Ordinary and Partial Differential Equations

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 visco- elasticity

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. Race

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

R. J. Amos:

G. Andrews:

F. V. Atkinson:

B. L. J. Braaksma:

K. J. Brown:

P. J. Browne:

J. Carr:

D. L. Colton:

A. Davey:

P. C. Dunne:

M. S. P. Eastham:

W. N. Everitt:

G. Fichera:

H. Grabm~ller:

P. Habets:

S. B. Hadid:

of Mathematics, Heriot-Watt University,

Riccarton, Currie, EDINBURGH EH14 4AS, Scotland

Department of Pure Mathematics, University of

St Andrews, The North Haugh, ST ANDREWS, Fife, Scotland

Department of Mathematics, Heriot-Watt University,

Ricearton, Currie, EDINBURGH EHI4 4AS, Scotland

Department of Mathematics, University of Toronto,

TORONTO 5, Canada

Mathematisch Instituut, University of Groningen,

PO Box 800, GRONLNGEN, The Netherlands

Department of Mathematics Heriot-Watt University,

Riccarton, Currie, EDINBURGH EHI4 4AS, Scotland

Department of Mathematics University of Calgary,

CALGARY, Alberta T2N 1N4, Canada


Riccarton, Currie, EDINBURGH EH14 4AS, Scotland

Department of Mathematics University of Delaware,

NEWARK, Delaware 19711, USA

Department of Mathematics University of Newcastle-

upon-Tyne, NEWCASTLE-UPON-TYNE NEI 7RU, England



Department of Mathematics Chelsea College,

~nresa Road, LONDON

Department of Mathematics The University, DUNDEE

DDI 4HN, Scotland

Via Pietro Mascagni 7,00199 ROMA, Italy

Fachbereich Mathematik, Technisehe Hochschule Darmstadt,

D 6100 DARMSTADT, Sehlossgartenstrasse 7, West Germany

Institut Mathematique, Universit~ Catholique de Louvain,

Chemin du Cyclotron 2, 1348 LOUVAIN LANEUVE, Belgium

Department of Mathematics, Chelsea College,

Manresa Road, LONDON

S. G. Halvorsen:

D. B. Hinton:

F. A. Howes:

G. C. Hsiao:

V. Hutson:

H. Kalf:

R. M. Kauffman:

H. W. Knobloch:

R. J. Knops:

I. W. Knowles:

M. K~nig:

R. Kress:

M. K. Kwong:

R. T. Lewis:

S. S. Lin:

M. Z. M. Malhardeen:

J. L. Mawhin:

XIV

Institute of Mathematics, University of Trondheim,

NTH, 7034 TRONDHEIM-NTH, Norway

Department of Mathematics, University of Tennessee,

KNOXVILLE, Tennessee 37916, USA

Department of Mathematics, University of Minnesota,

MINNEAPOLIS, Minnesota 55455, USA

Department of Mathematics, University of Delaware,


Department of Applied Mathematics, The University,

SHEFFIELD Sl0 2TN, England

Fachbereich Mathematik, Technische Hochschule Darmstadt,

D 6;00 DARMSTADT, Schlossgartenstrasse 7, West Germany

Department of Mathematics, Western Wastington

University, BELLINGHAM, WA 98225, USA

Mathem. Institut der Universit~t, 87 WURZBURG,

Am Hubland, West Germany



Department of Mathematics, University of the

Witwatersrand, JOHANNESBURG, South Africa

Mathematisches Institut der Universit~t MUnchen,

D 8 MI~NCHEN 2, West Germany

Lehrstuhle Mathematik, Universit~t G~ttingen,

Lotzestrasse 16.18, GOTTINGEN, West Germany

Department of Mathematics, Northern Illinois University,

DEKALB, Illinois 60115, USA

Department of Mathematics, University of Alabama in

Birmingham, BIRMINGHAM, Alabama 35294, USA




Riecarton, Currie, EDINBURGH EHI4 4AS, Scotland

Institut Mathematique, Universit~ Catholique de Louvain,

Chemin du Cyclotron 2, 1348 LOUVAIN LANEUVE, Belgium

E. Meister:

J. W. Mooney:

R. E. O'Malley Jr:

G. Pagan:

