t), t>O

u(s(t),t) = O; Ux(S(t),t) u(O,t) :

t>O

= - s(t); @(t);

(2.z)

t>O t>O

where it is assumed that ¢(t)~O, s(O) = O.

The function u(x,t)

is the temperature in the water component of a one-dimensional ice-water system, s(t) is the interface between the ice and water, and ¢(t) is assumed to be given.

The inverse Stefan problem assumes

that s(t) is known and asks for the solution u(x,t) and in particular the function ¢(t) = u(O,t), i.e., how must one heat the water in order to melt the ice along a prescribed curve?

If we

assume that s(t) is analytic the inverse Stefan problem associated with (h.i) can easily be solved using the results of Section II. Indeed if in (2.1h) we place the cycle It-~l = 6 on the two dimensional manifold x = s(t) in the space of two complex variables, and note that since u(s(t),t) = 0 the first integral in (2.1h)

75 vanishes, we are lead to the following solution of the inverse Stefan problem: u(x,t) = 12wilt-~l=6 E(x-s(T),t-x)s(T)dT

(4,2)

Computing the residue in (4.2) gives u(x,t) =

~ I Sn n=l ~ - - S t n [x-s(t)] 2n ,

(4.3)

a result which seems to have first been given by Hill ([15]).

The

idea of the inverse approach to the Stefan problem (4.1) is to now substitute various values of s(t) into (4.3) and compute ¢(t)=u(O,t) for each such s(t).

For example setting s(t) = /t gives n! ¢(t): Z ~ K , n=l

= constant ,

(4.4)

a result corresponding to Stefan's original solution. We now consider the Stefan problem in two space variables.

The

equations corresponding to (4.i) are now Uxx + u yy = ut; uI so x

¢(x,y,t)
= y(x,y,t)

(4.5) uI

= o ~=0 S¢

Su I

_ ¢=0

i

St ¢=O

where u(x,y,t) is the temperature in the water, ¢(x,y,t) = O is the interphase boundary, D is a region originally filled with ice, y(x,y,t) ~ 0 is the temperature applied to the boundary SD of D, is the normal with respect to the space variables that points into

76 the water region $(x,y,t) < O, and V denotes the gradient with respect to the space variables.

In this case from the analysis of Section !II

we see that in order to solve the inverse Stefan problem it is necessary to restrict the function #(x,y,t) to be more than just analytic, and we assume ¢(x,y,t) is given by an equation of the form (3.12) where D t D D for t ~ [O,T].

In this case we have from (3.11) and

(3.7) that Z/(~,~,~) = u(~,n,T) is given by

7 J (~, ~,T) : ~

1

!

i

I -~I:a

I

exp [(z-~)(z*-~q

t-~

i

~¢ ,dzidt

[ ~ h ( t i T ' ) ~ J IV$1 St

c() (~.6)

where C(t) is a curve lying on the surface $-l(z,t) = ¢ -l(z*,t) with endpoints on the characteristic hyperplanes z =~ and z* = ~. Computing the residue in (h.6) now gives

7j(~,L+)

i

~

: + z

i

n:0 hn(n!

)a

~n

~

{j

(z-~)n(z*-~) n

twl

~Tn

3¢ --

~

Id+l} • (~.7)

c(+)

Equation (h.7) gives the generalization to two dimensions of the series solution (h.3) for the one dimensional inverse Stefan problem.

We note that the integral in (h.7) is pure imaginary. We now turn our attention to the construction of complete

families of solutions for the heat equation.

By completeness we

shall mean completeness on compact subsets of a given domain D with respect to the maximum norm.

We first consider a solution

h(x.t) of the one dimensional heat equation (2.6) defined in a domain D of the form D = {(x.t):sl(t)<x<s2(t) , 0
o

If % c D

then we can assume

has an analytic boundary and

hence from the Corollary to Theorem 2.1 we have that for every

77 E ~ 0 there exists a rectangle R P

o

and a solution hl(X,t) of the

heat equation in R such that

(~.8)

max lh(x,t) -hl(X,t) I < o Since it is a relatively

easy matter to show that the heat poly-

nomials h (x,t) defined by (1.7) are complete for solutions of the n

heat equation defined in a rectangle, we can now conclude that the heat polynomials

are also complete for solutions of the heat

equation defined in a domain of the form described above. We next want to decide under what conditions on the separation constants

k

n

do the solutions + hn(x,t) = exp (+_ i~nX - ~2t)n

form a complete family of solutions From the above results

(4.9)

to the heat equation in D.

it suffices to show that every heat poly-

nomial can be approximated by a linear combination of the functions (4.9), subject to certain restrictions

on the constants

~ . n

From

the representation (2.1h) we see that it suffices to show that the 2 set {e -~nt} is complete for analytic functions defined in an ellipse containing

[O,to].

The type of restriction necessary is indicated

by the following theorem based on the theory of entire functions

(B@ ,p.219): Theorem h.l:

If {~n } is a sequence of complex numbers for which

the limit d =

lim n'~

n ~ ~n

> 0

exists, the syste~ {e ~nz} is complete in the space of analytic functions defined in anY region G for which every straight line parallel to the imaginary axis cuts out a segment of length less

78 than 2wd, and the system is not complete in any region which contains a segment of length 2~d parallel to the imaginary axis. From Theorem 4.1 we now see that the set (4.9) is complete for solutions of the heat equation defined in a domain D of the form described above provided lim n~

n --~2

> 0 .

(4.10)

n

Theorem 4.2:

Let D = {(x,t):sl(t)<x<s2(t),

s2(t) are continuous

functions.

O
Then the following sets are complete

for solutions of the heat equation defined in D:

I) h n(x,t) = [~]'.

[n/~ Z k=O

xn-2ktk ........ (n-2k)!k!

; n = 0,1,2 .....

+

2) h~(x,t) = exp (+ ilnX - k2t)~ -n

lim n > O. n ~-~ ~2 n

We now conclude this section by using the results of Section III to derive a result analogous to Theorem h.2 for the heat equation in two space variables defined in a cylindrical D is a bounded simply connected domain.

domain D x [O,T] where

From the proof of the Corollary

to Theorem 3.i we see that it suffices to obtain a set of solutions to (3.6) assuming data on the complex hyperplanes

z = 0 and z* = 0

that is complete for product domains in the space of two complex variables.

It is easily verified that one such complete set is given by + U~,m(r,e,t)

+ine = e-

m r2k+n tm-k Z k=O 4kk~(m-k)!(n+k)!

where (r,e) are polar co-ordinates, characteristic satisfies

(h.ll)

since on the above mentioned

(r,e,t) hyperplanes we have t h a t ~ / ~+, m (z,z*,t) = u ~ n,m

79 zntTM + (z , O , t ) . . . . . . n,m m'n'

(h.12) V

n,m ( O ' z * ' t ) : z*ntm m'n'

It follows from the uniqueness theorem for Cauchy's problem for the heat equation that another complete set for (3.6) defined in D x (O,T) is given by hn,m(X,y,t) = hn(x,t) hm(Y,t) where hn(x,t) is the heat polynomial defined by (1.7).

(h.13) On the

other hand if we separate variables for (3.6) in polar coordinates we find that

+ h-(r,e,t) = Jm(Anr)exp(+im8 - X~t) n ,m

(h.lh)

where J (z) denotes Bessel's function, is a solution of the heat m

equation satisfying the complex Goursat data z H~n+'m(Z'O't) --

Hn,m(O,z*,t ) =

Hn+--m(Z,z*,t)

exp(- ~ t) 2m m:

z*mexp(- ln2t) ..2m m' -.

.....

= h +(r,B,t) n~m

(h.15)

m

and hence from Theorem h.l we can conclude that (h.lh) is another complete set of solutions for the heat equation in D x (O,T) provided (~.10) is valid. Theorem h.3: plane.

Let D be a bounded simply connected domain in the

Then the following sets are complete for solutions of the

heat equation defined in D x (0,T): l) hn,m(X,y,t) = hn(x,t)hm(Y,t) ; n,m = 0,1,2,... O. 2) h+,m(X,y,t) = Jm(Xn r) exp(+_ ires - X2t)n;n'm= O,i ....limn~-Vn Xn

80 Theorem h.3 can also be extended to higher dimensions ([6~). For other methods of proving Runge's theorem for the heat equation see [18~ and [22].

References 1.

S. Bergman, Integral Operators in the Theor~ofLinear

Partial

Differential Equations, Springer-Verlag, Berlin, 1969. 2.

D. Colton, Partial Differential Equations in the Complex Domain, Pitman Publishing, London, 1976.

.

D. Colton, Solution of Bound ar~ Value Problems by the Method of Integral Operatgrs , Pitman Publishing, London, 1976.

.

D. Colton, A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem, Glas~ow Math ,. J. 18(1977), 125-130.

.

D. Colton, On reflection principles for parabolic equations in one space variable, Proc. Edin. Math. Soc., to appear.

.

D. Colton and W. Watzlawek, Complete families of solutions to the heat equation and generalized heat equation in

R n,

J. Diff. Eqns~ 25(1977), 96-107.

.

V. De Alfaro and T. Regge, Potential Scattering, North-Holland Publishing Company, Amsterdam, 1965. J

.

°

A. Erdelyl, Singularities of generalized axially symmetric potentials, Comm.:" Pure Appl. Math. 9(1956), hO3-hlh.

.

V. Filippenko, On the reflection of harmonic functions and of solutions of the wave equation, Pacific J. Math. lh (196h), 883-893.

81 i0.

A. Friedman, Part%al Differential Equations, Holt, Rinehart and Winston, New York, 1969.

ii.

P. Garabedian, An example of axially symmetric flow with a free surface, in Studies in Mathematics and Mechanics Presented to Richard yon Mises, Academic Press, New York, 195h, 149-159.

12.

P. Garabedian, Partial differential equations with more than two independent variables in the complex domain, J. Math. Mech. 9 (1960), 2hi-271.

13.

R . P . Gilbert, Function Theoretic Methods in Partial Differential Equations, Academic Press, New York, 1969. P. Henrici, A survey of I. N. Vekua's theory of elliptic partial differential equations with analytic coefficients, Z.An~ew.Mat h. Physics 8(1957), 169-203.

15.

C. D. Hill, Parabolic equations in one space variable and the non-characteristic Cauchy problem, Comm. Pure Appl. Math. 20 (1967), 619-633.

16.

C. D. Hill, A method for the construction of reflection laws for a parabolic equation, Trans. Amer. Math. Soc. 133 (1968), 357-372.

17.

F. John, Continuation and reflection of solutions of partial differential equations, Bull. Amer. Math. Soc. 63 (1957), 327-3hh.

18.

B. F. Jones, Jr., An approximation theorem of Runge type for the heat equation, Proc. Amer. Math. Soc. 52 (1975), 289-292.

19.

B. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, 1964.

82 20.

H. Lewy, On the reflection laws of second order differential equations in two independent variables, Bull. Amer. Math. Soc. 65 (1959), 37-58.

21.

H. Levy, On the extension of harmonic functions in three variables, J. Math. Mech.lh (1965), 925-927.

22.

E. Magenes, Sull' equazione del calore:

teoremi di unicit'a e

teoremi di completezza connessi col metodo di integrazione di M. Picone, Rend. Sem. Mat. Univ. Padova 21 (1952), I, 99-123, II, 136-170.

23.

M. Stecher, Integral operators and the noncharacteristic Cauchy problem for parabolic equations, SIAM J. Math. Anal. 5 (1975), 796-811.

I. N. Vekua, New Methods for Solvin~ Elliptic Equat%ons, John Wiley, New York, 1967.

25.

D. Widder, Th9 Heat Equation, Academic Press, New York, 1975.

26.

D. Widder, Analytic solutions of the heat equation, Duke Math. J. 29 (1962), h97-503.

Legendre polynomials and singular differential operators

W N Everitt

Introduction

The Legendre polynomials can be defined in a

number of different ways which we review here briefly before discussing the connection with differential operators.

For convenience

let (-I,|) be the open interval of the real line, and let N

o

= {0,1,2,. .... } be the set of non-negative integers. All the definitions given below are inter-connected,

as may be

seen in the standard accounts of the Legendre functions given in [2] by Erdglyi et al, and [9] by Whittaker and Watson. (i)

The Legendre differential equation For our purposes this is written in the form

(here

' ~ d/dx)

(I - x2)y"(x) - 2xy'(x) + n(n + l)y(x) = O

(x ~ (-l,l))

(I.I)

which is derived from the Laplace or the wave equation, considered in polar co-ordinates.

This equation has singular points at ±l where

the leading coefficient vanishes.

This equation has a non-trivial

solution which is bounded on (-I ,I) if and only if n £ No; the corresponding solution is P (-), i.e. the Legendre polynomial of order n. n See [2, chapter III] and [9 sections 15.13 and 15.14] (ii)

The Poisson generating function This definition takes the form

I = ~ Pn(x)hn /(I - 2xh + h 2) n=° valid for all h with I~ < I. (iii)

(x ¢ (-I ,I))

See [2, section 3.6.2] and [9, section 15.1]

The Rodrigues formula This definition has the form

Pn(x) -

1 2 n n.v

dn (x 2 -

1) n

(x

e

(-1,1),

dx n

see [2, section 3.6.2] and [9, section 15.|I]

n

~ No);

84

(iv)

The Gram-Schmidt

ortho$onalization

The Legendre polynomials may be defined by the GramSchmidt process applied to the set {x n : x ~ (-l,l), n E N } o

in the integrable-square

inner-product

space L2(-1,I).

and Glazman [1, section 8], [2, sections

See Akhiezer

10.1 and IO. I0] and

[9, section ll,6]. This definition

leads to the fundamental or thogonal

property of the Legendre polynomials 1

f Pm(X)Pn(X)dx = (n + ~)6mn -l

(m,n E N o )

(1.2)

where ~ is the Kronecker delta function. mn Whichever definition is adopted it is important subsequently to prove that the set of Legendre polynomials

{P (.):n e N o } n

is complete, equivalently closed, in L2(-1,1); and [2, section I0.2].

see [I, sections 8 and 9]

Once the orthogomal property of the polynomials

is known the completeness may be obtained by classical means, as for the completeness

of the trigonometrical

polynomial approximation intervals;

functions, using the Weierstrass

theorem for continuous functions on compact

for details see [] , section 11] and [2, section

10.2].

Our interest in this paper is to discuss the definition and completeness

of the Legendre polynomials

in L2(-I,I) from the

viewpoint of the Titchmarsh - Weyl theory of singular differential operators.

This theory is concerned with the differential equation -

(py')'

+ qy

=

%wy on

where p, q and w are real-valued

(a,b)

(1.3)

coefficients

on the interval

(a,b),

and % is a complex-valued parameter. If in (1.3) the coefficient w is non-negative

on (a,b)

then this is a so-called risht definite problem and is studied in the integrable-square

space ~ ( a , b ) ,

i.e. the collection of those functions

f f or which

l

b w(x) If (x) 1 2dx < ~

(1.4)

a

If in (1.3) it should happen that, whether w is of one sign or not, both p and q are non-negative

then the problem is called

85

left-definite and is studied in the space of those functions f for which b f {p(x) If'(x)I2 + q(x) If(x)12}dx < ~

(1.5)

a

For both the right- and respectively the differential limit-circle,

left-definite

cases

equation is classified as either limit-point or

at an end-point a or b, according as to whether not all or

all solutions of (1.3) are in the spaces in the neighbourhood the classification

(1.4) respectively

of the endpoint in question.

of the differential equation

(1.5),

For reference to (1.3) in the right-

definite case see Naimark [4, section 18.1] and Titchmarsh [8, sections 2.1 and 2.19]; and in the left-definite [6, sections

case see Pleijel

I . 4 and 5]. For the purposes of the study of the Legendre equation

(1.1)

in this paper we write the equation in the form -

((I - x2)y'(x)) ' + ¼y(x) = % y(x)

(1.6)

(x ~(-l,l)),

so that in comparison with the standard form (3.3) we have a = -I, b = | 2 p(x) = I - x q(x) = ¼ w(x) = | (x c (-1,3)).

and

Thus we may study the Legendre equation as right-definite

in the

space L2(-I,I), and left-definite in the space I

I

{(I - x2llf'(x) I2 + ¼1f(x) 12}dx < -I

which, for convenience in this paper, we denote by H2(-;,I). The original study of Legendre's differential equation in the right-definite 4.3 to 4.7]. limit-circle

case in L2(-l,l) is due to Titchmarsh;

see [8, sections

It should he noted that the Legendre equation at both end-points ~I in the right-definite

(1.6) is

case; for

details see [8, section 4.5] and [I, II, appendix II, section 9, II]. Here it is emphasized that the analysis of Titchmarsh is essentially 'classical' with no reference to operator theoretic concepts.

This

work was followed by the studies of Naimark [4] and Glazman as given in [3, appendix II, section 9]; in particular Glazman characterized elements in the domain of the differential

the

operator giving rise to the

Legendre polynomials. The first study of Legendre's differential equation in the left-definite

case is due to Pleijel;

see [5] and [6] in which may be

86

found references to earlier results of Pleijel and the work of his school at Uppsala.

In particular we owe to Pleijel the observation

that the Legendre equation (1.6) is limit-point at both end-points ±I in the left-definite case; see [6, page 398]. Our purpose in this paper is to study the right- and left-definite problems for the Legendre differential equation with the methods of Titchmarsh [8] in mind.

We link the Titchmarsh method

with operator-theoretic results in the Hilbert function spaces L2(-|,]) and H2(-I,I). The paper is in six sections; after this introduction the second section considers essential properties of the Legendre differential equation; the third and fourth sections are given over to r#mgrks. a study of the right- and left-definite cases; five and six to c e r t a i n / Notations ACIo c

R - real field; C - complex field; L - Lebesgue integration;

local absolute continuity; if D is a set of elements f then

'(f £ D)' is to be read 'the set of all f • D'; NO - the set of all non-negative integers. 2.

The Legendre differential equation In this section we consider certain essential properties of Legendre's differential equation required for consideration of the right-definite and left-definite cases. As before we write the equation in the form ~I - x 2 ) y ' ( x ~ ' + ¼y(x) = % y(x)

(x • (-I,])).

(1.6)

For convenience let p(x)

=

J - x2

(x ~

(-l,I)).

(2.1)

The standard form of the Legendre equation is given in (1.I) but for the purpose of considering both right- and left-definite problems the form (1.6) is to be preferred.

A detailed discussion of the classical

properties of the Legendre equation (I.I) is to be found in Erd~lyi et al

[2, volume I, chapter III]. The account of the Legendre equation in Titchmarsh

[8, chapter IV] is based on the Liouville normal form of (1.6) i.e. ~

2

-

y"(x) - ¼ sec x.

y(x) = ~ y(x)

(x • (-½~,½n~ .

However, as is evident from the results in [8, section 4.5], this differential equation does not of itself enjoy the property of having

87

polynomials

solutions.

Here we adapt the analysis

apply similar methods to the equation Legendre polynomials

of Titchmarsh

to

(1.6) which does have the

directly as solutions.

We write s = /%,

i.e. s 2 = ~, and determine

the /(.)

function by requiring O -< arg /% Following

<~

when

(2.2)

O -< arg ~ < 2~

the analysis

in [8, section 4.5]

of (1.6) above may be determined by -I COS X cos st dt Y(X, ~) = [ J -I (cos t - x) I/2 -COS x

(x £ (-l,l)

two solutions

~ ¢ C) (2.3)

½~ + sin-lx)

1

Z(x, ~) =

where the positive trigonometrical

+

cos st dt sin-lx,J (cos t + x) I/2

(x e (-l,l)

% E C)

square root is implied.

functions

are determined

In (2.3) the inverse -I by requiring cos x to decrease

from ~ to O, and sin-lx to increase from -In to ½~ as, in both cases, x increases

from-l

to I.

To show that the Y and Z, as defined by (2.3), are solutions

of (1.6) we follow [8, section 4.5] and write the integrals

as contour integrals cos sz

(2.4)

d z

Y(x, ~) = ½ [

J G (cos z - x ) | / 2

f

cos sg

Z ( x , ~) = ~ J F (cos z + x) I/2 where now the sign convention

(2.5) d z

of [8, lemma 4.4] is taken to hold for

the square root terms in the complex-valued

integrands.

In the integral

for Y the integrand is made single-valued by cutting the z-plane from -1 -I - cos x to cos x; similarly for Z the cut is from - (½~ + sinlx) to ( ~

+ sin-lx).

Here the contours G and F can be

taken as circles with centre the origin of the z-plane and of radius r and p respectively,

where cos-lx

<

r

< ~ and ~

+ sin-lx

< p <~.

88

We then follow differentiating order

under

the method

the integral

of [8, section

sign and integrating

to show that Y and Z are solutions

for all x c (-1,I) From trigonometrical

(2.4)

and

(2.5)

on using [8, lemma 4.4];

and the properties

(x ~ (-1,1)

The initial values we find,

p(x) (Y(x,%)

Z'(x,%)

(2.6)

= -Z'(O,%).

....

(2.7)

that the solutions

except when Y(O,%)

when % = (n + ½)2 for n = 0,I,2,

~ ~ c).

= Z(O,%)

25/2~;/2 = - r(¼ + ½s) r(¼ - ½s)

independent

of the inverse

for all % E C,

21/2~I/2 = r(~ + Is) r(~ - is)

is independent

(1.6)

of Y and Z at 0 may be calculated

We note from these results are linearly

in

equation

it may be shown that

Y(- x,~) = Z(x,~)

Y'(O,%)

by parts,

of the differential

and all % ~ C .

functions

Y(O,%)

4.5] by

= 0 or Y'(O,%)

Y and Z

= O, i.e.

In fact the Wronskian

- Y'(x,%)

of x and has the value,

Z(x,%)~

(2.8)

on taking x = 0 and recalling

that p(O) = 1~

Y(O ~) z' (o,~) - Y'(O,~)

z(o,~) = 2z(o,~) z'(o,~) = 8~ cos ~s

on following

the analysis

in [8, section

The asymptotic neighbourhood some details

of the singular for the point

forms

Y(x,k) These results (2.4);

~ ~ /2

4.5].

of the solutions

end-points

| and there

For the solution

are similar

lim x ÷ I-

at the origin Y(x,X)

of the plane

(~

£

C).

(2.]0)

for Y given by

the contour

and we obtain

= [ co~ s z dz JG (cos z - l) I12 ~

for -l.

(x + 1-).

representation

as x + I- the cut in the z-plane within

single point

calculations

we give

for all % E C ,

~ (% - ¼)~//2

follow from the integral

Y and Z in the

±I can be calculated;

Y we find,

Y'(x,X)

(2.9)

G tends

to the

80

Notwithstanding

the square root term the integrand

and on G except for a simple pole at the origin.

is regular within Using the calculus

of

residues yields lim x+l-

Y(x,%)

= 7/2

(~ e C).

Y'(x,~)

= ~ [ JG ( c o s z -

Similarly lim x + l-

COS

SX

dz = (~-¼)~//2

1) 3 / 2

also valid for all % e C. For the solution Z we find In ('/(l-x)

Z(x,%) ~ 2/2. cos ~s.

(x + 1-) (2.11)

Z' (x,%) ~ 2/2. cos ~s.

(I - x) -I

(x + I-)

both valid for all % c C except for the set of points {(n + i)2 : n = O, I, 2 . . . .

}.

The proof of the first of these results

follows from the analysis given in [8, section 4.5]; the second result then follows from the constant value of the Wronskian

of Y and Z as

given by (2.8) and (2.9). The asymptotic above results

forms of Y and Z at -| follow from the

at I and the relationship

Y(-x,%)

= Z(x,%)

It follows from these asymptotic results independence

of the solutions Y and Z, except for certain exceptional

values of %, that the Legendre equation which is bounded on the internal set of points

(x e (-1,1)). and the linear

(1.6) has a non-trivial

(-1,1) if and only if % lies in the

{(n + ½)2 : n = O, I, 2 . . . . }.

any non-trivial

solution

For all other values of %

solution of the equation is unbounded

at 1 or -I or

at both points. From Y and Z we now form solutions O and ~ of the Legendre equation 0(0,~.)

=

1

(1.6) which satisfy the following 9'(0,~)

= 0

(p(O,~)

= 0

initial conditions

~'(O,~k)

=

|.

(2.12)

In fact we have O(x,~,) = Y ( x , % ) + Z ( x , k ) 2Y(O,~) '~

¢(x,%)

= Y ( x , ~ ) - Z(x,~ .) 2Y' (0,%)

(x E ( - 1 , 1 ) ) ( 2 . 1 3 )

for all ~ ~ C (but with care needed at the set {(n + ½)2 : n = 0, I,2,...}).

90 From the general properties of the differential equation

(1.6) we know that the Wronskian p(@~' - 0'~) is constant

on (-I,I) and so (1 - x 2 ) ( e ( x , ~ ) ~ ' ( x , k )

- @'(x,~)~(x,~)

= 1

(x e ( - l , l ) ,

k ~ C)

The asymptotic forms of 0 and ~ follow from the earlier For

results for Y and Z given by (2,10) and (2.11).

x

~

I- these are (2.14)

e(x,X) ~ r(¼ ÷ ~s)r(i ~ iS) 2~ 1/2 e'(x,~)

~

r'(¼÷

]

½s)rd

-

~s)

"

1 -

*(x,k) ~ - 2F(~ ÷ ½s)r('% - ~ ; )

(2.16)

in

I/2 ~'(x,l) ~ - 2F(% +' ~s')'F(%

(2.15)

x

½s) "

1 1 - x

(2.17)

There are similar results at -I.

Note again that not

all these results are valid at the set {(n + 3) 2 : n = 0,1,2 .... } of the k-plane. With these results established it is now possible to look at the classification in the right-definite,

of the Legendre differential equation

and left-definite

(1.6)

cases, as introduced in

section ; above. In the risht-definite

case; from the asymptotic results

above for 0 and ~ it is clear that although these solutions are unbounded, in general,

in the neighbourhood

of the end-points

±1, both @ and

are in L2(O,I) and L2(-I,0) for all % e C; thus the equation is limit-circle

at both ! and -I in this case.

established independently

(This result may also be

on considering the solutions of (1.6) in

the special ease i = ¼, i.e. 1 and I n ~ l

+ x)/(l-

these solutlons are in L (0,I) and L2(-I,O)

x9

(x e ( - 1 . 1 ~

: both

and so from a general result

[8, section 2.|9] the equation is limit-circle at 1 and -I.) In the left-definite

case; here the spaces concerned

are, see section ! above, H2(O,I) and H2(-l,O);

the asymptotic results

for the solutions 0 and ~ and their derivatives,

as given above, show

that for all non-real ~ e C neither @ nor ~ is in H2(0,I) or H2(-;,O) and this implies that the equation is limit-point case.

(In the left-definite

at 1 and -I in this

case greater care has to be taken in looking

9~

at the nature of solutions at real values of %, in order to determine the classification of the equation; we do not discuss this point here but see the works of Pleijel [5] and [6], and references therein.) 3.

The risht-definite case.

In this section we consider the right-definite

case for the Legendre equation on the interval (-1,1).

This section

is dependent in part on the original analysis of Titchmarsh in [7] and later in [8, chapter II, section 4.4]. Consider the Legendre differential equation in the form (1.6) on the interval (-1,1) i.e.

- ((l-x2)y'(x)) ' + ¼y(x) = %y(x)

Cx £ (-1,1)).

For solutions of this equation in the neighbourhood of the singular point l, the general theory in [8, chapter 2] gives the existence of the Titchmarsh-Weyl m-coefficient;

the analytic function m(') : C ÷ C

and determines a p~rticular solution ~ of the equation in the form ~(x,%) = e(x,%) + m(%)~(x,%)

x £ (-;,I)

% £ C R ,

where e and ~ are the solutions determined by (2.13).

(3.;)

As the differential

equation is limit-circle at both ±I, the m-coefficient is not unique and, in order to determine the differential operator associated with the Legendre polynomials, we have to make a suitable choice from the family of m-coefficients belonging to the end-point

I.

We do this by

following the limit process which determines m(') from the 1-functions; for the general theory see [8, section 2.;]. With the solutions 0 and ~ determined from (2.;2) and with % ~ C

R, let ~ X be a solution of (1.6), here X •

~x(X,%) = e(x,%) + l(%,X,B)~(x,%)

(0,1), given by

(x • (-1,1) .

The function i is chosen so that ~X satisfies the following boundary condition at X @X(E,%) cos B + P(X)~x'(X,%)sin B = 0 for some B • (-½~,½~] ; thus, for all ~ • C~R, I(%,X,B) = -

e(X,%) cos ~ + p(X)e'(X,%) sin ~(X,%) cos B Y P ( X ) ~ ' ( X , % ) sin

Now let X + ;- and choose ~ as a function of X so that 1 tends to a limit m(.), where m(-) : C + C and is regular on c~R.

In this the

92 Legendre

case we can see how this is done explicitly

the form l(l,x,B)

_ O(X~I){In((I-X)-;)

and then choose B(X)

(X e (O,l)

so that for some y e ( - ~ , ~ r ]

the limits of all the remaining (2.14 to 17) .

we have

(Note that we can evaluate

terms in (3.2) from the asymptotic

In our case, guided by [8, section 4.5], we

take y = 0 and so choose ~(X) = 0 ( X e I(I,X,0)

(O,l));

it then follows that

= lira {O(X'X)/~(X,I)} X -~ 1

lira { Y(X,I) + Z(X,I) =X + I . . . . 2V(0~,I)

=

(3.2)

tan B(X).{In((I-X) -l)}-I

tan B(X).{In((I-X) -I) }-| + tan y as X ÷ I-.

m(l) = lira X -~ 1

I in

}-| + p(X)Ol(X,l ) tan ~(X).{in((l-X)-l)} -I

~(X,X){In((I-X) LI)}'I '+ p(X),'(X,X)

formulae

by writing

=

.

2Y'(O,I) } Y(XTi7 r ZT(X,~)

y'(o,k) 4F(% + ½ s ) r ( % Is) Y(o,------7)= - r( ~, + ½s)r(~ - ~s)

where we recall,

see (2.2), that s = /A.

From has the properties,

(3.3) we then find that the resulting

solution 4(.,%)

see (3.1),

(i) 4(x,l) = Y(x,I)/Y(0,%) (ii) 4(',1)

(3.3)

e L2(-I,I)

(x ~ (-I,])

I e ckR)

(I e CNR)

• .. lira (xXl)x + ] 4(x,l) = Y(I,I)/Y(0,1) =

. ,x lira ivJ ÷ ] 4'(x,l) (v) 4(.,i)

?r-I/2r(%

+

~s)r(%

-

½s)

= (l - ¼)F(% + ~s)F(% - ½s)/(2~I)

and 4'(',I)

are unbounded

in the neighbourhood

of -I

(1 c CxR).

A similar

analysis holds for the singular

there is a solution X(.,I)

end-point

-l;

of (1.6) which has the form

X ( X , 1 ) = O(x,%) + n ( t ) ¢ ( x , t )

(x e ( - 1 , I )

t E C~R)

where n(1) = + Z'(O,I)/Z(0,1)

= - m(1)

(% e C',R)

such that X(.,I) has the properties (i) (ii)

X(X,1) = Z(x,1)/Z(0,%) X(',I) ~ L2(-I,I)

(iii) x -lim >-I

X(X,l)

(x E (-l,l)

(I £ CNR)

= ~-I /2F(% + ½ s ) r ( % -

½s)

I ~ CxR)

(3.4)

83

(iv) (v)

lim X'(X,%) = - ( l x + -1 X(',l)

and X' ( . , l )

- ¼)1`(% + ~ s ) r ( % -

½s)/(2~)

a r e unbounded i n t h e n e i g h b o u r h o o d of

1

(X • C\R). The Green's function for this choice of m(-) and n(.) is given by G(x, ¢ ; X) ffi =-

(-1

_ ~ ! x , ~) ~ (~, ~) {p W(~,~)}(%) X(~)

(-1

~ (x,~)

<x


<x


< 1)

<1)

{p W(~,X) } (%) where, from the form of ~ and X, {p w(~,×)}(X)

= p(x)

(~(x,X)×'(x,k)

= n(~)

- m(~)

- ~'(x,~)×(x,l)

(~ •

(x • (-I,I)

C~R).

From the general theory of differential equations of the form (1.6) with two singular end-polnts, that the eigenvalues

it is known, see F8, section 2.18],

of the equation, with the particular

choice of

and X above, are given by the zeros and poles of n(') - m(-).

In this

case, from (3.4), n(%) - m(X) = - 2m(%) = 8 I'(% + ½ s ) F ( %

r(~ + Is)r(¼

-

½s)

(3.5)

- ½s)

and from this a calculation shows (recall s = ~;

see (2.2)) that the

eigenvalues are given by %n = (n + ½)2 Anticipating

(n • No).

the definition of the operator T below,

(3.6) let Po(T) denote

the set of eigenvalues given by (3.6). Following the analysis in [8, section 4.53 it may be shown that the eigenfunetlons

{~n(. ) : n ~ N o } corresponding

to the eigenvalues

P~(T) are given by ~n(X) = (n + ~)I/2en(x )

(x ~ (-I,I)

n • No)

where {P (-) ; n ~ N } are the Legendre polynomials, n o and 10.6] and [9, section 15.1].

(3.7) see [2, sections 3.6.2 ~

Here we leave the classical study of the differential equation of Legendre and turn to the study of the associated differential

94

operator in this, the right-definite, case. Let the symmetric differential expression M be defined by M[f](x) = -((I - x2)f'(x)) ' + ¼f(x)

(x ~ (-1,1))

for any f : (-I,I) -~ C with both f and f' E ACloc(-1,1). with these properties,

For any f and g

integration by parts shows that

B I J

{~ M[f] - fM[g]} = [fg](')l~

for all compact intervals

(3.8)

[~,B] c (-l,l), where

[fg](x) = p(x)(f(x)g'(x)

- f'(x)g(x))

(x c (-l,l)).

(3.9)

We define a differential operator T, later shown to be self-adjoint in the Hilbert function space L2(-l,l),

as follows:

firstly define the linear manifold A c L2(-I,I) by f ~ A if (i) f : ( - l , l )

-~ C and f ~ L2(-I,I)

(ii) f and f' c ACloc(-l,l) (iii) M[f] ~ L2(-I,I)

;

secondly define the domain D(T) c A by f e D(T) if (iv) f ¢ A and for some I c C\P ~ (T) (a)

lim x-+

If(x) ~ (x,l)] = 0 1

(b)

lim

If(x) X(x,I)] = O ;

x÷-I

thirdly define the operator T by Tf = M[f]

(f e D(T)).

Note that the limits in (iv)(a) and (b) exist and are finite in view of the Green's formula (3.8), and since f, ~ and X all satisfy conditions (i) and (iii) for A.

The genesis of (iv)(a) and (b) as the correct general

form of boundary condition at a singular end-point may be seen in [8, section 2.7], and receives its full development in Naimark [4, section 18]. conditions determined,

Note also that ~ and X themselves satisfy the boundary

(iv)(a) and (b) respectively, and that once ~ and X

are

as above, then D(T) is independent of the choice of I in (iv)(a)

and (b); these results follow essentially from the important result given in [8, lemma 2.3].

95

Similar analysis also shows that for all f, g E D(T) (but not for all f, g e A ) lim If g](x) = O

lim If g](x) = 0

x÷l

x'+-I

(3.|0)

and then from Green's formula (3.8) it follows that (Tf,g) = (f,Tg)

(f,g E D(T))

(3.11)

where (-,.) is the inner-produc~ in L2(-I,|). If Co(-l,l) represents all infinitely differentiable functions with compact support in (-],l) then clearly Co(-l,;) c D(T); hence D(T) is dense in L2(-|,I).

From this result and (3.11) it follows

that T is symmetric in L2(-],I). If also we define ~ : (-1,1) x C\R x L2(-I,]) ÷ C by, see [8, section 2.6] (x,l;f) =

I'

G(x,~;l)f(~)d~

(3.|la)

-I then ~ , % ; f )

e D(T) and MIni = % ~ + f on (-I,I);

(3.11b)

from this result it may be shown that (T i il)D(T) = L2(-I,;). Thus T is a self-adjoint (unbounded) differential operator in the space L2(-l,l). We now give a number of different but equivalent descriptions of the elements of the domain D(T), and, in particular, give a number of alternative forms of the boundary conditions (iv)(a) and (b) which reduce A to D(T).

For an alternative account of these boundary conditions see

[|, appendix II, section 9, example II].

finite.

We first show that if f £ D(T) then lim f exist and are both ±l We have from [8, lem~na 2.9], but with our sign convention, for

f e D(T) and % ~ C\Po(T) f(x) = ~ (x,l;M[f]) - ~ $ (x,l;f)

(x £ (-I,1));

it is essential for this result to hold that f satisfies the boundary condition (iv)(a) and (b). Next we prove the general result that if g ~ L2(-1,I) and E C\Po(T) then lim ~ (.,~;g) both exist and are finite; for ±I

(3.12)

96

(n(X) - re(X)) ~ (x,X;g) = ~(x,%)

fx

X(~,X)f(~)d~ + X(X,%)

11

-I

@(~,X)f(~)d~

x

and

IX (x,X) x*(~'X)f

I< " ' h - x '

• d~.xlg(~)l 2d

= O(ln ((l-x)-l). = o(1)

(x÷l)

on using the results in (2.10 and ll). lim Ix ~2 x-~l ~(x,X) _iX(~,X)g(~)d~ =Y-~-,X) which is finite.

(l-x) 1/2) Also

I~IX(~,X)g

(~)d~

There is a similar result if we take the limit at -I.

Thus it now follows from (3.12) that if f £ D(T) then lira f both exist and are finite; if f(±l) is defined by these limits then ±I f c C[-I,I] for all f ~ D(T). Now suppose that f e A and lim ± f exist and are both finite; then we state that lim + pf' = O.

To see this we can suppose, without loss

of generality, that f is real-valued on (-I,I); we have M[f] E L2(O,I), i.e. M[f] ~ L(O,I) and (pf')' ~ L(O,I); hence for some real k lim

1 Pf' = (pf')(O) +

I 1 (pf')'

(3.13)

= k;

O now if k ~ 0 then we can take k > 0 and obtain f(x) ~ f(O) + ~k I~p-I for x close to I, i.e. lim f = = which is a contradiction; hence k = 0 1 and lim lim 1 pf' = O; similarly -I pf' = O. lim Now suppose f e ~ and ±I Pf' = O; then If(x) l ~ If(o) l +

If'l = If(o) l +

Ipf'l

0 If(o) l + K in ( ( I - x) -I)

(x ( [ o , I ) )

for some positive real number K; this last result, and again using lim 1 pf' = O, together with the known properties of ~, proves that lira if ~ (. ,l)] = O for any % ( C\Po(T); similarly lim -I [f X (-,l)] = O;

I

hence f E D(T).

97

If now f E D(T) then, from the results overleaf, lira + pf,~ = O; hence from {plf'

+ ¼Ill 2} = pf,.?

+

.f

(3.14)

-X

it follows that pl/2f, e L 2 (-I,I).

Conversely let f E A and p |/2f, ¢ L2 t_i,I~; ~

then M[f] e L(o,I) and as in (3.13) lim pf, = k (say); if k ~ 0 then + for x close to I it follows that, for some k

> 0, o

iXp]f,[2

zk O

ix. P

and pl/2f, 4 L2(o,I); this is a contradiction and so k = O; hence, as above, f ~ D(T). Taking all these results together it follows that the domain D(T) can be described in any one of the follow%ng five equivalent forms f ~ D(T) if f £ A and either (~) lim 1 [f~] = lim -I [fx] = 0 or

(8)

lim +-I f exist and are finite

or

(y)

lim _+ pf' = 0

or

(8) pl/2f, e L2(-I,I)

or

(Z)

lim [fl] = lim [fl] = 0 ; -I

where in (Z) the notation 1 is used to represent the function taking the value 1 on (-1,I). We can prove a little more; if f c D(T) then from (3.14) above we see that {plf'

+ ¼Ill 2} =

-I

M[f]'f

(f c D(T))

-1

so that M satisfies the so-called Dirichlet formula on D(T) hut not, as may be readily shown, on the maximal linear manifold A.

From the Dirichlet

formula we see that the self-adjoint operator T satisfies (rf,f) > ¼(f,f)

(f E D(T))

with equality if and only if f is constant over (-I,I).

(3.15) This is a special

98

case of a general inequality for self-adjoint

operators which are

bounded below; in fact the first eigenvalue %o of T is ¼. We comment on the spectrum of the self-adjoint

operator T;

this consists of the set Po(T) = {%n = (n + ½)2; n E No}, see (3.6), each point of which is a simple eigenvalue with eigenfunetions Legendre polynomials and, in particular,

the

{Pn(') : n • No}; clearly Pn(') • D(T) (n • No) satisfies the boundary conditions

(iv)(a) and (b).

For any real ~ g Po(T) it is clear from the properties

of the

solutions ~(-,~) and X(',~), given earlier in this section, that no solution of Legendre's differential equation

(1.6), with % = ~, can be

found which satisfies the boundary conditions at both singular end-points +-l. Indeed at all points D e R\P•(T) it may be shown that (T - ~ I)D(T) = L2(-],l),

on using the result

(3. ;2); this shows that

is in the resolvent set of T; see if, section 43]. The general spectral theory of self-adjoint see [ 1, chapter VII now yields the completeness polynomials

in L2(-l,l),

as the set of eigenvectors

of a self-adjoint

operator T in L2(-I,I) with a simple, discrete spectrum. eigenvectors

operators,

of the set of Legendre

The normalized

of T, say {~n : n • N o } given by ~n = (n + ½)I/2pn(n ¢ No),

then give an orthonomal basis in L2(-l,l). One additional comment;

if we define the

co

operator S : Co(-l,;) ÷ L2(-l,l) by Sf = M[f]

(f • Co(-l,|))

then S is symmetric in L2(-l,l)

and satisfies the inequality

co

(Sf,f)

-> ~(f.f)

(f E Co(-l,l)),

see (3.15).

The general theory of semi-

bounded sy~netric operators then applies, see [I, section 85], and the operator T then appears as the uniquely determined Friedrichs extension of S; this relates to the form (6) of the equivalent boundary conditions, i.e. a finite Dirichlet condition. 4.

The left-definite

case.

We again consider the Legendre differential

equation in the form (1.6) M[y](x) = -((I - x2)y'(x)) ' + ~y(x) = %y(x)

(x c (-l,l)).

(1.6)

As in section I above we define H2(-I,|) = H 2 as the Hilbert function space H2(-l,l) = {f : (-l,|) -> C : f • ACloc(-l,l),

f • L2(-|,|)

and pl/2f, • L2(-I,I)}

99

with inner-product =

(f'g)H

-I

and norm llfllH;

{pf'g' + ¼fg}

(4.1)

here p(x) = 1 - x 2 (x c (-l,l)).

We noted in section 2 above that the differential expression M is limit-point in H2(-I,I)

at both the singular end-points ±I.

To obtain a self-adjoint

operator S, say, in H2(-I,I),

as generated by M and playing the same r~le as the operator T in section 3, we follow the method used in Everitt [3], using also, in part, the work of Atkinson,

Everitt and Ong in [ IO], There is a theory of the m-coefficient

for left-definite

problems which reflects some, but not all, of the properties

of the

Titchmarsh-Weyl m-coefficient

We again use

in the right-definite

theory.

the solutions O and ~ of (1.6) introduced in (2.13). Since the differential expression M is limit-point in H2(O,I) at the end-point H2(O,I) for any ~ e C\R.

I, neither solution @(',~) or ~(.,l) is in However there exists a unique coefficient m(')

(we use the tilda notation to distinguish the left-definite analytic (regular, holomorphic) = @ + m ~ ~ H2(o,I). independent

in C\R and such that the solution

Now for )~ c C\R there is only one linearly

solution of the equation

the asymptotic results

(1.6) which lies in H2(o,I);

from

(2.10 and 11), for the solutions Y(',%) and Z(',%),

it is clear that this solution in H2(O,]) must be Y(',%). ~(.,~.) = 0(-,%)

case) which is

+ m(%)~(.,%)

with k(-) to be determined.

= k(~)Y(-,%)

on [o,])

If we differentiate

both sides at O we find that 1 = k(l)Y(o,%) m(l)

Thus

this result and evaluate

and ~(%) = k(l)Y'(o,l),

i.e.

= Y'(o,I)/Y(o,I).

Similarly at the end-point -I the solution in H2(-I,o)

(4.2)

is

Z(.,%) and, writing ~ = @ + n q~,

~(~)

=

z'(o,~)/z(o,~)

=

- ~(~).

(4.3)

Note that m = m of (3.3), and similarly n = n, of the right-definite

case but note that m is unique whilst we had to select m in

section 3 as a consequence of the limit-circle classification.

100

These results give ~(.,%) = Y(.,%)/Y(o,%)

(= ~(.,%) of section 3)

X(',%) = Z(.,%)/Z(o,%)

(= X(.,%) of section 3).

(4.4)

As in section 3 m(.) and n(.) are meromorphic on C with simple poles only at the points {(n + ~)2 : n e N }. In particular these o functions are regular at O and we use this fact to construct the resolvent function $ as in section 3 above; in fact we can identify $ with ~ of (3.11a)

~(x,~;f) = ~(x,~;f)

(4.5)

but now defined for x ¢ (-|,l), % ~ C\{(n + i2) 2 :n e N o } and all f ~ H2(-l,l).

It is convenient to define ~ : (-;,I) × H2(-l,|) -> C by ~(x;f) = $(x,o;f);

(4.6)

it follows that, see (3.11b), M[~(x;f)] = f(x)

(x c (-1,1)),

(4.7)

Now define a linear operator A on H2(-l,l) by (Af)(x) = ~(x;f) for all f ¢ H2(-l,l).

(x £ (-I,1))

(4.8)

We shall show A is a bounded, symmetric operator

on H2(-l,l) into H2(-I,l);

also that A has an inverse A -I.

For this purpose we require Len~na

(i) (ii)

Proof

(i)

~(.;f) ~ C[-],|] lira +l p ~, (';f)g = O

(f £ H 2) (f,g E H 2)

This follows from the definition

(4.5) and (4.6) of ~ and the

asymptotic properties of the solutions Y and Z of Legendre's equation. (ii) We note that if g E H 2 then Ig(x) l = Ig(°) + i.e.

I-< Ig(°)l + [Jo

g(x) = O({In((l - x)-l)} I/2)

Hence, from (4.6),

op]g,12 i/2 (x -~ I).

