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

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 \

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
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-~) ~°-~ 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


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 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


~ + 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 , ~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 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 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)
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
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

2

g


~ 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 -lO

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) 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 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 - 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
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)

~
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


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)

/

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_


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 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

~
- 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 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~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


: =

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 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 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 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 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 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 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 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, l0

~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 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 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 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 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. =

,

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 O.

If g can be expanded in a series

co

g=

E gk tk, k=O

Itl 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

REFE&ENCES Aitken, A.C. (1946). A note on inverse central factorial series, Proc. Edinburgh Math. Soc. ~, 168-170. Bender, E.A. (1974). Asymptotic methods in enumeration. SIAM Rev. 16, 485-515. Bleistein, N. (1966). Uniform asymptotic expansions of integrals with stationary point and nearby algebraic singularity. Cormm. Pure Appl. .Math. . . . . . .19, . 353-370. Bleistein, N. (1967). Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, 533-560. Bleistein, N., and Handelsman, R.A. (1972). A generalization of the method of steepest descent. J. Inst. Math. Appl. iO, 211-230. Bleistein, N., and Handelsman, R.A. (1975). Asymptotic expansions of integrals. Holt, Rinehart, Winston, New York. Chester, C., Friedman, B., and Ursell, F. (1957). An extension of the method of steepest descent. Proc. Cambridge Philos. Soc. 53, 599-611. Corput, J.G. van der (1934). Zur Methode der stationaren Phase, I. Comp. Math. i, 15-38. Corput, J.G. van der (1948). On the method of critical points, I. Nederl. Akad. Wetensch. Proc. Ser. A. 51, 650-658. Corput, J.G. van der (1955/56). Asymptotic developments, I. Fundamental theorems of asymptotics. J. Analyse Mat h. ~, 341-418. Darboux, G. (1878). Memoire sur l'approximation des fonctions de tr~s-grands nombres. J. Math. Pures App !. (3) ~, 5-56, 377-416. •

Doetsch, G. (1950-56).

I!

Handbuch der Laplace - Transformation, 3v. B1rkhauser, Basel.

Erdelyl, A. (1947). Asymptotic representation of Laplace transforms with an application to inverse factorial series. Proc. Edinbursh Math. Soc. 8, 20-24. I

.

Erdelyl, A. (1956).

Asymptotic expansions. Dover, New York.

Erd~lyi, A. (1961). General asymptotic expansions of Laplace integrals. Arch. Rational Mech. Anal. ~, 1-20. Erdelyl, A. (1970). Uniform asymptotic expansions of integrals, in Analytic methods in'mathematical physics (R.P. Gilbert and R.G. Newton, eds) Gordon and Breach, New York, 149-168. Erd~lyi, A. (1974). Asymptotic evaluation of integrals involving a fractional derivative. SlAM J. Math. Anal. ~, 159-171. Erd~lyi, A., et. al. (1953). York. J

.

Higher transcendental functions, 3v. McGraw-Hill, New

Erdelyl, A. and Wyman, M. (1963). The asymptotic evaluation of certain integrals. Arch. Rational Mech. Anal. 14, 217-260.

269

Faber, G. (1922). Abschatzung von Funktionen grosser zahlen, Bayerischen Akad." Wiss. Munchen Math. -phys. Klasse, 285-304. Fields, J.L. (1968). A uniform treatment of Darboux's method. Arch. Rational Mech. Anal. 27, 289-305. Handelsman, R.A., and Bleistein, N. (1973). Asymptotic expansions of integral transforms with oscillatory kernels : a generalization of the method of stationary phase. SlAM J. Math. Anal. 4, 519-535. Handelsman, R.A., and Lew, J.S. (1969). Asymptotic expansion of a class of integral transforms via Mellin transforms. Arch Rational Mech. Anal. 35, 382-396. Handelsman, R.A., and Lew, J.S. (1971). Asymptotic expansions of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35, 405-433. Hardy, G.H. and Ramanujan, S. (1918). Asymptotic formulae in combinatory analysis. Proc. London Math. Soc, (2) 17, 75-115. Harris, B., and 8choenfeld, J. (1968). Asymptotic expansions for the coefficients of analytic functions, illinois J. Math. 12, 264-277. t~

TI

Hausler, L. (1930). Uber das asymptotische Verhalten der Taylorkoeffizienten einer gewissen Funktionenklasse Math. Zeit. 32, 115-146. Hayman, W.K. (1956). A generalization of Stirling's formula. J. Reine. Ang. Math. 196, 67-95. Ingham, A.E. (1941). A Tauberian theorem for partitions. Ann. Math. 42, 1075-1090. Jones, D.S. (1972). Asymptotic behaviour of integrals. SIAM Rev. 14, 286-316. Knopp, K. (1948).

Theory and application of infinite series.

Hafner, New York.

Levey, L., and Felsen, L.B. (1969). An incomplete Airy function and their application to diffraction problems. Radio Science 4, 959-969. Moser, L., and Wyman, M. (1956). Asymptotic expansions, I. Canad. J. Math. 8, 225-233. Moser, L., and Wyman, M. (1957). Asymptotic expansions, II. Canad. J. Math. 9, 194-209. Norlund, N.E. (1954).