R. B. Paris:

H. Pecher:

D. Rece:

T. T. Read:

B. D. Sleeman:

R. A. Smith:

M. A. Sneider:

D. C. Stocks:

C. J. van Duyn:

W. H. von Wahl:

J. Walter:

R. J. Weinacht:

XV

Fachbereich Mathematik, Technische Hochschule

Darmstadt, 6100 DARMSTADT, Kantplatz I, West Germany

Department of Mathematics, Paisley College,

High Street, PAISLEY, Scotland

Program in Applied Mathematics, Mathematics Building,

University of Arizona, TUCSON, Arizona 85721, USA

Department of Mathematics, Royal Military College

of Science, Shrivenham, SWINDON SN6 8LA, England

Centre d'Studies Nuclearies, DP4PFC/STGI, Boite

Postale No 6, 92260 FONTENAY-AUX-ROSES, Prance

Fachbereich Mathematik, Gesamthochschuie,

Gauss-strasse 20, D 5600 WUPPERTAL I, West Germany

Department of Mathematics, University of the

Witwatersrand, JOHANNESBURG, South Africa

Department of Mathematics, Western Washington

University, BELLINGHAM, Washington 98225, USA

Department of Mathematics, The University, DUNDEE

DDI 4HN, Scotland

Department of Mathematics, University of Durham,

Science Laboratories, South Road, DURHAM, England

Via A. Torlonia N.12, 00161ROMA, Italy

Department of Mathematics, Royal Military College

of Science, Shrivenham, SWlNDON SN6 8LA, England

Ryksuniversiteit Leiden, Mathematisch Instituut

Wassenaarseweg 80, LEIDEN, Holland

Universitat Bayreuth, Lehrstuhl fur Angewandte

Mathematik, Postfach 3008, D 8580 BAYREUTH,

West Germany

Institut f~r Mathematik, Universit~t Aachen,

51 AACHEN, Templergraben 55, West Germany

Department of Mathematics, University of Delaware,


J. Wimp:

A. D. Wood:

A. Zettl:

XVI

Department of Mathematics, Drexel University,

PHILADELPHIA, PA 19104, U S A

Department of Mathematics, Cranfield Institute of

Technology, CRANFIELD Bedford MK43 OAL, England

Department of Mathematics, Northern Illinois University,

DEKALB, Illinois 60115, USA

EXPONENTIAL BEHAVIOUR OF EIGENFUNCTIONS AND

GAPS IN THE ESSENTIAL SPECTRUM

F.V.Atkinson University of Toronto

I. Introduction.

In this paper we obtain conditions on the coefficients in

certain second-order differential equations which yield

conclusions regarding the spectra of associated differential

operators. Such results are particularly well-known for the case

y" + ( I - q)y = 0 , 0 ~ t ~ ; (i.i)

we shall consider also the weighted case

y" + (lw - q)y = 0 , (1.2)

and its vector-matrix analogue

y" + (IW - Q)y = 0 , (1.3)

again over (0,o~), where y is a column-matrix, and W, Q

are square matrices. It will throughout be assumed that

q, w, W and Q are continuous functions of t , with w(t)

positive, and W(t) hermitian and positive-definite. The

conclusions will mostly be of two kinds, either that the

spectrum contains the positive I - axis, or that certain

intervals necessarily contain a point of the essential spectrum.

As a typical result in the first vein we cite that of

~nol ( 3, p.1562), that for real-valued q(t), for which there

is a sequence of intervals (an '~ bn) with

b n - a n ~ ~ , (b n - an )-I ~q2(t)dt ~ 0 , (1.4-5)

the essential spectrum associated with (i.i) contains the

positive semi-axis. As to the second kind of result, we cite

the gap theorems for (i.i) obtained in the early paper of

Hartman and Putnam (I0); considerable extensions, covering also

higher-order scalar equations, have been given in a recent

series of papers by Eastham (4,5,6).