(4.5) and the asymptotic properties of solution of the

differential equation,

Ii[~[2}i/2 )

p(x) ~'(x; f)g(x) = O(p(x) Ig(x)|) + O(Ig(x) l{

= O~I - x){In((l - x)-l)} I/2)

÷

= o (l)

-

(x~

l).

101

Similarly at -1.

This completes the proof of the lemma.

We now show t h a t t h e o p e r a t o r A i s bounded on H2; i n f a c t

IIAflIH = ll~(';f)][H -<~llfl[H We have, f o r a l l x e ( o , l ) fXxf (~)

(f c H2).

(4.9)

(~;f)d ~ = Ix M[~(~;f)] ~ (~;f)d

--

--X

= p~'(-;f)~(-;f) xx + Ix {p(~)l~' (~;f112+¼1~(~;f) 12}d~. --

--X

Now l e t x -~ 1 to g i v e , on u s i n g t h e lemma above, ,[~(.;f),[2 = If I f(~)~(~;f)d~

<{I I 'f(~)'2d~Ij l'(~;f)12d~I I/2

-I

-I <

I

~o

since f £ L2(-I,I) (recall H 2 c L 2) and ~ (.;f) ~ C[-I.I], i.e. ~ (';f) c H2(f ~ H2). Also

if:

4f:

Hence I IAfl[H < 411fIIN

(f c H 2)

and from this (4.9) follows as required. I n t e g r a t i o n by p a r t s g i v e s (Af'g)H

=

fl -I

{p~flg, + ¼~g} =

= I I M[~]g -I =

f' f'

p~,g I -1

+

fl -1

M[~]g

(on using the lemma)

f g (f.g ~ H 2)

(on using (4.7))

-1

=

f M[~(-;g)] = (f,Ag)

-1 on r e v e r s i n g the argument; thus A i s symmetric.

follows that A is self-adjoint.

Since A i s bounded i t

102

Suppose now Af = O, i.e. ~(x;f) = 0

(x ~ (-!,l));

then from (4.7) 0 = M[~(x;f)] i.e. f = 0 in H 2.

= f(x)

(x E (-1,I)),

Thus A -! exists.

Now define an operator S : D(S) ¢ H 2 ÷ H 2 by D(S) = {Af : f e H 2} and -! S f = A

f

(f c D(S))

A standard

theorem in Hilbert space theory,

to theorem

1], implies that S is self-adjoint

H 2.

In fact S must be unbounded

(4.10)

see [1, section 4l, corollary (bounded or unbounded)

since, from the properties

in

of ~, it follows

that if f e D(S) then f' e ACloc(-l,] ) and so D(S) is strictly contained in H 2, even though the closure D(S) = H2; it may be seen that if S is bounded then D(S) = H 2 and this gives a contradiction. We conclude that S is an unbounded in H2(-1,1);

self-adjoint

we now show that S has a simple discrete

operator

spectrum with

O(S) = P~(S) = {(n + ½)2 : n ¢ No}; we note that this spectrum is identical with Po(T)

of the operator T introduced

in (3.6).

Suppose that % is an e i g e n ~ l u e eigenvector

of S, i.e. for some

f ~ O, Sf = Xf; then X ~ O, since S -! = A exists,

Af = A-!f; hence ~(.;f) Thus f is a non-trival that f ~ H2(-I,I);

= X-!f or, on using

and

(4.7), f = M[~(-;f)]

= ~-|M[f].

solution of the Legendre equation with the property

from the properties

of the Legendre equation given in

section 2 this can happen only when the solutions Y and Z are linearly dependent, dependent {Pn(')

i.e. when % c Po(S) on the corresponding

: n e No}.

with corresponding

and then the eigenvector

f is linearly

Legendre polynomial from the set

Conversely every point in the set Po(S) is an eigenvalue eigenvector

in {Pn : n £ N o }.

For all real numbers D @ Po(S) we can show that the range of S-ZI

is H2(-I,I),

from the properties

i.e. {(S - B I)f : f ¢ D(S)} = H2(-l,l); of the resolvent

Hence all these points are in the resolvent

set of S, and so O(S) = Po(S).

The spectral theorem for self-adjoint space,

this follows

function ~ as defined in (4.5).

operators

in Hilbert

see [I, chapter VII, now implies that the Legendre polynomials

form

103

a complete,

orthogonal set in H2(-],I).

The derived complete orthonormal

set in this space is then, say, {~n : n ¢ No} where ~n = (n + ½)-;/2P n (n E No). From this last result it may be shown that the Legendre polynomials

are dense in those vectors of L2(-I,;) which are also in

H2(II,I); however this set is d~nse itself in L2(-I,1) indirectly, yet another p r o o f ~ h e

completeness

and so we obtain,

of the Legendre polynomials

in L2(-I, I). 5.

Remarks on the operators T and S.

It is of some interest to compare the

operators T and S as defined in sections 3 and 4 respectively. T is an unbounded self-adjoint, differential

operator in

L2(-I,I) with a simple, discrete spectrum {(n + ~)2 : n E N o } and corresponding eigenvectors

{Pn : n e No}.

S is an unbounded

self-adjoint

operator in H2(-I,I) with

the same simple, discrete spectrum and eigenvectors; call S a differential

we may hesitate to

operator for the reasons given below.

The operator T, and its domain D(T), is defined directly in terms of the Legendre differential expression M; also we are able to give alternative

and simplified descriptions

of the elements of D(T), as

detailed in section 3. The situation for the operator S is different; we defined S as the inverse A -I of a bounded, symmetric operator in H2(-I,I).

Whilst

we can say something about the elements of the domain D(S), in particular as sets of complex-valued

functions on [-1,1] we have D(S) c D(T)

(see the alternative definition T below), it does not seem possible to characterize the operator S directly in terms of the Legendre differential expression M.

The definition

of S in section 4, i.e. S = A -I, depends

upon a general theorem in Hilbert space theory (see the result quoted in section 4 from [I]) which provides for the existence of S but, due to the generality of the theorem, general.

cannot give a constructive definition in

Thus S appears as a differential

operator only in an indirect

sense in comparison with the operator T. It is of interest to note that we could have defined the operator T in the same way as S is defined in section 4.

With the resolvent

function # defined by (3.|]a) let the operator B be defined on L2(-I,|) by (Bf)(x) = ¢(x,o;f)

(x ~ (-l,l))

104

for all f E L 2 (-I,I); compare with (4.8).

With arguments entirely similar

to those used in section 4 we may prove that B is a symmetric, bounded operator on L2(-I,I) into L2(-I,;), and that the inverse B -I exists; also that the operator T, as already defined in section 3, satisfies -1

T=B

Note that in the sense H2(-I,I) c L2(-I,I), we have A = B on H2(-;,]).

However, the inverse A -I has to be determined in

H2(-I,I) and B-; in the space L2(-I,I) so that there is no identification of S with T, even on D(S). It happens then that in the right-definite case, we are -I = T, a

able to give an explicit characterization of the inverse B

characterization which is not possible in the left-definite case. These remarks should also be read in the context of the general theory developed by Pleijel for both the right- and left- definite cases; see, in particular, the illuminating results given in the concluding remarks in [6, section 8]. 6.

H alf-ranse Lesendre series.

In the theory of Fourier series the two

collections of functions {sin n x : x ~ [o,~], n E N o } and {cos n x : x E [o,~], n e N o } give separate orthogonal sets in L2(o,~), both of which are complete in this space; these are termed half-range Fourier series.

The two collections conbined and extended to the interval

[-~,~] give an orthogonal set which is complete in L2(-~,~). The same phenomena is seen in Legendre series.

The two

collections of polynomials {P2n+l (') : n e N o } and {P2n (.) : n e N o } give two separate, complete orthogonal sets in L2(O,I). The associated self-adjoint, differential operators are given respectively, say, by T o and T 1 where D(T o) = {f:[o,|) ÷ C : f and fl E ACloc[O,l) f and M[f] E L2(O,I) f(o) = 0 and lim [f~] = O} I

and

Tof = M[f]

(f e D(To))

(in the notation of section 3); and D(TI) and T; are given by replacing the boundary condition at the regular end-point O by f4(o) = O.

It may

105

be seen that the spectrum (simple and discrete)

of T

o

is given by

{(2n + ½)2 : n E No}; similarly for T! but with spectrum 3 2 {(2 n + ~-) : n £ No}. There are similar half-range Legendre expansions in the left-definite

case.

106

References

I.

N.I. Akhiezer and I.M. Glazman.

Theory of linear operators in

Hilbert space: I and II (Ungar, New York, 1961; translated from the Russian edition). 2.

A. Erd~lyi et al.

3.

W.N. Everitt.

Higher transcendental functions: I and II

(McGraw-Hill, New York, 1953). Some remarks on a differential expression with an

indefinite weight function. differential equations.

S~ectral theory and asymptotics of

13-28.

Mathematical Studies 13,

E.M. de Jager (ed), (North-Holland, Amsterdam, 4.

M.A. Naimark.

1974).

Linear differential operators: II (Ungar, New York, 1968;

translated from the Russian). 5.

A. Pleijel.

On Legendre's polynomials New developments in differential

equations 175-180.

Mathematical Studies 21, W. Eckha~s (ed),

(North-Holland, Amsterdam, 6.

~. Pleijel.

1976).

On the boundary condition for the Legendre polynomials.

Annales Academiae Scientiarum Fennicae; Series A.I. Mathematica 2, 1976, 397-408. 7.

E.C. Titchmarsh. (Oxford)

8.

On expensi~s

E.C. Titchmarsh.

Quart. J. Math.

Eigenfunction expansions associated with second-order

differential equations; 9.

in eigenfunctions II.

ii, 1940, 129-140.

I (Clarendon Press, Oxford, 1962).

E.T. Whittaker and G.N. Watson.

A course of modern analysis (University

Press, Cambridge, 1927). I0.

F.V. Atkinson, W.N. Everitt and K.S. Ong.

On the m-coefficient of

Weyl for a differential equation with an indefinite weight function. Proc. London Math. Soc. (3) 29 (1974), 368-384.

SINGULARITIES AT

THE

OF

VERTICES

3-DIMENSIONAL .AND

AT

POTENTIAL

THE

Gaetano

EDGES

FUNCTIONS

OF

THE

BOUNDARY

Fichera

Icl raw,tory of Arthur Erd@lyi In which

this

have

paper

been

Let

shall

proved

E

be

in

the

space

X ~

of

the

terior

to

A

. We

smooth

enough

so

potential

we

extend [ i ]

boundary

point

P

function

the

a

~

as

Let a

general

particular

be

case

the

conducting

in

of

E

u

of

the

set

of

the

: R

and

of

u E

E

the a

A

surface

function

continuous

surfaces

domain

E

potential

is

more

bounded

.:: (×,v,z).

the

u

in

of

consider that

to

cube. cartesian points

ex-

suppose

does

and

results

r_

exist.

satisfies

The the

con-

ditions ~z~

:-

+

+

x a = t

Let

5*

be

a

tive

0

domain

If n

=

We

one

Let

(1)

arcs

of

conditions

of

assume

S

we

be

5 ~

L~ consider

be

coincide

H R

the

unit

sphere

with

IP-0

5

If

[k

I = t . Let

is

a

posi-

cone

~ S

P-O IP-OI

unit

: O.

the

not

,

9 5 ~

bali

the

-~ R o

is

a

IP-Of

},

P~O

for

~- }%

connected

circles,

Let

We

o- ~ IP-OI

~(P)

5

K

in

~

we

suppose

that

~-Ro •

great

point. )

R

the

that of

X 3 and

by

~

~R

0 < R

lira

does

denote

by

for

end

side

which

~z

,

of

IP-OI

denote

(h = ~,...,q the

5

o

shall

£

origin

shall

;

of

in

~h

IP

Ft R

collection only

we

we

B~

the

of

number, HR -

on

be

on

= 0

~y~

two

oJ I , .... '~I

size

of

denote

the by

~ P-O

of

them

be

the

angle

of

I h

the

Laplace-Beltrami

the

eigenvalue

vertices

: 0 ~r ~ 0

in of

a

finite

eventually, of

~S"

~h X ~

9S

~

and

let

, measured defined

bythe

) .

operator

),~

by

meeting,

on

problem f

formed

segment

= tP-Ot(~h-O

the

L~

set

in on

S

95

~

5

In

the

domain

5"

108

The

following

i) of

The

results

eigenvalues

of

functions

which

have

and

form

an

ascending

For

every

tive

2) geometrical

ing

the

be

the

problem

3)

The

eigenfunctions

4)

The

smallest

the

be

always the

to

of

is (i)

(or

the

the

~,(5

real

is

equal

continuous

in

tothe

the

negative) ~ H R

of

in H~

5~

correspond5"

defined

by

conditions P-0 ,

-

~ OS

P~o

for

-

IP-Ol

Let towards

~ E

be °

the

We

unit

shall

normal

field

following

grad u

edges

a regular

the

density

consider ~CQ)

The

in

theorem

:

-

_L

of the e l e c t r i c

I.

For

~

the

of

electric

(2)

grad

the b e h a v i o u r

density

~

near

of

charge

in

the

electric

the v e r t e x

and

the

P

(

H~

-

(i~

u .....

~ i~)

(o < R < R~

) we

have

I

,,,~ ~P h,t

where a<

:

~:

,

P-O

IP-OI

o~

,

=

-

2

-

J

IP-OI r_,

~:h(~)

The

H R pointing

~ 1 4 R-

of

THEOREM

(~)

~

~___! ~ .

describes

and

point of

symbol

:

Ira- ~0 k)

:

I~-

:

Im

- u~ h I

~

is

~h i log

the

S

Landau

> ~

if

~h

if

k~h : ~

if

~h

capital

0

4

,

~)

posi-

finite.

1 and

always

boundary

space

+

are

multiplicity

positive

part

in

multiplicity

problem

has

known:

La(S ~) ] are

tends

multiplicity

eigenvalue

is

~4 H R

of

to

algebraic

this

be

[considered

which

the

and

to

belonging

sequence

eigenvalue

eigenfunction

assumed

a gradient

multiplicity

Let

can

109

The the

asymptotic

symbol

G9

bitrarily

and

formula

in

it

fixed

cannot

point

Q

is

a point

~

~

FI~

( I h u

PROOF.

The

L'

of

For

=

0

0 < ~ ~

(5')

of

We

o

(z)

if

a suitable

i ~

) we

a) ¢ 5"

series

we

must

x

the

sense

tends

( ~ = ~"

that

to

neighborhood

0 m h ~I,+~

an

ar-

of

0

.

~ ; ~9+~ = ~

)

have

~:~(~;~:,,~,~'~)

,

in

]

'-~,~ ~ ~ : , ) '

have

be

understood

in

the

sense

have '~'U -:--

~2U

Multiplying

in

=

[~:

the

improved

by

plane

( I~

R o

be

replaced

the

[h+,)

convergence

cannot

I u ..- Iq

of

3(0)

(3)

be

of

If ~

(2)

both

+

2

sides

~U

+

by

~rk

(4)

~ -

h

and

U

~

o

.

integrating

over

5 ~

we

under

the

integral

needed

in

order

The

same

have

Zrk L,(oUdo)

where

"Uk(~ ) = f g~, U(tc, a') V k (CO)dCO .

(5) The in

the

are

differentiations

right

hand

permitted

apply

the

side

because Green

with of of

(5), lemma

respect which

I

of

to are

[ ~ ]

~

formula

Hence _a = _ U.

and

~- __ Z dU

_ _~_

U.

therefore

UK( ~): whe re

c

K

and

c -k

are

c k?

constan~

2

(~)

K

The

symbol

~

is

+ %(: and

k

the

Landau

small

o

=o

lemma

to

get

permits

sign (4)

,

to

110 Since

and

U(q~)

is

continuous

as

must

have

formal]y

(6)

U(qco)

grad

:

i

c

k:~

grad~

k

Since ~or every C~ ~notion we

, we

~ -) 0

zrk(co )

~ we have Igrad #I~ : ~ ,

get ~

(7)

I grad Mapping

graphic

u~(~)l

? the

~

o(~.I

: I% ~

domain

projection

and

(8)

5~ o n t o

applying

l~k(~)l

IZ

-i

20(

~

+ ~

~ I

I grad

~

a

bounded

planar

lemma

II

[I ]

of

2

domain (~) , we

by a s t e r e o have

~_ c 1~ , q

(9) where

grad C

"GK I z

is

a

C

constant

I kk~ ~ I hilt ~t (co) ~ (2 ~2 IS ~h(c°)

sup

only

Set

depending

on

~.

r,

~h(~ )

: l~-cohl

log I~ - coal 2

if From

(8),

(9)

we

Ph ( ~

"

deduce q

(io) where

Ivk(~)l -~ F ~ k ~ ~h (~) ~ Let

is ~

If w e

a constant be

only

a constant

depending such

q

q

h:~

h=¢ £

set

6

= ( rues

~)2

that

on for

~ co ~ 5 ~

then

i

(12)

(3)

{, I g r a % ~

(f

.............. ~

See,in

particular,

tU(R

~)i~d~

(1.7),(1.8),(1.9),(i.

)

_z R~ ~

I0),(I.II)

of

[I].

111

From

(7), (9), (i0),

(ii),

z It p.442)

is

that

Therefore, any

known

the

deduce

C~

asymptotic

k ~ ~k ~

there

we

--

from

lim

(12)

exist

~0.

two

k

It

theory

follows

positive

Jim

have

eigenvalues -± lim k ~

that

constants

po and

I

3

~

We

of

p~ such

in

the

is

justified.

the

lira

total

interval

We

logk

~-~ ( {

exp

in

~ I

4_

I grad

If

the

convergence

[0, R ].

have

that

for

P° ~.

+ Po l o g

i

J~ proves

(see [2] ± -- q~ ~ 0 .

0 4 ~ ~ R

for

k i k s {-~q )P°J~ :

which

z

Hence

the

H R - (i u...u 6 ~

M

is

the

series

development

E

~z~ (d

grad~

of

|~ ) , assuming

II rh(~) h.:t

constant

of

) = O.

Ro

~ ~ ~

by

such

that

(6)

,

4

¢~'+C

given

- -

k:~

such

that

for

0 ~ ~ ~

~R ~

~_ M we

have

in

H a- (I~u

,,. u i~ ) ~.~

I grad

ut I

<

M

--

6 ~

11 ~

-

i.e.

the

estimate

Suppose

We we the

h:~

have

can

that

there

exclude

c~ < 0

exists

(13)

Hence

~j~

a

~

the

since

estimate

exists

> 0

(~°) h

'

(2).

[grad -]p :o[~"~.-li

(13)

,

~

when O. such

P~: 0

>o

and

that

(13)

P?E

. In U<

0

implies holds

i I u-.

• u

(forP-'P~)

~(~)]

case

P~O If

a point

fact in

in

]q

, the

~ ~ . On

I~ ~-01 <

development the

other

(6) hand

¢~ = 0 . when

P-,

P~

,

then

~

there

112

a-~ q

(lap

l[grad~I~l

,_ 8~

]7 r (co) h:

for

HR-(I~u...~I~)

_P~

We may s u p p o s e

h

{

IP-@l

,

o < c~£ _~ ~..

8~,

<

For such

_P we h a v e

oo

Igrad L~I >. i c , ] l g r a d ~ ~ ( o o ) l (14)

[c,l la~

~ - -

(~5)

M~

~

be

~

Hence

%(c~)I

2.

+ ~

lgrad

a positive

~

for

~"

constant

such

]

[

~Y (co

* ~ )

r

< ~.

that

I

±

,2

20( - .2

(~),,r-:z :x,,(~g

_ o__~?°,=h Let

e( -~

~ ICi~l I g r a d ~ ( a ) )

"

for

0 ~ ~ e

)'

+

(13)',(14),(15)

]] ~(~) _> Ic, l~

]grad

I,.:~ h

[lC,]

grad

oa~r~(a))[- M t 6

~(co)

(~

g "~i (co)

h.:~

V1 ~(2a)

I-

11 =~,(~)]. h:l

The re f ore

grad Since lemma

If we

I.tr4(co)l z Ic,I

P~ ~ 0

III

of Q

]

and

~(a))>O

in

S ~, the

q

.

last

estimate

contradicts

[if. = ~w

is

a point

of

the

plane

0 cohogh~ ~

, then

for

~e~H~

have ~ (~)...

where the

L

is

a positive

estimate

R o)

PROOF.

II. that

From

the

and

that

than

the

smallest

can

Since

be

A necessary

is

..,~)

which

constant.

"- L

-.. V9 (~) Ic~(G)I

-~ I[ grad"]QIwe

obtain

(3).

THEOREM ( 0 < P, 4

~h.,(~)~t,~(~)

~

is

easily

and

HR ° should convexity

contained

eigenvalue computed

be of

in of and

sufficient

condition

a~

C*(HR}

convex. HRo

we

deduce

a hemisphere. the

for

problem

equals

2.

Hence (i) From

that

~h < ~ 11

is

for

the

the

inequality

(h={.

greater

hemisphere

, ~i > 2

113 we

deduce

~ > ~

is

continuous The

be

. Hence in

from

FI R

necessity

and

the

proof

vanishes

on

follows

from

shall

assume

of

Theorem

Hg

the

~

fact

I we

i h

that

see

that

( h ~ ~ , ""' ' @

the

grad

)

estimate

(2)

cannot

improved. From

have

Mh

now

on

> ~

(

THEOREM

we

HRo

is

III.

not

Let

that for

p

be

the

smallest

positive.

such that

value

the

numbers

of

h

we

2 IX~ J

Then I g r a d u l

,

•

.

j

e LP(H~)

( 0 ~ 1~ ~ Ro ) f o r

any

p

4 a p ~ ~.

PROOF. integral

one

of

2 1"1 )

are

least

convex).

3

that

at

is

We h a v e f r o m

(2)

that

the

following

finite:

FIR On

} g r a d ~ l c: L P(Ft R) i f

the

other

hand

following

integrals

are

where

second

h~ this

integral

is

finite

if

and

only

if

the

finite:

o

~h

the > ~T

. From

THEOREM

that ( 0

are < R

integral this

positive.

Then

the

Ro )

any

of

H R

with

ing

integrals

for Denote

the

(16) any h > ~

h

such

by

considered

proof

the

such

smallest

that

the

.We

any

of

the

h

such

that

numbers

density

~(Q)

~ 4

p .

planar

O~0hC0h,

for

follows.

electric

~h

plane

are

be

p

be

the

p

PROOF.

for

remark Let

~

IV.

must

p ~

sector

have

belongs

which

~(Q)E

is

LP(~

the

to

LP(g4H~)

intersection

H R) if

the

follow-

finite:

fZh~ (a'~) P l't::h(r~)Z'h~

( c u ) J P d Y'~

that

of

at

least

' ~h#~ > ~ " Assume in the plane

one

0 ~0 h ~oh+ ~

the

following

a polar

inequalities

coordinate

system

holds:

with

)

pole

0

and

polar

only

if

for

all

axis the

h

0 co h such

. The that

~h

integrals > ~

and

(16) for

0

are

finite

if

and

~

~ 4 2'E

we

ha~e

114

0

0

From

this

Theorem in

[3]

the IV

, which The

behaviour i, , ..

improves

states

a

result~obtainable

that

~ e Lz(94

developed

in

of

electric

field

the

, iq

to

o<

x < ~

the

~

numerical

follows.

theory

eigenvalue domain:

proof

the

,

problem

O < y ~

have

of

,

reduces the

the

In

obtained ~

ing

bounds

function

density

of

together

theoretic a

<

with

results

~

for

the

,i.e.

case

0.68025

~

:

<

0.4646. III

A

[ i]

and

electric

the

that

IV

field

of the

near

0

and

lowest

is

following

( [4],

~

theorems

density ~

, the ~

which l e a d t o t h e f o l l o w i n g bounds f o r

These

the

for

<

0.4335

developed

description

electric

o < z ~ i

0.62153

theory

the

constant

(i) o

i

been

paper

of

a

H R )

and

computation

of

results

this

from

the

cubic

rigorous ) :

lead

and

to

the

the

follow-

electric

cube 6000

where

£

is

an

arbitrary

number

R

e

such

f e

r e

that

n

c

D ~ a < I.

e

s

[ 1 ]

G. FICHERA, Asymptotic behaviour of the electric field and density of the electric charge in the neighborhood of singular points of a conducting surface, Uspekhi Mat. Nauk,30:3,1975,pp. lO5-124; English translation: Russian Math. Surveys,30:3,1975,pp.107-127; Ital.translation:Rend, del Seminario mat.dell'Univ, e del Politec. di Torino, 32,1973-74,pp. II~-143.

[ 2 ]

R. COURANT-D. HILBERT, Methods science Publ. New York, 1953.

[ 3 ]

M.A.SNEIDER, s_aa, Mem. Acc.

[ 4 ]

G. FICHERA-M.A. le voisinage Paris,278 A,

of

Mathematical

Sulla capacit& elettrostatica Naz. Lincei,X,3,1970,pp.99-215. SNEIDER, Distribution de des sommets et des ar~tes 1974,pp.1303-1306.

Physics,vol.

di

una

la charge d'un cube,

superficie

~lectrique C.R.Acad.

I;

Inter-

chiu-

dans Sci.

SINGULAR PERTURBATIONS

OF ELLIPTIC BOUNDARY VALUE PROBLEMS

P. Habets

. Introduction A major contribution boundary value problems [5]. An extension A. van Harten bibliography).

to singular perturbations

of linear elliptic

(BVP) is due to W. Eckhaus and E.M. de Jager

to nonlinear

[11][12]

equations has been worked out by

(see these references

for a complete

The present paper extends these results using monotone

methods and differential

inequalities,

As a first step, we consider

the nonlinear

elliptic BVP in a

domain ~ C ~2 eL2u + Liu = f(x,u)

in (1.1)

u = g(x)

on ~

where E > O, u E ~, x E ~2, Li is a linear operator;

i th

f(x,u) and g(x) are given functions

with respect to u, Using monotone

iteration

order differential

and f is increasing

[I], [8] we prove the

existence of solutions of (1.1) and a convergence

result to the

solution of reduced problem L1u = f(x,u)

in ~

(i .2) u

with F G 3~. Convergence

is

= g(x)

on

F

of order E except

neighbourhood of 3~ \ F, where exponential and in an arbitrary

in an arbitrary

boundary

small

layer can appear,

small neighbourhood of a set D where the solution

of the reduced problem

(1.2) is not C 2. We do not suppose the set

to be convex, which allows free boundary layers. Also, free boundary layers appear along the parts of the boundary of ~ which contain characteristics

of the reduced problem

we give no informations

a neighbourhood of these free boundary in the spirit of A. van Harten (I.I)

is supposed

(1.2). However,

in such cases,

on the behaviour of the solutions of (I.I) in layers.

Our work is very much

[II] chapter 6. In [II] a solution of

to exist as well as a better approximation

than a

solution of (1.2). This allows the use of the maximum principle together with a constant as a barrier function.

In our work, we

116

directly prove existence As a by-product

and the limiting behaviour

this also gives an explicit

Further our method

is not restricted

using monotone

iteration.

scheme to cumpute the solution.

to second order problem as it is the

case for the maximum principle used in []l]. Notice at last that the main tool of this section

(theorem 3.4) is a slight generalization

theorem on elliptic BVP (see H. Amann

In a second part, we allow nonlinearities More precisely,

we consider

with gradient dependance.

the BVP

EL2u = f(t,U,Ux,EUy) u = g Using a theorem of M. Nagumo

in on ~

[9] based on differential

extend the existence and convergence BVP could be investigated

of a known

['l] where other BVP are considered).

we

results of the first part. More general

using H. Amann

approach is that it is restricked

inequalities

[2]. The main drawback of this

to second order equation.

In the last part of the paper, we investigate

the fourth order BVP

E2u Iv - p(t)u" = f(t,u) u(o) = u(]) = n"(o) = u"(|) = 0 which can be interpreted to high order equations

as describing

a beam with pin-ends.

This extends

the type of results introduced by N.I. Bri§

[3] and

worked out in [6] [7]. Similar ideas appear in F.W. Dorr, S.V. Parter and L.F. Shampine

[4].

2. A fixed point theorem for increasing maps 2.1. Let ~ be a bounded domain in ~ n and

C(~)"the Banach space of continuous

functions u : ~ ~ ~ together with the norm Iluil = sup lu(x) l. The space C(~), IE.[i together with the order u] ~ u 2

iff Vx • ~

is an ordered Banach space

ul(x) ~ u2(x)

[]]

If ~ e C(~), B • C(~), ~ ~ B, let us define

[~,8] = {x ~ C(~)

An operator T : [~ B] ~ C(~) is said increasing

T(u l ) ~ T(u 2)

~<

if u| < u 2 implies

x < $~.

1t7

2.2. PROPOSITION

[I] Suppose that : I : [a,B] ~ C(~) is an inereas~ng map

which is c o . a c t and such that a ~Ta

TB~<6

,

then the iteration schemes

and u_~ = T ~ _ 1

Uk = TUk-I

converge to fixed points ~ and u of T such that
3.

~ = T(~) < ~ = T(~) < u k ~ B.

An elliptic boundary value Problem

3. I. Let ~ C fl~2 be a bounded domain whose boundary C 2+v manifold

for some v @ (o,I).

Consider

3~ is a I- dimensional

the differential

operators

L 2 = - all(X,y)DxDx - 2a 12(X,y)DxDy - a22(x'Y) DyDy L 1 = - p(x,y)D x + q(x,y) where the following assumptions (H-i)

are satisfied

:

aij , p, q E cV(~);

(H-ii) for some ~

> 0 and all (x,y) E ~, ~ E ~2, Z 2 lSjaij(x,Y)~i~j ~ Kol~ i ;

(H-iii) V(x,y) • ~,

q(x,y) i> 0.

Consider the linear BVP ~L2u + LlU = f

in (3.1) on 3~

u = g

where E > 0. It is well known [l] that under assumptions

(H-i) to

(H-iii), one can define the solution operators K : cV(~) ~ c2+V(~),

f ~ Kf

R : ~+v~)~

g ~ Rg

c2+V(~),

where Kf denotes the solution of the BVP (3.1) with g m 0 and Rg the solution of (3.1) with f m O. The solution of (3.1) reads then u = Kf + Rg.

118

3.2. Let f : ~ x ~ ~ ~ and g : ~ (H-iv) f E cV(~x~),

~ ~ be such that

fu E C(Hx~), g E c2+V($~);

V ( x , y ) e ~, u t > u 2 implies

(H-v)

f(x,y,u I) > f(x,y,u2) . In the following we consider

the non]inear BVP

gL2u + Llu = f(x,y,u)

in H (3.2)

u = g(x,y)

on ~

w h i c h can be s o l v e d u s i n g t h e f o l l o w i n g

Lemma ( s e e H. Amann [ 1 ] ) .

If assumptions (H-i) to (H-v) are satisfied, the BVP (3.2) is

3.3. LEMMA

equivalent to the fixed point equation u

=

Kf(.,u)

+

Rg

=

lu

,

in C(~l) and the map. I : c(~)-~c°(~)

, o ~ [0,2[,

is increasing and completely continuous.

3.4. We are now ready to prove the main tool of this section. THEOREM

Suppose there exist functions ~i C C2(~), i = 1,2,..,k and

Bj E C2(~), j = 1,2,.,,1 such that for any i and j, ai ~ Bj, £L2a i + LI~ i ~ f(.,a i) ,

£L26 j + LIB j ~ f(.,~j)

in H, (3.3)

l

j

Then if assumptions (H-i) to (H-v) are satisfied the iteration schemes u O = a = max a. (resp = B = min B ) i i j j Uk+ I = Tu k = Kf(.,u k) + Rg

converge in C2(~) to fixed points ~ (resp. u) of I i.e. to solutions of (3.2)

such that

119

Proo~

Let u = Kf(.,~ i) + Rg. Then (sL 2 + LI)(~ i - u) = eL2~ i + LI~ i - f(.,~i ) < O, in ~, ~. - u = ~. - g ~ O, on ~ , I

and from the maximum

i

principle [lO] a. - T~. = ~. - u ~ O i

Further

~ = max ~. > ~.. l

i

l

Hence from Lemma 3.3

i

Ta > Ta. ~ a . l

This proves

T~

~ = max a i. Similarly

From Lemma 3.3 and Proposition we only know u k ~ u in C(~). from Schauder's

the sequence

[uk] is bounded a subsequence

~

follows,

the convergence

in C2(~),

except

that

notice

that

< K(i f(.,Uk)l C N + llukll)

there exists

uk

2.2, everything

estimate I~Ic2+~

converging C2

To prove

i

one proves TB ~ @ = min @j.

in C 2+~ (~) and by Arzela-Ascoli's

theorem

[u.,] converging in C2(~). At last any 2 K ukC~ v converges in C(~) i.e. v = u. This implies

subsequence

u.

3.5. Consider

the reduced

problem L|u = f(x,y,u)

in (3.4)

u = g(x,y) with F = {(x,y) C $~ Direct

integration

I 3 h > O : [x - h, x [× {y} C ~ }.

of simple

u O of (3.4) has in general tangent

on F

to the boundary

examples

shows right away that the solution

singularities

at points

].

example

lines

of ~. Let D C ~ he such that u O ~ C2(~ \ D)

E fig.

on horizontal

of set D = {A,D,E}

U

120

Choose

~ > 0 as small as we wish and define ~o @ C2(~) u°(x,y)

3.6. Consider

if

(x,y) E ~ \B(D,~).

next a set = l(x,y)

such that ~ o @_, ~+

= u°(x,y)

such that

: a < y < b , ~ (y) < x < ~÷(y)

n ~

= {(x,y)

I a < y ~ b,

}

x = ~+(y)

C ~ \

B(D,6)

(3.s)

and

e C2([a,b]).

&

& fig.

Define

O E C~(~)

p(y)

2.

example

of a set ~o C

such that

= ]

if

y ~

[a,b],

p(y) = 0

if

y e

[a + 6, b - 6],

p(y) @

[0,1]

and y E C2(~) y(y)

such that

= ~_(y)

Consider

if

[a,b] .

the function

= ~o - Ae-%](x where A,B,C,%]

y @

- Y(Y))

- cB(e%2(d

and 4 2 are positive

- x) _ 1) - Cp(y)(d

constants

- x)

(3.6)

and d > sup {x I (x,y) @ ~} + 1.

121

Let us choose A large enough so that y E [a,b], x # ~+(y)

(x,y) E ~ ,

implies ~(x,y) < uO(x,y)

and C such that

y ~ [a,b] ,

- A ~ g(x,y)

(x,y) @ ~

implies

a(x,y) < ~°(x,y) - C < g. For such a choice of A and C

Let I be an interval large enough such that for any (x,y) E ~0 (x,y) ± A* ± l ± C*(d - c) @ l with

A* = sup {I: ° - g I : (x,y) E ~

, y E [a,b]} ,

C* = sup {I: O- g I / (d - x) , (x,y) E ~ c = inf {xl ( x , y ) E ~ assume

}. In order to prove

, y ~ [a,b]} and cL2~ + L1a < f(.,a) we shall

:

(H-vi) for some ~ > 0 and all (x,y) E ~ , u E I p(x,y) k ~ ,

q(x,y) - fu(X,y,u) k 0

One computes a m = u~ + %iAe -%l(x - Y) + %2EBe %2(d - x) + CO ay = ~

- hlY'Ae -%l(x - Y) - C0'(d - x)

~xx = USx

%~ Ae-%l(x - Y ) - h~gBeX2(d - x)

~xy = U~y + %~Y' Ae-Xl(x - Y) + CO' ~yy = U~y - (%lY"

+ %~X'2)Ae -%l(x - X) - CO"(H - x)

e2a < e2u° + K[(%~ + hl)Ae -%l(x - Y) + %~gBe %2(d - x) + C] with K larger than : sup (all - 2a12 Y' + a22 ~'2)

, sup (a22X")

, sup all ,

sup (-2a120' + a220"(d - x)). Next, one has gL2~

+ LI~ - f(x,y,~) < gL2:° + £K[(%~ + %l)Ae -%l(x - Y) + ~2 g~e~2 ~ %2(d - x) + C] _ P [:~ + %lAe-Xl(X - y) + %2~BeX2(d - x) + CO] - f(x,y,:0)

+ q:O

_ [q _ fu(X,y,e)][Ae-Xl(X

- X) + CB(e%2(d - x) _ I) + Co(d - x)]

122

with 0 E [a(x,y)

, uO(x,y)].

~L2~ + LI~

-

Using assumptions

(H-vi) one gets

f(x,y,~) < EL2u° + L1uO - f(x,y,u °)

+ gK [(%~ + %l)Ae -%l(x - Y) + %~gBe ~2(d ~ x) -

+

c]

~ [%iAe-%l(x - y) + %2EBe%2(d - x) + CO ]

+ gB(e%2(d - x) ~ I).

Let us choose %1 and %2 such that P - ~K)% I = 0 EK%2 - P%2 + ! = 0 i.e.

%1 = ~

- 1

and

! + O(£). %2 = ]~

EL2cz + LI~ - f(x,y,~) + EKC since if

It follows

~< gL2u° + LlUO

-

f(x,y,u O)

v_~i. -~I (x - y) 2 Ae - ~Cp - gB < 0

y ~ ]a,b[

gL2e + LI(~ - f(x,y,~) ~< gL2u° + L1u° - f(x,y,u O) - (~ - gK)C < O for g <~ ~/2K if

and C large enough

,

(x,y) e ~o ~L2~ + L1a - f(x,y,~) ~< sL2u° + gKC - CB < O

for B large enough, and if

y @ [a,h] , x ~< ~_(y)

sL2(~ + LI~ - f(x,y,~) _< gL2u° + L1u° - f(x,y,u O) + EKC - ~ 2 A < O for A large enough.

In a similar way, one proves that for appropriate constants,

choice of the

the function B = ~o + Ae-%l( x - ~) + ~B(e-%2( d - e) - I) + Cp(y)(d - x) (3.7)

is such that gL2B

+ LI~ - f(x,y,B) B ~ g

on ~

~ O

in ~ ,

123

3.7. Theorem 3.4 and the construction 3.6 prove the following theorem, THEOREM

Let ~ > o

be fixed and f~o = f~lU "'" U ~ k be such that each

~. is defined as in (3.5) i.e. i ~i = {(x,y) with ~-i ' ~+i

: a i < y < b i , ~_i(y) < x < ~)+i(y)} C ~ \B(D,~),i= l,..,k, e C2([ai,bi])

Assume assumptions (H-i) to (Hvi) are satisfied. Then the BVP (3.2) has a solution u such that for ~ small enough and any i u(x,y) = u°(x,y) + O(e - % I ( x - @ - i (y)) + e)

where

o <

%1 =

in ~. 1

0(½)

Further u can be computed using the iteration scheme u O = max ai (res~ min ~i ) , u k = Kf(.,Uk_ l) + Rg

with ei defined by (3.6) (resp. Bi

3.8. Notice that assumptions

(3.8)

defined by(3.7~.

(H-iv) and (H-vi) can be weakened by considering

only points (x,y) E ~ and u e [a(x,y),$(x,y)]

4.

Nonlinearities

with

gradient dependence

4.1. If one gives up the computing procedure

(3.8), one can consider a more

general BVP. Let ~ C ~2 be a bounded domain such that for each p E ~ , there exists a cone K with vertex p and a neighbourhood 0 of p such that (K ~ O) lies outside of ~. Consider next the differential

operator

L 2 = - al](x,y)DxD x - 2a|2(x,y)DxDy - a22(x,y)DyDy together with the following assumptions

:

(A-i) for some C and all (x,y),(x',y') e ~ , i,j = 1,2, l aij(x,Y) - aij(x',y')l

~ C H(x,y) - (x',y')ll;

(A-ii) for some K0 >°0 and all (x,y) e ~ , ~ e ~2 i~jaij(x,Y)~i~j ~ K O ~ ~

124

At last,

let f : ~ x ~3 ~

and g : 3q ~ ~ be such that

(A-iii)

f E C~(~ x R 3) , g • C ( ~ )

(A-iv)

there exists

a function

for some ~ e ]O,I[

c : ~+ ~ ~+ = [0,~[ such that

If(x,y,u,v,w)I

~ c(p)(l

for every p > O , (x,y) E ~ , u •

4.2. Consider

+ v 2 + w 2)

[-p;+p]

, (v,w) • ~2

the BVP eL2u = f(x,y,U,Ux,gUy)

in (4.1)

u = g(x,y) with c > O. The following

result

on ~

is then a consequence

of theorem

6 in

M. Nagumo [9]. Suppose there exist functions ~i @ C2(~)

THEOREM

k and ~j • C2(~)

' i = 1,2,,..

such that for any i and j, a i ~i ~j

, j = 1,2,...1

En2a i ~ f(.,~i,aix,g~iy)

, ~L2B j ~ f(.,Bj,Bjx,SBjy)

a. ~ g l

B. ~ g 3

in

on ~

Then if assumptions (A-i) to (A-iv) are satisfied there exists a solution u of (4.1)

such that max a . ~ u < min BJ i i j

4.3. As in 3.5 we assume

there exists a set D c ~ such that the reduced f(x,y,U,Ux,O)

= o

u = g(x,y) where F = {(x,y) E ~ 4 3 u°(x,y)

• C2(~

~o • C2(~)

on r C ~}, has a solution

~ > o as small as we wish and define

such that u°(x,y)

Using

in

h > o : Ix - h,x[×{y}

\ D). We choose

problem

= u°(x,y)

the notations

if

(x,y) • ~ \ B(D,~)

and assumptions

of 3.6 we consider a set ~o and

a function a

= ~o _ Ae-~1(x

- y(y))

_ ~B(e%2(d

- x) - I) - Cp(y)(d

- x)

125

Choosing A and C large enough ~
on 3fl .

Let I be an interval defined as in 3.6 and let us replace assumption H-vi by (A-v) for some ~ > o , Q > o a n d all (x,y) • ~ , u • I , (v,w) ~ ~2 - Q

~ fu < 0

, fv ~ ~

'

~fwI

~ Q ,

One computes then EL ~ - f ( x , y , ~ , ~ x , ~ y )

< ~L2u°

+ gK[(%~ + %1)Ae -kl(x - Y) + A2g~e-2 ~ %2(d - x) + C] - f(x,y,~O,~Ox,O ) + fu(X,Y,Sl,e2,@3)[Ae-%l(X

- y) + cB(e%2(d - x) - I) + Cp(d - x)]

_ fv(X,Y,el,e2,83)[%iAe-%l(X

- y) + %2sBe%2(d - x) + CO]

+ Sfw(X,Y,el,62,e3)[%iY'Ae -%l(x - Y) + Cp'(d - x) - ~Oy] with (el,e2,e3)

a point on the line segment joining the points

and (~,~x,e~y). Using assumption

(A-v) and if %1 > O and

(~o, ~ o , o )

%2 > O are

chosen such that gK% 1 - (~ - gK - ~Q sup :y'i) = 0 EK%~ - ~%2 + l = 0 one has gL2a - f(x,y,a,ax,Cay ) < EL2~° - f(x,y,~O,~Ox,O) - ~%IAe -%|(x - Y) - EB - pO___CC2

At last, using the type of argument used in 3.6, one proves EL2~ - f(x,y,~,~x,E~y)

Similarly,

<0

0

for an appropriate choice of the constants

it can be

shown that the function 6 = ~O + Ae-%l(X - y(y)) + EB(e%2(d - x) _ I) + Cp(y)(d - x) is such that ~L26 - f(x,y,6,6x,e6y) 6 ~ g

~ 0 on 3~

in ~ ,

126

4.4. Theorem 4.2 and the construction

4.3 prove

the following

theorem,

Let ~ > 0 be fixed and ~o = ~l U "'" U ~k be such that each

THEOREM

~. is defined as in (3.5) i.e. l ~'1 = { ( x , y )

: a i < y < b i , ~_i(y)

< x < ~+i(y)}

C ~ \

B(D,~)

with ~-i and ~+i E C 2. Assume assumptions (A-i) to (A-v) are satisfied Then the BVP (4.2) has a solution u such that for g small enough and any i u(x,y)

where

5.

= uO(x,y)

+ O(e-%l ( x ' ~ - i (y)) + ~)

in ~i

'

o < ~1 = O ( I / E ) .

The beam string problem

5.1. One interest

of Proposition

order problems. described

2.2 is that it can be applied

As an example

consider

to higher

an elastic beam with pin ends

by the BVP s2uiV - p(t)u"

= f(t,u)

u"(o)

= u"(1)

= o

u

= u

= o

(o)

(2)

with s > 0 , u E ~ , t ff [0, I] and let f : [0,2] x R + ~

be such that

(5.1)

p :[0,I] ÷ ~

,

:

(i) p is continuous and for some p > o and all t @

[0,I]

, p(t) ~ V 2 ;

(ii) f and fu are continuous and for all (t,u) e fu(t,u)

5.2. To solve

[0, I] x ~

,

> O.

the reduced

problem - p(t)u"

= f(t,u) (5.2)

u(0)

let us define

= u(1)

= o,

the operator K o : C([0, I]) ~ C([O,I])

where K o denotes

the solution

, f ~ Kof

of the linear BVP

127 - p(t)u" = f(t)

u(O) = u(1) = o . From the maximum principle

[I0] , K c is increasing.

Next we introduce

the Nemitskii operator V : u -~ Fu = f(.,u) .

(5.3)

It is then obvious that solutions of the BVP (5.2) are the fixed points of the increasing operator

To = KoF. Their existence can be obtained as follows

5.3. PROPOSITION

Suppose there exist functions so E C2([0,I]) and

So E C2([0, I]) s u c h t h a t - p(t)a~ < f(t,~o) %(o)


, - p(t)S~ ~ f(t,S O) , t e ]0,I[,

, So(1) < o

, So(O) ~ o

, So(1) > o

,

then the iteration schemes Us = Bo

~o = %

and

~k = To~k-I

~

= rouk-I

converge to solutions u and u of the reduced problem (5.2) such that % Proof

<$

< _ u < u < Uk-< SO .

Prom Arzela-Ascoli's

theorem T O is compact on [c~,S]. The inequalities

.(Toao. ~'' - ~o ~< O , To~o(O) - ~o(O) i> O , To~o(1) - as(1) >-- 0 and the maximum principle

imply To~ O > ctO. Similarly,

one proves ToS o ~< SO

and Proposition 2.2 applies.

U

5.4. To study the BVP (5.1), let us introduce Kg : C([O,I]) ~ C([O,I]) where KEfdenotes

the operator

, f~K~f

the solution of the linear BVP e2uiV - p(t)u" = f(t) u"(o)

= u"(1)

= u(O)

= u(1)

= o

.