Vorlesungen uber Differenzenrechnung.

Chelsea, New York.

Olver, F.W.Jo (1954a). The asymptotic solution of linear differential equations of the second order for large values of a parameter. Phil. Trans. Roy. 8oc. London. (A) 247, 307-327. Olver, F.W.J. (1954b). The asymptotic expansion of Bessel functions of large order. Phil. Trans. Roy. Soc. London. (A) 247, 328-368. Olver, F.W.J. (1956). The asymptotic solution of linear differential equations of the second order in a domain containing one transition point. Phil. Trans. Roy. Soc. London. (A) 249, 65-97.

270

Olver, F.W.J. (1958). Uniform asymptotic expansions of solutions of linear second order differential equations for large values of a parameter. Phil. Trans. Roy. Soc. London. (A) 250, 479-517 Olver, F.W.J. (1974).

Asymptotics and special functions.

Academic, New York.

Olver, F.W.J. (1975). Unsolved problems in the asymptotic estimation of special functions, in Theory and application of special functions (R. Askey, ed.) Academic, New York, 99-142. 11

Perron, O. (1914). Uber das infinitare Verhalten der Koeffizienten einer gewissen Potenzrei[~e. Arch. Math. Phxs. 22, 329-340. it

Perron, O. (1920). Uber das verhalten einer ausgearteten hypergeometrischen Reihe bei unbegrenztem Wachstum eines Parameters. J. fur Math. 151, 63-78. Pittnauer, Franz (1973). Zur Fehlertheorie positiver asymptotische Potenzreihen. Math. Nachr. 58, 55-61. Polncare, H. (1886). Sur les integrales irregulleres des equatlons lineares. Acta Math. 8, 295-344. Pollya, G. (1922). Uber die Nullstellen sukzessiver Derivierten. Math. Z. 12, 36-60. Rademacher, H. (1937). On the partition function p(n). Proc. London Math. Soc. (2), 43 241-254. Rademacher, H. (1973).

Topics in analytic number theory.

Springer-Verlag, Berlin.

Riekstins, E. (1974). Asymptotic expansions of certain types of integrals involving logarithms. Latvian Math. Yearbook 15, 113-130. Riekstins, E. (1974).

Asymptotic expansions of integra! s, v. I. (Russian) Riga.

Riekstins, E., et. al. (1965-75).

Latvian Math. Yearbook.

Rubin, R.J. (1967). Random walk with an excluded origin. J. Mathematical Phys. 8, 576-581. Rudin, W. (1966).

Real and complex analysis.

McGraw-Hill, New York.

Schmidt, H. (1936). Beitrage zu einer Theorie derallgemeinenasymptotische Darstellungen. Math. Ann. 113, 629-656. I

•

Stieltjes, T.J. (1886). Recherches sur quelques ser~es semi-convergentes. Ann. Sci. Ecole. Norm. Sup. (3) ~, 201-258. Szego, G. (1959).

Orthogonal polynomials.

Amer. Math. Soc.,New York.

Szekeres, G. (1953). Some asymptotic formulae in the theory of partitions. II Quart. J. Math. (2) ~, 97-111. Ursell, F. (1965). Integrals with a large parameter. The continuation of uniformly asymptotic expansions. Proc. Cambridge Phil. Soc. 61, 113-128. Ursell, F. (1970). Integrals with a large parameter : paths of descent and conformal mapping. Proc. Cambridge Phil. Soc. 67, 371-381. Ursell, F. (1972). Integrals with a large parameter. Several nearly coincident saddle points. Proc. Cambridge Phil. Soc. 72, 49-65.

271

Uspensky, J.V. (1920). Asymptotic expressions of numerical functions occurring in problems of partitions of numbers. Bull Acad. Sci. Russie (6) 14 199-218. Waerden, B.L. van der (1951). On the method of saddle points. Appl. Sci. Res. (B) 2, 33-45. Wimp, J. (1969) o On reeursive computation. Wright-Patterson AFB, Ohio.

Rep. ARL69 - O186, Aerospace Res.

Lab.

Wimp. J. (1970). Recent developments in recursive computation, in SIAM studies in applied mathematics , v.6,. SIAM Philadelphia, 110-123. Wong, R., and Wyman, M. (1972). A generalization of Watson's lemma. Canad. J. Math. 24, 185-208. Wong, R. and Wyman, M. (1974).

The method of Darboux.

J. Approx. Theory IO, 159-171.

Wright, E.M. (1932). The coefficients of a certain power series. J. London Math. Soc. ~, 256-262. Wright, E.M. (1933). On the coefficients of a power series having exponential singularities, I. J. London Math. Soc. 8, 71-79. Wright, E.M. (1949). On the coefficients of power series having exponential singularities. II. J. London Math. Soc. 24, 304-309. Wyman, M. (1959). The asymptotic behaviour of the Laurent coefficients. Canad. J. Math. II, 534-555.