We exploit here a little-used method, in which we argue

successively between:

(i) hypotheses on the coefficients, imposed in the most general

cases over sequences of intervals,

(ii) the exponential growth or decay, if any, of solutions of

the homogeneous equation, again over sequences of intervals,

for the A -value in question, and

(iii) a certain quantity @(A ), which in some sense measures

the distance of ~ from the ~ essential spectrum.

We make this last aspect precise, taking the general case

of (1.3). We consider operators in a hilbert space

measurable complex-valued column-matrix functions

that

" 2 d~f I ilfll f*(t)w(t)f(t) dt < co (1.6)

With (1.6) we associate a minimal operator T defined by

(Tf) (t) = w-l(t) (Q(t) f(t) - f"(t)) , (1.7)

of locally

f(t) such

with domain D(T) the set of continuously twice-differentiable

f(t) with compact support in (0,~). We then specify that

p (~) is the largest number with the property that

lim inf ii(T - I )fn II ~( ~ )' (1.8)

for every sequence {fJ in D(T) such that, as n -~o~ ,

i x 0 (1.9-10) fnl I = 1 , fn-

In practice we shall here bound ~(k ) by taking the fn

to have their support in intervals (an, bn), where a n ~

If ~(k ) = 0, we have a standard characterisation of the %

essential spectrum. If ~(~ ) > 0, and Q is hermitian, or

q in (I.I) or (1.2) is real, we have that ~ (~) is the

distance of ~ from the essential spectrum; this is the basis

of the method of "singular sequences" (see (6)).

Our approach differs from the direct singular sequence

method as used by Eastham (4- 6) in that we do not argue

from (i) immediately to (iii), but make a detour via (ii).

In breaking up the argument into two parts, we see the

different hypotheses involved, and bring to light the connection

with stability theory. The idea of this argument seems due to

~nol (see the text of Glazman ( 8 , 181-183).

Implications of the type leading from (i) to (ii) come under

the heading of stability theory and that of asymptotic

integration. The basic idea is to assooiate,~ in various ways

with a solution y of (1.3) a function such as

E(t) = y*'y' + i y*Wy , (I.ii)

of, roughly speaking, Lyapunov type. We do not attempt to survey

the vast literature on this topic.

Relevant results linking (ii) and (iii) are usually stated

in a form going from (iii) to (ii). Thus in the case of (i.i)

with real q(t) bounded below, and real l not in the essential

spectrum, it is known that there must be a non-trivial solution

satisfying

y(t) = 0(e- ~t) (1.12)

for some ~ > 0. This is due to ~nol (see(8) , p. 179), indeed

for the case of the multidimensional Laplacian. For the ordinary

case the conclusion is due to Putnam ( i~ with the more special

assumptions that q(t) is also bounded above, and that

> lim sup q(t) . The solution appearing in (1.12) will of

course be square-integrable, and may be viewed as an eigen-

function associated with some initial boundary condition.

We remark in passing that the condition that q(t) be

bounded below ensures that (i.I) is in the limit-point condition

at ~ . In what follows, when linking (ii) and (iii) we shall

need similar more general conditions with the same effect, so th

that the essential spectrum will be non-vacuous, and p( I ) finite.

We shall weaken the pointwise semi-boundedness imposed on

q above to integral analogues, of the type introduced by

Brinck (2); further extensions involve sequences of intervals,

and complex q

That (1.12) may fail for real unbounded q may be seen from

an example considered recently by Halvorsen ( 9 ), who proves

also an interesting result in the converse sense, without

boundedness restrictions, namely that if there exists a pair

of solutions of orders O(t-k-i/2), 0(t-k+i/2), where k~ 0.

then ~ is not in the essential spectrum.

We shall take up first the linking of (ii) and (iii). This

depends on an integral inequality, and simple inequalities

involving sequences. We shall then put this together with

stability-type information so as to get results concerning

~ ( I ), and so on the essential spectrum.