128

By applying twice the maximum principle, it is easy to show that f(t)

i> O implies u" < O and next u ~> 0, i.e. the operator K E is

increasing. Hence solutions of the BVP (5.1) are the fixed points of the increasing operator Tg = K~F. Once again, Arzela-Ascoli's theorem imply T E is completely continuous and as in 5.3, we can prove the following proposition.

5.5. PROPOSITION

Suppose there exist functions s E C4([O,I]) and

E C4([O,I]) such that g2siv - p(t)S" < f(t,s) , E2B iv - p(t)B" ~> f(t.$) , O < t < 1 , W'(o)

i> o

, W'(1)

i> o

, B"(o)

~< o

, S"(1)

~< o

,

s

<

, s

~< 0

,

~> 0

,

>

.

(0)

0

(1)

B (0)

Then there exists at least one solution

IB ( 1 )

0

u~ of the BVP (5.1) such that

S(t) ~< ug(t) ~< ~(t)

which can be computed iteratively as in Proposition 2.2. Proof.

One has to apply the maximum principle twice in order to prove Ts /> s

and

TB ~< B

and the proposition follows then from 2.2.

•

Suppose there e~r~st functions s ° E C4([O,i]) and Bo E C4([O,I])

5.6. THEOREM

such that Iv -

p(t)S o < f(t,S O) , - p(t)~ o > f(t,B o) , t e [O,1] ,

So(O)


,

%(1) ~
,

Bo(O) />

o

,

Bo(l) >/

o.

Then, for ~ small enough there exists at least one solution u~ of the BVP (5.1) such that a o ~< u

+ O(E 2) ~< flo "

129

Proof.

Consider the function =

ut ~o + c2 ( A e - ~ +

~(l-t) Be"

E

-

C)

where A,B and C are positive constants chosen such that ~"(O) = ~o(O) + ~2A + ~2Be -'~/~ > O , ~"(I) = ~o(I) + ~2Ae-~/c + ~ 2 B

>~ 0 ,

(O) = ~o(O) + E2(A + Be -~/~ - C) ~< 0 , (I) = ~o(I) + g2(Ae-~/E + B - C) ~< O. One computes next g2 iv _ p(t)~"

-

f(t,~) = E 2~Oiv + D 4 A e - ~

+ ~4Be ~(]~t)

,, Dt p 2Be_D(~-t ) - p(t)~ O - p~2Ae-~- _ - f(t,~o) - S2fu(t,~ O + ~(~ - ~O))[Ae

~(1-t) -C] ~ + B e - c-

with ~ E ]0,I[. Hence, for some K > 0 and c small enough E2~ iv - p(t)~" - f(t,~) ~ - p(t)~ O - f(t,~ o) + KE 2 < O. ~he theorem follows from Proposition 5.5 and a similar choice for the function B ~t _~(l-t) B = ~O - g2(Ae-~- + Be g - C).

a

References [I] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review 18 (1976), 620-709. [2] H. Amann, Existence and multiplicity

theorems for semi-linear elliptic

boundary value problems, Math. Z. 150 (1976), 281-295. [3] N.I. Bri§, On boundary value problems for the equation cy" = f(x,y,y') for small e, Dokl. Akad. Nauk SSSR 95 (1954), 429-432. [4] F.W. Dorr, S.V. Parter, L.F. Shampine, Applications principle to singular perturbation problems,

of the maximum

SlAM Review 15 (1973),

43-88. [5] W. Eckhaus, E.M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential Mech. Anal. 23 (1966), 26-86.

equations of elliptic type, Arch. Rat.

130 [6] P. Habets, M. Laloy, Perturbations singuli&res de problgmes aux limites : les sur- et sous-solutions, S~minaire de Math~matique Appliqu~e et Mgcanique 76 (1974). [7] F.A. Howes, Singular perturbations and differential inequalities, Memoirs A.M.S. 168 (1976). [8] M.A. Krasnosel'ski, Positive solutions of operator equations, Noordhoff, Groningen 1964. [9] M. Nagumo, On principally linear elliptic differential equations of the second order, Osaka Math. J. 6 (;954), 207-229. [10] M.H. Protter, H.F. Weinberger, Maximum principles in differential equations, Prentice Hall, Englewood Cliffs, N.J° 1967. []I] A. van Harten, Singularly perturbed non-linear 2nd order elliptic boundary value problems, PhD Thesis, Utrecht 1975~ []2] A. van Harten, On an elliptic singular perturbation problem, Ordinary and Partial Differential Equations, Ed. W.N. Everitt and B.D. Sleeman, Lecture Notes in Mathematics 564, pp~485-495, Springer Verlag, Berlin Heidelberg New-York 1976.

SINGULAR PERTURBATIONS OF SEMILINEAR SECOND ORDER SYSTEMS

by F. A. Howes* School of Mathematics University of Minnesota Minneapolis, ~innesota 55455

and R. E. O'Malley, Jr.* Department of Mathematics and Program in Applied Mathematics University of Arizona Tucson, Arizona 85721

I.

Problems with boundary layers at one endpoint Many physical problems can be studied as singularly perturbed two-point vector

boundary value problems of the form

I sy" + f(y,t,e)y' + g(y,t,g) = 0,

OJt_<

1

(1) y(0), y(1)

where

s

prescribed

is a small positive parameter (cf., e.g., Amundson (1974), Sethna and

Balachandra (1976), and Cohen (1977)).

Scalar problems of this form are analyzed

quite thoroughly in the forthcoming memoir, Howes (1978).

An enlightening case

history of such analyses was given by Erd~lyi (1975), and important early work includes that of Coddington and Levinson (1952) and Wasow (1956). For simplicity, let us assume that in

y

E ÷ O.

and

t

f

and

g

are infinitely differentiable

and that they possess asymptotic power series expansions in

e

as

We'll first consider the vector problem under the assumption that the

reduced problem

(2)

f(uR, t,0)u ~ + g(uR, t,O) = 0,

is stable throughout

(3)

there (i.e.,

0 < t < i

in the sense that

UR(1) = y(1) uR

exists and

f(uR(t),t,0) > 0

-f

is a strictly stable matrix having eigenvalues with negative

Supported in part by the National Science Foundation under Grant Number MCS 7605979 and by the Office of Naval Research under Contract Number N00014-76-C-0326.

132

real parts). to (i) near

We first realize that t = 0

uR

because we cannot expect to have

must expect boundary layer behavior nonuniform

convergence

from

"boundary layer jump" hypotheses

y(0)

T

with

0 <

il~li~

~ ÷ 0.

or for a constant

Instead, we

providing

the required

For a (very) small f(y,0,0),

no extra

however, we must require an additional namely that the inner product

(Here,

f(z,0,0)

the solution

UR(0) = y(0).

t = 0,

0, 0)ds > 0

along all paths connecting

fly(0) - UR(0)Jl.

to the condition

as

I ~ f(uR(0)+s, 0

~ + UR(0)

We note that if

equivalent

UR(0)

assumption,

(4)

~zTz.)

to

More generally,

"boundary layer stability"

for

to occur near

fly(0) - UR(0)il

are needed.

remains positive

cannot generally represent

T

represents

is the gradient

UR(0)

and

y(O)

the transpose and

VF(z - UR(0)) ,

llzli =

(4) is

that

~T(F(~) - F(0)) > 0

since the integral

is then path-independent.

(minimal) hypotheses the common assumption Pictorially,

Indeed,

(4) directly generalizes

the

used by Howes for the scalar problem and it is weaker than that

f(y,0,0)

> 0

for all

y.

the boundary layer stability assumption must hold within the

circle shown

y(1) = UR(1)

Figure 1

The results of Howes and others suggest that under such hypotheses, have a solution

y(t,c)

of the form

(5)

where the outer solution

(6)

y(t,~) = U(t,e) + H(T,e)

U

has an asymptotic U(t,e) ~

Z

j=0

expansion

U.(t)s j

J

(i) will

133

providing the asymptotic solution for

t > 0,

while the boundary layer correction

has an expansion 0o

(7)

Z K. (T)c j j=0 J

~I(T,E)

whose terms all tend to zero as the stretched variable

(8)

T = t/E

tends to infinity.

We would expect this solution to be unique.

smoothness assumptions on order approximations.

f

and

g,

Under weaker

we'd have to limit the expansions to finite

For the scalar problem, Howes doesn't actually obtain

higher order terms or complete boundary layer behavior, but they can easily be generated.

Applying his results to the boundary value problem for the remainder

terms, however, shows the asymptotic validity of the expansions so obtained. The outer expansion (6) must provide the asymptotic solution to (i) for t > 0,

since ~ is then asymptotically negligible. Thus, the terms U. can be 3 successively obtained by equating coefficients in the terminal value problem

f(U(t,e),t,c)U'(t,e) + g(U(t,e),t,e) = -cU"(t,~),

(9)

Evaluating at

c = 0,

then, shows that

(i0)

f(U0(t),t,0)U&(t) + g(U0(t),t,0) = 0,

(which has a unique solution j > 0,

UR(t)

U0

must satisfy the reduced problem

U0(1) = y(1)

under (2) and (3)).

Succeeding terms

Uj,

will satisfy linear problems of the form

(ll)

f(U0(t),t,0)U](t) + fy(U0(t),t,0)Uj(t)U&(t)

+ gy(Uo(t),t,O)Uj(t)

where

U(I,B) = y(1).

hi_ I

is known in terms of

t, U0(t),

= hj_l(t) ,

..., Uj_l(t).

U.(1)] = 0

The stability assump-

tion (3) implies that (ii) is a nonsingular initial value problem, so it also has a unique solution throughout ing the outer expansion

0 < t < i.

U(t,e)

with

The boundary layer correction

H

Thus, there is no difficulty in generat-

U(t,0) = uR(t). must necessarily be a decaying solution of

the nonlinear initial value problem

(12)

I _ _d2~ + f(U(~T,~) + ~(T,e), ~T e) ~d~ = -e[(f(U(~T ~) + ~(T,e) eT, ~) dT2 ' , ,

134

dU (e~ , c )

+ g(U(~'~,e) + n ( z , ~ ) ,

- f(U(ET,e),ez,e)) ~

(12)

~'v, e)

T > 0

- g(U(eT,e),er,e) ] ,

17(0,~) = y(0) - u(0,e). Thus, the leading term

d2~ 0

(13)

dr 2

~0

must satisfy the nonlinear problem

d170 + f(Uo(O) + ~O(T), O, O) d--~ = O,

n0(0) : y(0) - u0(0)

while later terms must satisfy linear problems d2~j (14)

+ f(U 0(0) + 170(T)

dT2

dN. J 0, 0) dr

d~ 0 + fy(Uo(O) + nO(Z), O, O)Hj(z) ~ = kj_l(T), where

kj_ I

dN~/dT,

is a linear combination of preceding terms

% < J,

with coefficients that are functions of

~

~j(0) : -Uj (0) and their derivatives

T

and

HO(T).

The

decaying solution of (13) must satisfy

d~ 0 dY +

fT

d~ 0 f(U0(0) + H0' 0, 0) d-~- d% = 0

and, thereby, the initial value problem

(15)

dn 0

f~0 (T)

f(U0(0) + w, 0, O)dw,

~ i 0,

10(0) = y(O) - UR(0),

d~ : -J0 Multiplying by

T ~0'

(16)

d ]l~0(T)l]2 = "H~(T) I 0 21 dT

the boundary layer stability condition (4) implies that

~0(z)

for nonzero values of

H0(T)

satisfying

f(U0(0) + z, 0, 0)dz < 0 [I~0(T)}] ! fly(0) - UR(0)H = IIH0(0)[I.

Thus, our boundary layer stability implies that ally as

T

Ultimately,

~0(T)

eigenvalues of than some

(17)

~0(r)~

increases until we reach the rest point

K > 0

will become so small that (3) (for

f(U0(0) + H0(T), 0, 0)

will decrease monotonic-

H0(T) = 0 t = 0)

of (15) at

T = ~.

implies that the

will thereafter have real parts greater

and (15) then implies that

H0 (T) = 0(e-
135 i.e.,

Y0

is exponentially decaying as

T + =.

Although we can seldom explicitly

integrate the nonlinear system (15), we can approximate its solution arbitrarily closely by using a successive approximations (1964)). wise.

Knowing

Rearranging

~0'

we next integrate

procedure on (15) (cf. Erd~lyi

(14) for

j = 1

and then proceed term-

(14) and integrating, we obtain

dH. 3 + f(U0(0) + ~0(T), 0 0)Hj + ~.(T) = 0 dT ' 3 where T

£j(T) =

dK 0 (r) {fy(U0(0) + N0(r), 0, 0)[~j (r) ~

fco

dK 0 -d~T . (r)H.(r)]3 + kj_l(r)}dr is known whenever

~. 3

and

dH0/dT

commute.

Thus,

~. 3

satisfies the integral

equation

IT (18)

H.(T) = 3

where

P(~)

P(T)U.(0) 3

-

P(T)P-l(r)%j(r)dr 0

is the exponentially decaying fundamental matrix for the linear system

d~ d-~ + f(U0(0) + K' 0, 0)K = 0,

In general,

T > 0,

H(0) = I.

(18) must also be solved via successive approximations,

directly provides the solution of (14) when the commutator We note that the boundary layer jump the minimum value of

II~II > 0

though it

[~j, dH0/dT] ~ 0.

lIH0(0)II = Hy(0) - UR(0)II

is limited by

such that the inner product

(19)

~T I ~ f(uR(0 ) + z, 0, 0)dz = 0. 0

The jump can, in practice, be quite large. ple, if

f(z,0,0) > 0

for all

z

It involves no restriction,

since then (19) cannot ever hold for a

It gives precise limits to the jumps for certain scalar problems reconsiders an example of 0'Malley

for exam~ # 0,

(Howes (1977)

(1974)).

We could also consider the reduced problem

(20)

f(uL,t,0)u L + g(uL,t,0) = 0,

Then the stability condition would be replaced by

UL(0) = y(0).

(3) and the boundary layer stability condition

(4)

136

(21)

for

f(uL(t),t,0) < 0

0 < t < 1

and the assumption that

(22)

eT

fe

f(uL(1) + z, I, 0)dz < 0

0 for all

8 + UL(1)

on paths between

UL(1)

and

y(1)

satisfying

0 < IlOlf

IIy(1) - UL(1)il. Nonuniform convergence of the solution to (i) would then take place near

t = i,

depending on the stretched variable

o = (i-

and the limiting solution on

0 j t < 1

t)/E,

would be

uL(t).

If

f

were nonsingular

with eigenvalues having both positive and negative real parts along an appropriate solution of the reduced system, we must expect boundary layer behavior near each endpoint (cf. Harris (1973) and Ferguson (1975) for discussions of problems where f (y,t,0) E 0). Y 2.

Problems with boundary layers at both endpoints Let us now consider the "twin" boundary layer problem

( (23)

~

~Y" + g(y,t,e) = 0, y(0), y(1)

under the assumption that

g

0 < t < 1

prescribed

is infinitely differentiable in the region

of

interest and that the reduced system

(24)

g(u,t,0) = 0

has a smooth solution

U0(t )

throughout

0 < t < 1

which satisfies the stability

assumption

(25)

gy(U0(t),t,0) < 0

there, i.e.,

gy

is a stable matrix when evaluated along

(U0(t),t,0) ,

0 < t < i.

Motivation for this assumption is obvious if one considers the linear scalar problems with

cy"± y = 0,

while generalized stability assumptions are sometimes

appropriate and necessary (cf., e.g., Howes (1978) or consider the scalar problem with

g = y2q+l).

converges to

With (25), one can hope that a solution to (23) exists which

U0(t )

within

(0,i).

Since we won't generally have either

137

U0(0) = y(0)

or

U0(1) = y(1),

we must expect "twin" endpoint boundary

(i.e., regions of nonuniform convergence and

t = i).

O'Malley

(1976), and Howes

stability" jumps gy

Our previous

assumptions.

experience

of thickness

(cf. Fife

(1978)) suggests

These generally

[[y(0) - U0(0)I;

and

limit the size of the boundary layer They'll certainly be guaranteed

the boundary layer regions

Indeed,

for small boundary layer jumps,

cient.

Under appropriate

assumptions,

t = 0

(1975),

that we must add "boundary layer

ily(1) - U0(1)[[.

remains stable throughout

0(~-e) near both

(1973, 1976), Yarmish

layers

(cf. Kelley

the stability assumption

if

(1978)).

(25) is suffi-

then, we can expect to obtain an asymptotic

solution to (23) in the form

(26)

y(t,c) = U(t,e) + 9(p,/~) + ~(o,/~) ~

where

U,

9,

and

Q

all have power series expansions

and the terms of the left boundary layer correction

in their second variables

~

tend to zero as the

stretched variable

(27)

p = t/~TEe

tends to infinity while the right boundary layer correction

(28)

Q + 0

as

o = (i - t)//~e

becomes unbounded. The outer expansion oo

(29)

U(t,E)

should therefore

Z U. (t)g j j=0 j

satisfy

(30)

eU" + g(U,t,s)

as a power series in tem (24) as

£ + O.

e

and converge

= 0,

0 < t < i

to the solution

U0(t)

of the reduced sys-

Higher order terms in (29) must satisfy linear systems of the

form

(31)

where

gy(U0,t,O)U. = C (t) j j-i ' C j_ 1

is known termwise

implies that the systems

(e.g.,

11

C O = -U0).

(31) are all nonsingular.

cients are simply and uniquely obtained

termwise.

j > 1 -The stability condition Therefore

successive

(Different roots

U0

(25)

coeffiof (24)

138

would,

of course,

under appropriate According unique)

result

in different

stability

to the Borel-Ritt

function

U(t,s),

sequences

of perturbation

terms

U., J

j > 0,

assumptions.) Theorem

holomorphic

(cf. Wasow

in

c,

(1965)),

having

there is a (non-

the outer

expansion

(29).

If we set

(32)

y(t,~)

we convert

the problem

(33)

= U(t,e)

+ z(t,g),

(23) into the two-point

cz" = h(z,t,~),

0 < t < i,

problem

z(0,~)

= y(0) - U(0,s).

Here ~

h(z,t,e)

= -~U"(t,s)

- g(U(t,£)

+ z, t, S)

satisfies

(34)

h(0,t,c)

since

EU" + g(U,t,e)

= 0(~ N)

= 0(eN).

for every integer

In particular,

h(z,t,0)

corresponding trivial

to the transformed

solution

and the outer

then, we shall deal with totic solution

(33) and,

the needed

ness assumptions

~EI,6

sI

system

= 0

(33) has the

(not necessarily

:for (33) is also

corresponding

z(t,c) = v ( p , ~ )

providing

the reduced

trivial.

unique)

Henceforth,

to (26), we shall seek an asymp-

of the form

(35)

where

problem expansion

N > 0

boundary

layer decay

will be required

= {(z,t,e):

i =

to zero within

number and,

0 < t < i,

for any

llz(0) - U0(0)II + 6, 6,

0 < t < i.

Our smooth-

in a domain

0 _< llz - U0(t)ll i d a ( t ) ,

is a small positive

d6(t)

+ w(u,~)

~ < t < 1 - 8

8 > 0,

0 < t <

0 < ~ < ~ I}

we define

139

l

ilz(1) - U0(1)li

i-

+ 6,

We shall determine the asymptotic behavior of of

[[zll = ~ z T z .

(36)

Here

~ < t < i.

z

by first determining that

IizI[ satisfies the scalar problem

ellzll" = [hT(z,t,s)z + e(llz'll2 - (llzll2)']/llzll,

0 < t < i,

where

llz(0,a)II and

llz(l,c)i[ are prescribed.

This follows via simple calculations, namely

d iizii2 = 211zlllIzii' = 2(z')Tz dt and d2 --

lizll2 = 21izllilzli" + 2(llzli')2

dt 2 = 2(z")Tz + 211z'il2

imply the differential equation for

(37)

llzll. Further,

llz'112 > (llzll')2

since the Cauchy-Schwarz inequality ((z')Tz/llzII)2 = (lizI[')2.

((z')Tz) 2 < llz'li211zll2

Thus, with a loss whenever

z

implies that

and

z'

llz'[i2 >

are not

collinear,

(38)

Jz]i" _> hT(z,t,E)z/llzll,

0 < t < i.

(Through the inequality (37), then, we eliminate the first derivative term from (38).

We note that (38) is an equality for scalar problems.) We'll now ask that for all

(z,t,s)

in

~

there exists a smooth scalar gl,~'

function

~(n,t,g)

such that

(39)

where

hT(z,t,s)z _> ~(llzll,t,e)]Izn

140

(40)

¢(O,t,c) ~ O,

¢(O,t,O) = O,

~

(O,t,O) > 0

and f

(41)

~(s,0,0)ds > 0

whenever

0 < ~ ~ Ilz(0,0)ll

if

z(0,0) # 0

~ ~(s,l,0)ds > 0 0

whenever

0 < ~ ~ Itz(l,O)li

if

z(l,O) ¢ O.

and

I

Existence of such a function cifically,

~ n (0,t,O) > 0

$

will constitute our stability hypotheses.

implies the stability of the trivial solution of the

reduced system within

(0,I)

endpoints.

(39)-(40) imply that

Hypotheses

(42)

where

Spe-

while (41) implies boundary layer stability at both

0 < ilz(t,~)il <_ m(t,e)

m(t,c)

(43)

satisfies the scalar two-point problem

cm" = ¢(m,t,e),

0 < t < I,

m(O,c) = iiz(O,c)ll,

m(1,g) = i~z(l,c)II.

The bounds (42) follow from the elementary theory of differential inequalities since zero is a lower solution for

lizil and

m

is an upper solution (cf. Nagumo

(1937), Dorr, Parter, and Shampine (1973), and Howes (1976)). = 0

and

~ n (0,t,0) > 0

~(m,t,0) = 0

Further,

~(0,t,0)

imply that the zero solution of the reduced problem

corresponding to (43) is stable and, according to Howes (1978), (41)

is the appropriate hypothesis for the needed boundary layer stability of this solution.

Indeed, the solution of (43) satisfies

(44)

where

m(t,g) = rO(P) + So(O) + O(~-c)

r0

is the decaying solution of the boundary layer problem

d2r 0

(45) while

dp2 sO

= ¢(r0,0,0)

P > 0,

r0(0 ) = lJz(0,0)il = fly(0) - Uo(0)ii

is the decaying solution of

d2s0 (46)

= ¢(so,l,O), do2

~ > O,

s0(0 ) = ;Iz(l,0)li = fly(1) - U0(1)ll.

141

The solutions to (45) and (46) are easily shown to exist and be unique. ing (45) by

dr0/dP,

for example, and integrating from

that

(dr0)ir0 =

d-7(by (41)).

Thus,

r0

to infinity implies

> 0

satisfies the initial value problem

dr°

(47)

~(s,0,0)ds

0

p

Multiply-

dp

2

S °(p)

~(s,O,O)ds,

ro(O ) = fly(O) - Uo(O)II.

0 Hence,

r0(P)

rest point

will decrease monotonically to zero as

r0 = 0

at

p = ~.

Since

~(s,0,0)~n

p

increases, reaching the

(0,0,0)s

for

s

small,

~_i Sn (0,0,0) > 0 implies that the decay of r 0 to zero is exponential as p ~ ~ (When r0(0) = 0, we have r0(P) E 0 since there is no need for a boundary layer correction.)

Continuing by solving linear problems, we could obtain an

asymptotic solution of (43) in the form

(48)

m(t,s) = r ( p , ~ )

+ s(o,~).

In terms of the original problem (23), our stability hypothesis (39) becomes the inequality

gT(u0(0) + z, t, e)z _< ~(ilzl),t,E)llzll

where

~

satisfies (40) and (41).

The expansion (44) corresponds to the expected

expansion (26) for an asymptotic solution for the vector problem (23). Now, we return to the vector boundary value problem (33) and its asymptotic solution in the form (35). totically negligible

(o

boundary layer correction

Near

t = 0,

w

and its derivatives should be asymp~

being infinite), so (33) and (35) imply that the initial v

should be a decaying solution of the nonlinear

initial value problem

(49)

v

pp

= h(v,~p,e),

p > O,

v(O,~) = z(O,e).

Thus, it is natural to seek an expansion

(50) by substitution into (49).

v(p,/T)

E v. (p)s j/2 j=0 J

The leading term

v0

must then satisfy the nonlinear

142

problem

(51)

d2v0 ......h(v0, 0,0) do2

Later terms

v., J

p ! 0,

j > I,

v0(0) = y(0) - U0(0),

v0 ÷ 0

as

O * ~.

must satisfy the linear problems

d2v. dP 2~ = hz(V0,0,0)v j + dj_l(p), (52)

where

dj_ 1

v.j(0) = 0,

j

v. -+ 0 3

p ÷~

as

odd;

p _> 0

v.j (0) = -Uj/2(0),

j

even

will be determined successively as an exponentially decaying vector.

Since (51) and our hypothesis (39) imply that

llv011p0 ~ ~(llv011,0,O),

we are guaranteed a decaying solution

(53)

(and

@ ~ 0,

v0(P)

llv0(O)ll = r0(0),

such that

0 ~ llv0(P)ll j r0(0),

v0(P) ~ 0

if

U0(0) = y(0)).

P ~ 0

No explicit solution

though an approximate solution can be obtained as usual.

v0

can be provided,

Introducing the matrix

= hz(0,0,0) > 0 (whose eigenvalues have strictly positive real parts by our stability assumption (25)), variation of parameters can be used to express the solution of (52) in the form

(54) --

(r)dr +

2

where

0

J

h(v0,0,0)

is linear.

e~(P-S)Fj (s) ds]

p

F.(p)3 = [hz(V0(P)'0'O) - ~]v (p)3 + dj_l(p).

tion to (52) whenever

f

This provides the exact solu-

Otherwise, the linear integral

equation (54) must also be solved by successive approximations.

In analogous

fashion, we could generate the terms of the terminal boundary layer correction w(o,~)

of (35).

Thus, we've formally obtained (35), which we expect is a locally

unique asymptotic solution.

143

We note that the assumptions -hz(z,t,0)

on

are everywhere stable.

~

automatically

Thus,

if we take

hold if

gy(y,t,0)

or

h (v,0,0) - ¥I > 0 z

(i.e., positive definite) llz(0,0)ll,

for some real

y > 0

the mean value theorem implies

v

and all

= zTh (~,0,0)z > yilzll2 g

for some "intermediate"

point

Thus,

> 0

> 0

and

/0 ~(s,0,0)ds

for

taking

~(n,0,0)

0 < D ~ Jlz(0,0)ll

We could also extend our discussion

0 ! livJi

that

hT(z,0,0)z

~.

satisfying

= yn, both

~

(0,0,0)

hold.

to systems of the form

sx" = F(x,x',t,s)

with SF/~x' 1 ~F ( ~ F ) T 2s ~x' Sx'

small.

Thus, Kelley

> 0,

just as Erd~lyi

more nonlinear

3.

(1978) considered

problems where

(1968) considered

SF _ ~x scalar problems somewhat

than semilinear.

Examples a.

A problem with an initial boundary Let us consider

layer

the vector equation

gy" + f(y,t,c)y'

+ g(y,t,~)

= 0,

0 < t < 1

where

y =

lyll f (yll Y2

1

In order to have a limiting

solution

YI+ ,

and

Y2 + I

Y2 uR

i ).

g = -

of the two-point problem which satisfies

the reduced problem

f(uR,t,0)u ~ + g(uR,t,0)

we must require

uR

to be stable in

= 0,

0 < t < ],

-f(uR(t),t,0)

must be a stable matrix,

UR(1) = y(1)

i.e.,

< 0

and we must also require boundary layer stability at

144

t = 0,

i.e., we ask that

f~

T

f(uR(0) + z, 0, 0)dz > 0

0 for all

$

such that

More specifically,

0 < lJCJl ! fly(0) - UR(0)ll. the reduced problem has the solution

UR(t) = ( D t t+ C

where

C = URl(0) = -i + Yl(1)

and

D = UR2(0) = -i + Y2(1).

Stability of

uR

requires the matrix -t

-

-i

C

-I to be stable throughout

-t - D

0 < t < i.

This is, however, equivalent to asking that

C + D > 0

and

CD > i,

i.e.,

Yl(1)Y2(1) > Yl(1) + Y2(1)

2.

Further, boundary layer stability requires that

s( wl+c w21l(dwll 0,

0

1

+ D

dw 2

i.e.,

~I3 + 2C~

for all

~ =

¢2

(Y2(0) - D)2. (C,D)

satisfying

+ 4~i~ 2 + ~

0 < ll~ll = lJy(0) - UR(0)ll =

Our initial values

y(0)

Setting

~2 = t~l'

(Yl(0) - C) 2 +

are thereby restricted to a circle about

with radius less than the least norm

cubic polynomial.

2 + 2D~ 2 > 0

such a

II~II of the nontrivial zeros of the ~

will satisfy

(i + t3)~l = -2(C + 2t + Dt 2)

and we minimize d(t) = ]l~ll = ~i-i+ t 2 I¢II.

145

(We note that the minimum for

~i = O,

then, determines an upper bound for For for

d(t)

C = D = 2,

i.e.,

corresponding

t = =,

2D.)

This calculus problem,

lly(0) - UR(O)~.

y(1) = (~),

to

is

we'd obtain the minimum value

tmi n = -0.291.

Thus, we're guaranteed that the

limiting solution of our two-point problem is provided by in the circle of radius

3.390

about

(~).

UR(t)

that boundary layer stability need only hold for y(O)

and

uR(0).

if

y(0)

lies

This is presumably a conservative

estimate for the "domain of attraction" of the reduced solution

tory joining

3.390

~ + UR(0)

uR(t).

We expect

on the actual trajec-

Finally, we observe that this example is quite

analogous to the simplest cases occurring in the analysis of solutions of the scalar problem

b.

ey" + yy' - y = 0

(cf. Cole (1968), Howes (1978), and elsewhere).

A problem with twin boundary layers at the endpoints Consider the vector problem

gz" = h(z,t,~),

0 < t < i

where 3 Zl and

z =

h = -z I + z 2 - z 2

z2 Here

U0 = 0

is a stable solution of the reduced problem

h(U0,t,0) = 0

since

the Jacobian matrix

hz(0,t,0) = ( -II

has the unstable eigenvalues mination of a scalar function

i i i. ~

ii )

Boundary layer stability involves the deter-

such that

hT(z,t,e)z _>

~(lJzll,t,s)IEz]l.

Here 2 2 4 4 hT(z,t,e)z = (zI + z 2) - (z I + z 2) _> IIzJl2(l - 11z~2).

Since

4 2 2 2 4 z I + z 2 < (z I + z2) ,

so we can take ~(n,t,~) = n(l - n2).

Clearly,

#(0, t,e) ! 0,

¢(0,t,O) = 0,

~n(0,t,0)

> 0

and

146

t 0°

1

~(s,i,0)ds

= ~ n2(l - n2/2)

Our preceding

results,

the two-point

problem which converges

(0,i)

provided

then,

> 0

guarantee

the boundary

values

0 < n < /2,

the existence

i = 0

of an asymptotic

to the limiting

solution

or

i.

solution

U0 = 0

within

satisfy

llz(0,0)l[ < ~

Indeed,

for

and

llz(l,0)il < ~ .

we then have

0 i l~z(t,a)lJ ~ m(t,¢)

where

m

satisfies

the scalar problem

gin" = %(m,t,¢),

The asymptotic

0 < t < i,

behavior

of

m

m(i,¢)

follows

= llz(i,¢)ll < /2,

from the scalar

results

i = 0

of Howes

and others.

c.

A problem with

internal

We now consider

transition

the very special

¢y" + f(y,t,c)y'

layers

problem

+ g(y,t,s)

= 0,

0 < t < i

where f2 (yl,Y2, t, ~) y =

,

f(y,t, ~) =

Y2

Y2

0 gl (yl,Y2) and

g = -Y2

This system decouples

into the two nonlinear

scalar

-

eY2 + Y2Y2 - Y2

equations

=

0

and

ey~ + fl(Yl,t,c)yl

+ [f2(Yl,Y2,t,E)y~

and

+ gl(yl,Y2,t,e)]

= O.

i.

(1978)

to

147

If Howes

Y2(1) > Y2(0) + 1

and

-Y2(1) - 1 < Y2(0) < 1 - Y2(1),

(1978) that the limiting solution for

UL( ~

- i) = 0,

UL(0) = Y2(O)

Y2

it follows from

will satisfy the reduced problem 1

on

0 < t < t* = ~ (i - Y2(1) - Y2(0))

and the reduced problem

UR(U ~ - i) = O,

UR(1) = Y2(1)

on

t* < t ! i,

i.e.,

f

uL(t) = t + Y2(0),

0 ! t < t*

U

Y2

uR(t) = t + Y2(1) - I,

t* < t ~ i.

Thus, the limiting solution is generally discontinuous (which is asymptotically increases monotonically

at

t*

and its derivative

one elsewhere) becomes unbounded there. near

t*

from

UL(t*)

relations between the boundary values

Y2(O)

to and

Indeed,

uR(t* ) = -UL(t* ). Y2(1),

Y2

For other

other limiting possi-

bilities occur (cf., e.g., Howes). One must generally expect the transition layer at corresponding discontinuity let's assume that

there in

f2(Yl,Y2,t,0)

to the equation for

YI"

YI"

= 0

t*

in

Y2

to generate a

To simplify our discussion,

however,

and attempt to apply Howes' scalar theory

Thus, consider the reduced problems

fl(VL,t,0)vL + gl(VL,U,t,0)

= 0,

0 < t < i,

VL(0) = Yl(0)

fl(VR, t,0)v~ + gl(VR,U,t,O)

= O,

0 < t < i,

vR(1)

and

The limiting solution for

Yl

will be provided by

= Yl(1).

if the stability con-

vR(t)

dition

fl(VR(t),t,O) holds throughout

0 < t < 1

> 0

and the boundary layer stability assumption

rVR(0) (vR(0) - Yl(0)) J

fl (s'0'0)ds

>

0

q for

q

between

vR(O)

and (including)

that the limiting solution is

VL(t)

on

Yl(O).

Similar conditions would imply

0 < t < i

with boundary layer behavior

148

near

t = i.

If, instead, we have

fl(VR(t),t,0) > 0

on

tR < t < i

fl(VL(t),t,0) < 0

on

0 < t < tL

while

with

tR < tL,

we can expect

Yl

to have a limiting solution /

Yl

as

c + 0

J VL(t),

0 ~ t < t

VR(t),

t < t ~ 1

V

provided we can find a

t

in

J(t) = O,

(tR, tL)

such that

J'(t) ~ 0

for r J(t) = J

VR(t) fl(s,t,0)ds VL(t)

(cf. Howes (1978)).

Pictorially, we will have limiting solutions

Y2

and

shown in Figures 2 and 3.

Y2 (i)

Y2 uR

i

..............

!

t

Y2 (0) Figure 2

Yl

as

149

yl

vR

b t A

t

t*

1

Figure 3

Note that

Y2

has a jump at

t

and

Y2'

has a jump at

Haber-Levinson crossing (cf. Howes (1978)).

t*,

corresponding to a

Much more complicated possibilities

remain to be studied.

Acknowledgment We wish to thank Warren Ferguson for his interest in this work and for calculating the solution to the first example.

References i.

N. R. Amundson, "Nonlinear problems in chemical reactor theory," SIAM-AMS Proceedings VIII (1974), 59-84.

2.

E. A. Coddington and N. Levinson, "A boundary value problem for a nonlinear differential equation with a small parameter," Proc. Amer. Math. Soe. 3 (1952), 73-81.

3.

D. S. Cohen, "Perturbation Theory," Lectures in Applied Mathematics 16 (1977) (American Math. Society), 61-108.

4.

J. D. Cole, Perturbation Methods in Applied Mathematics, Ginn, Boston, 1968.

5.

F. W. Dorr, S. V. Parter, and L. F. Shampine, "Application of the maximum principle to singular perturbation problems," SIAM Review 15 (1973), 43-88.

6.

A. Erd~lyi, "The integral equations of asymptotic theory," Asymptotic Solutions of Differential Equations and Their A~.!ications (C. Wilcox, editor), Academic Press, New York, 1964, 211-229.

150

7.

A. Erd~lyi, "Approximate solutions of a nonlinear boundary value problem," Arch. Rational Mech. Anal. 29 (1968), 1-17.

8.

A. Erd~lyi, "A case history in singular perturbations," International Conference on Differential Equations (H. A. Antosiewicz, editor), Academic Press, New York, 1975, 266-286.

9.

W. E. Ferguson, Jr., A Singularly Perturbed Linear Two-Point Boundary Value Problem, Ph.D. Dissertation, California Institute of Technology, Pasadena, 1975.

i0.

P. C. Fife, "Semilinear elliptic boundary value problems with small parameters," Arch. Rational Mech. Anal. 52 (1973), 205-232.

ii.

P. C. Fife, "Boundary and interior transition layer phenomena for pairs of second-order differential equations," J. Math. Anal. A ~ . 54 (1976), 497521.

12.

W. A. Harris, Jr., "Singularly perturbed boundary value problems revisited," Lecture Notes in Math. 312 (Springer-Verlag), 1973, 54-64.

13.

F. A. Howes, "Singular perturbations and differential inequalities," Memoirs Amer° Math. Soc. 168 (1976).

14.

F. A. Howes, "An improved boundary layer estimate for a singularly perturbed initial value problem," unpublished manuscript, 1977.

15.

F. A. Howes, "Boundary and interior layer interactions in nonlinear singular perturbation theory," Memoirs Amer. Math. Soc.

16.

F. A. Howes,

17.

W. G. Kelley, "A nonlinear singular perturbation problem for second order systems," SIAM J. Math. Anal.

18.

M. Nagumo, "Uber die Differentialgleichung Math. Soc. Japan 19 (1937), 861-866.

19.

R. E. O'Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.

20.

R. E. O'Malley, Jr., "Phase-plane solutions to some singular perturbation problems," J. Math. Anal. Appl. 54 (1976), 449-466.

21.

P. R. Sethna and M. B. Balachandra, "On nonlinear gyroscopic systems," ~echanics Today 3 (1976), 191-242.

22.

W. Wasow, "Singular perturbation of boundary value problems for nonlinear differential equations of the second order," Co_~. Pure Appl. Math_. 9 (1956), 93-113.

23.

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, WileyInterscience, New York, 1965 (Reprinted: Kreiger, Huntington, 1976).

24.

J. Yarmish, "Newton's method techniques for singular perturbations," S I ~ Math. Anal. 6 (1975), 661-680.

"Modified Haber-Levinson crossings," Trans. Amer. Math. Soc.

y" = f(x,y,y')," Proc. Phys.

J.

HIGHER ORDER NECESSARY

CONDITIONS

IN OPTIMAL CONTROL THEORY

H.W.Knobloch

1.1ntroduction The lecture non-linear

is intended systems

a dynamical (1.1)

to give a survey

theory.

Here the notion

law given by an ordinary x = f(X,U)

plus a constraint in u-space

on

u

of the form

and the control variable

by piecewise missible

u~U,

The state variable

of

functions).

control

t

x = f(x,u(t))

Attention

will be focused

and

x(.)

solution

is called a solution

(ii) characterization

of special

which has two advantages area.

It provides

mentary

tools.

essentially

where

U (adu(')

is

of the differen-

of (1.1).

solution

of (1.1),the

approach

tests for

of (1.1) . so called

to both problems

compared with other relevant work in this

more accurate

The background

in a careful

solutions

a unified

in

(i) Sufficiency

along a given arbitrary

We present

dimensional

of the control variable

(u(-),x(-))

on two topics:

set

x=(x 1,...,xn) T

are finite

local controllability

extrema!s.

to

an arbitrary

which assume values

A pair

function

tial eq.

singular

refers

equation

U being

u=(ul,...,um) T

C~-functions

an admissible

differential

We admit specializations

control

of "system"

in

= d/dt,

(control region).

column vectors.

on some recent research

results

and requires

work consists,

analysis

roughly

rather

ele-

speaking,

of the way in which the solutions

152

of the differential function.

eq.

(1.1) depend upon the choice

This analysis

will be carried

of the control

out in detail

in a forth-

coming paper. 2. Local controllability We consider tion)

a

fixed

and cones

solution

on some interval

controllable x I = x(t I )

(u(.),x(-))

[0,t I]

solution

of

x o = x(0).

xI

This concept

most common access

to sufficiency

systems uses cones

approximations

to the set

x ° . The usage

guments

~

tensions.

in linear systems

of all points

control

(c.f.

and associate

the since conex-

so far is the

To begin with we shortly (i.e.

in the sense of Hestenes)

ar-

of a

outline

a

a derived

which contains

cone ond the cones used in the work of Krener.

consists

fixed intermediate

involves

most attention

at

tI

the

have been made to find suitable

cone for

The procedure

e.g.

It was felt however

method for a cone of attainability

both the Pontryagin

at time

[2]) which leads to a statement

Principle".

x(t I )

The

with separation

theory,

construction d

theory.

attainable

Maximum Principle

The one which has received

"High Order Maximal

of the

cone does not yield the best possible

and attempts

recent work of Krener

initia-

These are convex

in connection

of a cone of attainability.

vex approximation

trajectories

is the local equivalent

tool in optimal

long that the Pontryagin

solu-

into any point of

of attainability.

standard proof of the Pontryagin construction

(reference

tests for local controllability

of such cones

is a familiar

tI

along admissible

controllability

of nonlinear

(1.1)

at the terminal point

if it can be steered within time

notion of complete

from

of

. The system is said to be locally

along the reference

a full neighborhood ting from

of attainability.

essentially

point

of two steps.

(u(~),x(~))

SteD I. We pick a

of the reference solution q7 with it a certain non-empty subset II = II~ of the

153

state space.

II can be described roughly as follows

tailed description

(for a more de-

cf. [1], full proofs will be given in the forth-

coming paper). We collect all n-dimensional vectors which appear as first non-vanishing coefficient in any formal power series which can be generated in the following way. Consider an admissible control function

u(-,k)

which depends upon some positive parameter

which is such that borhood of

~

u(t,k)=u(t)

(= reference control)

which shrinks to zero for

the solution of the differential has for

eq. (1.1)

except a neigh-

Let then

(with

and

x(',k) be

u=u(t,~)) which

t=0 the same initial value as the reference trajectory.

then the asymptotic expansion at Step 2. point

~0.

k

Each set xI

k=0

lit, 0 ~ t ~ tl,

of

x(t,k)-x(~).

is transferred to the terminal

by means of the linear mapping

of the variational equation.

Take

induced by the solutions

Then the union of the transferred sets

and finally the convex cone generated by its elements is taken. The result is then the cone of attainability which will be denoted by K and which provides the basis for the subsequent considerations. Theorem 2.1 Hestenes,

cf.

terior to

~.

K

is a derived cone for ~ a t [3]). In particular,

if

x(t I) (in the sense of

K = ~n

then

x(t 1)

is in-

If the reference solution is optimal then by standard arguments (state

augmentation technique and application of the generalized

multiplier rule, cf.

[3]) one can derive from Theorem 2.1 first

order necessary conditions.

These conditions can also be obtained

in a more geometric way via the following theorem. M will denote a subset of the state space which is defined in terms of equations and inequalities.

The notions

"regular point of M, relative interior

point of M, tangent cone T at some regular point of M" are used in the sense of

[4], Chapter VII, p. 320.

154

Theorem 2.2

Let

M

regular point of

be a subset of the

M. If there exists

and the tangent cone

R n)

. This implies

variable

y(-)

T

constraints

3. Singular

extremals.

application

extrema!.

ments of

K

necessary

conditions

a linear

control

problems form,

e.g.

u(')

assumes

the reference

solution

criterion)

Since in case of a singular Maximum Principle

there is particular

In the lecture

frequently

interest

all known second

type necessary

conditions,

in applications

and mostly

The first result is an inequality

known in the literature

as generalized

In case of a multivariable

in

special

turns

cone.

The

are then called

order conditions

in fact they are contained

completely

the

in those ele-

in the Pontryagin

in some detail.

one speaks

extremal

which arise from those elements

space which is contained

widely used Goh.

to some performance

which will be discussed

to equality

state

remarks.

which are not contained

will be touched upon, results

(in the

tE[0,t I] ,

and inequality

U. If in addition

of the Pontryagin

order".

M, then

hold

to optimality

that the reference

of

out to be of little use,

"higher

to

.

applications

General

(with respect

of a singular

adjoint

which

one is seeking for a Pareto-optimum.

in the interior

is optimal

of x(tl)

and all

both in equality

From now on it is assumed values

k~T

be a

are separable

relations

PE ~ t

for all

Note that this result allows

where

x(tl)

x(tl)

interior

of a non-trivial

for all

Y(tl)Tk- ~ 0

to problems

at

the following

y(t)Tp ~ 0

with terminal

is relative

M

the existence

such that

(2.1)

to

and let

a neighborhood

does not contain a point of ~ w h i c h K

Rn

in the two basic The second depicts IIt. It gives rise

cases of which are

attributed

to Robbins

and

type relation which is Clebsch-Legendre

condition.

control we obtain a conclusive

result

155

which seems not to be known so far (cf. related work by Kelley, Moyer,

Jaeobson,

The theory

Krener

application

also relies heavily linear

systems.

interest

and others).

of second order necessary

straightforward

conditions

algebraic

The formalism which brings

which are in complete

agreement

treat control

strictly

theory,

facts about non-

out of two ideas

with our general

line,

from the differential

Firstly we introduce

an analogue

bility matrix into the non-linear

theory.

namely

which arises

of the Kalman controlla-

Let us consider

(3.1)

x = f(x,u)

Let

us

v-th

assume

from the Hamiltonian y = -

for

simplicity

time derivative

(~H/~u)(x,y,u)

b

H(x,y,u)

the Hamil-

that is the

= y T . f(x,u):

fx(X,u)Ty that

u

is

scalar.