We use the symbol * to indicate the hermitian conjugate

transpose of a matrix, as in (1.6), (i.ii). For a column-matrix

f , we take its norm as Ifl = 4(f'f); for a square matrix S

we take its norm ISI in the operator sense, that is to say

I Sfl subject to If~ = 1 . The identity matrix will as max

be denoted by I ; we write Re Q = ½(Q + Q*)

In ~ 2 we prove the underlying differential inequality,

for which we need one-sided restrictions on the coefficients

over an interval;,this forms the second-order matrix analogue

of a result which, in the 2n-th order scalar case, was used

in (I) as a foundation for limit-point criteria. In ~3 we

use this inequality over a sequence of intervals, combined

with the hypothesis that ~(k ) ~ 0, to yield a sort of

exponential behaviour in an integral sense over sequences

of intervals. We then specialise 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, of solutions when

(~) > 0; our assumptions on Q(t) are weaker than the

usual pointwise bounds, and do not require that Q(t) be

hermitian (or q(t) real). In ~5 we continue this special

discussion so as to complete the argument of this paper in a

particular case; by imposing further assumptions, in fact

additional bounds on the coefficients, we can prevent the

solutions from behaving exponentially when ~ is real and

positive, and so force such ~ to be in the spectrum. In ~ 6

we go back to developing the full force of the argument, to

the effect that one-sided restrictions on the coefficients over

a sequence of "large" intervals, together with ~(~) ~ 0,

imply a certain degree of exponential behaviour over sequences

of "small" intervals, abstracted from the large ones. In 7

we develop more general criteria, than those of ~ 5, for the

to contain ~,o~), and in ~8 we exploit the relation spectrum

between the degree of exponential behaviour, over sequences of

intervals, the magnitude of C (~)' and stability arguments so

as to obtain order results for ~(~ ) as k -~ ; in so far

as they overlap, the results agree with those of (5, 10), but

provide a different approach together with variations on the

hypotheses.

Acknowledgements: It is a pleasure to acknowledge helpful

discussions with Prcfessors W. N. Everitt, W. D. Evans,

S. G. Halvorsen and A. Zettl, together with the opportunity

to take part in the 1978 Dundee Comference on Differential

Equations. Acknowledgement is also made to the continued

support of the National 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 W , Q we have

Lemma i. In the real interval [a, b] let, for positive constants

A 1 and A 2 , and for a continuously differentiable hermitian

matrix H(t) , there hold the inequalities

(b-a)2W(t) ~/ AII, Re Q(t)~/ H'(t), (b-a)IH(t) I ~A 2, (2.1-3)

Let the column-matrix z(t) satisfy

z" + (~W - 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

+2z'v' *w-l.zv"+2z,v, dt ~ t % ~ C z*Wz dt , (2.7) (zv

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 - 2~z*z'v'v"

Taking real parts, and using (2.1-2), (2.9) we have

Iz*'z'v '2 ~_ (9/4) :(b-a)-2 Iz*Wz - ~ z*H'zv '2 +

+ I z. zv2 + 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 ~ 36(b-a)-4 Iz*z <~ 36(b-a)-2Al-i I z*Wz .

Combining (2.11-12) we obtain

(lj2) Iz. zv2 {9 +j4 + A1 1(9A22 + 18A 2 +

• (b-a) -2 ~ z*Wz . (2.13)

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

The same result holds, of course, if v is replaced by l-v.

We supplement the above with a pointwise bound for z .

Under the conditions of Lemma 1 we have

z(t)v' (t) = ~(z'v' + zv")dT , v~

and so, by a vectorial version of the Cauchy-Schwarz inequality,

z*(t) z(t)v'2(t) ~ (b-a) I (z'v'+zv")*(z'v'+zv")dt •

Using the above estimations we get

z*(t) z(t) ~ (b-a) 5(t-a)-2(b-t)-2(A3 ~ + A 4) ~ z*Wz at, (2.14)

where A 3, A 4 depend only on AI, A 2 . Again, similar results

were proved in ( 1 , p. 170) for the 2n-th order scalar case.