Take

the

formal

of the scalar function

= yT-fu(X,U)

with respect

is easy to see that it can be represented where

to

equations

tonian system for the state and adjoint state variable, system

but

out these facts is of

in its own right and can be developed

viewpoint.

is not just a

of convex approximation

on some non-trivial

systems

Kopp,

is a n-dimensional

to the system

(3.1).

in the form

yT'b

vector having as components

It

certain /

functions of

x,u

and further

In case of a linear b

just coincides

system

independent

(i.e.

with the

v-th

variables

a system of the form column

AVb

It now turns out that in the general non-linear quence

of the

functions

b -provided

they are defined

of the independent

variables

role not only for the first order necessary not surprising)

x = Ax + bu)

of the Kalman matrix. situation

as above,

~,~,...

the se-

namely

as

- play the key

conditions

but also for the understanding

%

~,~,...u k~j

(which is

of the second order

conditions. The second idea - which so far seems to represent of our approach

- is to study systematically

with respect to the substitution

the real novelty

invariance

properties

156

(3.2)

u ~ u(x,v)

In other words,

we consider

which arise by making

the control

out that the quantities exhibit tution

along w i t h the given system all those

and relations

a rather transparent (3.2). With respect

vious from

contrast

properties

elaborated

behaviour

to the substi-

this is somehow

ob-

given in Section 2. The application

is our main technical

tool.

It allows

of

- in

to other relevant work in this field - to avoid the usage

of the machinery

of Lie-algebra-theory.

4. The oDerator

~

We introduce

.

a set of infinitely

ui, i=0,1,...

, each

ui

being

many independent a m-dimensional

U will be used in order to denote vector-valued ui

Ilt

It turns

by our approach

with respect

to the sets

the explanation

invariamce

depend upon the state.

functions

of

will be denoted by

tacitly be assumed rentiable

ned, where

(4.1)

(1.1)

lUo,Ul,... I ;

It will always

are infinitely

to all variables.

The symbol

many of the variables

) for shortness.

then the Lie-bracket

f=f(X,Uo)

tial equation

and finitely

g(x, U

vector.

the sequence

that these functions

with respect

column-vector,

x

variables

If

g

often diffe-

is a n-dimensional

[f,g] = gxf-fxg

is well defi-

is the right hand side of the given differen(with

uo

r(g) = [f,g] + ~

instead

of

u). Hence

(~g/~ui)'ui. I

i=O is also well defined. operator

The mapping

g ~ F(g)

represents

acting on the set of all n-dimensional

It is then easy to see that in the case we introduced by applying

vectors

Fv

and writing

g=g(x, U

the vectors

in the last section can be obtained

the operator

u o , Ul,U 2 ....

m=l

a linear

u,u,~

from

b

).

which

bo=fu(X,U o)

etc. instead

of

157

We now turn to the case of a multivariable of the

b

is then played by a sequence

where the n-dimensional

column vectors

as follows (the u-th component

(4.2)

~o = (~fl~u~)(X,Uo)

The elements of

~

are

they are polynomials in the case

m=l

if

Hu

in the components

of

ui

B

H(x,y,u)

= yT.f(x,u)

= yTfu(X,Uo)

for

and

1 ~ i ~ v •

As

and its Jacobian matrix

= (yTf 1 .... ,yTf m) u u v-times with respect to

t

and

is carried out according to the rules

dV dt ~ Hu(x,y,Uo)

yT.

=

, ui = ui+1' i=0,I ....

in the form

yTB (x,u)

along a given reference

(1.1),

i.e.

(4.5)

By(t) = Bv(x(t),

i.e. we have

,

By(x, U ) .

This remark leads to a further interpretation B

= IUo,Ul,.-.l

also can be obtained with the help of the

x = f(x,u o) , y= -fx(X,uo)Ty

Let us take the

defined

is denoted by u (~) henceforth)

x, U

then the result can be ~ i t t e n (4.4)

are recursively

= r v ( ~ o) , v=1,2, . . . .

is differentiated

if differentiation (4.3)

~

'

B~

of

Hamiltonian function

Indeed,

u

(m > 1). The role

of matrices By = (B~, .... B~),

C~° functions

the

Hu(X,y,Uo)

of

control

of the matrices solution

Bv

u(.),x(')

of

let us consider U(t)),

where

U(t)

= lu(t),u(t),...l.

It is then easy to see that the following relation holds (4.6)

d v (yTf. (x(t),u(t)) = y ~ v ( t ) dt v if the differentiation of y is performed (4.7)

y = - fx(X(t),u(t))TY

An obvious consequence

of this relation and the Pontryagin Maximum

Principle are then the well known"first If

u('), x(')

vector

according to the rule

order necessary conditions":

is a singular extremal and

(i.e. a non-trivial

solution of

gin Maximum Principle holds)

then

y(')

an adjoint state-

(4.7) for which the Pontrya-

y(t) ist orthogonal

to the

158

columns

of

By(t),

v=0,1 . . . .

These conditions

of the following

more general

optimal,

the assumption

however

Theorem 4.1

~t

of the matrices $(t)

contains

result.

Here

u(t)Eint

the linear

Bv(t),v=0,1, . . . .

x('), U

u(')

need not to

has to be made

space spanned by the columns

This space will be denoted by

henceforth.

There are good reasons solutions,

for introducing

i.e. via the relation as functions

(4.6),

U(cf.

(4.3),(4.4)).

one to express not only the first

tion

in the next two sections.

is the following

Theorem 4.2

The importance

forming

of

B

relation

as we are going to step in this direcin its own right.

holds identically

in

x, U

for

m=O of this result lies in the fact that it links two which can be performed

with the

with respect to the control variable

Lie-brackets

out of its columns

B v - namely uo

in connection

with

and the

(which involves x

differen-

only).

A proof of Theorem 4.2 which is based on the invariance

paper.

This

but also all the

An important

tiation with respect to the state variable

mentioned

them

p,o = 1,...,m ~+I

operations

differentiation

order,

result which is of interest

The following

~,v ~ O,

different

in terms of the

not along given but introducing

enables

demonstrate

x and

B

(4.7)

formally

second order conditions

of

the

instead

all

are also a consequence

principles

(3.2) will be given in the forthcoming

159 5. The generalized Given a reference us assume that

Clebsch-Legendre solution

matrices

B (t)

Theorem 5.1. ~1,...,g m

then

Rn

which is spanned by the columns of the

given an integer

~ ~ O

such that the following conditions (

~.E

¢(t)for

is even and m

~i~ ~( bBi/bu~ ) ) (x(t), U (t))E

and real numbers

are satisfied

Ci~j(bB~/~UoJ))(x(t)' ~(t)) L ' ~ ~ t )

(-1)~/~

I and let

for all tEI. We denote as before by ~ ( t )

Let there be

i

~

on some interval

(cf. (4.5)).

m i,0=1

u(.), x(')

u(t)EintU

the linear subspace of

condition.

all

for some

tel

if

~<

~

tel

if

~ =,

~

i,j=l Assume now that

u('),x(-)

plier rule of the form state vector orthogonal

y(-).

is a singular extremal and that a multi-

(2.1) holds with some non-trivial

It follows then from Theorem 4.1 that

to ~ ( t )

for every

to have the maximal rank

are orthogonal given this

to

tel. If ~ ( t )

n-1

determined up to a positive order conditions)

(i.e.

I

is

in addition happens

if the multiplier

y(')

is

then c o n v e r s e l y ~ t )

consists

of all vectors which

y(t). The statement of the theorem can then be

simple form. is constant and equal

then for each choice of the m-tuple

non-vanishing m

(5.1)

y(t)

scalar constant by means of the first

Corollary I. If the dimension of ~ ( t ) n-1 on

adjoint

~

~I' .... '~m

to the first

among the numbers

~i~Y(t)T(bB~/bU~J))(x(t),

U(t)),~=0,1,2, . . . .

i,j=1 carries an even subscript has the sign

(-I) v/2

~

(which may depend upon

~1,...~m )

and

,

160

As one observes from (4.3) and (4.4) the coefficient of

Ci~j

can

also be interpreted as the quantity (5.2)

~

# d~

~

H(x,y,u O)) : : hi'J(x,y,~ U )

°

taken "along" the singular extremal, i.e. taken for U =

x=x(t),y=y(t),

U(t). Hence the corollary can be rephrased in a way which is

more close to what is known as generalized Clebsch-Legendre condition in the literature. Note however that the standard version of this condition in case of a multivariable controlconGerns the quadratic form (in indeterminates zl,z2,...Zm) i zizoh~'J(x(t)'y(t)' i,0 as such, whereas each

U(t))

our corollary yields the analogous statement for

i n d i v i d u a i

value of this quadratic form.

We state a further result which is an immediate consequence of Theorem 5.1. In the following corollary the reference solution is not required to be optimal and the rank of

~(t)

need not to be maxi-

mal. Corollary. 2. (5.3) for

~ >- 0

be an integer such that

bB~/bu(J))(x(t), U ( t ) ) + i,j=l,...,m,

that (5.3) is Then

Let

~

all

n o t

tel

and

v=0,...,~-1

true for all

tel,

Assume furthermore

and all

i,j

if

V=~

is an even number and we have

(-1)~/2~bB~/bu(J))(x(t), U ( t ) ) + for all

(bBJ/bu(i))(x(t), ~(t))E ~ ( t )

tel , and

(bB~/b/u(i))(x(t), U ( t ) ) ~ T

i,j=i,...,m .

t

161 6. Higher order equality-type These are conditions

of the form

according to the multiplier which are such that ~(t)

A aE I~t . All elements

u(.),x(')

and they arise, An

of the linear subspace

but there may be more,

to show in the next theorem.

tEI; but otherwise

y(t)Ta = 0

rule (2.1), from elements a of the

have this property,

solution

conditions.

as we are going

We assume again that the reference

satisfies the condition it can be arbitrary.

u(t)Eint(U)

In particular

for all

it need not

to be optimal. Theorem 6.1. Let

~ _> 0

be an integer such that the following ele-

ments belong to the space (i)

(bB~/bUo(J))(x(t),

for

U(t))

for every

for

tEI:

v ~ ~, i,j=l ..... m

i,j = I .... m.

Conclusion:

The elements

(6.1) belong

~(t),

~ (~B~/bu~J))(x(t), to

"Ift

for every

tEI

U(t)) and

i,j=1,...,m.

There is an important special case of Theorem 6.1 which is known (or rather its consequences

Corollary

1.

Let

f(x,u)

for singular extremals are known).

be a linear function in

u

Then + (~B~/~Uo(J))(x(t)' U(t))E for i,j=l,...,m

Proof.

If

f

and all

~t

tEI

is linear in

u, then

Bo = fu

is independent

from

162

U and hence

Bi, {j)

b o/bU o

is zero identically in

x, •

. It follows

then from Corollary 2 to Theorem 5.1 that the hypotheses 6.1 are all satisfied if one takes

~=I

of Theorem

.

One observes that the conclusion of Theorem 6.1 can also be phrased in this way: The convex cone generated by the elements of ~ t tains the linear space generated by the union of ~ ( t ) ments

con-

and the ele-

(6.1). Let us now return to the special situation considered

in the first corollary of Theorem 5.1. Since every linear subspace of "lit

is orthogonal to the multiplier

the maximal linear subspace of the elements

IIt

y(t),~(t)

is necessarily

if it has dimension

(6.1) actually belong to ~ ( t )

n-1. Hence

and a straightforward

induction argument leads us to the following result. Corollary 2. Assume that

~(t)

is the maximal linear subspace con-

tained in the convex cone spanned by the elements of every

~It , for

tEI. Assume furthermore that (5.3) holds for all

i,j=1,...,m

and for

v=O,...,a

,~

tEI, for

being some non-negative integer.

Then we have (~B~/bu~O))(x(t), U ( t ) ) E ~ ( t ) for every

tel, i,j=l,...,m

and

~=0,...,~

7- ApPlication to sensitivity analysis. We wish to touch briefly upon a further application of the foregoing results which underlines a certain advantage of our approach. the standard techniques of general properties the cone

K

Since

sensitivity analysis are based on the

of derived cones only, they can be applied to

which was introduced

in Sec. 2 - the same cone from

which we have deduced all necessary conditions discussed in this lecture. Thereby one arrives on sensitivity results which take the

163

higher order variational crease

effects

This leads to an in-

of accuracy.

Sensitivity

analysis

in general

changes which the value performance

criterion)

are changed.

is concerned

function undergoes

We confine

fold consists direction

an optimal

function

derivative

V(k)

V'

of

can be estimated

y

Maximum Principle

satisfy

xI

at

k=O

is discussed.

by

x1+kp,

xI

in a certain

k being a positive small

k~O

and that a right-hand

exists.

side V'

for which the statement

holds

true.

The estimate

of

set of all

of the Pontryagin V' now remains

vary instead on the set of those multipliers

of

also a lowering

the

It is then known that

on the (suitably normalized)

the first of the conditions

a restriction

can be found in

x 1. We now change

is well defined

assumes

those multipliers

y

account

of a typical

from above in terms of the values which a certain

linear functional

if we let

outline

that for each sufficiently

V(k)

of the

control problem where the terminal mani-

ef a single point

Let us assume

for the

if the data of the control problem

example

p, that is we replace

paramter.

(i.e. the optimal value

A more detailed

[5] where also an illustrative Let us consider

with estimates

ourself to a sketchy

result and its extension.

value

into account.

y

V'

follows

cone is contained

which

That this indeed means

to a subset of the original

of the bound for

that the Pontryagin

(2.1).

valid

set and therefore

simply from the fact

in the convex cone

K

References.

[I] H.W.KNOBLOCH, Dynamical Press

Systems,

1977, pp.

[2] A.J.KRENER, to singular pp.

Local controllability

256-293.

A.R.Bednarek

in nonlinear

and L. Cesari

systems,

eds.,

Academic

157-174.

The high order maximal principle extremals,

SIAM J. Control

and its application

Optimization

15 (1977)

164

[3] M.R. HESTENES, Calculus of Variations and Optimal Control Theory, Wiley, New York 1966 [4] H.W.KNOBLOCH und F.KAPPEL, Gew~hnliche Differentialgleichungen, B.G.Teubner, Stuttgart, 1974. [5] B.GOLLAN, Sensitivity results in optimization with application to optimal control problems. To appear in: Proceedings of the Third Kingston Conference on Differential Games and Control Theory 1978.

Author' address: Mathematisches Institut, Am Hubland, D-8700 WGrzburg, Fed.Rep.Germany.

RANGE OF NQNLINEAR PERTURBATIONS OF LINEAR OPERATORS WITH AN INFINITE DIMENSIONAL KERNEL

J. Mawhin and M. Willem

I n s t i t u t Meth~matique Universit~ de Louvain B-134B Louvain-la-Neuve Belgium I . INTRODUCTION Much work has been devoted in recent years to the s o l v a b i l i t y o f nonlinear equations o f the form (1.1)

Lx - Nx = 0

in a Bsnach space, or to the study of the range of L - N, when L is a Fredholm mapping of index zero and N satisfies some compactness a s s u ~ t i o n . graphs[ 8 ] , [ 1 1 ]

and[13]

.

Basic

for

this

study

is

the

5ee for example the mono-

reduction

of

equation

(1.1)

to

the fixed point problem in the Banach space X (1.2)

x - Px -

(JQ + KF~)Nx = 0

or to the trivially equivalent one in the product space ker L x ker P,

(u,v) = (u + JQ~u+v), KpQN(u+v)) where P and Q are continuous projectors such that (1.3)

Im P = ker L ,

Im L = ker Q ,

KpQ i s the associated generalized inverse o f L and J : Im Q -~ ker L i s an isomorphism. The compactness assumption on N generally implies that (JQ + KpQ)N i s a compact mapping on some bounded subset o f X and, P being by d e f i n i t i o n of f i n i t e rank, (1.2) i s a fixed point problem for a compact operator in X and degree theory is available in one form or another.

If one replaces the Fredholm character of the linear mapping L by the mere

existence of continuous projectors P and Q satisfying and is at best non-expansive,

(1.3), then P is no more compact

which makes the study of the fixed point problem (1.2)

very d i f f i c u l t even f o r (JQ + KF~)N compact (see e . g . ~ ] In a recent paper, Br~zis and Nirenberg [ 4 ]

, chapter 13).

have obtained interesting results

about the range of L - N when X is a Hilbert space, N is monotone,

demi-continuous and

verifies some growth condition, and L belongs to some class of linear mappings havif~g in particular compact generalized inverses.

Those assumptions are in particular

satisfied for the abstract formulation of the problem of time-periodic solutions of semi-linear wave equations.

The proof of the main result in [4] for this class of

mappings is rather long and uses a combination of the theory of maximal monotone operators, 5cheuder's fixed point theorem and a perturbation argument.

166

In this paper, which is the line of the recent work of one of the authors ~B]for the case of a Fredholm mapping L, we consider problems with dim ker L

non finite

by an approach which is closer in spirit to the continuation method of Leray and Schauder [14], although we still have to combine it with other powerful tools of nonlinear functional analysis like the theory of Hammerstein equations and of maximal monotone operators.

We first obtain a continuation theorem for Hammerstein equations

(Section 3) whose proof requires an extension of some results of De Figueiredo and Gupta ~ g i v e n

in Section 2.

This continuation theorem is applied in Section 4

to obtain an existence theorem for equation

(1.1) in a Hilbert space under regularity

assumptions slightly more general than the ones of Br@zis-Nirenberg growth restrictions replaced by a condition of Leray-Schauder's of some set.

This existence theorem is then applied in Section 5 to abstract problems

of Landesman-Lazer [9]

and with the

type on the boundary

type, the first one corresponding

when dim ker L
to a result of Cesari and Kannan

, and the second one being essentially a version of the

result of Br~zis-Nirenberg.

In Section 6, the main theorem of Section 4 is applied

to a second order periodic boundary value problem of the form -x" = f(t,x) i n a H i l b e r t space,

and the e x i s t e n c e r e s u l t which i s

obtained i s s f i r s t

but s t i l l

partial answer to the question raised in 6 0 ] about the solvability of this periodic problem for differential equations in infinite dimensional spaces. we consider equation

In Section 7,

(1.1) in a reflexive Banach space X and obtain a Leray-Schauder's

type theorem when the assumptions of monotonicity and compactness are replaced by some assumptions of strong continuity. Another problem in differential equations and leading to equations of the type (1.1) with an infinite dimensional kernel is the periodic boundary value problem for first order ordinary differential equations of the form x' = f(t,x) in an infinite dimensional Banach space. in [5] and Z 6 ] a n d

Such a problem was considered by Browder

will not be treated here.

The interested reader can consult the

recent paper ~ 9 l w h e r e some of the results of Browder are generalized, particular to existence theorems of the Landesman-Lazer's

leading in

type for this situation.

167

2. SOME RESULTS ON H.AMMERST.EIN E~UA.TI.ONS IN HILBERT SPACES Let H be a real Hilbert space with inner product DEFINITION 2.1. compatible

( , ) and corresponding norm ~.~ •

A pair (M,N) of mappings from H into H is said to be Hammerstein-

(shortly h-compat.~ble) with constants

a

an___.dd b

if the following conditions

hold :

(i) (ii)

0 ~ b ~ a . N is demi-continuous,

i.e. if

(iii)

(V u ~ H ) ( V

v~H)

x ~x in H, then N(x )--~N(x) n n : a|Mu - My| 2 ~ (Mu - Mv,u - v) •

(iv)

(V u ~ H ) ( V

v ~H)

: -b{u

-

vl 2 _~ (Nu

-

Nv,u -

v)

.

.

This class of pairs of mappings is related to the unique solvability of the abstract Hammerstein equation (2.1)

x + MNx = f

for every f ~ H, by the following results, which generalizes and completes a theorem of De Figueiredo and Gupta [ 1 0 ] PROPOS3TION 2.1. constants

.

Let (M,N) be. a.. pair .of h-compatible mappinqs from H t__ooH, with

a an__.d.d b •

Then.~ for every f ~ H, equ..ation (2.1) has a unique sol.u.tion.

If, moreover , (2.2)

M(O)

the unique s.o..lution x o f

(2.1) s a t i s f i e s

(2.3)

Ix - f |

Pro pf. paper.

= 0 ,

the foll.pwinq estimate. ~

(a - b ) - I I N f l

.

We only sketch the proof, details of which will be given in a subsequent

Equation

(2.1) is trivially equivalent to equation y + MN(y + f )

= 0 ,

and hence to O~Ty,

(2.4) if

T : H .-~ 2 H

i s d e f i n e d by Ty = - M - 1 ( - y ) + N ( f + y ) .

By the assumption on M and N and basic results of the theory of maximal monotone operators

(see for example [ I]

and strongly monotone, prove ( 2 . 3 ) , l e t -blf

- x~ 2 ~

), it is easy to show that T is maximal monotone

so that (2.4), and hence (2.1) has a unique solution.

us n o t i c e t h a t ( f - x, Nf - Nx) = (MNx, Nf - Nx) = -(MNx,Nx) + (MNx,Nf) - a~MNx~2 + (MNx,Nf)

and the result follows.

~ -

~Ix

- f { 2 + ~x - f ~ N f ~

,

To

168

Under some more assumptions, one can obtain results about the continuous dependence of the solution of equation (2.1) with respect to M, N and f . The formulation of the following Proposition 2.2 is modelled after a theorem of Brezis and Browder (~2] PROPOSITION 2.2.

, Proposition 5) and the proof will be given elsewhere.

Let (M,N) .be a pair of h-cqmpatible m appinqs, f ~

H, ( M n , N n ) n ~ N .

be a sequence of pairs of h-compatible mappinqs with .c.onstants a and b , and (fn)n~_N . be e sequence of elements of H which converqes to f.

AssuMe that the followinq

conditions hold : (I} (2) (3)

(4)

for each bounded subset

5 C H,

U N (5) n n ~ N* F o r each u ~ H, M N u - - ~ MNu i f n--P=o . n For each uC¢. H, N u - - ~ N u if n---~=~. n For each n ~ N*, Mn(Oi = 0

is bounded .

Then, if u = (I + MN)-If

,

u

n

= (I + M N ) - I f n n n

,

(n E N * ) ,

one has u

If we call a mapping

n

--~u

F : H --PH

if

n-.~=o.

bounded when it takes bounded subsets of H into

bounded subsets of H, we have the following immediate consequence of Proposition 2.2. COROLLARY 2.1. H--~H bounded.

Let (M,N) be a pair of h-compatible mapp.~nos, with M(O) = D s n d N

:

Then (I + MN) -I H --~H is continuous.

If we now call a mapping F : H --~H

closed when F{S) is closed for every closed

subset S of H, we deduce at once from Corollary 2.1 the following COROLLARY 2.2. a

Under the es@umptions of Cuorollary 2.1,

I + MN : H--p H

i~s

cl,psed mappinq.

3. A CONTINUATION THEOREM OF LER.A..Y.-SCHAUDER'~ TYPE Let still H be a real Hilbert space with inner product ( , ) and corresponding norm I.| , let

I = [0,1]

let us denote, for each Let now and let

~. C

H

and, if ~Q

E C_H

I, by T ~

and

T : E x I -~H

the m a p p i n g

, (x,~) w-pT(x,~),

T ~ : E --m H, x ~ T ( x , ~ )

be an open bounded set, with closure c l ~

us consider the mappings

M, N : H x I --~H

and

and boundary

C : cl~.

. fr ~'~ ,

x I --~ H .

The following result is a continuation theorem of Leray-ichauder's type for mappings which are not necessarily compact perturbations of the identity. H

Recall that

F : E--P

is called ~ compact on E if F is continuous on E and F(E) is relatively compact.

169

THEOREM 3.1.

Assume that the mappinqs M, N and C satisfy the followinq conditions.

I. There exist real numbers is h-compatible

a

enid b

such that, for each ~ _

with constants a an_..~db , and M~(O) = 0

2. N : H x I - ~ H

is bounded . and, for each

x ~_H,

I, the pair ( M

, N~)

for each ~ I .

the mappinq

p ~. N(x, p ) i s continuous. 3. For .each

x~

H

and each

V ~ I, the mappinq

~ ~-~ M(N(x,y ) , ~ ) is continuous.

4. 0 ~

(I + MoNo)(D~)

5. C : c l ~

x I ~

H

f o r every ( x , ~ ) Then, for each

. is compsct

~

fr~.

~I,

and

CO = 0 .

x I .

equation

X + MpN~x + C x has at least one solution Proof.

Let us define,

0

x ~ ~L . for each ~

TM = It follows therefore

=

I

I, the mapping + MpNp

from the assumptions

T~:

H --~H

by

.

(I) to (3), Proposition

2.1 and Proposition

2.2 thst the mapping

i s defined, continuous and bounded on H x I . U = {(TBx,~) i s an open bounded subset of

Consequently,

: (~,~)~

the set U defined by

~.x I]

H × I , and, using assumption (5), the mapping K defined

by

i s compact on c l U •

Moreover, i t

follows from condition (4) that (O,O) ~

Now, by Corollary 2.2, T ~

U .

is a closed mapping for each

cI(TF(~L)) c

T (cl ~

~E

I, and hence

) .

Therefore, T~ being moreover one-to-one, one has f r T (-~)

= c l ( T B ( ~ ) ) k i n t ( T M ( ~ ) ) = cI(TM(3"L)) ~ C

T~(cl~)

~- T~.(J'L)

= T (fr-~)

T (~L)

,

which, together with assumption (6), implies that, f o r every ( y , ~ ) ~ y + K(y,~) ~ 0 .

C

f r U, one has

170

As, by assumption (5), K0 = 0 , all the conditions are satisfied to apply the usual Leray-Schauder's continuation theorem [14] to the family of equations

(3.2)

y + K(y,~)

implying that, for each

= 0

,

~ 6

I,

~ E I, equation (3.2) has at least one solution y such that

(y,p)

~

U,

and hence, l e t t i n g y = Tpx

which i s possible by d e f i n i t i o n

,

xE~C

,

o f U, we see t h a t

x

satisfie

equation (3.1) and

the proof is complete. REMARK.

Theorem 3.1 is distinct from but in the spirit of generalized Leray-Schau-

der's type continuation theorems due to Browder [ 73 and Br@zis and Browder

[2].

4. A CONTINUATION THEORE M FOR SOME NONLINEAR PERTURBATIONS OF LINEAR MAPPINGS

WIT H

AN INFINITE DIMENSIONAL KERNEL Let still H be a real Hilbert space with inner product ( ~ ) and corresponding norm

I'~

We shall be interested in existence results for equations in H of the

form

(4.1) where

Lx = Nx L is a linear and N a not necessarily linear mapping from H to H satisfying

some regularity assumptions we shall describe now. work by Mawhin [16], Br~zis and Haraux ~ ]

Following the line of recent

and Br~zis and Nirenberg ~ ]

, and in

contrast with most of the literature devoted to equations of type (4.1) (see for example the s u r v e y s t B ]

, D I ] , D 3 ] ) we shall not assume t h a t

ker L

is of f i n i t e

dimension. More precisely, let dense domain

dom L

L : dom L ~

H -~PH

and closed range

(4.2)

be a closed linear operator with

Im L

such that

Im L = (ker L)"L

Consequently, using the closed mapping theorem, the restriction of L to dom L ~ Im L is a one-to-one linear mapping onto Im L

with a continuous inverse

K : Im L--P dom L ~ Im L . We shall denote by P the orthogonal projector in H onto the (closed) subspace ker L , so that condition (4.2) is equivalent to Im L = ker P . Let now

N : H--~ H

i.e. such that

Nx

n

be a bounded mapping which is moreover demi-continuous on H,

--~ Nx

for every sequence (Xn) n

~ N*

which converges to x in H.

171

Recall that a mapping F : H - ~ H

is called monotone if, for every x E

H and every y ~ H,

ore has (Fx - Fy,x - y) ~ THEOREM 4.1.

Let us assume that the mappinqs L, N, K, P defined abpve verify the

fol.lowin.q F onditions

:

I. I - P - N : H--~ H

is monotone.

2. Ther# exists an open bounded set i. K(I - P)N ii.

Then, equation

~).C H such that

0 ~ ~

and

is compact on cl ~ - .

Lx + (I - ~)Px - ~ N x

Proof.

0 .

~

0

for

all ( x , ~ ) ~

(don L t3 fr ~.

x ]0,I~ .

(4.1) has at least one solution

First step. Let us first show that, for every

(4.3)

Lx = - ( 1

has at least one solution

x ~

~]0,I~

, the equation

- ).)Px +~Nx

dom L (~ ~L

By a result o f ~ 5 3

(see a l s o ' t 3 3

),

equation (4.3) is equivalent to the equation in H

(4.4)

x +%P(-P - N)x - % K ( I - P)Nx = O

and hence, by assumption

(2.ii), one has

(4.5)

x +~P(-P - N)x - ~ K ( I

for every in dom L .

(x,~) ~ Let

- P)Nx ~ 0

fr ~. x ]0,1[ , because the solutions of (4.4) are necessarily ~

~0,I~

be fixed; then, one immediately obtains , using the

orthogonal character of P and assumption

(I), for every x G H and y E

(%(Px - Py),x - y) = ~ | P x - py|2 =

H, and each ~ ( I ,

X-1]%px _ %py]2 ,

(F(-Px - Nx + Py ÷ Ny),x - y) ~ - ~ I x - y l 2 ~ - I × - y|2 , so t h a t the p a i r ( ~ P , ~ - P - N ) ) i s h-compatible with constants P(O) = O, the mapping continuous f o r every x ~

(x, ~ ) ~-~ ~(-Px - Nx)

~-:

F(-Px-Nx)

(~, ~ ) ~ K ( I

I + ~pP(-P - N)

- P)Nx

reduces to I f o r ~ = 0 .

i s c l e a r l y compact on c l J~L x I , vanishes f o r

0, ands, by ( 4 , 5 ) ,

x + ~.'.XP(-P - N ) x - f f " ~ K ( I for avery ( x , ~ ) ~

fr ~

x I, because, when

this is implied par the condition 0 ~ - ~ . Second step. K(I

bounded, the mapping ~ -

I , the mapping ~ ~ - ~ P ( y ( - P x - N x ) ) constant and hence c o n t i -

nuous f o r every (x, ~ ) ~ H x I , and Now the mapping

~-I and I f o r every ~ 6 I ,

.

#~ 0

0 (the only case not covered by (4.5)},

The result follows therefore from Theorem 3.1.

being bounded, it follows from the first step, the compactness of

P)N on c l ~ ~. and the properties of L that one can find sequences

(~n)n~.N . , (4.6)

~

- P)Nx ~=

auch t h a t

)~n--.~ 1

n--~oo, X h ~ ~ 0 ~

if

Lx

n

= -(I

, n~-N~,

- Xn)PXn + ~ Wx n

n

,

n ~-N*,

(~

) n n~N*

'

172

and, if we write Yn =

such that, for some (4.7)

y~

yn~y

H ,

(I

-

P)x

and z E Zn---~

,

n

z

= Px

n

H,

z

and

LXn = Ly n ~

Now, from the monotonicity of I - P - N, we deduce, (Yn -

and hence, using

(4.8) Using

NXn -

(I

-

P -

,

n

N)v

,

x

n

-

v)

Ly

if

n--~.

for every n ~ _ N * and

~

O ,

v ~_-H, that

n ~_N*,

(4.6),

(Yn - A n 1(1 -

)~n)zn - ' X n l L Y n - (I - P - N)v, Xn - v) ~' 0 , n ~ N*.

(4.7), we can go to the limit in (4.8), which gives (y - Ly - {I - P - N)v, x - v) ~

for every v ~

H.

0

Since I - P - N is cZearly maximal monotone, y

-

Ly

=

(I

-

P -

N)x

= y

-

Nx

this implies

,

and hence Lx = Ly = Nx and the proof is complete. REMARK 4.1.

It is immediately checked that -L has the same properties as L, with

the same orthogonal projector P.

Consequently,

the assumption

I - P - N is

monotone can be replaced by I'. I - P + N : H ---~H

is monotone

if (2.ii) is replaced simultanuously by 2.ii'.

Lx - (I - ~ ) P x - % N x

Of course, the monotonicity of

~ 0

for all (x, ~ ) ~

I - P - N

(dam L ~

x ]O,i[

.

(resp. I - P + N) is implied by the

m o n o t o n i c i t y of -N (resp. N), as it is easily checked because 5. A P P L I C A T I O N

fr~L)

TO SOM E A B S T R A C T LANDESMAN-LAZER

I - P is monotone.

PROBLEMS

We shall show in this section how Theorem 4.1 allows simple proofs for abstract Landesman-Lazer

problems,

some growth restrictions.

i.e. results about the range of L - N when N satisfies The first theorem is modelled after a result of Cesari

and Kannan Z 9] for the case where ker L is finite-dimensional. is suppressed here at @ e exp.nse of a m o n o t o n i c i t y condition on C O R O L L A R Y 5.1,

This assumption I - P - N or I - P + N.

Let us assqme that the mappinqs L, N, K, P verify the conditions

listed at the beoinninq of Section 4 an d that the followinq assumptions hold; a. I - P - N : H--~H

is mop£tone.

b,' K(I - P)N : H--~H

is completel.y continuous,

K(I - P ) N ( B ) ~ s c. There exists

i.e. continuous and such that

relatively compact for every bounded subset B o f H. r~

0

such that, for all x ~ iK(I

-

e)Nx~

H, r

.

173

d. There exists R > 0 such that I for all x E H for which

IPxt = R and

|(I - P)x~ ~ r

,

one has (Nx,Px)

~

0 .

Then t equation (4.1) has a t l e a s t one s o l u t i o n . Proof.

We shall apply Theorem 4.1 with the open bounded s u b s e t ~ ~xEH

Q- =

:

IPx|~R

and

|(I

-

of H defined by

P)x~
so that fr ~'L = 51 U 5 2 , with = R

and

l(I

- P)xl_~r~

,

S2 = {X ~ H : IPxl ~- R

51 = ~ x G H

:

and

l ( I - P)xI= r }

•

IPxl

By applying P and I - P to equation

(4.4), we see that, for each

~]0,I[,

equation

(4.3) is equivalent to the system (5.1)

( I - P)x = ~ K ( I - P)Nx

(5.2)

-(I

- ~)Px + ~PNx = 0 ,

and hence every possible solution of (4.3) is necessarily such that

(5.3)

l(I - P)xI = ~ K ( I

i.e. such that

x ~

5 2.

Now, -(I

- P ) N x ~ Xr

<

r ,

(5.2) implies that, for every ~ ] 0 , I ~

because, by the o r t h o g o n a l i t y o f P, (PNx,Px) = (Nx,Px) = (PNx,x). assumption

(d) and (5.3), one has

satisfied with our choice o f - ~ REMARK 5.1. conditions

By using

,

- X) IPxl 2 + k(Nx,Px) - 0 ,

x ~

5 I.

Thus, assumption(2)

Therefore, by of Theorem 4.1 is

and the result follows.

Remark 4.1, one easily obtains that, in Corollary 5.1,

Ca) and the inequality in condition

(d) can be simultanuously and respectively

replaced by a'. I - P + N : H ~ H d'.

(Nx,Px) ~

is monotone.

0

w i t h the same conclusion f o r the equation ( 4 . 1 ) . We shall now deduce in a very simple way, from Theorem 4.1, a recent result of Br~zis and Nirenberg[4]

.

Assume that L : dom L C

the beginning of Section 4.

H --~H

satisfies the conditions listed at

Then, since, for every

x E dom L,

l(I - P)xI = ~KLx ~ ~; I K I | L x | , one has 2 (Lx,x) = (Lx,(I - P)x) ~ for every

x ~ dom L.

-$Lx| l(I - P)x~ ~

Let us denote by

- IKllLx|

@ the largest positive constant such that

174

(5.4)

(Lx,x) ~

for all x ~

dom L.

Let now B : H - ~ H

- ~-I--|Lx~2 ,

be a bounded demi-continuous mapping.

We have

the following result. COROLLARY 5.2.

Assume that the mappinqs L a n d s

satisfy the above conditions and

that the followinq conditions hold; a. I - P + B : H ~

H

b. K(I - P)B : H--~ H c. There exists

(5.5)

~

~s monotone. ..i..s...complete.l.E..continuous.

, with

O f ~

~,

such th..a.t,,for all x ~ H

and y ~ H,

(Bx - By,x) >~ ~-I~Bx%2 - c(y)

where c ( y ) depends only on y. Then, i n t ( I m L + cony Im B) Proof.

~

Im(L + B) .

Let

f ~ i n t ( I m L + cony Im B) and let us apply Theorem 4.1 with

N = f - B.

By the assumptions of regularity made

on L and B, it suffices to prove that the set of solutions of the family of equations

(5.6)

Lx + (I - X ) P x + k B x = %f

is a priori bounded independently of ~ ] 0 , I [ 4.1 will be satisfied with ~ For all

h ~ H

an open ball of center 0 and radius sufficiently large.

of sufficiently small norm, we can write n f

+ h = Lv +

~-

i=I where

v~

don L, wi ~-H, t i ~ 0

(5.6) that (1

, because then condition (2) of Theorem

t . Bw I l

,

( 1 ~ i ~ _ n ) and

n ~ t i = I. i=1

It

f o l l o w s then from

n -

~)Px

+ ~(Bx

-

~

t.Bw,)

l

i=1

z

* ~h

= ~Lv

-

Lx

.

Taking the inner product of this equality with x and using (5.4) and (5.5), we obtain

(1 - X ) l P x l z + A ~ t i ( ~ r - l l B x l 2 - c(w.))~. ÷ X ( h , x ) i=1 -~

~lLv|

|(I

- P)x~ +

~ X(Lv,x) + e - l l L x I 2 -~

@-llLxl 2 ,

and hence

(5.7)

~-1~Bxl 2 -

~ i=I

t.c(w.) i i

But, by (5.6) and

(5.1) with

N = f - B , one gets e a s i l y

~Lx~(~( ~x~ + If~ )

,

+ ( h,x)~Lv|

l ( I - P)x{ +

(~@)-IILx~2.

~(I - P ) x | 6 XIK~(~BxI + |f~) ,

which, together with (5.7), gives, for all h E H of sufficiently small norm,

(h,x)

~

(8-I

~'-l)[Bx~2 + c'(h)IBx~ + c"(h)

175

where c' and c" only depend from h. (h,x)

~

As

@-I ~ - I

, this implies that

c"'(h)

for some c"t(h), and hence, by the Banach-Steinhaus

theorem, there will exist R > 0

such t h a t

Ixl~

R,

which achieves the proof. REMARK 5.2.

More general versions have been given by Br~zis and Nirenberg in [4]

and they could be treated similarly.

We have restricted to this one for simplicity.

Let us also notice that Br~zis and Nirenberg assume that B is monotone instead of I

-

P + B.

COROLLARY 5.3. a'. B : H - ~ H ~hen

L + B

Proof.

If r in Corollary 5.2, condition is monotone and

(a) is replace d b ~

B is onto,

is onto. By Corollary 5.2, int(Im L + cony Im B) = H ~

In particular,

Im (L + B) ~

H .

by the theory of maximal monotone operators, B will be onto if

~Bxl--*~

Ix~---~,w~

if

.

We shall refer to [ 4 ] for an interesting application of Corollary 5.3 to the existence of generalized solutions in L 2, 2Tt-periodic

in t and x, of the nonlinear

wave equation

utt - Uxx

=

f(t,x,u)

.

A generalization of some of the results of [17]is obtained which replaces some Lipschitz condition in u for f by a monotonicity assumption. 6. AN APPLICATION TO A SECOND ORDER PERIODIC BOUNDARY VALUE PROBLEM. IN A HILBERT SPACE In this Section we shall apply Theorem 4.1 to the second order periodic boundary value problem (6.1)

-x"(t)

= f(t,x(t)),

(6.2)

x(O)

x(1)

where x ' = d x / d t

and

-

= x'(O)

t ~ -

x'(1)

I = [0,1] , = 0

,

f : I x HI--~ HI, with H I a real Hilbert space with inner

product ( , )I and corresponding norm

|'~ I "

One could consider similarly the

Neumann boundary conditions x'(O)

= o = x'(1)

which share with the periodic ones the feature of furnishing a non-invertible part in the abstract formulation of (6.1-2).

linear

When the boundary conditions make this

linear part invertible and f is completely continuous,

rather general results for

the solvability of the corresponding boundary value are available,

even for H I a

176

Banach space (see e.g.[20] ).

But, in contrast with the situation where H I is finite-

dimensional, the general case with periodic or Neumann boundary cnnditions is much more difficult and ~ e s not seem to have been explored.

Hence, even the very special

results we obtain here may be of interest. Let H = L2(I,HI) with the inner product

(u,v) = ~

(u(s),v(s))ds

JI

and the corresponding norm

~.~ , end let us define N on H by

(Nx)Cs)

for

x ~ H.

= f(s,x(s))

a.e.

on

I

To avoid lenghty technical discussions, which will be given in a

subsequent paper, we shall directly make our regularity assumptions on N instead of on the original mapping f .

Let us assume that :

I. N maps H into itself in a completely continuous way. 2.

(=1

r >O)(V

x ~H)

:

INx I ~

r

.

3. N is monotone, i.e. (f(s,x(s)) - f(s,y(s)),x(s) - y(s))ds

~

0

I for all x ~

H and y ~ H

Let us now define dom L

= {xEH

. dom L ~ H by : x is absolutely continuous in I together with x', x"~= H and

x(O) - x ( 1 )

= x'(O)

-

x'(1)

so that dom L is a dense subspace of H~

let

= o ~

,

L : dom L C H --~H~ x~-P-x", so

that L is closed and ker L = ~ x ~ dom L : x is a constant mapping from I into H I } ,

Im L = ~x ~ H

ker L = Im P

:

~I x(s)ds = 0 } =

with P : H - - ~ H

(ker L) "L

the orthogonal projector onto ker L defined by

Px = ~

x(s)ds

.

I

THEOREM 6.1. R~

Assume that the condit~pns above hold for N and that there exists

0 such that~ for a.e. t E 1

(6.3)

and all x ~

(f(t,,x),x) I

~

H I with

~ x | 1 ~ R, one has

(r/21¢) 2 .

Th.en~ problem (6.1-2) has a t l e a s t a (~aretheodor.v) s o l u t i o n . .Proof.

We s h a l l apply the v a r i a n t o f Theorem 4.1 mentioned i n Remark 4.1 to the

equivalent abstract equation in dom L ~ H

177 Lx = Nx . C l e a r l y , the sssumptions

completely continuous

we have made imply t h a t I - P + N i s monotone and K(I - P)N

(one shall notice that for H I infinite-dimensional,

continuous but not s compact mapping).

K is a

It suffices therefore to show that the

possible solutions of the family of equations (6.4)

Lx = (I - A)Px + $~Nx

are a p r i o r i bounded.

,

XE]O,I[

,

By applying I - P and P t o both members o f ( 6 . 4 ) , we o b t a i n Lx = ~ ( I - P)Nx ,

(6.5)

O = (I - %)Px + APNx ,

and hence, Ix"l

Letting

= ILxI(INx|

(r

x = u + v, with

.

u = Px and using elementary properties of Fourier series,

this implies that (6.6)

Iv|

~

(2T~) - 1 | x ' l

~ (211;)-2 ~x"l

~

(2TC)-2r

and max t~.I Therefore,

Iv(t)|

_~ ItrI((21~)23 I/2)

lu~ = |u | 1

if

= r' •

I

~

Ix(t)~l~lUll

R + r', one has, for all t ~

-

max t~I

I,

l v ( t ) l 1 ~, R ,

and therefore, by (G.3), for a.e. t & I,

(?(t,x(t)),x(t)) I >~ (r/2~) 2 so t h a t (6.7)

(Nx,x) ~. ( r / 2 ~ ) z .

But (6.5) impliesp after having taken the inner product with x, O = (I - ~ ) l u | 2 + %(PNx,x) , i.e.

0 = (1 - ~ ) l u | 2 + ~ ( N x , x )

- ~(Nx,v)

.

Consequently, using (6.G) and (6.7)~ one gets

0 >~ (1 - ~ ) | u 1 2 + a contradiction.

A(r/2~) 2- ~(r/(2It)

2)

= (1 - % ) l u l

2 ~

(1 - ~)(R + r ' ) 2 ,

Therefore lul 1 <

R + r'

and hence, by ( 6 . 6 ) , Ixl 1 ~ lul 1 + Iv~ 1 < and the proof is complete.

R + r' + ( 2 T c ) - 2 r

,

178

7. A CONTINUATION THEOREM FOR SOME NONLINEAR PERTURBATION OF LINEAR MAPPINGS IN REFLEXIVE BANACH SPACES Let X be a real reflexive Banach space, Z a real normed space, L : dom L ~

X --~ Z

a linear mapping such that there exist continuous projectors P : X - ~ X ,

Q : Z-~Z

for which ker L = Im P, Im L = ker Q ~ ~ P | = ~ and such that there exists a linear homeomorphism denote by

Kp,q : Z --~ dom L ~

ker P

J : Im Q--~ker L .

Let us

the linear mapping Kp(I - Q) where Kp : Im L

dom L ~ ker P is the inverse of the one-to-one and onto restriction of L to dom L ~ ker P .

Let

I~LC- X be a bounded open convex subset with

let N : c l ~ L ~

Z be a (not necessarily linear) mapping such that the mapping (dq + KpQ)N : c l ~ l - C

is stronqly continuous on c l ~ -

X --~

and

p

, i.e. such that

(JQ + Kpq)N(xn) ~ f o r every sequence ( x ) N* n n~

X

O~L,

(JQ + Kpq)N(x)

in cl3~- such that x ~ n

x

if

n---~

Let us r e c a l l that the strong c o n t i n u i t y on c l ~

.

implies the compactness on c l ~ . ,

for X a reflexive ~anach space, but the converse is not true (see e.g. L 1 2 ~ THEOREM 7.1.

).

Assume that L and N satisf~ the conditions above and that

(7.1)

Lx ~ -(I - ~)J-Ipx + ~ N x

for every (x, % ) ~

(dom L ~ fr &"L) x ] 0 , 1 ~ .

(7.2)

Then equation

Lx = Nx

has at least one solution.

Proof. As shown in [153 (see also [133 ), equation (7.3) is

Lx = - ( 1

equivalent

to

- ~)J-1px

+ ~Nx

the equation x - Px = (JQ + K p ~ ) ( - ( 1

- ~)J-1px

+~Nx)

,

i.e. to the equation

x - ~Px = X(JQ + Kp,Q)NX , in c l ~

(7.4) For each

.

Let us fix ~ ] 0 , I [

and consider the family of equations in c l ~

x - ~ P x = ~ ( J Q + Kp,Q)Nx , ~ ~ I, the mapping

I -~P

~ ~

,

I = [0,I~ .

is a linear homeomorphism of X onto itself

with ( I -~XP) - I = (I - ~ ) - I P

+ ( I - P) ,

and hence the family of equations (7.4) is equivalent to the family of equations

179

x = (I - h ~ ) - 1 ~ J Q N x + ~KpQNx = T A ( x , ~ ) , in cl~rL .

~

i ,

By our assumptions and the r e f l e x i v i t y o f X, T~ : cl~'Z x I ~ X

a compact mapping, and by (7.1) with ~

replaced by ~

is

and the equivalences of

equations described above, x - T~(x,/~) ~ 0

for every (x,~) E

fr~.x

condition

Therefore, using the Leray-5chauder's continuation theorem,

O~

.

I, the validity for

T (.,I) has at least a solution in ~ solution in dom L A ~ sequence in

~=

0 being a consequence of the

t and hence equation (7.3) has at least one

, and that for every

]0,I[ which converges to I

~]D,I[

.

and (Xn) n ~ N .

Let now (~n)n ~ N* be e a sequence in dom L ~

such

that Lx = - ( I n

- X )J-Ipx + ~ Nx n n n n

,

nEN*.

,

n ~ N* .

Therefore,

(7.5) If

we w r i t e

Xn " ~nnPx = x

n

= Yn + z

n

~ n (JQ + KpQ)NXn

with

Yn = PXn ' n ~

N*,

then

(Yn)n~N.

and ( Z n ) n ~ N .

are

bounded and, hence, the compactness of (JQ + KpQ)N and the reflexivity of X imply that, going is necessary to a subsequence, PXn = Yn .~b y ~

one has

ker L , (JQ + KpQ)NXn --~. z E X ,

so t h a t , by (7.5), z

Consequently,cl~being

n

--~

z Eker

P

if

n-~¢~.

weakly closed, x

~ y + Z = x i f n--~o~ , and x ~ c l ~ n and then, by the strong c o n t i n u i t y o f (JQ + Kp~)N, one gets

,

(JQ + KpQ)N(xn) --~(JQ + KpQ)N(y + z) • The uniqueness of the limit implies that z = (JQ + K p Q ) N ( y + z ) , i.e. x - Px = (JQ + KpQ)Nx

J

which is equivalent to (7.2) as noticed above. REMARK 7.1.

Thus the proof is complete.

It is easy to check that Theorem 7.1 can be used instead of Theorem 4.1

to prove the existence of a solution for the periodic boundary value problem (6.1-2) is assumptions I and 3 on N are replaced by the assumption I'. N maps H into itself and is strongly continuous on every bounded subset of H , the proof being entirely similar to that of Theorem 6.1.

It is an open problem to us

to know if the result still holds if one replaces in (I') the strong continuity by the complete continuity without the monotonicity condition used in 5ection 6.

180

REFERENCES

I. H. BREZIS, "Op~rateurs maximaux monotones et semi-groupes de contractions dens les espaces de Hilbert", Mathematics Studies 5, North-Holland, Amsterdam, 1973. 2. H. BREZIS and F.E. BROWDER, Nonlinear integral equations and systems of Hammerstein type, Advances in Math. IB (1975) 115-147. 3. H. BREZI5 et A. HARAUX, Image d'une somme d'op~rateurs monotones et applications, I s r a e l d. Math. 23 (1976) 165-IB6.

4. H. BREZI5 and L. NIRENBERG, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. Spuola Norm. Sup. Pisa, to appear. 5. F.E. BROWDER, Existence of periodic solutions for nonlinear equations of evolution, P~Rc- Nat. Ace d. 5ci. U.S.A. 53 (1965) 1100-1103. 6. F.E. BROWDER, Periodic solutions of nonlinear equations of evolution in infinite dimensional spaces, in "Lectures in Differential Equations", vol. I, A.K. Aziz ed., Van Nostrand, New York, 1969, 71-96. 7. F.E. BROWDER, "Nonlinear Operators and Nonlinear Equations of Evolution in Benach Spaces", Proc. S~mp. Pure Math., vol. XVII, part 2, Amer. Math. Soc., Providence, R.I., 1976. 8. L. CESARI, Functional analysis, nonlinear differential equations and the alternative method, in "Nonlinear Functional Analysis and uifferential Equations", L. Cesari, R. Kannan and J. 5chuur ed., M. Dekker, New York, 1976, 1-197. 9. L. CESARI and R. KANNAN, An abstract existence theorem at resonance, Proc. Amer. Math. Soc. 63 (lg77) 221-225. 10. D.G. BE FIGUEIREDO and C.P. GUPTA, Non-linear integral equations of Hammerstein type with indefinite linear kernel in a Hilbert space, Indaq. Math. 34 (1972} 335-344. 11. 5. FUCIK, "Ranges of Nonlinear Operators", 5 volumes, Universites Carolina Pragensis, Prague, 1977. 12. S. FUCIK, d. NECA5, J. SOUCEK, Vl. 50UCEK, "Spectral Analysis for Nonlinear Operators", Lecture Notes in Math. n ° 346, Springer, Berlin, 1973. 13. R.E. GAINES and J. MAWHIN, "Coincidence Degree and Nonlinear Differential Equations", Lecture Notes in Math. n ~ 568, Springer, Berlin, 1977. 14. J. LERAY et J. 5CHAUDER, Topologie et @quations fonctionmelles, Ann. Sci. Ec. Norm.

,5,~. 51 (1934) 45-7B. 15. J. MAWHIN, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, ~.

. D i f f e r e n t i a / .E~uations 12 (1972) 610-636. 16. J. MAWHIN t Contractive mappinge and periodically perturbed conservative systems, Arch. Math. (Brno) 12 (1976) 67-73. 17. J. MAWHIN, Solutions p~riodiques d'~quations aux d@riv~es partielles hyperboliques

non lin~aires, in "M~langes Th. Vogel", B. Rybak, P. Janssens et M. dessel ed., Presses Univ. de Bruxelles, Bruxelles, 1978, 301-315. 18. J. MAWHIN, Landesman-Lazer's type problems for nonlinear equations, Confe~ t Sam. ~ t . Univ. B ari n= 147, 1977.

181

19. J. MAWHIN and M. WI|LEM, Periodic solutions of nonlinear differential equations in Hilbert spaces~ in "Proceed. Equsdiff 78",Firenze 1978, to appear. 20. K. 5CHMITT and R. THOMPSON, Boundary value problems for infinite systems of secondorder differential equations, J. Diffe!en~al Equations 18 (1975) 277-295.

SOME CLASSES OF INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS OF CONVOLUTIONAL TYPE~)' E. Meister (TH Darmstadt

Abstract. S t a r t i n g w i t h c l a s s i c a l c o n v o l u t i o n a l i n t e g r a l equations on ~ n , t r a n s l a t i o n i n v a r i a n t operators and t h e i r symbol r e p r e s e n t a t i o n according to H~rmander are introduced. The various g e n e r a l i z a t i o n s concerning the domains G o f ~ n t e g r a t i o n lead to Wiener-Hopf i n t e g r a l and i n t e g r o - d i f f e r e n t i a l equations on ~+ and on cones. Compound i n t e g r a l and i n t e g r o - d i f f e r e n t i a l equations of the p r i n c i p a l value and L1-kernel type are discussed on ~n using r e s u l t s by Rakovsh~ik. Simonenko's theory of local type operators permits us to i n v e s t i g a t e g e n e r a l i z e d t ~ a n s l a t i o n i n v a r i a n t operators. Wiener-Hopf i n t e g r a l equations w i t h s t r o n g l y s i n g u l a r kernels correspond to equations w i t h piecewise continuous symbols in both v a r i a b l e s . Convolutional equations on quadrants and wedges are studied via the theorey of operators of b i - l o c a l type. O.

Notation.

In the sequel the f o l l o w i n g a b b r e v i a t i n g terminology is used: ]Rn : = {x : x = (x I . . . . . Xn), x EIR} : n - dim. Euclidean space with n

(0.I)

<x,y> : =

(0.2)

Ixl

: = <x,x> I / 2

:

length of vector

(0.3)

IRn

• = ]Rnu{~}

:

one-point-compactification

(0.4)

~

• = iRnu ~r

:

r a y - c o m p a c t i f i c a t i o n by adding one ideal

infinite ~n

:

Z xvy v ~=1

: s c a l a r product

element to each d i r e c t i o n

x

X

e : = -~

on

n-sphere = {xE]Rn : I x l = I }

and completing the t o p o l o g i e s by the usual way

" = (~1 . . . . .

(o.5)

Un)eIN n : m u l t i - i n d e x

n

Ipl

:=

z G

(0;6)

D~

" = Dl

(0.7)

D

: = i ~

~=1 ~1

Cmo ~n

...'D n

where

; ~ = I, .... n

*)Extended version of a General Lecture at the Dundee Conference on D i f f e r e n t i a l Equations, 31st March 1978.

183

(0.8)

(0.9) (0.10)

~P

Pl

= ~t

""'~n

~n

' ~ : dual v a r i a b l e G, Ec~ n : measurable subsets :

XE(X)

: characteristic

LP(o;G)

: = space of e q u i v a l e n c e classes of f u n c t i o n s

II f I l

function

" = IS P ( X ) ' I f ( x ) I G

L=(o;G)

• = ess sup p ( x ) . I f ( x ) I xEG

iIfll

O(X) - i

( i n case of

ck(G)

and

ck(~)

:

< ~

space of f u n c t i o n s

put

fcck(G)

max O~l~l~k

C~(IR n

• = {fEck(IR n )

• = {mEC~(IR n )

sup x~

: n-dim.

(Fm)(~) = 8 ( ~ )

: = IIfll

differentiable

such t h a t

at ~

-

is c o n t i n u o u s l y

: lim D'#f(x) : 0 f o r O
l e a s t on ~ f

< =

f o r k E ~ o , ~ e ~ n}

decreasing f u n c t i o n s dual to

d e f i n e d by

f ei<X '~>m(x)dx

i

is d e f i n e d on

since (0.16)

F

(FD~m)(C)

~'

,

~E~ n

-

e-i<x,~>~(C)d~

by

• -

maps

)

functions

2J2~-~n ~n

(0.15)

)

o

: sup(l+Ixl2)k/21D'~m(x)l

~ in

: =

f

'

Fourier transformation

(F-I~)(x) F

LP(IR n

< ~ " kE~

the space of tempered d i s t r i b u t i o n s

w i t h the i n v e r s e (0.14)

, respectively,

llfiIp

I D~ f(x)l

Schwartz'sspace o f r a p i d l y

(0.13)

, or

G : = GU~G w i t h

• =

F

l~p<~

such t h a t

p(x) ~ 0

G = ~Rn

II f l l ck(~ )

:

,

f

G ; k c ~ o u{~}

extendable to

tr

E

pdx]l/p<=

: space of k-times c o n t i n u o u s l y on

(Q.12)

of set

LP(p;G)

f o r a measurable weight f u n c t i o n

(0.11)

to

bijectively = ~(Fm)(C)

for

f E ~ ' and onto i t s e l f ,

= ~(~)

for

mE~, with property mE~

and

184

(0.17)



(0.18)

GM

=

(-l)lhI

• =

where

{~EC~(~n)

for

: lDU~(x)l~Ipu(x)l

p~ is a polynomial depending on

"space of m u l t i p l i e r 3E , ~::

m(m,9)

fE ~'

~ and

Functions of J~ and

xE~ n}

~ :

~' "

Banach spaces with norms I I ' I I z

: Banach space of a l l

for all

and

II'ii~

linear and bounded operators

A : ~÷I~ ~(~,t~)

: subspace of a l l compact operators

~(~,~)

: set of Fredholm-Noether operators

ac ~(Z,~) i f f (i) ~(A)

~(~,~)

: = A(~C) is closed in

(ii)

~(A) : = dim ~(A) = dim ker A <

(iii)

~(a) : = dim ~/R(A)= dim coker A < ~(A) = ind A : = ~(A) -~(A) : "index of A".

In case of 1.

~

= JC

one w r i t e s

~(~)

etc.

Introduction.

D e f i n i t i o n 1.1: AI,A2E~(~,~) are called "equivalent" i f f AI-A2E~(~,V) Or: = {BC~(~,~) : B~A} gives the quotient-space with norm (1.1)

llc~l

: =

inf

II A+VII

There is the well-known theorem by ATKINSON (1951) [2 ] Theorem 1.1 : E~(~,~) , iff BrE~, ~, (1.2)

and

:

Let ~ , ~ be B-spaces then AE~(~,~) is Fredholm-Noether, i . e~ there e x i s t a " l e f t - r e g u l a r i z e r " B~E~(~,~) ~ a " r i g h t - r e g u l a r i z e r "

V I E ~ Z ) , and

V2E~ )

B~A = IZ + V1 and

are Fredholm-Riesz operators.

such t h a t ABr = I~ + V2

t85

Remark 1.1 : This has been generalized to pairs of Fr~chet spaces ~,l~ and linear, densely defined, closed operators A (cf, eg, GOHBERG& KRE[N (1957) [42] or PRUSSDORF(1974) [71]. There you may find the properties of Fredholm-Noether operators listed. Let kELl(~n) , ~ELp(~n) , 1~p~ , then

(1.3)

-

k(x-y)~(y)dy

e x i s t s a, e~ on

~n

and

(1.4)

II kll

Ii k~mllp

I"

II ~lIp

holds "

The Fourier transformation may be continued from

~ onto

Lp ( ~ n )

, 1~p~2 , such

that FE~(LP(~ n), LP'(~n)) where ~I ÷ ~, i = I . F is then u n i t a r y on The "convolutional integral equation of the second kind" (1.5) (Am)(x) : = re(x) - (k*m)(x) = f(x)ELP(m n) is algebraized by F to

:

[1-~(~)]~(~)

(1.6)

1<~<2

~({)EL p' (m n )

( c f . e@. TITCHMARSH [ 101]).Asolution e x i s t s - then uniquely - i f f (1.7)

L2(~n)

~a(~) : = Z-k({) # 0

on

the "symbol of A":

~n

I t is given by ;(~)

: ~(~) + ........k(~) .~(~)

(1.8)

1-C(~) A

: [1 + ~(~)] . f ( ~ )

where 1 + ~(C) is an element of the "Wiener-algebra" ~ : = { ÷ FLI(~n) the Wiener-Levi theorem. On the other hand let (1.9)

(p(D)@)(x) :

be a d i f f e r e n t i a l

S

due to

a .(D~m)(x) : f ( x ) e ~'

equation of order

m with constant c o e f f i c i e n t s . Applying the

F-transformation t h i s y i e l d s (1.10)

p(~).~(C)

= ~(~)E ~ ' .

This leads to the "problem of d i v i s i o n " which asks f o r conditions under which

(I.Ii)

~(~) = ~

E 3~'

186 gives the transform of a s o l u t i o n . Taking

f = ~c9'

of existence of the "fundamental s o l u t i o n s "

one is led to the question

E E 9 ' such that

~(~) = [p(~)]-I E~,. Convolutional i n t e g r a l equations and d i f f e r e n t i a l equations with constant c o e f f i cients are a l l special types of the f o l l o w i n g operators: D e f i n i t i o n 1.2a) Let 3E be a B-space of functions or d i s t r i b u t i o n s on let

Th : ~ + 3 E

(1.12) then then

and

be defined by

(Thm)(x) : = m(x-h) ~h

Rn

for

mc~

and hER n

is called a " t r a n s l a t i o n operator on ~

". I f

f C 3 E ~, the dual o f 3 E ,

!

Th : 3(1' ÷ 3( ~ is defined by

(1.13)

< ~ f,m>

: =

b) A e ~ ( 3 ( , ~ )

for

fc~'

and

me~ .

- or at least l i n e a r , densely defined, and closed

- is called " t r a n s l a t i o n i n v a r i a n t " i f

on

(1.14)

(AThm)(x) = (ThAm)(x)

In t h i s case one writes

for

AC~(JE,~). H~RMANDER [51]

Theorem 1.2 :

Let ~ c 3 E , ~ c ~

duals - where

~

is dense in the topologies of

Al~m

hC~ n . proved in 1960

be B-spaces of functions on

Then there e x i s t s a uniquely defined (1.15)

mc3E and

~AC ~'

= F- I OA.Fm

~

and ~

~n _ or t h e i r , resp., AE ~(~,~).

such that for all

~

~.

~A is called the "symbol of A". I f JL~E,~) denotes the set of a l l s~T~ibols to p l i e r f u n c t i o n s " - then i t is known that [51] J~(k2(~ n ) ) = L~(~ n)

AE~)

- also called ' ~ u l t i -

~ ( L P ( ~ n) c L~(~ n) ~V~(LP(mn) , Lq(mn)) = {0} Extending to Frech~t-spaces one has

or f o r i t s dual

~'

:

if

l~q
.

187

Now, e. g. on (1.16)

L2(• n) , the s o l u t i o n to the t r a n s l a t i o n

(am)(x) = (F-loA.Fm)(x)

i n v a r i a n t equation

= f(x)Ek2(~)

is found via F-transformation: (1.17)

~a(~). ~(~) = ~(~)

e L2(~)

.

I t exists f o r every f (uniquely) iff A and hence [OA(E)]-Ic L~(R n ) , given by

is " e l l i p t i c " ,

(1.18)

~(~)

=

~1(~).~(5)

(1.19)

m(x)

=

(F-l~AZ(~)'Ff)(x)

A-I

: F-IGA1.F e ~ ( L ~ n)) and

i. e.

infI~A(~)l ~c~ n

> 0

, thus , i . e.

C~(L2(~n))

too .

Remark 1.2 : For 1~p<2 this argument works only i f OAIEj{(LP(LRn)) which is the case f o r s u f f i c i e n t l y smooth OA(~) (cf. e.g. MICHLIN (1965)[61]!). Examples:

I. The " H i l b e r t transformation"

(1.20)

(Hm)(x) : = _1 ~S m(y)dy y _x IT

for

in one dimension is given by mc LP(m) , l
=co

i t s symbol is (1.21)

OH(~) = - i - s i g n

(cf. e. g. TITCHMARSH [101 ], p. 120, ( 5 . 1 8 ) ) . 2. The "Calder6n-Zygmund-Michlin-operator (1.22)

where

(1.23)

(M~)(x) : = ( a l + A f / . ) ( x ) ae~ , f(O)

f(~I)

: = a.~(x) + p.v. f ~Rn Ix-Yl n ~(y)dy

eLq(zn)

~M(~)

, l
(cf. e.g. MICHLIN [61]

, p. 100).

Modern theory of convolutional ential

(CMO)" given by

integral,

integro-differential,and

equations aims into three main d i r e c t i o n s :

pseudo-differ-

188

A. (!)

To admit more ~eneral domains ~n

+

: = {xE~n

: Xn ~ O}

G of i n t e g r a t i o n instead of

kELI(~ n)

~n

, e

g.

:

Wiener-Hopf i n t e g r a l equations (WHIEs), (ii)

Fc~ n

cones defined by smooth (n-2)-dimensional manifolds on

smooth s e m i - i n f i n i t e (iii)quadrants

domains

sn

and other

c~ n ,

like

~2++ : = {xE~2 ' Xl ~X 2 => O} or wedges W(~)::{xE~ 3 , x l + i x 2 = r ~ , X3ER} or cones w i t h edges: polyhedral domains.

0 g~,

A l l of these are special cases of Definition a

"General Wiener-Hopf operators (WH0s)" : Let

1.3 :

P = P2E~(3E)

B-space, and

a linear,

AE~(]E) , 3£

continuous p r o j e c t o r on 3C . Then i t

is

given by (1.24)

(Tp(A)~)(x)

: = (PAlm(p)~)(x)

Remark 1.3 :

In the " c l a s s i c a l

l < p < = , and

P = ×G"

B.

cases" mentioned above we got

the "space p r o j e c t o r "

for

Gc]Rn

and

~_= LP(~Rn)

,

A E~((3E)

To admit " v a r i a b l e kernel f u n c t i o n s " , e. g. the "generalized L l - c o n v o l u t i o n

i n t e g r a l equations" (1.25) or

(A~)(x) : = a ( x ) ~ ( x ) - ~ k ( x , x - y ) ~ ( y ) d y = f ( x ) E L P ( ~ n ) IRn - even more -

Definition

1.4 : Equations w i t h "generalized t r a n s l a t i o n

(1.26)

(A~)(x) : = ( g i l x ~ A ( X , ~ ) . F y ~ ~ + ~ ) ( x

where~,,e, g.

OA(X,~) = j =sl a j ( x ) .OA.(~ O ) , aj

or

being m u l t i p l i e r

Rn'

~A

operator on ~

symbols, and

Combinations of

A and

V

on

~n

being a completely continuous

B

lead to important classes of s i n g u l a r i n t e g r a l

~(~ , e. g.

30 = ~ c ~ , a Ljapounov-curve: equations"

(1.27)

being continuous c o e f f i c i e n t s

.

equations on manifolds (j)

i n v a r i a n t operators ( G T I ~ ) "

"classical

Cauchy-type s i n g u l a r i n t e g r a l

(K~)(x) : = a ( x ) ~ ( x ) + ~ i f k ( x , y ) ~ ( y ) d y x - y

= f(x)

in various spaces ~ = C~(r) , mE~ , O<~
with

189 (jj)

CMOsor singular integral equations with "Giraud kernels" with

f(x,e) C.

instead of

ac-{ or

f(o) eLq(En)

in

a(x)

and

eq. (1.22), respectively.

To admit d i f f e r e n t i a l operators which means to study " i n t e g r o - d i f f e r e n t i a l

equations of convolutional or principal value type" l i k e m

(1.28)

S (a (x)l+b (x)H+K)D~m(x) = f ( x ) ~=0

in Sobolev-spaces (1.29)

Wm'P(IR)

~

or

Wm'P(IR+)

or even

(A D~# )(x) + (Vm)(x) = f ( x ) e ~ =

LP(8) , GclRn

where the

K are one-dimensional generalized Ll-convolutions (cf. eq. (1.25)), n the A~ = F-I OA(X,~). F , ue]li° , are a r b i t r a r y generalized t r a n s l a t i o n i n v a r i a n t operators and V : Wm'P(G) ÷ LP(G) a compact operator, respectively. A l l these operators, of course, are special versions of "pseudo-differential operators", a term introduced by KOHN & NIRENBERG in 1965 [ 5 2 ] in standard use as (1.30)

where

L : 3£ ÷ I ~ defined by

(Lm)(x) : = (F -10L(X,~)Fm)(x ) OL(X,~ )

, and, since then,

for

mE~

may be expanded asymptotically with respect to

of functions

~ _k(X,~)

homogeneous with respect to

o _k ( x , t ~ )

=

k(X,~)

(i.31)

°L(X'~) ~k=oE ~m-k (x,c)

t ~-k ~

for

~

C of degree

into a series m-k. Here

t > 0 , kE]No , and

co

for

I~I ÷ ~

We shall desist from entering into the discussion of the r e s u l t s on pseudod i f f e r e n t i a l equations since there are many excellent survey a r t i c l e s (cf. e. g. FRIEDRICHS [36], SEELEY [83], CORDES [12], ESKIN [31], KUMANO-GO[57]). Here we shall keep close to the lines of SIMONENKO's theory started in (1964, '65)[91,92 ] and shall report mainly on results of our research groups. Acknowledgement.

This paper has been prepared partly during the time when the

author was a visiting professor at the Depar~ent of Mathematics, University of Regina, Saskatchewan. He wants to thank the Head of the Department, Prof. E. Lo Koh, for arranging ideal wo~ki~ conditions and for the great hospitality there. The author wishes also to thank Dr. F.-O. Speck, TH Darmstadt, for his valuable criticism and substantial remarks during the preparation of the manuscript.

190 2.,~,,,Integral- and i n t e g r g - d i f f e r e n t i a l equat!ons of the Wiener-Hopf type

WIENER & HOPF studied in 1931 [ 107 ! certain types of homogeneous convolutional integral equations on the h a l f - l i n e R+ in connection with problems of radiative transfer. They developed the function-theoretic method, a f t e r applying the Fourier transformation. This is now called the "Wiener-Hopf-technique". KREYN in 1958154] completed the classical theory of WHIEs in LP-spaces (2.1)

(W~(x) : = ~(x) - I _ ~

k(x-y)~(y)dy = f ( x )

E ~+

0

where

kELI(~)

and

fE3(÷= LP(~+) , 1~ps= , or certain closed subspaces of

L~(~+) such as Co(~+) for instance, are given and me~.+ is sought. GOHBERG & KRE~N [43 ] extended these functional-analytic investigations to systems of WHIEs. Making use of the convolution theorem of the F-transformation arrives at the image equation to eq. (2.1) (2.2)

(1%p~2) one

h (~) = fP'*(~)e L P ' ( ~ ) [i-~(~) ] • ~+(~) - ^-

where (2.3)

~÷(~) : = (FP+m)(C) : : ( F × ~ m ) ( ~ )

and A

(2.4)

h-(~) : = (FP_h)(c) : = (Fx N -h)(~)

are one-sided F-transforms which may be continued holomorphically into the upper, H+,

and lower, H-, complex half-plane, respectively. Here we put

[

--~f

(2.5)

h(x) : :

k(x-y)m(y)dy

for

x < 0

for

x > 0 .

o 0

Equation (2.2) a c t u a l l y denotes a "Riemann boundary value problem for the l i n e " which is a special form of

(2.6)

¢+(t)

= G(t)-¢-(t)

+ g(t)

,

tEF ,

F being a smooth contour c C,G(t), g(t) given data and ¢±(t) the unknown boundary values of a (sectionally) holomorphic function ~(z) vanishing for z-~. In solving such problems (cf. e. g. the books by MUSHKHELISHVILI (1953) [62 ] or GAKHOV (1966) [37]) one of the crucial steps is the p o s s i b i l i t y of " f a c t o r i z a t i o n "

191

of

^ -1 G(t) - or in our example of [ 1 - k ( ~ ) ] - into

(2.7)

G(t) = a - ( t ) - f t - P ÷ 1 < A+(t) ~t-p_ z

with functions in

Am(z)

holomorphic in

~ , being bounded and

# 0

on

denotes the "winding number of (2.8)

<:=~

1 [arg

This makes sense only f o r

G along

G(t)]

i.e.

Ow(~) : = I - ~ ( ~ ) # 0

Theorem 2.1 : (WHO)

W =

(2.9)

Let

inf

with

under c e r t a i n smoothness assumptions - at l e a s t A

GELS(?). In case of

1-k(~)E~),

is always p o s s i b l e f o r " e l l i p t i c " on

kELI(~),

P+(I-k~)P+

F "-

r"

G(t) # 0

one has to know a b i t more than Wiener-algebra, f a c t o r i z a t i o n

D± , the i n t e r i o r and e x t e r i o r domains of F

D± : = D~F . p+~ D ± are chosen a r b i t r a r y and

convolution operator,

~ E . The r e s u l t by KRE~N[34] is given in the f ~ l w i n g

fEX~as

above) be given. Then the Wiener-Hopf-operator

P+ : = x~+-

IOw(~)I = i n f

the one-dimension~

Ii-~(~)I

is

E ~(3C~+) (Fredholm-Noether) i f f

> 0 .

I f t h i s c o n d i t i o n holds then (i)

m(W) = max(O,<) , 8(W) = max(O,-<)

(ii)

for

< > 0

there e x i s t s a base

~ok E L l ( ~ + ) n C o ( ~ + ) d mo,k+l(X) = (2.10)

(iii)

for

(2.11)

mo,j(+O)

=

%,<(+0)

=

< < 0

and

ind W = u(W) = K

{mol . . . . . mo<}

of

ker W such t h a t

and mok(X~.,

0

, k = 1 ..... <-i , j = 1 . . . . . <-1

the orthogonal r e s o l v a b i l i t y

/ f(x).~ok(X)dx = 0 o

for

c o n d i t i o n s are

k = 1.....

I
where (2.12)

(W"~ok)(x) : = ~ok (x) -

7 k(y-X)~ok(Y)dY

= 0

0

(iv)

for

< ~ 0

a s o l u t i o n to the inhomogeneous WHIE is expressed by means of

the r e s o l v e n t kernel (2.13)

minh(X ) = f ( x ) + I ~ ( x , y ) f ( y ) d y o

where we have the r e s o l v e n t equation

192 oo

(2.14)

y(x,y) : Y l ( x - y ) + - { 2 ( Y - X ) + ~ l ( x - t ) Y 2 ( Y - t ) d t 0

with uniquely defined (2.15)

A

l-k(~)

Remarks: 2.1 :

Ll(IR)-functions

5,1,y 2 via the factorization of -i

K

: (I+(FP+YI)(~))(~)

.(I+(FP+~2)(-~))

.

The reasoning which leads to theorem 2.1 has been carried over to n,

the case of ~ + , n m2; instead of ~+ by GOLDENSTEIN & GOHBERG in 1960 [48] where the f a c t o r i z a t i o n applies to the n-th F-variable Cn in ~. In this case A [ a r g ( l _ k ( ~ l , " " " ,~n))] is always 0 f o r an e l l i p t i c WHO since due to K : : T ~i ~n=-~ the Riemann-Lebesgue l e n a , continuously on

~(~i . . . . . ~n)ECo ( ~ n )

~' : = (~I . . . . . ~n-I )

and tends to

implies that 0

for

K depends

l~'I ÷ ~

•

2.2 : Concerning the group B of generalizations {AHBAGJAN in 1968 [7] ] and RAKOVSH~IK in 1963 [ 75 ] t r e a t e d "variable kernel WHOs" (2.16)

(Wm)(x) = m(x) - f

k(x,x-y)m(y)dy = f ( x )

E3E+

IR+ n

co

where k(x,.)ELI(IR n) and satisfying the conditions: (2.17)

k(x,t) = z a i ( x ) k i ( t ) with the i=1

a i ( x ) e C ( I R n)

such that

sup l a . ( x ) I

X#~n

1

< ~

ai(x ) and

for all

ki(t )

ielN

oo

(2.18)

kieL1(IR ) such t h a t

(2.19)

inf Ii-k(-,~)I EIRn

= IIk i II 1 < i=i

A

> 0

in ~AHBAGJAN's paper, while RAKOVSH~IK[75 ] assumes ~o

(2.20)

m(x)~+ I k(x,x-y)m(y)dy

to be a bounded operator on X = LP(IR), ISp<~ , be compact from f o r any f i n i t e (2.21)

[ a,b] c]R , lira

k ( x , t ) =; k+(t) CLI(]R)

X-~_+~

(2.22)

Ik(x,t) and

- k+(t) l ~ _ + ( t ) - n + ( x )

~+(t)ELI(IR),

lim +X_~+oo

A

(2.23)

inf

~elR

_II-N+(C)] _ > 0

i

f o r large

n+(x) = 0 -

+x > 0

LP(N) ÷ LP(a,b)

193 Here A

oo

__

1-k+(~) -~ The proofs are worked out by inserting terms, e. g. (2.25)

(W~)(x) = ~(x)

i~lai(~).~nki(x-y)m(y)dy +

-

s [ai(x)-ai(~)]. I ki(x-y)~(y)dy i=l Nn +

= (l-W=lm(x) + (wlm)(X)

= f ( x ) E3E~+

where I-W~ c~(3E+) and even boundedly i n v e r t i b l e f o r ~ = 0 ( i . e . always' for n~2!) and W1 is the sum of a compact operator and one of "small norm", so being E~(~+) too. Thus the index of

W is the same l i k e that of the f i r s t

term.

2.3 : WHOs, p a r t i c u l a r l y for ~+ , have been studied also f o r the whole scale of Sobolev-Slobodezki-spaces ws'P(~+) and w~'P(~+), the functions of ws'P(R) : = { f ~ ' : F-I(1+I~I2) s/2 FfcLP(~)} ; l~p<~ , sE~, also called "spaces of Bessel p o t e n t i a l s " , and the subspace of them having supports, supp f , in ~+ while Ws ' P ( ~ + ) denotes the r e s t r i c t i o n s P+f of fEwS'P(~) to ~+ Due to the Sobolev embedding theorem i t is well-known (cf. e.g. TALENTI (1973)[100], p. 28) that Wo' s,2,~+)^wS,2(o ~ - ) = {0} for s ~ - 1 / 2 but span { ~ , 6 ' , . . . , ~ ( n ) } f o r ,s,2,~ .s,2, s < -1/2 and the largest integer n < - s - i / 2 while wo ~ + ) m w o (~_) = wS'2(~) for Is~ < 1/2 and = { mEwS'2(~) : m(k)(o) = O, k = 0 . . . . . n - I } for n-1/2 < s < n + 1/2, nE~. The defect pair WHO then depends on the number n

(~(W),B(W))

being related to

of the one-dimensional

n-I/2 < s < n+1/2 (see f o r the

d e t a i l s the paper by TALENTI, pgs. 63 - 71!) 2.4 : first

The theory of WHOs on the h a l f - l i n e has been generalized to operators of the kind where the symbol is given by

C(¢)

vanishing at i n f i n i t y and to more

general " n o n - e l l i p t i c or non-normal WHOs" whose symbols may be written in the form ~-i < N ~-~. n. aw(C) = ( c - i ) -n~ G'(C)(~-~-~) G+(C)'j=~ 1 (~-TT ~) J

(2.26) with

~jER

being d i s t i n c t , n , NE]~° ; n j E ~ . Considering the d e t a i l s to these

questions being studied on a functional a n a l y t i c basis since about

1965 by

SAMKO [78] and mainly by PR~SSDORF (1965, '67, '69) [68~69~70 ] look at PR~SSDORF's book (1974)[71]! TALENTI (loc. c i t ) is involved mainly in WHOs of the f i r s t kind (pgs. 71 - 77) which play a dominant role in the study of mixed boundary value problems in

~+2 (cf. e.g. PEETRE (1963)[64] , SHAMIR (1962, '63) [84,85 ]

and

194

f o r the higher dimensional case: ESKIN's book (1973)[31]!) 2.5 :

In quite a s i m i l a r way "two-part composite convolutional equations" and t h e i r

adjoints the "dual i n t e g r a l equations" may be treated via the F-transformation and the Riemann boundary value problem: 0

(2.27)

(W2m)(x) : = ~ ( x ) ' ~ ( x ) - ~ k l ( X - y ) ~ ( y ) d y - ~ k2(x-y)m(y)dy = f ( x ) , x ~ "~

where

u(x) = u I

for

x < 0

and

0

u(x) = u 2 f o r

x > O, r e s p e c t i v e l y .

(.~)(x)={

U l ' # ( x ) " 7 kI(Y-X)~(y)dy : g_(x) , x < 0

(2.28)

~2"~(x)

- 7 k2(Y-X)~(y)dy

= g+(×)

, x > 0 .

A good survey on t h i s subjects with a p p l i c a t i o n s to other dual i n t e g r a l equations is provided by the book by ZABREYKO et a l .

(1975)[108] in Chap. V I I I , §§ 4 - 6.

We are now going to review the r e s u l t s on i n t e g r o - d i f f e r e n t i a l

eqs. of the Wiener-

Hopf type, i . e. some kinds of general WHOs acting between suitable Sobolev spaces, viz. m

(2.29)

(Lm)(x) : =

S {a (x)D~m(x)+b (x)~ c (x)k (x-y)D~m(y)dy} = f ( x ) ,

x > O.

BANCURI in 1969 [3] treated only the case of constant c o e f f i c i e n t s a , b , c assuming b • c 1 or - O , k ELI(]IR), ' f E L I ( ~ ) given and q)~'O ~ , l ( m ) ~L =~+ ~ sought, i.e. m. . . . . m(m)ELI(]R+) and m(+O) . . . , ~(m-I )~ +0 ) = O. He applied the F - t r a n s f o r mation and a r r i v e d at the Riemann boundary value problem (BVP) a f t e r some manipulations: (2.30)

m ^ ^ ^ ^ ~, [a u + ku(~)] ~u m+(~) _ h-(~) = f + ( ~ ) E F L l ( ~ + ) ~=0

Set (2.31)

~+(~) : = am.(i+~)m~+(~) ^_ ^ (~1 : = h - ( O

A

(2.32)

G(~)

: = 1÷~(~) : ={~=om am " ~ +k + ~ m(~) ja ~-i

(2.33)

g(~)

: = f ÷ ( ~ ) , g(~)

(2.34)

h-(~) : = (Fx]R ( x ) . z

and

A

^

where m

oo

f k (x-y)D~m(y)dy)(~)

!J=O 0

we get the equivalent BVP A+

(2.35)

A

~ (¢) = G(~)~-(~) + g(~)

in ~]~(]R)

.

195 which is e l l i p t i c i f f G(~) # 0 on JR, GERLACH (1969) [38] admits a l l the Kre~'n spaces and the subspaces )[~m) : = {q)e)E+ : Du m ~ f o r u = 0 . . . . . m} or ]E[(+m) : = {mE~(+m): (Djm)(+O)= O; j = 0 . . . . . m-l}. by w r i t i n g

F i r s t he t r e a t s the constant c o e f f i c i e n t s '

D = (D-il)

case too and reorders

+ il m

(2.36)

(Lm)(x) = P+

Now, introducing

~(x) : = ( D - i l ) - m m ( x )

GmC~C(~+,3E~m)) (2.37)

= (Gmm)(x) = (gm*m)(x)

where

(kgm~)(x) = (W+Vm)~(x) = f(x)c3E+

(2.38)

where

W = P+ + P+(km,l, +

is a classical

WHO, since ( . . . )

a c t u a l l y having f i n i t e (2.39)

symbol

= f(x)c~+

he gets

as a perturbed WHIE on ~+

Then

~ (a , l + k p , l ~ ) ( D - i l ) U m ( x ) p=o

iff

~w(~) # 0

is a L l ( ] R ) - k e r n e l

convolution and

Vm is compact,

rank, given by

(Vm~)(x) : = LGmE~(J~ F)

m-1 ~ Gm_ .[a ,1-1+kp,l*])P+ ~=o

m-1 v ~ [(D-il)Vgm~](+O). E G~-j+IX]R+'Kj,I ( x ) l v=o j=o

WE~'(~c+)

on

~

"

which is the case, according theorem 2 . 1 ~ i f f

the

. Now, this is given by

Ow(~) = OA(~)'gm(~ ) =

m ^ E (a +k (~))~'¢[2/~(~-i)] -m p=o

Then Gerlach treats the case of continuous c o e f f i c i e n t s ap(x),b ( x ) , c p ( x ) using a simpler version of RAKOVSHCIK's compactness theorem (1963)[75] of convolutions multiplied

by functions:

Put

m ,Bp,yu

f o r the l i m i t i n g

x + ~ and rearrange eq. (2.29) to read m (2.40) (km)(x) = P+ ~ (~ l+~p~ .k~*)(D~m)(x) p=o

values f o r

ap,bp,c

( : : ~)m(x))

m

+ P+ ~] [(a ( x ) - ~ ) I p=o + ~ .k~(c~(y)-~) Then

~)

+ (b (x)-6p)k * c ( y ) ' I

+

• l](DUo)(x) .

is a WHIDO of the type above and the last term of eq. (2.40) being

compact from

3(~(m) into ~F.. The same argument applies to the subspaces ~ )

which coincide with

o'P(~+) Wm

for

l
as

196 GERLACH's r e s u l t s may be summarized in Theorem 2.2 : on ~ + , ~ eq. (2.29)

Let

a ,b , c -~ C ( ~- )- ,

k f~Ll(~)_

for

u= 0,1 . . . . . m with

: = lim a~(x) ; 6 similarly___ defined. Then the WHIDO L is x + + ~ e ~ + , )~ + ) or' Y ~E( ~ ) , ~ o + ) i f f i t s "main symbol"

(2.41)

~ )

: =

m ^ z [a +8 ~ "k (~)]-~ ~ # 0 ~=0

~

on

am(X ) ~ 0 in

N~

P p

I f t h i s is true one has the r e l a t i o n s (2.42 a)

0 s e(Lo) ~ ~(L) s ~(Lo) + m

(2.42 b)

0 ~ 6(L) g 6(Lo)

where

Lo : = Ll~(m ) ~0+

and (2.43 a)

ind L

= v(~

= < + m/2

(2.43 b)

ind Lo = v(Lo) = v ( ~

= v(L)

= K - m/2

with the winding-number (2.44)

<(L) = ~(Lo) = ~ L a r g

The spectrum of (2.45)

L

Ow(~)]~=_~

is given by

s(L) : = {zE¢ : z = q L ( ~ ) , ~ e ~ } u { z c C

Remarks: 2.6 :

:(~(A-zl),6(A-zl))

The theory of WHIDEs has been generalized from ~+

one takes (n-l)-dimensional F-transformation with respect to one a r r i v e s at an equation in the remaining v a r i a b l e x n > 0 ~' : = (¢1 . . . . . ¢n-i ) equations (2.46)

to

R n+. I f

x' : = (x I . . . . . Xn_ I ) but containing now

as parameters. So i t is quite natural to i n v e s t i g a t e

P+A(e~Dn)m+(Xn) = f(Xn) , x n > 0

where to

# (0,0)}.

8' : = ~ ' / I ~ ' I

~+.

Here

(2.47)

A(8',Dn)

has been f i x e d and

P+ denotes the r e s t r i c t i o n

is defined as a p s e u d o d i f f e r e n t i a l

A(e',Dn)~+(Xn)

operator

operator (PDO) by

_i ^ : = (F n A(e',~n)Fnm(Yn))(Xn)

A

where

A(e',(n)

is homogeneous of degree

m with respect to

(n

or even a more

general one ( c f . the work by VlSHIK & ESKIN (1965,'67,'73) ~i03,104,105] RABINOVI~ (1969 , ' 7 1 , ' 7 2 ) [ 7 2 , 7 3 , 7 4 ] (1971,'73)

[17,1~ and others!).

, BOUTET DE MONVEL ( 1 9 6 9 , ' 7 1 ) [ 5 , 6 ] ,

DIKANSKI[

197

2.7 :

WHIEs and WHIDEs may be considered not only for C-valued or sN-valued

functions but in a more general context as B-space-valued equations. Then the problem for ~ +n , n~ 2, may be f i t t e d into the theory too. Concerning a general theory of operator WHeqs. FELDMAN investigated several cases (1971)[32,33,34] in connection with problems of radiative energy transfer. GRABMOLLER(1976, 1977) [49,50] discussed such i n t e g r o - d i f f e r e n t i a l equations on ~+ of f i r s t order with a linear, closed operator (2.48)

- A generating an analytic semi-group involved

m'(t)+c(Am)(t) + 7 ho(t-s)(Am)(s)ds + ~7 hl(t-s)m'(s)ds O

where h o , h l E L l ( ~ )

= f(t)E~.~

O

are scalar-valued functions, ~ +

a r e f l e x i v e B-space and

denotes the strong d e r i v a t i v e , the integrals to be understood in the Bochner sense of LP(R t;31[+), %aEt. He is mainly interested in the asymptotic behavior of the solution as t ÷ ~. 3.

Compound integral and i n t e g r o - d i f f e r e n t i a l and Ll-kernel type on ~n

equations of the princ!pal value

MICHLIN [ 60 ]introduced in 1948 the notion of the symbol f o r singular Cauchy-type integrals along curves r c £ and also f o r operators on ~ n , nm2 . He and mainly CALDERON & ZYGMUNDstudied the mapping behavior of the CMOs since 1956 (cf. e.g. [ 8 ]). The f i r s t systematic treatment of the corresponding integral equations probably was published in MICHLIN's book (1962) whose English translation appeared in 1965 [61]. A more recent account, also on i n t e g r o - d i f f e r e ~ t i a l equations with CMOs as c o e f f i c i e n t s , may be found in Chap. IX of the book by ZABREYKO et al. (loc. c i t . ) . AGRANOVI~ (1965) treated equations of the following type in his extensive survey a r t i c l e [1 ]: (3.1)

(Am)(x) : =

as an operator

s (MuDUm)(x) + ( T ~ ( x ) : f ( x ) l~l~m

A : wm+~'2(~ n) ÷ W~'2(~ n ) , or

~n

replaced by

~+n

or a smooth

compact manifold ~O . The symbols (3.2)

oM (x,~) : = au(x) + (p.v. Fy~ fu(X'ly_] y -_) )(~) lyl n

> n-I are assumed to be EcP(]R n ,Hq(sn) ) where pc]No and q ~ such that, by Sobolev's embedding theorem, they form an algebra of continuous functions on ]Rn>~ En which are homogeneous of degree zero in ~ . He shows that to every such function o(x,~) there corresponds a characteristic f(x,O) EcP(IR n,Hq'n/2(sn )) (theorem 7.12). The operator T is one of order almost m-i which would be n compact for a compact manifold ~9 instead of ]Rn or JR+ by Rellich 'S c r i t e r i o n .

198

AGRANOVI~ proves a couple of theorems which give necessary and s u f f i c i e n t conditions f o r

A to be Fredholm-Noether by means of the e l l i p t i c i t y

condition of the

symbol or the existence of a - p r i o r i estimates (theorem 12.1). He obtains then the well-known properties f o r e l l i p t i c

operators, such as r e g u l a r i t y , s t a b i l i t y with

respect to parameters etc. (cf. his theorems 12.2, 12.3, 12.4L). But, in the case of

IRn or IR+ n he does not give regularizers in the sense of theorem i . I above. SEELEY[82] investigated at the same time (1965) singular i n t e g r o - d i f f e r e n t i a l

operators on vector bundles of smooth manifolds ~P and tensor-products of such. We are not going to enter into t h i s detailed material but j u s t want to give two d i f f e r e n t approaches: one relying on DONIG's work in (1973,'76) [19, 21 ] and the other on SIMONENKO's (1964,'65)!91,92] , RABINOVI~'s (1969-'72)[72,73,7'4] and SPECK's approach (1974-'77)[96,98]. (n=i) (3.3)

In 1973 DONIG[ 19] treated the case of ~R

with the singular i n t e g r o - d i f f e r e n t i a l operator (SIDO) m (A~)(x) : = ~ {au(x)l+b (x)-H+c (x).ku~}(DP~)(x) = f ( x ) p=o

on Sobolev-Slobodezki spaces

ws'P(IR) , sEIR, l
am(X), bm(X), Cm(X)CC(]R ) and a (x)-a (=) etc. CLo(]R ) = {~L=(]R): mes {Ix~i>n : I~(x)l > E} +O,n + ~ , for a l l c > O} and where fEWs-m'P(~R)

is

given. He applies the RAKOVSH~IK technique (1963)[75] by defining the symbol of A through A

(3.4)

OA(X,~ ) : = [am(X)_ibm(x)sig n ~ + Cm(~)km(~) ] ~m + m-1 + Z ~a ( ~ ) - i b (~)sign ~ + c#(~)C ( ~ ) ] ~ p=O )~

~

I]

:=~u{-~}u{+=}. ~A(X,~) # 0 on ]RxIR , where Using the Bessel potential operators jm : = F-1(1+I~I2)-m/2F he gets the relations

which is called " e l l i p t i c "

(3.5)

iff

Dmjm : ( i l l ) m (l+rmw)

with a rmcLl(]R).

Then he treats the case of constant coefficients f i r s t . r i g h t - h a n d side by

A is m u l t i p l i e d from the

jm _ s i m i l a r to GERLACH's approach - leading to the equation m

(3.6)

(3.7)

(A dm~)(x) = (B'm~)(x) +

B"

: =

~ ~ lJ=o

r ~( p

where

I + b H + k ~ ) " { (iH)m (a

u

~

I

for

~ =m

for

O~ u-<m-I

and some r~ELI(1R). He succeeded in c o n s t r u c t i n g a bounded i n v e r s e operator CmC~(LP(IRn)), l
Cm = Mml(iH) m + hl,m~ + h2,m~H

199

where Mm : = am.l + bm.H, H is the Hilbert transform and hl, m , h2, m E L I ( ~ ) e x i s t due to the Wiener-Levi theorem. He gives an e x p l i c i t formula, though a

complicated, for finding the a - p r i o r i estimate

(3.9)

^

hl,m(~ )

and

h2,m(~ ). The existence of

Cm then implies

¥ li~ II m,p ~ II Aml] o,p

with a

y > 0 , that means "strong coerciveness". In the general case the constant

c o e f f i c i e n t operator with ~m : ~m(=)' Bm : = bm(=) is s p l i t off and is inverted with Cm as the inverse (A .J m) on L P ( ~ ) . The product with the perturbed terms, and p a r t i c u l a r l y the lower order ones, gives a compact operator on LP(L~) (Rakovsh~ik's r e s u l t from 1963!). This leads to the coerciveness inequality:

(3.10)

Y'IImllm,p ~ IiAmtlo,p + IIV~llo,p

with a certain

VE~(Wm'P(R),

LP(IR)). This one implies that

A~'(Wm'P(IR),

LP(IR)) having an ind A = v(am(x)+bm(X)H ) which may be calculated from the winding number K : = [arg ~a_iblx~l~j ~ . In case the data are smoother, e. g. -oo

am, bm~S(~R)~sE~ , av,bvccs-l(~ )

and a (s)

b(S)CLo(]R)

one obtains for any

fEwS'P(]R) smooth solutions mEWm+s'P(IR). After discussing the dual operator on W m,p (IR), 1/p' +l/p = 1, --

c

m

(3.11)

(A*~)(X) : :

v

S (-1)~DV{au(x).l - H b ( . ) . l + k v ~ } ~ ( x ) p=O

where ki(x ) : = kv(-x ) i t is possible to show that A and A* may be extended continuously - for s u f f i c i e n t l y smooth coefficients a (x), bv(x ) - from Wm'P(~) to J)Lp(N) and from w-m'P'(N) to j ) ~ p , ( ~ ) . DONIG in 1976 [ 21]extended his reasoning in order to include the n-dimensional version of the singular i n t e g r o - d i f f e r e n t i a l eq. (3,3) to the case of coefficients M + K where the M are CMOs and the K of L l ( ~ n ) . kernel type, both being x-dependent. In contrast to AGRANOVI~ in (1965)[ 1 ] h e succeeded in arriving at an e x p l i c i t r i g h t and l e f t regularizer in the case of e l l i p t i c operators A . I t is impossible to give all details here, but his main result is formulated in the following Theorem 3.1 :

Let

n ~2, l
go (q) : = > ~o(q).

n_@_

such that for

for

q ~ p-P1

i
q~2

and

200 Furthermore assume that S f~(x,~)d~

a (x)EC(~ n) , f (x,o)EC(~ n, Wg°'q(sn))

= 0 for a l l

xEIR n

and

k (x,t)EC(IRn, L I ( N n ) ) ,

with

I~I < m.

Let the symbol of

f (x,~) (3.12)

(Am)(x) : = lu!~-~m~{a~(x)(D~m)(x) + p.v, mn~ u Ix Ix[yl(Dum)(y)dy-yl'' + + f k~(x,x-y)(D~m)(y)dy} IRn

= f(x)ELP(~R n)

be defined by

f (x, - ~

(3.13)

aA(X,~) : = I~ SI~m {a (x)+ p.v.F Y~+~(

and i t s "principal (3.14)

lim

Fourier m u l t i p l i e r I.

symbol" by

oAo(x,~) : = i~l=m{a~(x)+(P'V'Fy~

Then putting

A is e l l i p t i c

aA(X,~ ) = : OA(~)

f~(x,T~- F) ]y[n ) ( ~ ) } ~

and assuming [a~(~)]'l.(1+l~12)m/2

to be a

symbol on LP(~ n) the following statements are equivalent: on

~n

i

e

(3.15)

inf I (x,~)l > 0 x C~ n aA° ~dR n

and

(3.16)

i n f lOA(¢) I > 0 ~E~ n

or

2.

(~) + (FY'+~k~(x 'Y)) ( ~ ) } ~

iYl n

A is coercive on Wm'P(~Rn ) , i . e .

there exists a T > 0 and a compact semi-norm

x'l[~llm, p <_ IIA~LIp +

(3.17)

L~I

l.]

for a l l

such that eEWm'P(IR n ) ,

or

(3.18) where

3.

AE~(Wm'P(IR n ) , LP(~ n ) )

ind A = ind (

s

M • Ru) = 0 and the

the "Riesz-operators" (3.19)

Rj : = F - I T ~ F

, j=l ..... n .

R~ = R1

1.

~n

...'R n

are powers of

201 B may be calculated

I f one of the conditions holds then a two-sided r e g u l a r i z e r by (3.20)

B = B~.M -M°

where the symbol

M=, M°

are CMOs. M°

aAo(x,~)-l~I-m

(3.21)

is a bounded r e g u l a r i z e r to

being homogeneous of degree

M~ : =

z

M~. R~

where

0 ,

f (@) : =

M "Ru with lui=m

lim f (x,o) IxF~ ~

and ~ 12)-m/2)-iF B® : = F- i (OA(~).(l+I~

(3.22)

Remarks : 3.1 :

DONIG (1977)[ 22] applied his method, in case of

n = 1 , to give

conditions f o r the Fredholm property of a transmission problem f o r s e c t i o n a l l y holomorphic functions

~(z) , z c C - i ~ ,

conditions on the common boundary

i~.

which have i n t e g r o - d i f f e r e n t i a l

transmission

On the other hand (1974) [20 ]

he solved the Cauchy-problem (3.23 a)

~

(3.23 b) with e l l i p t i c

+ A(t)~

= f(x,t)EC°([o,~],L2(N n))nCz((o,~],k2(N n))

m(x,+o)

= mo(X)cL2(Rn)

operators

even depending on

t

A as before of the compound Ll-kernel and CMO-type, now

but with r e g u l a r i t y conditions such as the usual ones f o r

parabolic evolution equations. The p r i n c i p a l symbol is assumed to be "uniformly strongly e l l i p t i c " : f (x,t, (3.24) for all

Re(-l) m x c~ n

and

z {a ( x , t ) + (p lul =2m ~ .V.Fy~

lyln

4 1 ) ) ( ~ ) } ~ > cl~ 12m =

tE [o,~].

The r e s u l t s are quite s i m i l a r to those in FRIEDMAN's book (1969)[ 35 ] 4.

Simonenko's theory fgs. generalized t r a n s l a t i o n i n v a r i a n t operators

We want to give now a short outline of SIMONENKOs approach which he established in 1964,'65 [91,92]

and which was applied by RABINOVI~ (1969-'72)~72,73,74] and SPECK

(1974-'77)~96,98] iff

A-BE~T(~L,I~)

in the theory of GTIOs. For abbreviation we shall w r i t e for

A,BG~(3(,~)

where

~,~

A~B

are Banach spaces. ATKINSONs

theorem then may be formulated in the f o l l o w i n g way ( ~ = ~

for simplicity)

:

Ac~(3E) i f f there are RL, RrE~(~ ) such that R~A~ARr~-I. In what follows we shall mainly be interested in the spaces ~ = Wm'P(IRn), mcINo , l
202 1 W-m'p ' (JRn ) , ~1 + ~, : I. So, for the following d e f i n i t i o n s and results, l e t

duals

us assume ~c~_c~' being dense in the topology of the bigger space. Let the translation operator ThE~(~ ) and the multiplication operator m(x).l with a C~(]R ~)-function be a bounded operator on ]E , too. ]Rn may stand for ~n or ]Rn (see sec. 0). Definition 4.1 : (i) iff

m1.A~2.1~O

AE~(3E)

for all

is said to be "of local type (with respect to with

]Rn) ''

AE.A,(L~]. xo ( i i ) A,BE~(~E) are said to be " l o c a l l y equivalent at Xo~]Rn , written as__ A~B , i f f for all ~ > o there exists a neighborhood %(Xo) and an ~IzEC=(]R n) with

mlL(x) m 1 on ~ (Xo)

ml,~2~C~(]R n)

~im2 = 0 . We then w r i t e

such that

(4.z)

inf II V~0

< ~

and

(4.2)

inf II (A-B) ~z" I-V 1[ZL~) < c VN)

, too.

~-I(A-B)-VlI~(z)

-if'', written as A E ~xO(~) ( i i i ) AC£(~) is said to be " l o c a l l y Fredholm at Xoe-R i f f there exists a neighborhood ~(xo), and a pair of operators RL,x o and Rr,xo such that (4.3)

(R~,xoA-l)w~.l~O

and

~ .l(ARr,xo-l) ~ 0 .

Remark 4.1 : I t may be shown (cf. e. g. SPECK (1974)[96], p. 50) that in case of ~[= LP(A n)___pproperty ( i ) is equivalent to ( i ' ) [ A , w . l ] : = A ~ . I - ~.I A ~ 0 for all ~CC~(R n) and ( i " ) X E . A xG.I-~O for all E , G c ~ n with EnG = ~. For this case also ~ # C ~ ( ~ n) may be replaced by ×~ in ( i i ) , ( i i i ) , see also RABINOVI[ (1969)[Y2]. SIMONENKO in 1964 [91]proved the following theorems. Theorem 4.1 :

Let

AEy~(X). Then A E ~ ( ~ )

Theorem 4.2 :

Let

A,B c~(~)

iff

AE~×o(~_)

Xo for a fixed and A-~B

Xo~

for all

XoEIRn.

. Then

A~x ° (3~)

iff

Bc ~Xo(~" Some of SIMONENK0's additional results had been generalized by SPECK (1974)[96] ,viz. Theorem 4.3 : AE~Xo

Let

A E~0E), where

for an i n f i n i t e remote point,

~_= Lp ( ~ n ) , xo E ~ I ~ - R n

vertible. Theorem 4.4 :

Let

X = LP(~ n) , l
and

lz:Jp<~ or iff

A

.3E= Co(~n ). Then is (continuously)

in-

203

(4.4)

(Am)(x) = am(x) + p.v. [ Rn

Ix-y

i n- m(y)dy + [ k(x-y)m(y)dy ~ n

a compound CM- and Ll-convolution operator, (cf. eq. (1.22)); then finite

point

Remark 4.2 : a) Z :

[#~{n),

xo E ~ n

iff

al + Af/.

AE~Xo

for a

is i n v e r t i b l e .

These results c~rry over to more general situations as e. g~ ~n 1
+

%,

O

being homogeneous of degree zero and c o tending to zero at i n f i n i t y , or b) 3C_= L2(R n) , ~ being an a r b i t r a r y L'-function (cf. SPECK [97]!)~ Thus we are led to the following generalization of Definition 1.4. D e f i n i t i o n 4.2 :

( i ) A~(3E)

is called a "generalized translation invariant

operator (with respect to ~-T~),, i f f for any XoE-~ there is an AxoC~(~) NA (3~) ~ch that AX~°A Xo -I then(ii) I f A is a GTIO "enveloping" the family {Ax}xE-~n where Ax ~ = F CAx(~)-F,

(4.5)

o(x,g) : = aA (g) , (x,~) E~n

is called a "presymbol of

~n

A ,,

X

Remark 4.3 :

Denoting the presymbol space by

corresponding to compact GTIOs A by ~o (4.6)

A ~ o ~ ~ +

~

and the subideal of all functions

we see, that the map T defined by

~o

is a homomorphism of the space of GTIOs onto the quotient space ~/~'o with kernel O~, the compact operators (the sequence {o} + d~ ÷ GTIOs +~/I~"o ~ {o} is "exact"). I T Thus d e f i n i t i o n ( i i ) makes sense. The main results in this context may be summarized in the Theorem 4.5 :

(SIMONENKO (1964) - SPECK (1974-77)):

Xo

I. Let A'~A x E ~ ) equivalent, o

N A(~)

for all

for all

XocNn;

XoC~"~. Then the following conditions are

(i)

Ac~(~_)

(ii)

AxoE~Xo(~( )

I I . For (iii)

p = 2 at the same time holds sup ess inf l~(x,~) - Co(X,~)i > O. ~oe ~o (x '~)e-~Trxmn

Thus we get (iv)

essinf Lo(x,~)I > o (x,~)e~n~ n

asa s u f f i c i e n t condition, while a necessary one in any case is given by

204 (v)

ess i n f l{(x,~)[ > 0 ( x , ~ ) e ( ~ - ] R n ) x IRn

where x ~ a ( x , . ) III.

For

(vi)

~l(xo,~ )

has to be continuous from

~'T~_ ~Rn

into

L=(~n).

p ~ 2 , A being of type (4.4), in formula ( i v ) we have to add is a ~ - F o u r i e r m u l t i p l i e r symbol for a l l

XoE-~-11-~n

( t h i s leads to more complicated conditions). Applying t h i s to the compound pole-dependent Calder~n-Zygmund-Michlin and L1integral operators on ~ = L2(]Rn) given by x-y

f(x,

(4.5)

(Am)(x) = a(x)m(x) + p.v. I in ]Rn Ix-Y

m(y)dy

+ f k(x ,x-y)m(y)dy ]Rn

one gets the Fredholm c r i t e r i o n by the symbol conditions (4.6)

i n f laA(~,~)i ~ 0 ~e~ n

(4.7)

min xE~ n

i n f Io ~EZn a ( x ) I + A f ( x , . ) / .

since the symbols of the

(~)I > 0

,

CMOs are homogeneous of degree zero.

These are exactly the same conditions l i k e those got by DONIG [21]- in the case of

m = O. As we had seen he proved furthermore the equivalence to coercive inequal-

i t i e s , while SPECK [97] gave the complete extension f o r a r b i t r a r y (non-stabilized) GTIOs in the case of Ac~"x

p = 2. Due to the kind of the characterization theorem for

at f i n i t e points

xo

there occur rest-classes of functions in the d e f i n i -

tion °of the symbol. Now, a l l the considerations above may immediately be generalized to integrod i f f e r e n t i a l equations (4.8)

(Am)(x) = l ~zl ~

(ApDUm)(x) = f ( x ) E

where the c o e f f i c i e n t s A are generalized translation i n v a r i a n t operators according to d e f i n i t i o n 4.2 ( i ) and ~ c ~ m ) c ~' be d i s t r i b u t i o n spaces such that D~mc~ f o r

~E~ o , 0 ~ lul ~ m. F i r s t we notice that the concept of t h i s section

can be completely transferred to the case of

A : ~ ~

operating between d i f f e r e n t

B-spaces. Then choosing l~ = ~C(m) we can see that the Bessel potential operator jm = ~1(1+i~12)-m/2.F wi~l reduce eq. (4.8) to one in ~ . The essential f a c t here is that the js are not only t r a n s l a t i o n i n v a r i a n t operators from ~ ( t ) = wt,p(~n) onto ~ ( t + s ) = wt+s,p(~n) for t,sEIR, l
~ n , cf. RABINOVI~ (1972)[74 ]and SPECK (1974)[96]. So we can state the

205

Theorem 4.6 : (i)

Let

AE ~(wS'P(IR n ) , wt'P(]R n)), s,tG]R ; l
A is a generalized translation invariant operator form ws'P(]R n) to wt'P(]R n) i f f ~ : = j - t A j s is one on LP(~ n) (with respect to ]Rn!) AE~(wS'P(]Rn), wt'P(IRn)) i f f ANc~(LP(]Rn)),

(ii)

I t turns out that "generalized convolutional i n t e g r o - d i f f e r e n t i a l operators" like in eq. (4.8) are generalized translation invariant operators from Wm'P(]Rn) into LP(]Rn), l
(cf. RABINOVI~ [ 7 2 ] , p. 87, Th. 4.1; SPECK [96] , p. 89) A : = i ~ l ~ A Du be a generalized convolutional i n t e g r o - d i f f e r e n t i a l

Let I t is

c~:'(Wm,p(~n), L P ( ~ n ) ) ,

where

{Ax }

. xem n

l
iff

Ax j m E ~ x ( L P ( ~ n ) l

is the family of operators

(4.8)

xE~ n

~ A xDUc~NA(Wm'P(~ n ) , LP(~ n)) I~ I <-m

(operator with "frozen c o e f f i c i e n t s " ) . I t is necessary that A= be b i j e c t i v e and i t is s u f f i c i e n t that for all

for all

operator.

Ax

be b i j e c t i v e

xE~ n. This holds for

in# I #~rn~A (x,~)'~U(l+l¢I2)-m/21 ×c~ n ~

>0

~cR n

i f p = 2 in which case the pseudo-inverse of AJm can be constructed as an enveloping operator to the local inverses (AxJm)-IE#NA(L2(~n)). For p # 2 the local existence of these operators must be assured (cf. theorem 4.6). Remark 4.3 : The concept of local equivalence of operators has been generalized by SIMONENKO (1964,'65)E91,9~ from the very beginning to the following one D e f i n i t i o n 4.3 : Let ×, Y be Hausdorff-spaces being homeomorphic by ~ : X ÷ Y and l e t (T~u)(x) : = (u,~)(x) be a non-distorting transformation for all ueLP(Y) , l
(4.9)

T _IP A

Xo' Yo"' written as

AXe9 ° Y~°B i f f

Yo P~B P~r

in the sense of d e f i n i t i o n 4.1 ( i i ) with space projectors respectively, u and n denote complete respectively.

P~ = ×~.I

and

P~= 7~I,

~-additive, e - f i n i t e measure5 on X and Y

This notion allows us to t r e a t semi-infinite

smoothly bounded regions

One has the corresponding results for half-spaces

~n

n >2

Gc~ n

if

206 D e f i n i t i o n 4.4 : Let ~ = LP(]Rn), l_~p<~ , and NEj, j = 1 . . . . . N, be a f i n i t e set of d i s j o i n t measurable sets in ~Rn such that u E. = ~ n I f A j , j = 1 . . . . N , are generalized t r a n s l a t i o n i n v a r i a n t operatorsJo I J ~ (with respect to ~I~) then N

(4.10)

WN : =

is called

a "component GTIO" or an "N-part composite WHO". I f

presymbol of (4.11)

z AjPE. j=l j

Aj, j = 1 . . . . . N, then the "presymbol of ~WN(X,~) : = OA( x , ~ ) j

for

min inf j = l . . . . . N (x,~)cAj

denotes the

is def~lned by

(x,~)CEj~
Theorem 4.8 : (SIMONENKO (1967)[94], p. 1322) : LetW N be a component GTIO as above where Ej = Fj GTIOs of L l - t y p e , and IR~ = ~ n Then W N ~ (4.12)

WN"

~A-

are smooth conical sets, Aj are iff

IOWN(X,~) ! > 0

where (4.13)

Aj : = [ ( ~ N } ~ j ) ~

In this case the index of

n]u[Fj~{~}]

, j = l . . . . . N.

WN equals zero.

Remarks: 4.4 : This theorem can be generalized in various ways - e s p e c i a l l y for p = 2 - by the following more general assumptions:l).Let Ajx , the l o c a l l y quasico

equivalent t r a n s l a t i o n invariant operators in xcIR n have symbols ~Jx = ~J x+ ~jx where 'fjx are " r e l a t i v e l y t h i n " L ~ - functions instead of being EFLiCCo(]Rn), i. e. (4.14)

sup l~oi~2r

~ l~jx(~)Id~ = o ( r n) I~- o Isr

(cf. SPECK [98]. 2).Take f o r Ej smooth cones at i n f i n i t y , i. e. (4.15)

for

r÷ ~ ,

measurable sets which are only asymptotically

mes(Ej-Fj)n{~cIR n:I~O-~l

~ I} ÷ 0

for

l~oI ÷

where the Fj are smooth cones, cf. SPECK [98], too, 3). Let r j be piece-wise smooth cones, cf. ~!EISTE~ & SPECK [58]. 4).Assume that l o c a l l y at every 9oint xE~ n more than two sets 4.5

:

~j

are allowed to i n t e r s e c t ,

cf. HEISTER & SPECK [59],

SIMONENKO (1964)[ 91] generalizes to composite WHOs with space projectors

pEjE~(L2 ( ~ n ) )

where the

manifolds of f i n i t e

Ej

are domains

area and the operators

c~ n Aj

bounded by smooth Ljapounov

are singular having homogeneous

207 symbols (or symbol matrices) of ( p o s i t i v e ) order

0

proves that the symbol ~w(X,~) : = ~A(X,~)

(x,~) c E j x ~ n

values as

x ÷ x oC~Ej

d i t i o n to be f u l f i l l e d

for

being continuous on

~n ~ He

and t h e i r l i m i t i n g

from both sides has to be zero and a certain index-conin order that

WN be Fredhlom-Noether. This can be done

by reduction to a two-part composite problem l o c a l l y at every boundary point since the quasi-equivalence is established by the local mapping of ~

= ~n-l.

~Ej

xo

onto

This idea of local coordinate mapping is inherent to a l l e l l i p t i c

boundary value problems, f o r smoothly bounded domains. SIMONENKO's method has been applied by RABINOVI~ in (1969)[72,§ 5] to boundary value problems for generalized convolutional i n t e g r o - d i f f e r e n t i a l equations - and systems of such - in semi-infinite domains GC~n

with smooth boundaries behaving l i k e a cone f o r large distance

from the o r i g i n . Again the non-vanishing of the symbol on AG : = [ # N ~ R ~ ] U [ G ~ { ~ } ] is necessary and s u f f i c i e n t for (4.18)

PGA(X,D)PGm + TPGm = fcHS-~(G)

to be Fredholm-Noether. Here PG denotes the r e s t r i c t i o n operator on Hs+C (G) : = Ws+c,2 (G) and TE~(H°s+c (G), Hs-z(G) where ~s+~(G) is the closure of

C~(G) with respect to For the case of

Fredholm property, on

II.llHS+~(G)-norm.

G = R n , or a bounded domain, a thorough discussion of the n p a r t i c u l a r with boundary conditions or potentials carried V

~G , and the related conjugate problem has been performed by DIKANSKII(1971,'73)

[17,18]. RABINOVI~ (1972)[74] then studied pseudo-differential operators on classes of noncompact manifolds with boundary conditions. Quite recently CORDES (1977) stQdied C*-algebras of e l l i p t i c

boundary problems [10]. Shortly before he derived

a global regularizer - or even parametrix - to pseudo-differential operators on R n (1976)[14].

208 5.

Wiener-Hopf type integral equations with strongly singular kernels

So far we have been interested in convolutional

equations having smooth symbols on

G~R~admittingc.l~x for the x-dependence of the factors or kernels more general domains or having symbols being continuous on IR~ n_{0} but x varying in the whole of ] ~ n We want to look now into equations combining stronly singular kernels, of the Cauchy p r i n c i p l e value or Calderon-Zygmund-M1chlin type, with piecewise constant coefficients on ]Rn. The simplest case arises from the well-known x " a i r f o i l equation" +I (5,1) _1 f q~(y)dy -1 'y - X = (m(-1,1)m)(x) = f ( x ) , XE(-1,1) J

•

involving the " f i n i t e Hilbert transformation Hr- , . j I~"" l BETZ in (1920) [,4 ] gave an inversion formula for s u f f i c i e n t l y smooth f E C # o c ( ( - 1 , 1 ) ) N L I ( ( - I , I ) ) 0<),<1, by (5.2)

m(x) = - - - - ~ _ 1

with a r b i t r a r y

"

CE~ (or ~). Now, this is completely in contrast to

H2 = -I

acting on L P ( ~ ) , l
#(z)

=

1 +} my_~d z

g'~3- _1

,

[-1,1]

Z

then the Sochozki-Plemelj formula

r = IR: ~+(x

- G-(x)

=

{

for

~(x)

0

xE(-l,1)

for .xEIR-[-1,1]

and (5.4)

~+(x

T(H(-1,1)m)(x)

for

xE(-1,1)

+i __1 f < y )dY Ti L1 y - x

for

xcIR-[-1,1]

+ ®-(x) :

So eq. (5.1) is equivalent to the Riemann boundary value problem with a piecewise continuous factor only: (5.5)

@+(x) = G(x).~'(x) + g(x) ,

XE~-{-1,1}

where =I -I (5.6) and

G(x)

l

+I

for

x E(-I,1)

for

xeIR - [ -1,1 ]

209

1 f(x)

for

xe(-1,1)

0

for

xE~-[-1,1]

x = ±i

must be the reason for the non-uniqueness of the

T (5.7)

g(x) :

So the jumps of

G at

solution given by eq. (5.2). This f i n i t e H-transformation, and the s e m i - i n f i n i t e along

~+

as w e l l , have attracted the attention of many mathematicians up to now

concerning one- and multidimensional integral equations involving them or t h e i r multidimensional counterparts. While the Russian school

around MUSHKHELISHVILI (cf. his book (1953)[6211 has

mainly looked into Hoelder space solutions by the classical method TRICOMI in 1951 [102], NICKEL in 1951 [63], and SUHNGEN in 1954 [95]

applied the

LP-theory, in

the l a t t e r case by a transformation which diagonalizes the f i n i t e H-transformation. KOPPELMAN & PINCUS in 1959 [53] and J. SCHWARTZ in 1962 [81] derived spectral representations for

H(_I,I)

on

L2((-I,I)).

WIDOM in 1960 [106] and SHAMIR in

1964 [86] studied singular integral equations and systems of them, respectively, on measurable subsets Ec~ and on ~+ or [ - I , + i ] , respectively. They treated the LP(E) - and ws'P(~+) - , s ~ O, l
L2(~+)

of purely Cauchy-type and of compound ones with additional Ll-kernels. IndependenCy of the l a t t e r paper KREMER in his thesis (1969) [55] treated the equation (5.8)

(W,)(x) : = a(x)~(x) +

.

y - x o

+ c ( x ) . f k(x-y)~(y)dy o = f(x)EL2(N+)

where a,b,cEC(~--T) , kELI(~). The studies were continued, for LP(~+)

c(x) z O, to

with the symbol calculus by GERLACH & KREMERin (1972,'73)[39,40 ] •

Following quite a different line GOHBERG & KRUPNIKstarted in (1968) [44] to study Cauchy-type singular integral equations along curves rc{

with piecewise

continuous coefficients anew and introduced 2 x 2-matrix symbols for scalar equati~ons in 1970 [45]. The whole theory, even under much weaker conditions on the coefficients a ( t ) , b(t), tEr , in the equation (5.9)

(K~(t)

: = a(t)~(t)+b(t).(Sr~)(F ) = f(t),

tErc~ ,

has been displayed in a l l d e t a i l s in GOHBERG & KRUPNIK's book (1973) [47]. DUDU~AVA applied t h e i r theory to Wiener-Hopf type equations and compound equations with symbols having weak smoothness assumptions in a number of papers in 1973,'74,'75,'76 [24,25,26,29] .

210

SHAMIR in 1966,'67 [87,88] studied systems of singular integral equations in LP( ~+n ) , ~UBIN in 1971 [90] and ESKIN in 1967,'73 [104,30] applied the method of matrix-factorization

with respect to the n-th Fouriertransform variable

~n

in

order to solve general boundary value problems for pseudo-differential operators being homogeneous of degree ~ + iB in ~n+ with additional potentials and/or boundary conditions on ~ = R 'n-1 added• In his book (1973) ~SKIN is also concerned with mixed-type boundary conditions on ~, n-I so that boundary pseudo-differential arise.

operators with piecewise continuous coefficients on

~,n-I

A main part in the technique is played by the "Mellin transformation" and the "Mellin convolution". This enters in a natural way by studying certain operatoralgebras acting on LP(R+). To show this we are going to follow the lines of KREMER (1969) whose work strongly parallels CORDES'(1969). In order to build up the algebras he introduces for ~EL2(IR+) : (5.10)

S : = SIR+

defined by ( S ~ + ~ ) ( x ) :

(5.11)

T

defined by (T~+m)(x) : = i

(5.12)

Kk : = P+k~P+ defined by (Kkm)(x)

• _ 1 ! k(x-y)~(y)dy

(5.13)

Ck

• _ 2 ~1o 7 k(x+y)m(y)dy

: = T]R+

defined by (Ckm)(x)

= ~Ti ! ~

'

i y+xmlY)d~

Then the basis for the whole theory are the compactness relations given by the following [55], p. 4 & 11: Lemma 5.1 :

Let

a(x)cC(~+)

(5.14)

S2

=

(5.15)

[S,T]~O

, i.

(5.16)

[al,S]~-O ,

[al,T] ~ 0

(5.17)

a.K k --- 0

in case of

(5.18)

Ck ~ 0

(5.19)

[S,Kk]~O

(5.20)

[T,Kk]~-O

(5.21)

Kkl.Kk2-Kkl.k 2

and

S,T,Kk,C k with

I + T2 e.

0

C~(L2 ( ~+ )) lim

a(x) = 0

kcLl(~)

as above• Then

211 Then KREMERstudies, as does CORDES (loc. c i t . ) , the algebra (~, generated by S and T which is commutative, modulo the compact operators, with an i d e n t i t y . D e f i n i t i o n 5.1 : (5.22)

~(t)

If

mcL2(~+)

: = l.i.m. ~ ~0

then

1 }/~x - ( z + i t ) / 2 m(x)dx 2d~-

is called the "Mellin transform of I t has the property (5.23)

(S~o)(t) = tanh

(5.24)

(Tm)(t) =

~

- ~'(t)

1,11

(at Re s = ~) .

and

- i • ~'~t) cosh ~ t 2 (cf. e.g. CORDES [11], p. 897!). The algebra ~ I proves to be isometric isomorphic to the function algebra C ( ~ ) ( ~ = ~ in this context ) where ~(t)EC(~) corresponds to K EC~1 with (K~)(t) = ~(t).~'(t). ~1 is called the "generalized Mellin convolutional algebra" since i t contains an algebra O~ of integral operators of a "quotient convolution" type (525)

:

which transforms

dy

o mELl(x-i/2; ~ + )

, x > o

into i t s e l f when the kernel

qcLl(x-I/2;

~+

(cf. e.g. CORDES [11] , p. 897/88!),The general Mellin transform (Mm)(s) : = i x S - l ' m ( x ) dx transforms such a quotier~ convolution into an algebraic product: o

(5.26)

M(gem)(s ) = (Mg)(s).(Mm)(s)

(cf. e.g. TITCHMARSH's book, p. 304). CORDES & HERMAN in (1966) [15] introduced the algebra ~ generated by Ct I , the m u l t i p l i e r s a ( x ) . l EC(~'~) and ~ ( L 2 ( ~ + ) ) such that ~ / ~ is a commutative B*-algebra and the space of i t s maximal ideals is homeomorphic to the Shilov boundary ~ ( ~ x ~ ) = : v ~ . I f • denotes the Gelfand homomorphism of ~ / ~ o n t o C(~() then we have the symbols OA(x,t) ECOl,) given by aal(X,t ) = a(x) ~Kq(X,t) = ~ ( t )

on on

0 ~ x ~ ~ , t = ±~ -~ ~ t ~ +~, x = 0 or

+~

and Os(X,t) and ~T(x,t) as in eqs. (5~23), (5,24). This has been generalized to the corresponding subalgebras of ~ ( L P ( ~ + ) ) , l
212 = tanh {~ [~t + i ( ~i _ ~ ) ] }

~P)(x,t) changes.

Now, i f we apply these r e s u l t s we get Theorem 5.1 :

(CORDES (1966), KREMER (1969)). Let

a,bEC(R +)

be given.

Then (5.27)

(Kom)(x) : : a(x)~(x) + b(x)(S~)(x)

defines a Fredholm-Noether operator on (5.28) on

L2(R+)

~Ko(X,t ) = a(x) + b ( x ) . t a n h ~

sE where

E : = ~+~t

iff

0

or, e q u i v a l e n t l y , i f f

(5.29)

a2(x) - b2(x) ~ 0

(5.30 a)

alOl+b(O) -~ < arg a 0 -b(O) < ~ and

(5.30 b)

-~ < arg a

~

on

~+

and

< ~

The index is given by (5.31)

ind Ko = v(Ko) = ~

fDE darg ~Ko(X,t).

In order to s i m p l i f y the w r i t i n g in what follows l e t us assume be Fredholm-Noether and the c o e f f i c i e n t s to be normalized s. th. on

~...

I f we define

bI = -b(a2-b2) - I = -b

KI : = a l l + b l S then

KI

a I = a(a2-b2) - I = a is Fredholm-Noether a l i k e with

i ind K1 = - ind Ko = - ~

(5.32)

and the two products

KoK1 and

where

Ko = al + bS to a2(x)-b2(x) = 1 and

f darg DE ~Ko(X't)

KIKo

have zero indices but are in general not

Fredholm-Riesz due to (5.33)

KIK o = I + blbT 2 + V1

(5.34)

KoK1 = I + bblT2 + V2

KREMER's idea (th. 23, p. 51) [55] is to construct a two-sided r e g u l a r i z e r L = I + blbT 2 by i n v e r t i n g the symbol OL(X,t). He then i n v e s t i g a t e s the Wiener-Hopf-algebra 6(2 ators

K-k , I

and the compact ones on

ator norm of ~ ( L 2 ( ~ + ) ) .

~2 = ~ 2 / ~ i s

L2(~+ )

U to

being generated by a l l oper-

and being closed under the oper-

commutative semi-simple with an i d e n t i t y ,

i t s space of maximal ideals being homeomorphic to

~

A

and ~2

is isomorphic to

213 C ( ~ ) . The Gelfand theory then allows us very easily to characterize the Fredholm operators, that are the i n v e r t i b l e elements in ~ 2 ' of the type by the condition

A

i + k(~) # 0

on

~

I + Kk

on

L2(R+),

as we know already.

To t r e a t the compound integral equation on L2(~+) the algebra ~N. being gener~ ating by OCI and 02 is constructed. Again ( X / ~ i s commutative and semi-simple with an i d e n t i t y . The space T~C(~J of maximal ideals is homeomorphic to a compact subset of J~, (cf. KREMER[55] (ths. 19,20) where ~:

= ~

u{(t,~)

: t¢~,

~ =0}

: = (m--~)

Then the compound integral equation with constant coefficients Km e (I +aKk+~SKk)m(x ) = f(x)CL2(IR+)

(5.35)

may be treated. There holds the Theorem 5.2 : (KREMER, th. 24, p. 54). Let m,BEC , k E L I ( ~ ) , Then Ke ~c ~, and ~(o) = o , I f the function G(~) : = I + (~+B'sign ~)~(~) @ 0 on ~ then (i) KE~(L2(~+)) (5.36)

(ii)

ind K = v(K) = - ~

5.37)

(iii)

R : = I + ½(I+S)Kkl+ 7( 1 I_S)Kk 2

is a two-sided regularizer to

[arg G(~)] =

K where

A

(5.38)

k 1, k2ELI(~)

are defined from

^

1 + k1,2(~) = [1 + (~±B)k(C)] -1

as elements of the Wiener-algebra

i~(~),

Now, similar to the argument by GERLACH [38] (cf. p. 27/28!) one is ready to discuss the general case of continuous coefficients a,b,c on R+: (5.39)

(W~)(x) : = (a(x)l+b(x)S+c(X)Kk)~(x) = f ( x ) E L 2 ( ~ + )

making use of Rakovsh~ik's r e s u l t of (1963). So KREMERarrives at his main r e s u l t , viz. Theorem 5._33 i

([55], th. 26, p. 62). Let

Fredholm-Noether operator on : = c(~)a(~), B = -c(~)b(~). (5.40)

i +(~±B)k(~) # 0

a,b,cEC(~'--~), k e L l ( ~ ) ,

L2(~+). Assume a2(x) - b2(x) = i Let on

and

al + b.S on T+~

be

a

214

(5.41)

1 + k ( 0 ) { ~ - s . t a n h T~t - ~p(t) + B tanh

p(t))

# 0

on

]Rt

wh~re p(t)

(15.42) Then

: = b2(=).[b2(=)

+ cosh 2 _~]-1 .

•T(L2(m+))

W is

oo

C.£rollary 5 . 1 - -

Let

a,b,c,k

be as above but a d d i t i o n a l l y

~(0) : / k(x)dx = O.

Then the second c o n d i t i o n f o r the symbol is a u t o m a t i c a l l y f u l f i l l e d a two-sided r e g u l a r i z e r (5.43)

Ro

-~ ( l f 0 )

and

is given by

R° : = U3U2UI

where (5.44)

UI : = al - bS

(5.45)

U2 : = I - b ( x ) . b ( o )

(5.44)

1 I+S)Kk I + ½(I-S) Kk2 U3 : = I + ~(

[ l - t a n h ~] • Kq1- b(x)b(~) tanh •

. Kq2

where

K • ~ c ~ 1 ; i = 1,2 ; and qi q1(t) : = [cosh 2 3_~ + b2(0)] - I (5.45 a) q2(t)

(5.45 b)

: = [cosh 2 ~

Furthermore the index of (5.46)

+ b2(~)] - I

W is given by

ind K : v(K) = ~

+ ~1~indl~ mo+m~

(cf.

= -~ [arg G ( ~ ) ]

th. 1.4!) where a(O)+b(O) a o = arg a(O)-b(O) = Co + 2~ mo

(5.47 a)

defines (5.47 b)

-a

= arg

such t h a t

iCol,

al + bS

and

mo,m~

m

Ic I <

C o r o l l a r y 5.2 : ( [ 5 5 ] , let

= -c-2~

th.

I + ~K k

eq. (l+V)m' = f ' where sign of ind(al+bS).

2.7), Let be

V~O

a, b, c, k

c~(L2R+)).

be as above but

b(o) = b(~) = O,

Then eq. (5.39) is e q u i v a l e n t to an

v i a a c l a s s i c a l Wiener-Hopf equation depending on the

Now, we shall sketch the theory developed by GOHBERG & KRUPNIK [45] f o r s i n g u l a r

215 Cauchy-type integral equations involving piecewise continuous c o e f f i c i e n t s . While the whole theory has been developed f o r spaces LP(p;c), l
rcC

(or s = ~ )

and

L2(F). Ne are following mainly along the lines of t h e i r paper in (1970) ~5 ]. Let

PC(F) denote the algebra of a l l piecewise continuous functions

being l e f t continuous

from the r i g h t

a(to) = a(to-O ) =

lim

a(t)

and

a ( t ) on F

having l i m i t i n g values

lim a ( t ) where ~ and ~ shall denote that the point t t ÷to t >t o is before or behind t o , respectively, in the sense of the orientation on F . G : = F ~ [ O , I ] shall denote the cylinder { ( t , u ) : tEF , O~u~1} and M(t,p)#~2x2(F), i.e.

a(to+O ) =

a 2x2-complex matrix whose entries

O l l ( t , ~ ) , ml2(L,u), m21(t,~), and

~22(t,1-~)~C(G) where ~12(t,0) = ~21(t,0) = ~12(t,1) = ~21(t,1) = 0 for tCF . Now, the algebra ~ of a l l such function-matrices becomes a B-algebra by introducing the norm (5.48) l l M ( t , ~ i i : = max Sl(M(t,~)) where s~(M(t,~))

denotes the largest eigenvalue of

H(t,~).M*(t,u)

being

semi-positive d e f i n i t e . They prove then the following theorems in several steps: Theorem 5.4 : ~(L2(F)) a,bEPC(F) (i)

( [ 4 5 ] , t h . 0.2, p. 194). Let OL(PC(F))

containing a l l operators of the form and ~

(~t(PC(s))/~

be the smalle~subalgebra of

a ( t ) l + b(t)S F with coefficients

the ideal of compact operators on

L2(F). Then

is isometric and isomorphic to the matrix-algebra

~ . This iso-

morphism transforms the operator C~mA = a , l + b-S F + V , Vc~, into the "symbol" aA(t'~)=~(t'~):= I(1-~)c(t)+~,c(t+O)

, ~[d(t+O)-d(t)]l

(5.49) ~[c(t+O)-c(t)],

~d(t)+(Z-~#)d(t+O)

/

where c(t) : = a ( t ) + b ( t ) , d(t) : = c ( t ) - b ( t ) (5.50) ( i i )

ll~(t'~)ll

= Vc~]infIIA+V]I

Remark 5.1 :

This may be generalized to the case of NxN-systems

equations introducing then

(L2(r)) of integral

2Nx2N-symbol matrices (cf. GOHBERG& KRUPNIK [45], § 1).

Since they show (th. 3.2) that for a matrix-function M(t,~) e ~ with det M(t,~) ~ 0 on G ( M ( t , ~ ) ) - I E ~ too, they can prove the following essential (th. 4.2, p. 199!):

216

Let

Theorem 5.5 : det j ~ ( t , ~ )

~ 0

AE~(PC(F))c~(L2(F)),

on

Then

AE ~ ( L 2 ( ? ) )

or

e~-_(L2(F))

iff

G~.

I f t h i s c o n d i t i o n holds the f u n c t i o n (5.51) and

fA(t,~)

AE~(L2(F))

(5.52)

-1

: = det ~ ( t , u ) . [ ~ 2 2 ( t , O ) . ~ 2 2 ( t , l ) ]

e C(G), G : = F ~ [ 0 , 1 ]

having the

ind A = v(A) = - ~

[arg f A ( t , ~ ) ] m

Remarks : 5.2 : In t h e i r paper 1971 [46] they carry over t h e i r arguments to the ---m Bk spaces LP(p;?) , l
fixed

5,3:

(cf. also t h e i r j o i n t

SCHOPPEL (1973,'76)[79,80]

l i k e above in

book (1973)[47],

treated n o n - e l l i p t i c

v ~ ' P ( p ; r ) - s p a c e s where

i s o l a t e d zeros of f i n i t e

order

DORF's f o r closed curves with non-elliptic

chap. I X ! ) .

det~(t,~)

s i n g u l a r i n t e g r a l operators

may have a f i n i t e

number of

s m . The discussion is s t r o n g l y r e l a t e d to PRUSS-

F w i t h continuous c o e f f i c i e n t s or f o r Wiener-Hopf equations

symbols

i-~(~)

on

~

( c f , e.g. PRUSSDORF's book (1974),

chapts, 5,6 !). 5.4:

GOHBERG& KRUPNIK discuss In sects. 6 and 7 of t h e i r book (1973) [47] the cases

of F = ]R and .IR+in LP(Po;? ), i< <=, and the weiaht f u n c t i o n Po(t ) ( l + x 2 ) ~12 N ;I JX-Xkl 6k ,- 1 < N z pBk+B
(5.53)

A = a.l+b.S]R

where

a =

I

a I c~

for

x < 0

a 2 e~

for

x > 0

for

x < 0

for

x > 0

IblEC b b2 E C in the spaces

L P ( I t l B, ~ )

, l
-1
DUDU~AVA in (1973) [24] s t a r t s o f f with the t r a n s l a t i o n i n v a r i a n t operators (5.54) where

Wal~ : =

F-laFim and

~j

Wa : = P+Wa IR~P+)

P+=

x]R+

.I ,

P~I~(~) denotes the class of a l l f u n c t i o n s on ]R of type n ^ z ak(~)-×k(~ ) where the ak(~) = c k + g k ( ~ ) E ~ ( ] R ) , the one-dimensional k=1 Wiener-algebra, and the x k being c h a r a c t e r i s t i c f u n c t i o n s to i n t e r v a l s EkC IR

a(~) =

217

o

such t h a t

n

EkFIE ~ J = (~ and

U Ek = IR. Wa is then the unique c o n t i n u a t i o n to a k=1 i n v a r i a n t operator on L2(IR) ( c f , [24], chap. 2, th. 2 and 1°!)

bounded t r a n s l a t i o n To each f u n c t i o n

a(~) E P ~ ( ] R )

he associates the f u n c t i o n

{ a(~-O)[l-u]+a(~+O).u (5.55)

a(2)(~,~)

a(+~) [1-~]+a(-~) having a closed curve gular" if

f o r I~I < ~ and O~u~l for ~ = ±

: = cC

"u

as numerical range. This f u n c t i o n is c a l l e d " ( 2 - ) n o n s i n -

inf__la£2)(~,~)l" " > 0. He proves then the 0~I

Theorem 5.6 : at

([2~, th. 1, p. 1002 ) . Let

c I . . . . . c n. Then

iff

a(2)(~,~)

left

invertible,

aEP~(R)

Wa = P+F-IaFP+E~,(L2(~+)

is n o n - s i n g u l a r .

is

having the points of jumps

E ~+(L2(~+~

I f t h i s c o n d i t i o n holds then

or r i g h t i n v e r t i b l e

Wa

or

E ~'_(L2(R+))

is i n v e r t i b l e ,

if

v(Wa) : = Ind Wa : = - K ( a ( 2 ) ( ~ , ~ ) ) is zero, p o s i t i v e , or negative, r e s p e c t i v e l y .

Here

~ denotes the winding number

n

(5.56)

<(a(2)(~,u))

Remarks : 5.5 : result (cf. 5.6:

: ~

[arg a(~)] + s ~ -~ k=l

In the case of continuous

aE~(~)

l

a(P)(~,~)

(5.57)

LP(~+)

a(~-O)[1-gp(~)]+a(~+O).gp(~)

where now f o r

r = p gr(~)

(5.59)

or

: _

.gq(U) ; ~ = ±

q = p_-~l ~ 2 : sin [email protected]~O I@ sin @ .e

" @ : = ~-2~

r-lr

He constructs an algebra of operators A =

r s k=l

s ~ Wa j=l kj

where akj

• E P~]~( ~R )

w i t h symbols given by (5.60)

; I~I < ~ O~u~l

ak(~ ) = c k + ~k(~ ) , g k ( X ) c L l ( ~ ) n L q ( ~ ) .

5.7 :

by i n t r o d u c i n g

: =

[a(+~) [ 1 - g q ( ~ ) ] + a ( - = )

and

t h i s corresponds to KRE~N's

th . 2 . 1 ) .

The whole theory may be generalized to

(5.58)

[arg a(2)(Ck,U)] I ~=o

oA(P)(~,~ ) : =

r ~ k=l

s a(p) ~ (~,~). j=l kj

218

Then a s i m i l a r theorem ( [ 2 4 ] , th . 2, p. 1003 ) character of

A

if

i n -f - ~(P) A (~'~)J > 0

guarantees the Fredholm-Noether

holds. Then

v(A) = ind A = _K(~p)(~,~)).r

Os~l A d e t a i l e d representation is given by him (1975) [26 ]. 5.8 :

DUDU~AVA in (1974) [25] generalizes his theory to include m u l t i p l e - p a r t

composite Wiener-Hopf equations with strongly singular t r a n s l a t i o n i n v a r i a n t operatorsinvolved, viz. N

S aj(x)-(Wbj.cj(y)m)(x) (Am)(x) : = j=1

(5.61) where the

bjeP?A)(~)

as before, a j ( x )

and

piece-wise constant functions with a f i n i t e I I ~ l l ~ (Lp( multipliers

)

, i . e. f o r

on

L2(~).

p = 2

in

cj(x) •~ C(R),

the closure of the

number of jumps in the operator norm of

L~(~)-~Jorm, since a l l

L~-functions are

The author constructs complicated 2 x 2-symbol matrices along

the l i n e s of GOHBERG & KRUPNIK and formulates necessary and s u f f i c i e n t conditions for

A to be

e~(L2(~))

and calculates the index. The d e t a i l s are too lengthy to

be w r i t t e n down here! In

his paper (1976) [29] DUDU~AVA gives a d e t a i l e d account of the whole theory

extending i t to the quarter-plane case, permitting Sobolev spaces, and systems as w ~ l . 6.

Convolutional i n t e g r a l equations on the 9uadrant

In accordance with

A (iii)

in Chap. 1 l e t us consider the f o l l o w i n g "Wiener-Hopf

i n t e g r a l equation on the quadrant" (6.1) where

(W++ m)(x) : = m(x) - f kcLI(R 2)

and

f

are given,

k(x-y)m(y)dy = f ( x ) E L P ( ~ + ) 2 mcLP(~++)

, x2~0}.

If

sought, ~ + ^ d e n o t i n g ~(~) : = 1-k(~)

the f i r s t

quadrant

= {x = (Xl,X2)E~ 2 : x I ~ 0

c III0(~)

, the two-dimensional Wiener-algebra, and i f we assume i t to be

denotes the symbol # 0 on ~2

then we may f a c t o r i z e i n t o four continuous functions which are holomorphically ex+ ~< H± tendable into the four respective products of half-planes H~i ~2 such t h a t (6.2)

~(~)

(6.3)

a±,±(~) = A

=

P±,± : = quadrants of

~++(~)~_+(~)~_~(~)~+_(~)

where

1 + ~+,+(~) . . . =. exp{Pt, . +~ log ~(~)} FX 2 . I . F -1 ±±

are the F-transformed projectors onto the four

~ 2 . A f t e r grouping the factors in the r i g h t way one recognizes t h a t

(~_+~__)(~)(o++o+_)(~)corresponds to a f a c t o r i z a t i o n of ~ ( ~ ) i n t o symbols belonging

219

to a WH problem f o r the l e f t and r i g h t half-plane of into

(~+j__)(~).(~++o_+)(~)

R 2 while the grouping

corresponds to one f o r the lower and upper half-plane~,

r e s p e c t i v e l y . Denoting the half-plane WHOs by TD (W) and Tp (W) , r e s p e c t i v e l y , ~m where W : = l - k ~ is the two-dimensional L 1-convolution on u ~ L , we have the inverse by GOLDENSTEIN & GOHBERG ( c f . remark 2.1!) e x i s t i n g f o r (6.4)

aW(~) # 0 on ~2 as

[TPr(W)]-I = F-I[~++~+_(~)] -1FP r F-I[o_+~__(~)]-IF

and a s i m i l a r formula f o r

[TPu(W)]-I . Due to a r e s u l t by SIMONENKO (1967) [94]

assuming that

is compact on

XE.I.k,XGI

Lp ( ~ 2 )

, l
E,Gc~ 2

lying opposite on the same diagonal we a r r i v e at the Theorem 6.1 :

(STRANG (1970)[ 99 ]) • Let kELI(R 2) and inf.~ I I - ~ ( ~ ) I > O. Then 2 the WHO W++EBm(LP(~++)), l
R = P++{[TPr(W)]-I + [TPu(W)]-I - W-I}P++

The present author and F.-O. SPECK [59] r e c e n t l y studied the WH integral equation with (6.6)

kELI(~ 3)

and

(WGm)(x) : = m(x) - } k(x-y)m(y)dy = f(x)EL2(G) G

where G = G(~) : = {x = (Xl,X2,X3) : x l + i x 2 = re i ~ , o ~ _ _ ~ , r ~0, x3E~}= S ~ denotes a wedge with x3-axis as i t s edge and with an opening angle equal to ~ . F i r s t they proved the Theorem 6.2 :

( [ 5 9 ] , th. 1). Let

there e x i s t s a vector Let iff

hE~ n

A be t r a n s l a t i o n i n v a r i a n t i t is i n v e r t i b l e on

Remarks :

6.1 :

Zc~ n , n ~ 2 ,

such t h a t f o r a l l xcZ E ~ ( L 2 ( ~ n ) . Then

be a c y l i n d r i c a l also

region, i . e.

x + phEZ f o r a l l p ~ O.

Tp(A) : = PzAI~9~(pz)~(L2(Z))

~(P2) ~ L2(Z)"

The same statement holds f o r

LP(z) , 1~p~, f o r certain Sobolev

spaces and spaces with a weight function. 6.2 :

The r e s u l t carries over to cones

Fc~ n instead of

i n v a r i a n t " operators, e. g. p s e u d o - d i f f e r e n t i a l behaves l i k e

~A(p~) : ~A(~)

for

p > 0

and

Z admitting " d i l a t a t i o n

operators of order zero whose symbol ~c~n-{o}.

Now, the f o l l o w i n g r e s u l t is true Theorem 6.3 :

( [ 5 9 ] , th. 2). Let

WG be given as above. Then the f o l l o w i n g r e s u l t s

are equivalent: (i)

WGE~(L2(G))

is i n v e r t i b l e

(ii)

WGE ~(L2(G))

and

220

(iii) every operator W in the family defined by F-1 1,2 ~( " " ' x 3 ) F I , 2 ' ~,x3 X3E~, is i n v e r t i b l e as two-dimenslonaL WHO on the sector S~ with angle ~ at the vertex. While in the case of

G = ~n

or

= ~+n , n ~ 2 , the e l l i p t i c i t y

i n f I~w(~)I > 0 ~wE}q@(~n) is necessary and s u f f i c i e n t for the i n v e r t i b i l i t y of ~E~n ' , W operating on LP(~ n) or L P ( ~ ) , l
Let

A = l - k , , kELI(Rn), E 9 r

taining a cone. Then for ellipticity

Tp(A) : = PEAI~(pF)

is necessary and "strong e l l i p t ~ c i t y " ,

(6.7)

infn([Z-~(~)].ei~)

for suitable

~E[0,2=)

Cor011ary 6.1 : i t is

a)

be a measurable subset in

and

~n

con-

to be i n v e r t i b l e on "~.(PE) ~ L2(E) i . e.

a ~ >0 ,

~ > 0 , is s u f f i c i e n t .

([59], Corollary 1). For the WHO WG of eq. (6.6) to be i n v e r t i b l e

necessary that

l-k(~)

be e l l i p t i c

and

b)

s u f f i c i e n t that i t be point-

wise strongly e l l i p t i c (6.8)

i n f {Re e i ~ [ l - ~ ( ~ l , ~ 2 , x 3 )]

:

(~I,~2)ER 2 } = ~(~,x3) > 0

In order to admit strongly singular convolutional operators on quadrants depending even on the variables

Xl,X 2 we introduce the theory of operators of " b i - l o c a l type"

by PILIDI (1971) [66] which has a predecessor in SEELEY's paper (1965) [82] Chap.13~, in a sense (see also DOUGLAS & HOWE (1971)[23]!). D e f i n i t i o n 6.1 : Let ~1' ~2 denote two B-spaces of Lp-functions on ~m and ~ n , respectively. Ajc ~ ( ~ j ) the operators of local type on Xj • ~j the set of operators which are t r a n s l a t i o n i n v a r i a n t

c~(~j),

"~j

ideals of the Fredholm and compact operators on ~ by

BI®B 2

the subalgebra and ~

, respectively. Then denoting

the topological (and algebraica]) tensor product of two B-spaces

B2 we have: a) A,BEA : = AI~ A2 are called " i - e q u i v a l e n t " , A~B , i.ff b)

the BI

and

A-BE~I@A 2

AEA is called "l-Fredholm-Noether", AE~ 1, i f f there e x i s t

R~,RrEA such that

R~A ~ A R r ~ I c)

A,BCA are called " l o c a l l y I-equivalent at exists neighborhood such that

(6.9)

~ (Xo)C~ m

o m XlE~ "

and ~uEC~(~m)

inf II ( A - B ) ( ~ u ' I I ~ I 2 ) - T I ~ TE~A 2

<

iff for all ~ > 0

with

there

~L(x) ~ 1 on ~ (Xo)

221

d)

AEA is called "locally 1-Fredholm-Noether at x~E~ m'' i f f there is a neighborhood 1~L(x~)c~m , a~Lcc~ ( ~"m ) as above, and R~,x~ , Rr,x~ E A such that

(6.10)

R~ ~,x oI a(o~Iz® 12)-~i ( ~ l l ~ 1 2 ) A R r , x ~

,~1 ~ I i ~

12

Remark 6.3 :

Analogous notations hold with respect to the second component.

Theorem 6.5 : (PILIDI (1971)[66]) : AEA = A1(~(1)®A2(~2) is c~(JEI~)E2) i f f A is locally l-Fredholm for all x~c]Rm and is locally 2-Fredholm for all x~E]Rn. Theorem 6.6 : (PILIDI (1971)[66]): Let A,BEA and A be locally l-equivalent to B at xOERm . Then A and B are at the same time locally l-Fredholm at x~ or 1 not. These oheorems are applied to "bisingular Cauchy-type integral equations on ~2 or ~ m ~ n , , and to Wiener-Hopf integral equations on the quadrant. Theorem 6.7 : Let a (Xl,X2)EC((~)2), ~ = 0,1,2,12, and the bi-singular Cauchytype operator be defined by (Lm)(Xl,X 2) : = ao(Xl,X2)~(Xl,X2)+al(Xl,X2).(Sl~)(Xl,X2) (6.11)

+a2(xl,x2)(S2~)(x1,x2)+a12(Xl,X2)(S12m)(Xl,X2 )

where (6.12)

(Sl~)(Xl'X2) : = ~ i f

(6.13)

(S2m)(Xl'X2) : = ~ i f

• (Yl,X2)dY 1 Yl - Xl ~(xl,Y2)dY 2 Y2 - x2

~2 denote the "partial Cauchy transforms" and (6.14)

(S12~)(Xz,X2) : = (Sz(S2~))(Xz,X2) = ~

i

m(Yl,Y2)dY2dY 1 S f ]RI JR2 (Y2-X2){Yz-Xl) "

Then the following statements are equivalent: LE~(L2(IR2)) (6.15 a) for all (6.15 b)

or

[ao(Zl,X2)+al(Zl,X2).sign ~ i ] . I + a2(zl,x2)+a12(Zl,X2)sign ~i ] S2 ZlE~ I

and fixed

~i = ±I

and

[ao(Xl,Z2)+a2(xl,z2)-sign C2]-I+ a1(xl,z2)+a12(x1,z2)sign ~2 ] SI

for all z2E~ 2 with respect to

and fixed C2 = ±1 are invertible one-dimendional singular operators x 2 and x I , or the symbol of L :

222 (6.16)

aL(Z1'Z2;~l'~2)

: = ao(Z1'Z2)+al(Zl'Z2)sign ~1 + + a2(z1'z2)'sign ~2+a12(z1'z2 )sign ~I "sign ~2

is e l l i p t i c

i. e.

(6.17 a)

# 0

on

I~2 ~ { - 1 , 1 } 2 and a d d i t i o n a l l y

[arg aL(Zl,Z2;~l,~2) ]~

= 0 for all

ZlC~l

; ~i'~2 = ±I

z2=-~ and (6.17 b)

[arg aL(Zl,Z2;~l,~2)] = = 0 Zl=-~

Remarks : 6.3 :

In case of

avEC(~2 )

for all

z2c~ 2 ; ~i,~2 = ±1

the last two conditions are superfluous!

6.4 : This theorem holds also for R I ~ ~2 replaced by Ljapounov-curves rl,r2cC and B-spaces LPl(pl;F1) and LP2(p2;F2) instead of L2(~1 ) and L2(~2 )" The coefficients a may even be piecewise continuous only or systems of equations with matrices a may be involved. (cf. mainly PILIDI & SAZANOV (1974)[67] and DUDU{AVA (1975) [27,28]. They calculated a two-sided regularizer and the index of the operator

L.

Corollary 6.2 : Let LEA = AI(LPl(pl;FI) ) ~ A 2 ( L p 2 ( p 2 ; ~ ) ) be a Fredholm-Noether operator. Let R1 and R2 denote I - and 2- partial regularizers, respectively. Then a regularizer (6.18)

R for

L

is given by

R : = R1 + R2 - RILR2

and the index

L by

(6.19)

Ind L = [
where

Kj, j = 1,2 , denote the winding numbers with respect to

xjeFj

of the

functions (6.20)

a±±(Xl,X2) : = (ao±al±a2±al2)(Xl,X2)

Remarks : 6.5 : Applying the two-dimensional F-transformation to the Wiener-Hopf equation on the quadrant ~ + - or more general: studying four-part-composite equations

223 (6.21)

(Am)(Xl'X2) = ~ ( X l ' X 2 ) -

IR2

L kI(Xl-YI'X2-Y2)m(YI'Y2)d(Yl'Y2) ~++'~

N2.~(Yl,Y2)d(YI,Y2)-S N3-~(Yl,Y2)d(YI,Y 2) ]R2

-~r

~2

=-

k4.~(yl,Y2)d(Y1,Y2) = f(x],x2)EkP(m 2) , l
4--

where ~ = ~++E$ ~ leads to Riemann boundary value problems for two complex variables in the four products H±± of half-spaces in C . This is then equivalent to an equation like (6.11) with operator L . The problems have been thoroughly discussed mainly by DUDU~AVA in 1976 [ 29] ' §§ 2,3, in Sobolev spaces (Hs'P(~ 2+ + ))m .' mE~", i. e. including systems. 6.6 : PILIDI & SAZANOV (1971,'74) [66,67] treated also operators A of the bi-singular CMO-type where AI~XAxEQm®A2(~ n) for all x ~ m and A2'~YBy~AI(~m)®Qn for all yE~ n, where the Qm' Qn denote m u l t i p l i c a t i o n by homogeneous functions of degree zero Ec(~m-{o}) and E c ( ~ n - { o } ) , respectively. 2 where 6.7 : KREHER(1976) [56] studied n o n - e l l i p t i c Wiener-Hopf operators on ~++ the symbol may have a f i n i t e number of zero lines of f i n i t e order in each of the two variables 7.

~i,~2.

Concluding Remarks

SHINBROT in 1964 [ 89], DEVINATZ & SHINBROT in 1969 [16], REEDER in 1973 [76], and PELLEGRINI in 1973 [65], j u s t to mention a few papers, were involved into the study of general WHOs on H i l b e r t spaces ~ ; Tp(A) i~(p ) where AE~(~) and P = P2E ~(~.) a projector. They derived c r i t e r i a for the i n v e r t i b i l i t y of Tp(A) on ~(P) and the connection to f a c t o r i z a t i o n . Concerning e l l i p t i c systems of singular integral operators having symbol matrices being (~+l)-times continuously d i f f e r e n t i a b l e positively homogeneous functions of ~ : ( 6 1 . . . . ~n) , ~ > ~n , the i n v e r t i b i l i t y of the n

corresponding WHOs on ~+ has been discussed by SHAMIR in 1966,'67 ~7,88] in w S ' P ( ~ ) - s p a c e s . He derives a p r i o r i estimates and makes use of the f a c t o r i z a t i o n of matrices according to GOHBERG & KRE~N (1958)[43]. See also the paper by ~HUBIN in 1971 [90]! These problems have been displayed and treated by the Banach f i x e d point principle for strongly e l l i p t i c translation invariant A by HEISTER & SPECK (1977/78) [58]. They also discussed the more general problem of an "N-part conjposite WHO" N

~JN : = z Aj,Pj j=l

on

LP(~ n) , l
224 where the N

zP.= j=1J

AjE~(LP(IRn))

are i n v e r t i b l e and

PjPk = 6jkPk E~(Lp(IRn))

with

I.

In t h i s paper one may find also a couple of examples of mathematical d i f f r a c t i o n theory leading to such operators (cf. [58], chap. 2!). The present research centers around the following questions, at least in the mind of the author: (1)

Which are the necessary conditions for an AE~(3C) and a given continuous

projector P on ~ for Tp(A)_ _1~l~,p) to be invertible? The same question applies N to ~ AjPj to be i n v e r t i b l e on ~. j=l In order to allow x-dependent symbols

~A(X,~)

of generalized translation

invariant operators or pseudo-differential operators one wants to know: (2)

Which are the necessary conditions on

is of local type, i . e. [~I,A]

~A(~) EL=(~ n)

is compact on

what are the conditions on ~,aEL=(IR n)

such that

A = F-I~A(~)F

LP(IRn ) , l
- or subspaces - so that

~EC(]Rn)

or

~(x).F-I~(~)F

is

compact on ws'P(~ n) scIR , l
Prof. Dr. Erhard Meister Fachbereich Mathematik Technische Hochschule Darmstadt Schlo~gartenstr. 7 D 6100 - Darmstadt

have to be

225

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22 23 24 25 26

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227 54 55 56 57 58

59 ( 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

Kre~n, M.G. Integral equations on a h a l f - l i n e with a kernel depending upon the difference of the arguments. AMS Transl. 22, 163-288 (1962) Kremer, M. Ober eine Klasse singul~rer Integ~Igleichungen vom Faltungstyp. DisS. TU Berlin 1969 Kremer, M. Ober eine Algebra 'nicht normaler' Wiener-Hopf-Operatoren I & II. Math. Ann. 220, 77-86 & 87-95 (1976) Kumano-Go, H. Pseudo-differential operators ' (japan.), lwanami-Shoten Publish., Tokyo 1974 Meister, E., Speck, F.-O. Somemulti-dimensional Wiener-Hopf equations with applications. FB Hathem., TH Darmstadt, preprint Nr. 373 (1977); to appear in Proc. 2nd Sympos. on Trends in Appl. Pure Math. Mech. Kozubnik, Sept. 1977 Pitman Monographs, vol. 2 (1978) Meister, E., Speck, F.-O. Wiener-Hopf operators onthree-dimensional wedgeshaped regions, FB Mathematik, TH Darmstadt, preprint Nr. 389 (1978)~ to appear in Applicable Analysis. Michlin, S.G. Singular integral equations, Uspehi mat. Nauk (N~,) 3, 29-112 (1948) (russ.),~IS transl. 24, 84-198 (1950) Michlin, S.G. Multidimensional s-Tngular integrals and integral equations. Pergamon Press, Oxfor~ et al. 1965 Mushkelishvili, N.I. Singular integral equations. Noordhoff, Groningen 1953 Nickel, K. L~sung eines Integ-r'algleichUngssystems aus der TragflUgeltheorie, Math. Zeitschr. 54, 81-96 (1951) Peetre, J. M i x ~ problems for higher order e l l i p t i c equations in two variables I. Annali di Scuola norm. Pisa (3) 15, 337-353 (1961) Pellegrini, V. General Wiener-Hopf operators and the numerical range of an operator. Proc. Amer. Math. Soc. 38, 141 - 146 (1973) P i l i d i , V.S. On multidimensional~isingular operators, Sov. Math. Dokl.l_~2, 1723-1726 (1971) P i l i d i , V.S., Sazanov, L.I. A priori estimates for characteristic bisingular integral operators. Sov. Math. Dokl. 15, 1064-1067 (1974) Pr~ssdorf, S. Operators admitting unfunded regularization (russ.). Vestn. Leningrad Univ. 20, 59-67 (1965) Pr~ssdorf, S. Ei~imensionale singul~re Integralgleichungen und Faltungs91eichungen ni%-ht normalen Typs i n lokalkonvexen R~umen-~.~abil.schrif{,Karl-Marx-S'tadt (1967) " Pr~ssdorf, S. Zur Theorie der Faltungsgleichungen nicht-normalen Typs. Math. Nachr. 4_22, I03-131 (1969) Pr~ssdorf, S. Einige Klassen singul~rer Gleichungen. Birkh~user Verlag, Basel-Stuttgart 1974. . . . . . . . . . . . . Rabinovi~, V.S. Pseudo-differential equations in unbounded regions with conical structure at i n f i n i t y . ~lath. USSRSbornik 9, 73-92 (1969) Rabinovi~, V.S. Pseudo-differential equations in unbounded megi'ons. Soy. Math. Dokl. 12, 452-456 (1971) Rabinovi~, V.S. Pseudo-differential operators on a class of non-compact manifolds. Math. USSRSbornik 18, 45-59 (1972) Rakovsh~ik, L.S. On the theormof integral equations of convolution type (russ.). Uspehi Matem.Nauk 18, 171-178 (1963) Reeder, J. On the i n v e r t i b i ~ t y of general Wiener-Hopf operators. Proc.Amer. Math. Soc. 27, 72-76 (1971) ~ahbagjan, R-~.L. Convolution equations in a half-space, AMS Translat.,Ser. 2, 75, I17-148 (1968) Samko, S.G The general singular equation in the exceptional case. Diff. Equat. l , 867-874 (1965) SchUppeT, B. Regularisierung singul~rer nicht e l l i p t i s c h e r Integralgleichungen mit unstetigen Koeffizienten, Diss. Univ. MUnchen 1973 SchUppel, B. Regularisierung singul~rer Integralgleichungen vom nichtnormalen Typ mit stUckweise stetigen Koeffizienteno 'Function theoretic methods for partial d i f f e r e n t i a l equations': Proc. Int. Symp., Darmstadt; 4 3 0 - T , Springer Lecture Notes 561, Berlin-Heidelberg 1976

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Schwartz, J.T. Someresults on the spectra and spectral resolutions of a class of singular integral operators. Comm. Pure Appl. ~,lath. 15, 75-90 (1962) Seeley, R.T. I n t e g r o - d i f f e r e n t i a l equations on vector bundles. Trans. Amer. Math. Soc. 117, 167-204 (1965) Seeley, R.T. Topics in pseudo-differential operators. Pseudo-Diff.Operators (C.l.M.E.,Stresa 1968). Edizioni Cremonese, Roma, 169-305 (1969) ~ir, E. Evaluation dansW~,Ppour des probl~mes au l i m i t e s e l l i p t i q u e s mixtes dans le plan. C.R. Acad. Sci. Paris 254~ 3621-3623 (1962) Shamir, E. Mixed boundary value problems for e l l i p t i c equations in the plane. The Lp theory, Annali Scuola Norm.Sup. Pisa (3) !7, I17-139 {1963) Shamir, E. Reduced H i l b e r t transforms and singular in~-~gral equations. Journ. Analyse Hath. (Jerusalem)f2, 277-304 (1964) Shamir, E. ~Jiener-Hopf type pr~lems for e l l i p t i c systems of singular i n t e gral equations. Bull.Amer.Math.Soc 72, 501-503 (1966) Shamir, E. E l l i p t i c systems of singular integral operators,l. The h a l f space case. Trans. Amer. Hath. Soc. ]_~_7_, 107-124 (1967) Shinbrot, M. On singular integral operators. Journ. Math. Mech. I_33, ~95-406 (1964) Subin, M.A. Factorization of matrices depending on a parameter and e l l i p t i c equations in a half-space. Math. USSR Sbo~nik 14,65-84 (1971) Simonenko, I.B. A new general method of investigat~-n-nq l i n e a r operator equations of the type of singular integral equations. Sov. Math. Dokl. 5, 1323-1326 (1964) Simonenko, I.B. A new general method of investigating linear integral operatorequations of the type of singular equations. (russ.). Izvest. Akad., Nauk, Ser. Matem., 29 567-586 (1965) Simonenko, I.B. Convolution operators in cones. Sov. Math. Dokl. 8, 1320-1323 (1967) S~monenko, I.B. Operators of convolution type in cones. Math. USSR Sbornik 3, 279-293 (1967) S~hngen, H. Zur Theorie der endlichen Hilbert-Transformation. Math. Z e i t schr. ~ 31-51 (1954) Speck, F.-O. Ober verallgemeinerte Faltungsoperatoren und eine Klasse von Integrodifferentialgleichungen. Dissert. TH Darmstadt 1974 Speck, F.-O. Ober verallgemeinerte Faltungsoperatoren und ihre Symbole. Function theoretic methods for p a r t i a l d i f f e r e n t i a l equations: Proc. I n t . S-~p., Darmstadt;459-471, Springer L e ~ r e Notes 561'~, Berl~nJHeidelberg 1976 Speck, F.-O. Eine Erweiterung des Satzes von Rak--~vsh~ik und ihre Anwendung in der Simonenko-Theorie. Math. Ann. 228, 93-100 (1977) Strang, G. Toeplitz operators in a quarter-plane. Bull. Amer. Math. Soc. 76, 1303-1307 (1970) Talenti, G. Sulle equazioni i n t e g r a l i di Wiener-Hopf, Boil. Un. Mat. I t a l . ~, Suppl. fasc. ~, 18-118 (1973) 2nd Titchmarsh, E.C. Theory of Fourier I n t e g r a l s . e d . , Clarendon Press., Oxford 19~8 Tricomi, F. On the f i n i t e Hilbert-transformation. Quart. J. Math. (2), 2 199-211 (1951) , Vishik, M . I . , Eskin, G.I. Equations in convolutions in a bounded region. Russ. Math. Surveys 20, 85-151 (1965) Vishik, M . I . , ~skin~--G.l. Normally solvable problems for e l l i p t i c systems of equations in convolutions (russ.). Mat. Sbornik 74, (116), 326-356(1967) Vishik, M . I . , Eskin, G.I. E l l i p t i c equations in c o ~ o l u t i o n in a bounded domain and t h e i r applications. Russ. Math. Surveys 22, 13-75 (1967) Widom, H. Singular integral equations in Lp. Trans'-. Amer. Math. Soc. 9_7_7, 131-160 (1960) Wiener, N., Hopf, E. Ober eine Klasse singul~rer Integralgleichungen. Sitzg.-ber. Preuss. Akad. Wiss.; Phys.-Math. KI.30-32, 696-706 (1931) Zabreyko, p.p. et a l . Integral Equations - a reference t e x t . Noordhoff Intern. Publ., Leyden 197-~. . . . . . .

MULTIPARAMETER PERIODIC DIFFERENTIAL EQUATIONS

B. D. Sleeman Dedicated to the memory of Professor Arthur Erd~lyi

§I.

Introduction. The most widely used method of solving boundary value problems for linear

elliptic partial differential equations is the classic separation of variables technique.

Usually this approach leads to the study of eigenvalue problems

for linear ordinary differential equations containing a single spectral parameter, namely the separation constant.

Indeed it is fair to say that this is perhaps

the prime motivation for the study of spectral problems associated with linear ordinary differential expressions.

In many cases, however, when the separation

of variables method is effected it is found that the separation Constants cannot themselves be decoupled and several may appear as spectral parameters in each of the attendant ordinary differential equations. A classic example of this is the problem governing the vibrations of an elliptic membrane with fixed boundary.

An application of the separation of

variables technique leads to the study of solutions of the following pair of linked eigenvalue

problems for the Mathieu's equations.

d2yl dx~

+

(-Xl + x2c°sh2xl)y I = 0,

Yl(O) - regular,

0 ~ x I ~ ~,

yl(~) = 0,

d2y2 dx~

+ (%1 - %2 cos2x2)Y 2 = O,

0 ~ x 2 N 2~,

Y2(X2) - periodic of period ~ or 2~. In this example %1 and %2 are the spectral parameters.

Another example, recently

considered in [ 15], is the problem of diffraction by a plane angular sector. lying

App-

separation of variables in this case results in the study of the following

pair of Lame equations.

230

d2y 1

dx]

+

{%1 - %2 k2 sn2xl}yl

-K -< xI -< K,

= 0,

yl(-K) = Yl(K) = 0, d2y2 dx~ + {%1 - %2 k2 sn2x2}Y2 = O,

Y2 (K) = Y2' ( K + 2 i K ' )

x2

=

K + 2i~,

0 < ~ < K' _

--

,

= 0.

Once again %1' %2 are spectral parameters and k, the modulus of the Jacobian elliptic function s n ~ is related to the semi-apex angle ~ of the sector by k = sin ~. One can cite a wide range of problems leading to linked systems of ordinary differential

equations containing 2, 3 or more spectral parameters.

During the early decades of this century the study of multiparameter

eigen-

value problems gave way to some extent to the study of the "special functions" generated by such problems.

Thus in the 1930~s and 40's there was much interest

in the properties of Mathieu functions, Lame functions, Heun functions and the like.

ellipsoidal wave functions,

The leading instigator of these researches was

undoubtedly Arthur Erd~lyi who not only advanced our knowledge of these functions to the degree that we know them today, but also gave an encyclopaedic account of them in the well known volumes on higher transcendental arose from the Bateman manuscript project.

functions [12] which

These special functions are being

actively studied today and apart from their obvious interest to the mathematical physicist they do arise in other areas as well.

Let us consider for the moment

one example which leads to an as yet unsolved problem. The problem we have in mind is of importance in the study of quasi-conformal mappings[5]

and may be described as follows.

In Lam~'s equation let %2 = -

and consider the equation

d~ (h + 1 k 2 dx 2 + ~ sn 2 x)y = 0 and denote two linearly independent solutions by ~I and ~2-

Classical

I

231

uniformisation theory asserts that for each value of the modulus k and consequently for each fundamental pair of periods of the Jacobian elliptic function sn x,

there

is precisely one value h such that

n_l (C

- periods of sn x) = ~ a simply covered d i s c }

n2

L or

J

a half plane

However the actual value of h is known in only a few special cases.

In general

we ask; for which values of h is

n~ (C - periods of snx)

=

n2

Clearly

~

simply covered 1 ?

L Jordan domain

one method of attacking this problem is to investigate thoroughly the

solutions nl and N2"

In the slightly more general situation in which %2 = V(V + I)

1

V = ~ + m (m an integer) there are results due to Halphen LI4j, Ince [16J

where

and quite recently by Pearman [|8j.

Indeed Ince [16j observed that when k 2 = ~ 2

some of the eigenvalues of Lam~'s equation giving rise to periodic solutions of periods 8K, 8iK' were rational and conjectured that this may be true for infinitely 1

many values of v = ~ + m. l

We may confirm this conjecture by observing that when

!

l

= ~ + m, %1 = ~ ~(~ + l) and k 2 = ~.

nl'

~2 = ( s n x

Lam~'s

± d n x / k ) 1 / 2 2FI (- ~]m , ~ m1

This can be verified by direct substitution.

equation has solutions of the form

3. ~1 (snx ± dnx_k.2../) +I; ~, ~

Clearly if m is even and positive

then there are infinitely many rational values of %1 each giving rise to solutions of periods 8K, 8iK'.

There must be a large number of results of this kind relevant

to the problems above. An important feature of the multiparameter eigenvalue problems which arise in mathematical physics is that the differential equations often have either singly or doubly periodic coefficients and it is to such systems that we now turn. §2.

The Problem Consider the linked multiparameter system of periodic ordinary linear differ-

ential equations of the form

232

d 2Yr (Xr) dx 2 r

+ qr(Xr)Yr(Xr) +

n ~ %sars(Xr)Yr(Xr ) = O, s=l

r = 1. . . . , n. (2.1)

Here x r £ [0, ~0rj, (0 < 0or < ~) and the coefficients ars, qr are continuous real valued functions with period ~r"

We make the basic assumption that the

determinant det{ars(Xr)} n r,s=! > O.

for all x r E [0, ~r j,

r = I, ..., n.

(2.2)

Several other structural hypotheses may be

applied, see for example [20] or [4J. In addition to (2.2) we impose periodic boundary conditions of the form

(P)

Yr(0) = yr(~r), y'(0) = Yi(~ ), r Lr

(2.3) r = l. . . .

n.

or semi-periodic boundary conditions of the form (S-P)

Yr(0) = -yr(~r), (2.4) Yr '(0)

= -y~(~r ),

r = l, ..., n.

As is now standard we say that the n-tuple of necessarily real numbers % = (%1 . . . . .

%n ) is an eigenvalue of the system (2.1, 2.2, 2.3) (or 2.4) if this

system has a non-trivial set of solutions Yr(Xr; %), particular n-tuple.

r = l. . . . .

n, for this

The algebraic product Yl(Xl; %) ... Yn(Xn; %) is called the

corresponding eigenfunction. The study of the above system and similar Sturm-Liouville problems has captured the interest of several authors recently.

Indeed such problems may be

formulated in terms of linear operators in Hilbert space.

The book of Atkinson

[4] and the monograph of Sleeman [20J are suitable references to this and other matters.

From these works it may be readily verified that eigenfunctions and

eigenvalues do exist for the above problems and that the eigenfunctions form a complete orthonormal set in the weighted Hilbert space of Lebesque measurable functions on

233

E0, ~] ~

the Cartesian product of the n

n x [0, ~r ], r=l

compact intervals

[0, ~r j,

r = I, ..., n.

The

norm in this space is given by

]lull2 = f

[ul 2 det{ars}dX

1 "'"

[0,~]

In the following one parameter

dx

(2.5) n"

section we show how a classical argument well known in the

case may be used to assert

the existence of eigenvalues

and eigen-

functions. A question of particular (2.1,

2.2, 2.3) or (2.1,

interlacing parameter

properties

the eigenvalues

of system

2.2, 2.4) or any combination of these problems

enjoy any

thus generalising

periodic case.

of eigenfunctions.

interest is to ask whether

the results well known in the one-

In the same way one would like to discuss

Recently Browne [8j has studied the interlacing problem and

Browne and Sleeman [9j have obtained some results concerning eigenfunctions.

the stability

To describe

the stability of

these results we shall need to introduce

some

notation. Let o = (oi, ..., ~n ) £ ~n, Or = 0 or I can then speak of boundary conditions

r = I, ..., n be an index set.

of type o applied

to the system (2.1) to

mean that periodic conditions are to be applied to the r-th equation if O semi-periodic

conditions

mnltiparameter conditions

eigenvlaue

(P) or (S-P)

if o

r

= I.

problem consisting of (2.1)

n},

That is each ir,*

r = I, ..., n, is a

This multi-index provides an ordering of the eigenvalues.

Notice that there are 2 n possible of boundary conditions.

(2.2) with boundary

as

where ~i = (i 1, ..., in) is a multl-index. integer.

= 0 and

In this fashion it is appropriate

i i X~(O) = {%r(O)[ r = | . . . . .

non-negative

r

Thus for each fixed o we have formulated a

as indicated by O.

to label the eigenvalues

We

spectra corresponding

to the 2 n different

Thus to begin with we would like to know how these

spectra interlace each other.

types

234

Consider

vectors

r = I, ..., n.

a, b c ~ ~

~n

w h i c h are partially ordered by a ~ b if a -
Now for a collection of functions

llyrlI = ! we denote by V(y)

Yr(Xr)

~ L2(O, ~r ), with

the n × n m a t r i x whose entries

are

w i 0r ars(Xr)lYr(Xr)

Next introduce

the set C c ~ n

12dXr,

r, s = l, ..., n.

defined by

C = {a e i~n I V(y)a -< 0 for some Yr £ L2(O' C°r)' IIYrll = |, Clearly C defines the 2 n possible

a cone and, as demonstrated

by Binding

r = l.....

and Browne

n}.

[6], each of

spectra is ordered by C.

Finally we define

the set

i = {~~(~) l i-multi-index; ~

o

~

= 0 or I,

r = 1.....

n}.

~r

Thus I is the collection of all eigenvalues

from all possible problems

periodic

or semi-periodic. The m a i n interlacing

theorem of Browne

LSJ can now be stated;

T h e o r e m 2.]. Let T ~ ~ n be such that ~ multi-indices

= 0 or

r

with 0 N e N (l,|, i

(~~(~)

|, r = |, ..., n.

...,

Then if i and e are

l). the set

i+e + c)

n

(~~ ~(~)

-

c)

n

~,

J contains (i) (ii)

any eigenvalue

X~(~) for w h i c h

i -< j -< i + e and O r = T r except that if e r = ; and (ir, T r) is (even, 0) or (odd, then ~

r

To illustrate

m a y be 0 or l,

r = ;, ..., n.

this result consider

the following

elementary

Let x l, x 2 e [O, lj and take Sl, s2, ql' q2 to be real valued functions

on EO, IJ with p e r i o d one.

The two parameter

system is

|)

example for n = 2. continuous

periodic

We further assume s I > O, s 2 > 0 on [O,l].

235 d2yl(x 1) + ql(xl)Yl(Xl ) + (-%1 + %2)sI(xl)Yl = O,

dx~ d2y2(x2 )

+

dx~

The definiteness

condition

det

For this example

q2(x2)Y2(X2 )

(2.2) is satisfied

-Sl(X 1)

Sl(X 1)

-S2(X 2)

-s2(x 2)

the cone

+ (-~'1

C

-

%2)s2(x2)Y2 = O.

since

= 2Sl(Xl)S2(x 2) > O.

is given by

C = {(a,b) ~

~2

I -a ~ b ~ a}

and the spectral diagram is shown in figure I. If o = (0,0) the eigenvalues i ~ are marked • and denoted by %~. If o = (0,1) the eigenvalues are marked o and i ~ ~ i denoted by ~~. If o = (I,0) the eigenvalues are marked m and denoted by ~~. ~

~

Finally if o = (1,1)

i

the eigenvalues

~

are marked • and denoted by N~.

In order to treat the stability question we must first define what we mean by stability as applied to the multiparameter

Definition (i)

problem.

2.1 A point % £ ~ n is said to be a point of stability for the system (2.1)

if all solutions Yr of the r-th equation are bounded over (ii)

(-~,~),

A point % E ~ n is said to be a point of conditional

system (2.1) if each of the equations has a non-trivial

r = I, ..., n.

stability

for the

solution bounded over

(-~,~). (iii)

A point % £ ~ n which does not satisfy either

be a point of instability

for the system

(2.1).

(i) or (ii) is said to

236

o~

O~

• ~0 ~

",.

\

/'

¢e\

i

%

r

%

s 1 1

i

/

>, n

i

s

s

I

p i z i

r i

Stability regions for Example I. Figure I. We now require to modify our definition of a cone as follows:

Given a

multi-index i = (i|, ..., in) we define the cone C(i) to be the collection of all points a E ~ n for which there are non-zero points fr

L 2 (0, ~ r ) such that

287

[V(f)a]r; the r-th element of the column vector V(f)a, satisfies [V(f)a] r~

~ 0 0

if ir is even, if i

r

is odd.

This defines 2n cones, i.e. 2n-I cones and their negatives.

Finally we denote

by S the set of all points of conditional stability of the system (2.1).

Our

stability result can now be stated as

Theorem 2.2. i S ¢ u [{X~(0) + C(i)}

i n {X~(1) - C(i)}],

(2.6)

I

where

I =

(1,|,

l)

e

This result is illustrated in the case of the above example by the shaded regions of figure I.

For this example we have equality holding in (2.6).

However

in [9J we give an example involving Mathieu's equation for which the inclusion in (2.6) is strict.

§3.

Existence of Eisenvalues Here we apply a classic approach [I0] to the system (2.1) (2.2) (2.3) or (2.4)

to obtain some information regarding the existence of eigenvalues. Let ~r(Xr; X), ~r(Xr; X), r = ]. . . . .

n be linearly independent solutions of

(2.1) satisfying the initial conditions ~r(0; ~) = I,

~r(0; X) = 0,

~'(0; r

~'(0; r

~) = 0,

(3.1)

~) = i,

r = I, ..., n. The general solution to (2.1) can be expressed as a linear combination of the functions ~r and ~r"

That is

Yr(Xr; X) = Cl, r ~r(Xr; X) + C2, r ~r(Xr; X).

(3.2)

It is well known that a necessary and sufficient condition for the existence of two-linearly independent solutions each satisfying the periodic boundary condition (P) is

238

~r(mr; X) = I,

~r(~r; X) = 0,

Cr(mr; X)

~'(~

=

0,

r

r

;

X)

=

(3.3)

1.

Similarly a necessary and sufficient condition for the existence of two-linearly independent solutions each satisfying the semi-periodic boundary condition (S - P) is ~r(~r; %) = -I

~r(Wr; %) = 0,

T

(3.4)

!

~r(~r; X) = 0,

~r(~r; X) = -I.

In general of course a necessary and sufficient condition for (2.1) to have a solution satisfying

(P) or (S -P) is that Dr(h) = ~r(Wr; X) + ~r(~r; X) =

r

=

|~

...,

±2,

(3.5)

n.

The problem of existence of eigenvalues solvability of the system (3.5).

is thus reduced to the question of

By a standard use of the variation of parameters

method we find Dr(X) ~X

r r

2

JO {~r(T; %)~r(~r;

S

X) +

+ ~r (T; X)~r(T; %)[~$(~r; - ~$(T; %)¢~(mr; r~s

=

I,

...~

X) - ~r(~r; %)]

(3.6)

X)}ars(T)dT,

n.

Now if D (%) = i2 r

then the term in { } is a perfect square and since

det{ars} > 0 it follows from the inverse function theorem that (3.5) is uniquely solvable provided the eigenvalues

are simple.

If some of the eigenvalues are

not simple, in the sense that for some range of r, (3.3) and or (3.4) holds then we must work with the n x n derivatives of D

r

Hessian matrix constructed from the second partial

and make use of the inverse function theorem again.

As in the one parameter case, wherein

stability and interlacing

theorems

may be obtained from a study of D and its first derivative, one could study the gradients of D

r

for each r = I, ..., n to arrive at the multiparameter

analogue

239

of these results.

However the technicalities

appear complicated.

In the following section we outline a different approach to the study of periodic multiparameter

eigenvalue problems.

This approach is based on the

calculus of variations.

§4,

The Variational Approach In this section we consider the case of two-parameters

(n = 2) and study

the eigenvalue problem defined by 2 2 d Yr dx 2 + qr(Xr)Yr + s=I~ ~s ars(Xr)Yr = 0, r r = 1,2,

x

r

(4.1)

e [0, ~ J r yr(mr) = Yr(0)exp i ~ t r,

(4.2)

y$(Wr) = y$(O)exp i ~ t r, -I < t ~ I, r

r = 1,2, together with the definiteness

condition

(2.2).

For this problem the existence of a countably infinite set of real eigenvalues can be established either from [20] or by the method outlined in the previous section.

If we consider the eigenfunctions

as being periodically

extended to the whole of ~2 as continuously differentiable conditions

functions the boundary

(4.2) may be rewritten in the form Yr(Xr + ~r ) = Yr(Xr)eXp i~ t r,

(4.3)

r = 1,2. In addition to the condition

(2.2) we shall assume, without loss of generali~,

that al2(X I) < 0

on [0, ~i j,

a21(x 2) > 0

on [0, ~2 j.

(4.4)

This can always be arranged by a suitable scaling or affine transformation applied to the parameters %1' X2' As is well known [20] the eigenvalues and eigenfunctions (4.2) are simultaneous

eigenvalues and eigenfunctions

of the system (4.1)

of the following periodic

240

problems for partial differential equations; viz.: ~2y - al2(X I) ~

~2y + a22(x 2)

~x 2

~x!

+ La12(x])q2(x 2) - a22(x2)ql(Xl)~Y

= % det{a 1

rs

}Y

(4.5) (4.6

Y(x + ~r ) = Y ( x ) e x p i ~ tr, r = 1,2. 82y ~2y al1(x;) ---~ - a21(x 2) ----~ - [all(Xl)q2(x 2) - a21(x2)ql(Xl)]Y ~x 2 ~x I = %2det{ars}Y, together with the boundary condition (4.6).

In this condition the vector ~~r is

defined as ~l = (el'

0),

~2 =

(0, w2).

It should be noted that because of the assumed positivity conditions on a12

and a21 the left hand side of (4.5) is elliptic and it is to this equation that

most of our remarks are addressed. Let the eigenvalues of (4.5) (4.6) be denoted by An(t),

(t = (t I, t2)) and

let the corresponding eigenfunctions be denoted by ~n(X; An(t)).

It is readily

proved that the An(t) are real and form a countably infinite set with -~ as the only limit point.

They may be ordered according to multiplicity as Ao(t) e Al(t) e A2(t) e ....

(4.7)

and the corresponding eigenfunctions are orthonormal in the sense of (2.5). Notfce that since the eigenvalues %l(tl) of the given problem form a subset of the A (t) they exhibit a similar ordering to (4.7). n ~

We also have the completeness

theorem. Theorem 4.1. ) (r~a = 0, I .... , k~ - I) be the kj°-th roots J Let ~R (I -< R -< klk 2) denote the pair (trl, tr2)

For each j = 1,2 let e x p ( i ~ t r of unity where -I < tr. <- I. J in any order.

Then the set of all functions ~n(X; tR ) where n ~ 0 and

241

1 ~ R ~ klk 2 is a complete set of eigenfunctions for the periodic problem over the rectangle

Xl, x 2 E [0, WlklJ x [0, ~2k2 j. Again completeness here means completeness in the sense of functions Lebesque measurable on [0, ~Ikl j × [0, ~2k2 ] with respect to the weight det{ars}. The proof of this theorem is a simple modification of the proof of Theorem 6.2.1 in Ill].

With the aid of theorem 4.1 we can now argue as in [19j to prove

that the eigenfunctions of the given problem defined by (4.1) (4.2) are complete in the same sense. Now let

Q(Xl , x2) = a l 2 ( X l ) q 2 ( x 2 )

- a22(x2)ql(Xl),

(4.8)

and let F be the set of complex valued functions f(x) which are continuous in A ~ [0, ~i j x [0, w 2]

and have continuous first order partial derivatives in A.

Define the Dirichlet integral ~u

D(u,v) = fA{a22(x2) ~x ~u I ~x ~V 1

~

a l 2 ( X l ) ~x2 ~ x 2 (4.9)

+ Q(x 1, x2)u(x)V(x)}dx I dx 2 for all u,v ~ F.

This Dirichlet integral forms the basis for a variational approach to the multiparameter periodic problem.

It is discussed, at least for SchrSdinger

operators, in Eastham [I|] and its application to multiparameter Sturm-Liouville problems is considered in [19, 20]. Suppose we impose Dirichlet conditions on the boundary of A, i.e. Y(x) = 0

on

~A,

then for the associated Dirichlet problem we have corresponding eigenfunctions and eigenvalues which may be denoted by

@n(X; 6n) and 6n respectively.

In a

similar way, if we impose Neumannconditions on ~A then the associated Neumann problem will give rise to corresponding eigenfunctions ~(x; Nn ) and eigenvalues ~.

242

Now, arguing as in Eastham [II, p 101] we have the interlacing theorem Theorem 4.2. For n e 0 and all t ~n ~ An(i) ~ Nn"

(4.10)

This simple interlacing result may be employed to give a further interlacing

i theorem for the two-parameter problem as follows.

i

Let (%7(t), ~%(t)) be an eigenj

i

~

value for (4.1) (4.2) then since the %7(t) form a subset of the An(i) theorem (4.2)

gives, in an obvious notation, the result

i

i

i

for all t. ~

i

Substituting %7(t)into the first member of the system (4.1) we have the classic one parameter problem.

d2y I i dx~ + ql(Xl)Yl+ all(Xl)X~(t)Yl

+ X2al2(Xl)Yl = O,

~

(4.12)

Yl(X! + e l) = Y l ( X l ) e x p ( i g t l ) , where of course al2 < 0. If we let the eigenvalues associated with (4.12) and the conditions

yl(wl) = yl(0) = 0, i be denoted by 62 and those associated with (4.12) and the conditions y~(~l ) = y~(0) = O, i be denoted by n2 then again appealing to Ell, p 39] we have, for those eigenvalues

i %~(t) which are simultaneous eigenvalues of (4.1) with r = 2, i

i

i

(4.13) These arguments may be summarised in Theorem 4.3. Let the eigenvalues for the Dirichlet problem over A for (4.5) be denoted by

243

i i ~~ and the eigenvalues for the associated Neumann problem be denoted by n~, i i i where i = (i|, i 2) is a multi-index. If k~(t) = (%~(t), %2(t)) is an eigenvalue for the two-parameter t-periodic problem then i i i 8~ ~ k~(t)~ ~ ~n~. i For this value of k~(~) we also have i

i

i

i

i

where B2, 62 are the Neum~nn and Dirichlet eigenvalues for the one parameter equation (4.12). There are a number of results to be obtained via the variational approach all of which extend in one way or another the results known for periodic Schr~dinger equations and promise interesting applications to the multiparameter periodic problem.

We close this section with the remark that hidden in the

variational approach outlined above is the fact that the partial differential operator appearing in (4.5) is elliptic.

For n ~ 3 we cannot always arrange for

any of the associated partial differential operators to be elliptic in general. In this case the arguments have to be modified and for details at least in the abstract case we refer to the monograph [20].

§5.

The Alsebraie Approach In the study of periodic differential equations some useful insights are to

be gained by the study of various algebraic forms of the associated periodic differential equations.

There is a strong connection here with the study of

Monodromic groups, the famous Riemann problem arising from Hilbert's 21st problem (see the article by Nicholas Katz in [7]) and also the study of quasi-conformal mappings, the subject raised in the first section of this paper. Here we outline some of the results which are to be obtained in this setting and in particular the case of differential equations, like Lam~'s equation, which have doubly-periodic coefficients. [17, I, 2] as suitable references. when

For a background to this work we cite

The essential feature to be noticed is that

expressed in algebraic form a singly-periodic differential equation

is

244

distinguished by having only two finite singularities whereas a doubly-periodic equation has three. The basic concept is that of a solution which is multiplicative path in the complex plane.

for a given

That is a solution which when continued analytically

along a closed path is merely multiplied by a constant.

Consider the differential

equation _

dw + q(t)w = 0, d2w + p(t) ~-~ dt 2

(5.1)

_

where t E ¢ and p(t), q(t) are rational functions of t.

Let the singularities

(not necessarily regular) of (5.1), i.e. points where p(t), q(t) or both are singular, be denoted by tl, t2, ..., tn, ~.

If t o is an ordinary point of (5.1)

then we denote by F i a simple closed path from t o encircling the singularity t.

i

oncepositively

(clockwise) and enclosing no other singularity of (5.1).

further denote by rij the path consisting of order and similarly for rij k etc. in the negative

(anti-clockwise)

provided the singularities

ri, rj

We

described successively in that

The path which is the same as F i but described direction is denoted by -F i.

tl, ...

t

Observe that

are labelled in an appropriate order then

~ n

FI2 " • "n is effectively a path making a negative circuit about infinity. FI2 ... n is equivalent

That is

to the circuit -r .

Since p(t), q(t) are rational functions of t the singularities of equation (5.1) are isolated.

Hence if y(t) is a solution valid in a neighbonrhood of the

ordinary point t o it can be continued analytically hood of to giving a solution y*(t).

along r i back to the neighbour-

Symbolically we write

y(t) ÷ (Fi)Y*(t). In the case that y*(t) is a constant multiple

(5.2)

s i of y(t), i.e. y*(t) = siY(t) we

can write y(t) + (Fi)siY(t). In this case we say that y(t) is multiplieativ e factor.

for the path r i and s i is the path

It is easy to prove that there exists at least one multiplicative

of (5.1) for the path F.. I

(5.3)

If (5.3) holds then also

solution

245

y(t) ~

(-ri)

l

Y7 y(t). l

A p a t h Fi f o r which t h e r e e x i s t two l i n e a r l y ,,independent m u l t i p l i c a t i v e is called non-degenerate.

I f o n l y one such s o l u t i o n e x i s t s

solutions

the path is d~generat,~

Thus for a non-degenerate path r. there are linearly independent multiplicative l solutions Yi(1)(t), y~2)(t) which can be combined into a solution vector Yi (t) = {Y l)(t)' Yi(2) (t) }

such that Yi + (ri)siYi

(5.4)

where S. is the constant diagonal matrix <s! l), s! 2)>. l i i Let us now consider the case in which (5.1) has two finite singularities tl, t2 in the complex t-plane and a singularity at infinity.

We wish to consider

the behaviour of solutions when continued analytically around a composite path such as FI2.

To this end we introduce the following notation. 2Pi = s(1)z" + s(2)'i

2qi = s(1)i - s.i(2),

r. = s! I) s! 2), i i i M(012) =

i = 1,2,

cos @12

sin Ol2 I '

sin Ol2

-cos 012

where el2 £ C in general and is called the "link" parameter for the path FI2. Consider now a solutiQn vector YI such that

(5.5)

YI ÷ (FI)S]YI"

The analytic continuation of YI about r2 yields a solution vector of (5.1) so there is a constant matrix L = (%ij) such that Y| ÷ (r2)LY | . The eigenvalues of L must be the path factors s~ I), s~ 2) for F 2 and so (1)

trace (L) = s 2

+

s~2)

= 2P2

det (L) = s2(I) s~2) = r2" This then shows that for some constant ~ the entries in L must be such that

246

%11 = P2 + ~'

%22 = P2 - ~'

2 2 %12%21 = q2 - ~ " As is demonstrated in [2j there is no loss in choosing = q2 cosel2 with -~ ~ Re ~12 ~ ~ and taking L to be

LI2 = p21 + q2M(Ol2 ), where I is the 2 × 2 identity matrix. To summarise these results we have Theorem 5.1. Let F I, r 2 be non-degenerate paths.

Then there exists a link parameter el2

uniquely determined in the region R = {O I {0 < Re 0 < ~} u {Re 0 = 0, Im e < 0} u {Re e = ~, Im e < 0} such that there is a unique vector Yl2 determined up to a constant scalar multiplier with the properties Yl2 + (FI)SIYI2 Y12 ÷ (F2)LI2YI2 ° From this result we deduce Theorem 5.2. The path factors for FI2 are the roots of the equation s 2 - 2(plp 2 + qlq2 cos e12)s + r ir2 = O.

(5.6)

Since (5.1) in this case has only 2-finite singularities r12 is equivalent to - F

and so the path factors for - r

determined from (5.6).

must be precisely the path factors

Indeed as we shall show this provides a means of deter-

mining the link parameter el2 or the characteristic exponents of 5.1 at infinity. Consider for example Hill's equation in algebraic form, i.e. __

1

|

dXw + 2 {~ + dt 2

~

dw

~(t) w

} d-t + t(t

l)

_ 0,

(5.7)

247

where ~(t) is an integral function of t.

In this case it is known that t = 0,I

are regular singularities while the singularity at infinity is irregular.

It

turns out that with t I = 0, t 2 = I Pl = P2 = O,

ql = q2 = 1,

r I = r 2 = -l,

sll) = s~ I) = 1,

s12) = s~ 2) = -I.

Here (5.6) reduces to s with roots s = e x p ( ± i e ) .

- 2s cosel2 + I = o,

But since FI2 is equivalent to - r

characteristic exponents ± ~ precisely ± el2.

2

it follows that the

at infinity, in the sense of Erd~lyi [13] are

In another way it follows that el2 is related to the number t

in the definition of the t-periodic problem (see (4.1) (4.2) above), to the descriminant D discussed in section 3.

or if t = 0

In this way the determination of

characteristic exponents at infinity is related to the solvability of the equation D = 2 cos ~t. This interplay between characteristic exponents and the descriminant D has been exploited considerably in the study of such special functions as Whittaker functians, Mathieu functions, spheroidal wave functions etc. and in the older literature to the study of the so-called Hill determinants. Arscott Ill.

See for example the book of

How successful these ideas are in the study of general multiparameter

periodic problems is yet to be investigated. It should be noted that the assertion that there is at least one multiplicative solution of (5.7) for FI2 is simply the algebraic form of Floquets theorem. If (5.1) has three singularities in the finite part of the plane then the situation becomes much more complicated.

However the results can be simply stated

and have a nice geometrical interpretation,

see [2].

Theorem 5.3. The path factors for FI23 are determined as the roots of the equation s2 - 2s{PlP2P 3 +

I qlq2P3 cos012 - 2iqlq2q 3 sin012 sin @23 sine3l} 1,2,3 + rlr2r 3 = O,

(5.8)

248

where cos e23 = cos 012 cos 813 + sin el2 sin 013 c o s ~ 2 3 .

(5.9)

In this result (5.9) has, at least for real 8ij, the following geometrical interpretation.

Suppose e12, 013, e23 are the three sides of a spherical triangle,

then ~23 is the angle between the sides 012, 613.

Equation

the symmetry one would expect between the indices I, 2, 3.

(5.8) appears to lack However returning to

our geometrical analogy we observe that N = sin~23 sin @31 sin @12 is simply the "norm" of the our spherical triangle and so is also given by N 2 = sins

sin(s - Ol2)sin(s - 023)sin(s - e31) ,

where 2s = 812 + 023 + 031. The syn~netry of N and so of (5.8) is now obvious. As an illustration of theorem 5.3 consider the algebraic form of Lam~'s equation,

i.e.

__ I I I I d2w + ~ {t + + _t- - ~ - dt 2 t- ] -

dw

dt

Here we take t I = 0, t~ = I, t 3 = k

s (I)~. = I,

-2

s (2)~. = -I,

+

h - ~(V + 1 ) k 2 t

t(t-

w = 0.

(5.10)

l ) ( t - k -2)

and find that

Pi = O,

qi = I,

r.~ = -1,

i = 1,2,3.

Thus in this case (5.8) reduces to s But FI23 is equivalent to - F exp-i(v

+ l)n.

2

- 4iNs

- I = O.

and so the path factors of FI23 must be exp i ~ ,

Consequently I

N = ~sinv~.

(5.11)

Thus if two of the link parameters O., are known then the third is determined by lj (5.11).

Using ideas related to the above Arscott and Wright [3] have essentially made a study of the link parameters when ~ of the resulting solutions.

is rational and have discussed the uniformi~

This in a sense brings us full circle as these ideas

249 and results may have direct bearing on the outstanding problem of quasi-conformal mappings introduced at the beginning of this paper.

References [l]

F. M. Arscott,

Periodic Differential Equations.

Pergamon Press, London

(1964). [23

F. M. Arscott and B. D. Sleeman, ential equations.

[33

J. London Math. Soe. 43

F. M. Arscott and G. P. Wright, ential equations.

Multiplicative solutions of linear differ(1968), 263-270.

Floquet theory for doubly-periodic differ-

Spisy P~irodov Fak. Univ. J. E. Purkyn~ V. Brn~

(]969), I]]-124. [43

F. V. Atkinson, Operators.

[5]

Multiparameter eigenvalue Problems:

Matrices an d Compact

Academic Press, New York and London 1972.

Lipman Bers,

Quasi conformal mappings with applications to differential

equations, function theory and topology. Bull. Amer. Math. Soc. 83 (1977) 1083-1100.

[6]

P. Binding and P. J. Browne,

A variational approach to multiparameter

eigenvalue problems in Hilbert space.

SIAM J. Math. Anal.

(1978) (to

appear). [7]

F. E. Browder (Ed.),

~thematicalDevelopments arisin$ from Hilbert Problems.

Proceedings of Symposia in Pure Mathematics, Vol. 28 Part 2.

American

Math. Soc. Providence R.I. (1976). [8]

P. J. Browne,

The interlacing of eigenvalues of periodic multi-parameter

problems. Proc. Roy. Soc. Edin. A (]978) (to appear). [9]

P. J. Browne and B. D. Sleeman,

Stability regions for multi-parameter

systems of periodic second-order ordinary differential equations (submitted), [I0]

E. A. Coddington and N. Levinson, McGraw-Hill, New York (]955).

Theory of Ordinary Differential Equations.

25O

[11]

M. S. P. Eastham,

Th e Spectral Theory of Periodic Differential Equations

Scottish Academic Press, Edinburgh and London, (1973). [12]

A. Erd~lyi et al, New York

Higher Transcendental Functions Vol 3.

Mc Graw-Hill,

(1955)

[13]

~Erd~lyi, Asymptotic Expansions.

[14]

G. H. Halphen,

[15]

B. A. Hargrave and B. D. Sleeman, The numerical solution of two-parameter

•

Dover (1956).

Y

Tralte des Fonctions Ell iptiques

Vol 2. 1888.

eigenvalue problems with an application to the problem of diffraction by a plane angular sector. [16]

E. L. Ince.

J. Inst. Maths. Applics.

Periodic Lame functions.

14 (1974) 9-22.

Proc. Roy. Soc. Edinbursh A

60

(1940) 47-63. [17]

E. L. Inee,

Ordinary Differential Equations.

[18]

A. E. Pearman,

Lam~ functions in scattering problems with particular

emPhasis on the elliptic cone. [19]

B. D. Sleeman,

Dover (1956).

PhD Thesis, Dundee (1974).

Completeness and expansion theorems for a two-parameter

eigenvalue problem in ordinary differential equations using variational principles. [20]

J. London Math. Soc. (2) 6 (1973) 705-712.

B. D. Sleeman,

Multiparameter Spectral Theory in Hilbert Space.

Pitman Press, London (1978).

Department of Mathematics The University DUNDEE Scotland

DDI 4HN UK.

UNIFORM SCALE FUNCTIONS AND THE ASYMPTOTIC EXPANSION OF INTEGRALS by Jet Wimp Drexel University,

Philadelphia

and University of Strathclyde,

Glasgow.

NOTATION

S, sector : S E S(~l,~2.z o) = {z ~ ~ i~ I < arg(z-z o) ~ ~2' z # Zo} , SA = S ( - 7

+ A,

~-

A, 0),

0 < A < ~ .

UR(Zo) = interior of circle, radius R, centre Zo," UR(O) = UR; U 1 = U. CR(Z o) = circumference =

CR;

C1

~R(Zo) = UR(Zo) U

=

of UR(Z o) with clockwise orientation;

C.

CR(Zo).

H 2 = Hardy class. n,z = asymptotic parameters, n ÷ ~ in J+, z + ~ or z ° in S. ~(A)

= class of functions analytic in A.

P = {(~l,a2 ..... ap) = ~laj e ~}. ~(z) =

io o

e -zt f(t)dt

(Laplace transform).

CR(O)

252

I. Introduction Asymptotic alysis.

series are today one of the next important

In their original

J

care (1886) and Stieltjes embodiment

to pre-existing

about the mathematical

almost simultaneously

- though vague and non-rigorous

analysis

series have not always been spoken of clearly.

in general - is a subject that

to clear the dead wood of heuristic

Even recently,

thinking from this subject.

scale goes back at least to Schmidt

asymptotic

I feel that Arthur Erd~lyi did much

pointed out, his ideas were not new (what mathematical

As Erd~lyi himself

idea is ?).

His idea of an

(1936), but Erd~lyi was the first to

basis.

In no other area of mathematics crippling.

- ideas already floating

series.

series - in fact, asymptotic

exploit the idea on a systematic

by Poin-

and which can be traced back at least to Euler

use of divergent

has never quite managed to escape the taint of mystery.

asymptotic

an-

It is possible these writers were simply giving

community,

and his free-wheeling Asymptotic

form they were discovered (1886).

tools in mathematical

has intuition been both so vitalizing

In the early 1900's it was clear that in asymptotic

analysis

and so the good

had become the enemy of the best.

What were needed - among other things - were reas-

onable definitions.

And Erd~lyi's

definitions

were very reasonable

and useful ones.

of asymptotic

equivalence

and scale

Erd~lyi was one of those who recognized

the asymptotic

properties

the individual

terms, let's call them fk(z,~), where ~ is in some parameter

c ~P and z E S.

For certain values of z individual

could not become arbitrarily

The asymptotic measures

- which "closely" bounded the fk but which

small on compact

(z,~) sets.

idea turned out to be easier to define and implement The effect of these definitions

permissible

Fortunately,

was to greatly enlarge our concept of a

term - a base function - in an asymptotic

larger the parameter expansion

in~,

space GL

expansion.

Now the fk(z,~)

its application

of uniformity

of the

One does not have to

general theory - see our discussion of Darboux's method and

. i nor Stieltjes would have recognized series that neither Polncare

the behaviour

jects of mathematical how general asymptotic functions

and the greater the requirements

than the

In fact, the

to functions having the unit circle as a natural boundary - to en-

counter asymptotic Inevitably,

functions

in the classical Poincare theory.

the more arcane will he the functions fk(z,~).

travel far in Erd~lyi's

the essential

than this clumsy formulation

were allowed to be members of a much larger class of mathematical simple inverse powers of z occurring

set

terms could be O, thus

conveying a false sense of the precision of the expansion. needed were functions - call them pk(Z,~)

suggests.

that

of an expansion shouldn't be judged by the magnitude of

involved.

and properties

interest

of these new base functions became sub-

in their own right.

expansions

In this paper, I wish to discuss

arise, and to describe

some of the special base

253

2..... Poincar~ asymptotic series We differentiate between two kinds of Pozncare asymptotic series. Definition By

(i) Let f be defined in S. co

(I) we

f ~

m~an

If -

E akz k=O K ~

-k

z->=

,

akz-k I = o(z-K),

in S,

z ÷ = in S,

6=0

K = 0,1,2, . . . .

(ii) Let f be defined in S with vertex z . E a (Z-Zo)-k , k-O

f~ w e mean

By

O

co

z ÷ z

in S, o

K

If -

E

a {Z-Zo)-kl. = o[(Z~-Zo)-

z -> z° in S,

k=O

K = 0,1,2, . . . . .

As is well known, asymptotic series in common sectors may be added, multiplied (Cauchy product),

(synthetically) divided.

Any function has at most one asymptotic

expansion in a given S, but different functions may have the same asymptotic expansion. If f is analytic at Zo, its Taylor series is an asymptotic expansion.

For all this

material see Knopp ~948~,

Olver

or better yet, a modern treatment, such as

This definition is good, as far as it goes•

C1974)o

However it does not cover the many

cases of interest where f has "asymptotic-like" series for which the definition fails. A simple example is furnished by the Legendre polynomials #

= (-l)n Pn(X)

2n nl

an-- [(l-xm)n],

where z is real, z = n e J+, n ÷ ~.

I will write

_, (2)

n = 0,1,2,....

dx n

Pn(C°S 8) ~ (

)½

~ ( k)( k=O

cos{(n-k 2~0 + (n - ~k - ¼)~} (2 sin 0) 0 < @ < ~,

but the "~" notation can't be that of the previous definition, since individual terms are not of the required form. In fact, an even simpler example was the one that motivated Erd~lyi's first use of an asymptotic scale.

Aitken, in a 1946 paper, studied some curious series.

They

t The definition of all traditional special functions used in this paper will be the same as in Erd~lyi [1953].

254

were called inverse central factorial series totic-like in their properties. 1 (3)

, and were both convergent and asymp~

One example he gave was

I =

t

-

(a 2

_

+

E k2 a2 k---n+~ -

-

i

~)

-

(a 2

_

I) (a 2

_

~) 9

+

3~ (v2-1)

5~ (v2-1) (~2-4)

r (~-k) £ (a+k+~) k=O

(2k+l) F (a-k+l) £ (v+k+l) 1

This series converges slowly.

(The general term is 2~-k-2(i + o(I))).

But considered

as function of the asymptotic variable n, the terms - as Aitken points out - become small, reach a minimal value, and then begin to increase again. of certain Poincar~ asymptotic expansions, see Knopp (1948).

This is a feature

Aitken showed how such

series could be used to accelerate the convergence of infinite series.

For example,

the remainder on approximating ~2/6 by n E k=O

i (k+l) 2

may be expanded in an inverse central factorial series and for large n

computed quite

accurately. In closing, Aitken says that Erd~lyi has pointed out to him (both men were at the University of Edinburgh at the time) that when a function f(t) has an expansion in t 2 k, powers of (2 sinh 7) then its Laplace transform f(z) will have an expansion (not necessarily convergent) (2k)~ P(z-k) £(z+k+l)

in the functions I= e -zt (2 sinh ~) t.2k dt. o

If z is replaced by ~, these are the functions occurring in the series (3). Gradually,

in a series of papers that started with an investigation of such series,

Erd~lyi adopted the following definitions. In what follows, let ~k' ~k' fk be sequences of functions. both sequences will be defined for Izl > R,

For a given problem

or IZ-Zol < ~ in some sector S withvertex

z . (This allows us to combine z ÷ ~ and z + z in one definition). o o ~k and ~k may depend on ~ s ~ c [ P .

t

Factorial series

see Norlund series.

ak/(Z+l)(z+2)...(z+k)

In addition,

had already been discussed by many writers

(1954) and his references - but not from the point of view of asymptotic

255

Definition: (i) {~k } dominates {~k } if ~k = 0(~k)' k = 0,1,2, .... (ii) {~k } weakly dominates {~k } if ~n = O(~k) for some n, (iii) ~k and ~k are equivalent

k = 0,1,2 .....

if each dominates the other.

(iv) #k is an asymptotic scale if ~k+l = °(~k)' k = 0,1,2,.... (v) the series

k=O

fk

is an asymptotic expansion of f with respect to the scale {~k } if ~ K f - E fk = ° ( ~ k ) ' K = 0,1,2, .... k=O We then write oo

(4)

f ~

E fk; {~k }" k=O

(The {fk } are called base functions.)

(vi) if any of the underlined words in (i) - (v) are preceded by uniformly in ~ this means the "0" or "o" signs involved held uniformly i n ~ . (vii) the series (5)

f ~

E k=O

Ck ~k ; •

J

{~k } '

is called a Polncare asymptotic series.

(Then the term on the right

is usually deleted.) For the basic properties of asymptotic sequences and expansions, (1956),

(1961), and particularly Erd~lyi and Wyman (1963).

of these definitions,

see Erd~lyi

Because of the generality

an asymptotic expansion (4) loses the uniqueness property enjoy-

ed by the Polncare expansions

(I) or (5), see Erd~lyi (1956).

A given function may

have the same asymptotic expansion with respect to different scales. tice, does not seem to be a drawback.

This, in prac-

However, despite the flexibility inherent in

the new expansions, we cannot expect them to do our thinking for us.

For some warn-

ings, see Olver (1974, p.26). The reader can now make sense of the previous examples.

The Legendre polynomial

expansion is an asymptotic expansion with respect to the s c a l e ~ - ½ - ~

(see the dis-

cussion in section 6) and the inverse central factorial series (3) is asymptotic with scale {n-l-2k}. 3. Choice of scale, an example This example shows how a change of asymptotic scale can make an intractable problem easy.

I

wish to find an asymptotic expansion for the coefficients a n in

256 co

r(l+t) =

Z (-I) n an t n, n=0

Itl < i.

I have an = b n + Cn,

I1

(-l)n n'

bn =

e -t

(£n t) n dt,

(-l)n cn = ~

I i

e -t (Zn t) n dt.

o

Expanding

e

-t . in its

Taylor series in the first integral and integrating

termwise

gives b

=

E k= 0

n

(-l)k k~(k+l)n+l

But this is also an asymptotic -

bn

K (-i) k E k=O k!(k+l) n+l

series, with scale ~k = (k+l)-n since 1

e

~

I

- o ~ K + I ) -n]

(K+2) n+l

In the integral for c , using the fact that 2t ½ n ~n t ~ - - ~ - , i < t ~ ~, shows Cn

and so a

~ n

Z

(-l)k

k=O

;

{ (k+l)

the same result obtained by Riekstins It is interesting

-n

} ,

k: (k+l) n+l (1974) from a general theory.

that the series converges

Another approach is to use Laplace's method,

an =

(-I) n n!

e-t

(but not to Cn~). see section 5, on the integral

(~n t) n dt.

O

The location of the critical point t* depends on the large parameter, of a transcendental

and is the root

equation

t* £n t* = n. Everything

can be carefully estimated,

but when all is said and done, it is hardly

possible to give more than the first one or two terms of the expansion.

257

4z Algebraic

and logarithmic

The result generalized Theorem:

scales

called Watson's

initial

(Watson's

Let ~ > 0, R e B

f ~

; Laplace

transforms

lemma is historically

and final value

theorems

the first of a large number

for the Laplace

of

transform.

lena) > -i, and k

Z k=0

t *O + .

ak t ~

Let f exist for some z. Then %

Z k=O

a k r ( ~ + B +I) k --+ B + I

,

z ~ =

in S h.

z~ The applications For example,

of Watson's

lemma are many.

For a discussion,

see Olver

(1974).

if I(z) = [ e zh(t) ~F

where g,h c ~ ( B ) ,

g(t)dt ,

F c B, then the method

of I for large z is determined

primarily

where h' (t) = O, called critical will not give conditions,

of steepest

by the values

points.

descents

supposes

that the value

of h near those points

Assume h has just one critical

which are difficult

(again,

see Olver

(1974))

= (t-t*) 2

h''(t*) 2

+ ... = -w 2 ,

h''(t*) This transformation

is at least locally

< O.

invertible,

so in a neighbourhood

t = t* + [-2/h''(t*)] ½ w + .... One would expect

the major contribution

l(z) % e zh(t*) [--2/h''(t*)]~ I~

to I to occur at w = O.

e-zw2

u(w)dw,

0o

e zh(t*) [-2/h' ' (t*)] ½

Z C2k F(k+½)z k=O

-k-I 2, Z

->

oo

where u(w) = Co+ClW+C2W2

+ .... ,

lul < 6.

in S A ,

Thus

I

but the gen-

eral idea is to make the substitution h(t) - h(t*)

in B

point,

of 0

258

In his first paper cept of an asymptotic Laplace

transform,

in the area of asymptotics scale to handle

including

(1947),

other initial

the result mysteriously

This work is sun,ned up and vastly extended

Erdelyl

introduces

and final value alluded

the con-

theorems

to in the Aitken

in his 1961 paper.

Generally

for the paper.

speaking,

+

if {#k } is an asymptotic

sequence

z ÷ ~ in SA and vice versa. then {~n } is an asymptotic paper,

Erd~lyi

states

in the earlier

paper.

ween asymptotic

as t ÷ 0 , then {$k } is an asymptotic

And if ($ k } is an asymptotic sequence as t ÷ ~ in R and vice versa.

sequence

2 theorems

showing when this is true,

Such a relationship,

expansions

of course,

sequence

as z ~ O + in R, In the latter

and correcting

induces

as

a result

a correspondence

bet-

for f and for f.

We quote two of the main results: Theorem:

(generalized

initial value

Let O < Re~ ° < Re~ I < ....

(6)

f ~

E fk ; {t k=O

Let f, fk' k = 0,1',2,...,

theorem)

and

~k-I

} , t ÷

0 +.

exist for some z.

Then (7)

~ ~

Theorem:

E fl& k=O

(generalized

; {z

-~k

} ,

final value

z ÷ ~ in some S A. theorem)

Let Re% ° > Re% I > ... > O and (8)

f ~

Z fk ; {t k=O

Let f, fk' k = O,1,2,...,

k-l}

,

t ÷ ~

in R +.

exist for each z > O.

Then (9)

f %

Z ~k ; {z k=O

-~k}

The word "some" preceding formulation Specific

each SA is a bother.

in terms of the scale { (Re z) examples f ~

(io)

, z ÷ 0 in some S A.

But Erd~lyi

k} for which

gives an alternative

(8) and (9) hold in any S A.

are:

E Ck(l-e-t)k k=O

; {tk},t ÷ 0 +

E k~Ck/Z(z+l)...(z+k); k=O (this is a factorial

series;

{z -k}

,

z -~ ~ in some S A

see the footnote

following

equation

(2))

259

(II)

i

f~

E k=O

ck(et-l)k

f~

E k=O

k:Ck/Z(z-l)...(z-k);

f ~

I k=O

t 2k ; {t 2k} , t + O + ; Ck(2 sinh 7)

~

E k=O

(2k) l Ck/(Z-k)(z-k+l)...(z+k)

(12)

• ,

{tk},

t + 0+

{z -k-1 } z +

; {z

in some SA

-2k-l~ ~, z ÷ ~

in some S A.

Next, Erd~lyi gives a general theorem, similar to (6) - (7), for asymptotic expansions with respect to the scales {(in t) Bk t ~k-l} and {(in z) ~k z -~k} . orem generalizes

This the-

a number of results given in Doetsch (1950-1956).

Many authors have discussed other generalizatio~of Olver (1974), Bleistein and Handelsman above provide much information.

Watson's lermna. The books by

(1975) and the Doetsch volumes referenced

Of special interest are two early papers by van der

Corput (1934, 1938) where integrals of the form ib e x h(t)-yt

(t-a) -% g(t)dt,

a

x-~

, y-~

,

are treated, and also a much longer survey article (1955,56) by the same author.

See

also vander Waerden (1951), and Wong and Wyman (1972). For a discussion of the numerical error involved in using Poincare type asymptotic series, Olver's book (1974) is excellent.

See also the recent paper by Pittnauer

(1973).

5. Darboux's method Let f e ~(0). (13)

It is no loss of generality here to assume f e ~(U)

Z fn tn' Itl < I. n=O An important problem is:how does f behave as n ÷ ~ ?

at least, so

f =

If f is entire the problem is

n

usually handled on an ad hoc basis by applying the method of steepest descents, or one of its variants,

to the integral I

(14)

fn

=

~

[ f (t) ~ -r t

where F c U is homotopic to C.

dt, Such an approach does not usually yield a complete

expansion, and the details may be very messy. The kind of argument used is well-illustrated p. 329) where f(t) = exp[et]. f = Pe Q, P,Q polynomials.

in an example given by Olver (1974,

P~lya (1922) gave the lead term for fn when

The case f = e Q, Q a polynomial, was more fully treated by

Moser and Wyman (1956, 1957), who give references

to earlier work.

have been given by Rubin (1967) and Harris and Schoenfeld formulasobtained

(1968).

Other examples Often the asymptotic

from (14) depend in complicated ways on the roots of transcendental

260

equations involving n and seldom is it possible to do more than derive a leading term for f • n On the other hand, when f has singularities on the circle of convergence and a function g can be found which matches the behaviour of f at these points and whose Taylor's series coefficients gn are known, then a very elegant method due to Darboux (1878) provides an asymptotic estimate of fn in terms of gn"

In practice, what / results is often a complete asymptotic description, but not one of Poincare type, for f . Since Darboux's method has not received full attention in any of the available n texts on asymptotics and is a rich source of general asymptotic expansions, I will discuss it in some detail. Let f c ~ ( U )

and put

M(f,r) = I~-~ I I~

1f( reiO) 12d0}'2,

0 < r < I.

Definition: If lira M(f,r) < o~ r÷l then we say f ~ H 2

(the Hardy class H2).

Example: Let f = h(~-t) O, ~ e C,

Re o > -I,

h E ~(U);

then f g H 2. Definition: Let f, g g ~(U) and for a fixed m = 0,1,2,..., f(m) - g(m) E H 2 . a comparison function of order m (to f). In what follows let g =

Theorem:

E n=O

gn tn"

(Darboux's method)

Let g be a comparison function of order m to f. Then (15) Proof:

(16)

fn = gn + °(n-m)' n ÷ ~. I may write

fn-g n

I 2~i(n_m+l) m

I CR

h(m)(t) dt tn+l.m ,

h = f-g, O
Then g is called

261

But the radial limit of h (m) exists almost everywhere and eL2(C), see Rudin (1966, p.366).

Thus the integral on the right of(16) maybe expressed as an integral around C

(again, see Rudin). The use of the Riemann-Lebesgue l e n a then gives the theorem. A simple but important case is when the singularities of f on C are finite and algebraic in nature, so (17)

f

r = 1,2,...,R;

=

Z ak(r) (I k=O Re m r > O,

-

t) r

Br+kY r '

~

~

C,

r

t near ~r ,t and the ~r are distinct.

Then the function formed by adding together the first (K+I) terms of each of the R series (17) will be a comparison function of order m provided (18)

m < ½ + min {(K+I) Rear + Re8 r} r

I have thus demonstrated Szeg~'s result (1959, p.205) it. Theorem: (Darboux's method for algebraic singularities) Let f be as in (17). Then (19)

fn =

K R Z k=O r=l

~r)~r+~Tr)(_~r)-n

+ o(n-m)

for all m satisfying (18). It is easy to show this formula is uniform for ~ = (el,~2 .... ,=k) e ~[whenever ~ i s a Cartesian product of compact disjoint subsets of ~. The general term of the sum in (19) is [( r)] -k~r-Br-I Br + ~Y = O(n ), 0 n •

JV

but we have no way of interpreting the formula in terms of Pozncare s definition of an asymptotic series.

However when K -~ ~ the series yields an asymptotic expansion in -Pk }~Pk = l+min{k ReYr+ReBr}. r (The reader will verify this is, in fact, an asymptotic scale, since Rey r > O).

Erdedlyi's mare generalized sense with respectto the scale {n

Formula (19) has been very fruitful in classical analysis.

It provides asymptotic

expansions for the Jacobi polynomials (SzegS, (1959)), the generalized Bernoulli polynomials, and even for more exotic polynomials, such as the Pollaczek polynomials,

t

For this expansion, we make the branch cut along B =[~r' ~ r ] " t s Ud(~r), t ~ B.

tt There is a minor error in this reference.

Thus (17) holds for

262

P% (x; a,b), Szeg~ (1959, p.390). Darboux himself (1878) originally applied his n method to the Legendre polynomials. Darboux's method says if we can solve the problem of finding an asymptotic formula for the Taylor coefficients

of a comparison function, we can find, to within algebraic

terms, the Taylor coefficients

for f.

This sometimes suffices to describe f n

completely. An enormous amount of work has been done on deriving asymptotic formula for gn for certain typical g.

One of the simplest functions having an essential singularity on

C is exp {%/(l-t)}.

Perron (1914), in fact, considered the function (l-t)-aexp{%/(l-t)}

and gave the leading term of gn"

Wright (1932) gave the complete asymptotic develop-

ment. Perron generalized his own work in (1920) to find that if (l-t) -a ~Qa;c l--%t) = n ~ 0 gnt n ' then gn =

~

I %4

F(c)

c 2

e

% 2

n

% > O,

c 3 a - ~ - ~

e

2 %~n

1 X[I + O(n- ")j. Faber (1922), Ha~sler

(1930), Wright (1933, 1949) continued this kind of research,

the last reference giving the leading term of gn for the very general g(t) = (l-t)-a[£n(l-t)] b e P(t) h(t), where

h ~ ~(U),

h(1) # O, and M

c

m=l

(l-t) dm

m

e(t) =

Recently, Wong and Wyman (1974) have done work on functions with logarithmic singu ~ larities on the circle of convergence. Suprisingly, mathematicians

have obtained results even in cases where f has singu-

larities which are dense on C, i.e. when the unit circle is a natural boundary for f, Such cases are of great interest to number theoreticians. (1937), Ingham (1941), Szekeres have contributed

Uspensky

(1920), Rademacher

(1953), and Bender (1974) are some of the workers who

to this research.

In particular,

I want to talk about Rademacher's

work, since many analysts seem to be unfamiliar with it, though it is one of the triumphs of analytic number theory and a tour de force of complex analysis. Let Pn be the number of ways n can be written as a sum of positive integers $ n, Po = I, Pl = I. and

Then the infinite product I write below converges for It] < 1

263

n

f(t) =

see Rademacher

H (l-tJ) -I = E pn t , j =i n=O

(1973).

Itl < i.

Clearly, C is a natural boundary for f. Rademacher

the previous ,~ork of Hardy and Ramanujan

(1918).

extends

He integrates Cauchy's integral

(14) over a doubly indexed sequence of so-called Farey arcs adjoining a circle which approaches the unit circle from the inside.

His analysis involves the transformation

theory of modular functions and a lot of bone breaking estimates of integrals, but the result is worth it. (20)

Pn = ~

I

He finds

kZl A~,n

k½ d

-I

sinh

)',_

C =

,

N = (n - ~i ~

The Ak, n are bounded functions of n, =

Ak'n

~

e

-2~in/k

h mod k ~h,k (h,k) = i

where ~h,k is a 24k th root of unity. The series (20) is asymptotic in n (perhaps the reader can verify a scale is N-2e cN/k) and also convergent, can thus be used to compute Pn"

a very unusual feature in series derived this way,

It

In fact Rademacher computes exact values of P599 and

P721 and shows they agree with earlier estimates baked on the work of Hardy and Ramanuj an. What about uniform asymptotic expansions? ~ ~C

~P,

all interesting choices o f ~ ? ~[i = ~ 2

in (19)?

In other words, if f E f(t,~),

can Darboux's method be made to yield expansions which are uniform for

=

"'"

=~r

A typical question is, what happens if = C

As the reader may suspect, such questions are very difficult to answer, and

the requirement of uniformity of the expansion in a parameter set invariably raises the hierarchy of the base functions in the expansion for f . See Olver's very nice n (1975) discussion of this "you-can't-get-something-for-nothing" principle of asymptotic analysis.

The only effort I know in this area is the substantial work of Fields

~968),

who treats the case when f in (19) has two singularities which are allowed to coalesce. Finally,

can anything at all be said when f belongs to some general class of func-

tions more interesting than just those with algebraic singularities? yes.

Hayman (1956), Wyman (1959), and Harris and Schoenfeld

Surprisingly,

(1968) have all worked on

the problem of defining what general properties of classes of functions ~ , adequate to enable one to make asymptotic statements about fn for f g ~ . call the elements of ~

admissible functions.

The authors

In any case, the properties of ~-f are

not easy to describe, but generally they involve restrictions of the elements.

are

on the growth indicators

264

6. Higher transcendental

scale s

Let us return to the problem of estimating (21)

I(z) = [ e zh(t'~) g(t)dt, ~p

~ e~C~

P.

It would seem that if I could do the analysis for certain simple representative

choices

of h then I have really solved the problem for a wide class of integrals, namely, those which can be reduced to the representative

form by a change of variable, just as the

method of steepest descents enabled me to reduce an integral with one stationary critical point to an integral which could be handled by Watson's lemma (h(t) = t). The simplest function having a movable critical point is h(t,~) = st + t 2, and the representative

i

I(z) =

integral is

=o e_Z (~t+t2)

g(t)dt.

O

Assume L~. = [O,r] for some r > O.

If g can be expanded in a series

co

g=

E gk tk, k=O

Itl <~,

then termwise integration generates the expansion co

(22)

E k=O

gk fk'

(23)

fk =

I

~ e_ z (at+t 2)

tk dt.

O

St seems fk cannot itself be uniformly estimated in ~Lby simpler functions. this is to be expected:

as the requirements

But

of uniformity of the approximation or

the dimensionality of the parameter space ~ p increase, so will the complexity of the base functions involved in the asymptotic expansion. Nevertheless,

the functions f k can be considered known.

terms of parabolic cylinder functions.

They can be expressed in

Integration by parts shows they satisfy a 3-

term recurrence relation, and they can be very easily generated on a computer by applying the Miller algorithm,(see Wimp (1970) and the references given there.) have 2zfk+ 2 + z~fk+ 1 - (k+l)f k = O,

k = O,1,2, .....

A possible normalization relationship for the application of the Miller algorithm is

We

265

k z

Z k=0

f2k (2k)~

Furthermore

I ~ > 0,

z~

aN

F(k_k_k_k~l)e ,,7 2 fk -

k+l 2z

~I + O(k-½)]

,

k ÷ ~,

2

and using this I can show if ~ E (0, ~) the Miller algorithm for the computation of fk will converge for z e SA•

J

One would expect the expansion (22) to be of Polncare type since Theorem: +

fk is a uniform

asymptotic

scale

in a as z ÷ ~ in R .

Pf : Let t = t*(l+u) where t~ satisfies 2zt ~2 + ~zt*-k = 0, or

I [//2 t* = ~ 4

8k +---

~]

.

z

I get O

Ik = ~k

l+u) e

du,

-I

~k = (t*)k+l e

-~zt*-zt .2

= 4k/E(a2z+4k) + ~z½ Ja2z + 8k],

0~
i.

Thus the integral above may be bounded and bounded away from zero uniformly in ~ and z, so it follows that fk+l fk

z ~ ~ > O, 7

'

independent of a and z. •

J

Rather than show for which conditions the expansion C22) is a Polncare expansion, I will use an asymptotic scale simpler than fk (but still uniform i n k ) . result is in Erdeflyi (1970). Theorem: Let g e ~(0) and let I exist for some z ° > 0 and all ~ e ~ then I ~

~ ~g~ fk ; {(~z + 2/~z)-k} , z + ~ in R +. k--O

The following

266 i

•

I have taken T = ~ in Erdelyl's result, and also assumed g independent of z. Note that the absolute convergence of the Lebesgue integral guarantees that his hypothesis (d) is satisfied, as an integration by parts of -(z-z o)(~t+t 2) I

e

it G(t)dt,

e

G(t) =

e

-z (~u+u 2) o

g(u) du

o

will show. For additional material on other such expansions, see Erd~lyi (1974). The next level of difficulty is encountered when h in the integral (21) has two movable critical points, the dimension of the parameter s p a c e ~ still being I: (24)

h(t,~) = a(~)t + b(~)t 2 + t 3,

e ~ ~.

Under suitable conditions l(z) may be expressed as a sum of two asymptotic series with scales

2 Ai(cz ~) 2k

2

'

Z

Ai'(cz ~) 2k Z

respectively, where c depends on ~ and Ai, Ai' are Airy functions. They may be i expressed in terms of modified Bessel functions of the second kind, order ~ and 2 order 7' respectively, see Olver (1974, p.392 ff.) An analysis of integrals which can be reduced to this form by a change of variable constitutes the famous method of Chester, Friedman and Ursell (1957), (CFU). exposition of this method, see the survey by Jones (1972) orOlver (1974).

For an

Olver~ in

a series of papers that are now considered classics (1954a, 1954b, 1956, 1958) encountered their same functions in determining asymptotic expansions for the solutions of sound order linear differential equations with large parameter in the neighbourhood of a turning point.

For those functions to which it applies, Olver's

theory has the advantage that z may approach ~ in sectors S A other than R.

The CFU

theory establishes a nice relationship between the asymptotic expansion of integrals and the asymptotic expansion of the solutions of differential equations. Determining the precise s-region of uniformity of the CFU expansions, and finding conditions guaranteeing that an integral may be transformed into one which can be handled by the CFU technique are very difficult problems, and Ursell devoted two subsequent papers to these investigations (1965, 1970).

At least the base functions

in the expansion, the Airy functions (25) are well understood and can be easily calculated on modern computers. If one wants to analyze the integral

(25)

l(z) = Ipe-ZH(w'~ ) G(w,~)dw

267

where H is to be transformed into the general polynomial h(t,~) = ~i t + ~2 t2 + ... + ~ tp + tp+I, p then one will have to live with incomplete results;

~ = (~i,~2 .... ,~p

),

justifying the reduction of

(25) to the representative integral f=

Jo e-Zh(t'~)

g(t,~)dt

involves difficult-to-verify hypotheses, and some of the work is only formal. who have treated this problem are Bleistein (1966, 1967) and Ursell (1972). case the base functions are called generalized Airy functions.

Authors In this

They satisfy a

differential equation of order p+l (see Bleisten (1967)), possess an asymptotic expansion in z (Levey and Felsen (1969)) and the techniques Wimp uses on similar integrals (1969) will work to show the functions satisfy a(p+2) term recursion relationship to which the Miller algorithm can be applied to compute the functions.

The real

problem, though, is not the analyzing the properties of the base functions, but justifying the transformation of the given integral to representative form. Obviously, precise information about the asymptotic expansion of the very general integral l(z) = IrH(z,t,g)dt

is even more fragmentary. special results available.

For integrals such as these, there are a large number of Often it is assumed that z is real, and F = ~ 0 , ~ , and

often the integral is analyzed by transform methods.

Over the last decade an

enormous number of relevant articles by E. Riekstins and other authors have appeared in the somewhat obscure publication Latvian Mathematical Yearbook. book

See also the

(1974) by Riekstins, the book by Bleistein and Handelsman (1975) and papers by

the authors Handelsman, Lew and Bleistein (1969, 1971, 1972, 1973).

It is my

personal feeling that a unified treatment of such integrals will involve a large number of complex and all but unverifiable hypotheses on the function H.

Perhaps

the whole of asymptotic analysis of integrals (the same could be said of differential equations and difference equations) has reached the point of diminishing returns. The physicist waves his hands and obtains an asymptotic expression which he uses with confidence because he "knows" it must be ture.

For difficult problems the mathemat-

ician has no way of codifying the physicist's intuition.

Perhaps for those problems -

say, integrals with coalescing multiple critical points and singularities - we are couching the answer in the wrong terms, and it is tempting to hope that there might exist a choice of base functions - such as in the example in section make the impossible easy.

3 - that would

268

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