3. Exponential behaviour.

We now suppose that the hypotheses of Lemma 1 are satisfied

over a sequence of intervals (ar, br), with

0 ~a I ~ b I ~ a 2 < b 2 ~ . . . . (3.1)

with the same constants AI, A 2 throughout. We will suppose

also that 1 is such that ~( I ) ~ 0. We take some non-trivial

solution y of (1.3), and write

Let

function u rs

Wr = I y*Wy dt . ~T

~i £ (0, ~ p(l )), and, for 0 ~ r < s , let the

(t) be defined by

and

(3.2)

Urs(t) = 0 , t ~ (a r, b s) , Urs(t) = y(t), t 6 [b r, as],

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

in (a r, b r), (a s , b s) respectively, where, as in (2.6), %

P

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

By the definition of ~! i ) we have that, for given

and sufficiently large r , say for. e ~ r O,

II(T - I )Ursll 2 ~/ ~12 ~ ~u *Wurs rs dt

~ ~12 ~ y*Wy dt

2 ~ . >1 rl w

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

vanishes in (b r, a s ) , we have that

IIcT ~)Ursll 2 .< CCWr + w s)

(3.3)

(T - t )Urs(t)

where C is as in (2.8). Hence, using (3.3), we have

W r + w s ~ ~12C -I ~ wj, r~ r O . (3.4)

Writing

M 1 = 1 + FI2C -I ,

we then deduce from (3.4) that

wj

(3.5)

(3.6)

We distinguish the two possibilities

In view of (3.4) the first implies that

wj -.--> co . (3 .7 )

while the second implies that

wj -~ 0 . (3 .8)

Thus either (3.7) 6r (3.8) is the case.

We next note that it follows from (3.6) that

h__ wj >i Mlk w m , m-k r o

and so, by (3.4), that

Wm_ k + win+ k ~/ (Ml-l)Mlk-i w m • (3.9)

Suppose now that k o is an integer such that

(MI-I)MI k°-I ~ 2

It then follows from (3.9) that if k ~ k o the sequence

w m . win+ k , Wm+2k . . . . ( 3 . 1 0 )

is ultimately convex, and so is either ultimately increasing

or else ultimately decreasing, according to the cases (3.7-8).

Suppose for definiteness that we have the case (3.7). We

claim that there exist m I , k I such that

Wm+ k > w m , if m ~/ m I , k ~ k 1.

We choose m I so that the sequence (3.10) is increasing if

m~ m I and k = k o or k = ko+l; this involves testing only

a finite number of such sequences, We can then take k I = ko2 ,

since any k~/ ko2 can be represented in the form

nlk o + n2(ko÷l) , with non-negative 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 ~ k+k I , 2k ~ k I . (3.12)

Similarly, in the case (3.8), we can choose integers

m I , k I so that (3.10) is decreasing if m~ m I, k~ k I , and

can then deduce from (3.9) that, subject to (3.12), we have

Wm_ k ~/ ½(M 1 - l)Mlk-lwm . (3.13)

We now introduce a quantity ~ = ~ (k), which measures

the exponential growth or decay of the w m | we set

\<~ = lira sup k -1 lim sup !I in Wm+ k - in WmI.~ (3.14) ~-9.~ ~

It follows from (3.12-13) that

in M 1 = in (1 + ~12C -1) (3.15)

Our argument may be summed up, with a slight loss of

precision, in

Theorem 1. Let, in the intervals [a r, br~

(br-ar)2W(t) ~ AlI, Re Q(t) ~/ Hr'(t), l(br-ar)Hr(t)l~ A2,(3.16)

where Hr(t) is hermitian and continuously differentiable,

and let y(t) be a non-trivial solution of (1.3). Then, with

w m given by (3.2) and ~ by (3.14), we have

(~) ~/ in (1 + ~2(~ )c-l). (3.17)

This follows from (3.15), which holds for any ~l { (0,~).

We have discussed the case ~> 0 only| the result is

otherwise trivial.

4. Corollaries regarding global exponential behaviour.

We have now completed an argument to the effect that if

suitable conditions are imposed in sequences of intervals, and

is not in the essential spectrum, then the solution exhibits

exponential behaviour in an integral sense over this sequence

of intervals. We shall later use this formulation to derive

results about the essential spectrum. At this stage, however

it is appropriate to give some more specialised and simpler

formulations, including information on pointwise behaviour.

We may work in terms of intervals of fixed length if

10

the coefficient of k has a positive lower bound, and so

certainly for (1.1). Taking the more general case (1.3), let

us assume that, for some positive A 1 ,

W(t) >/ AlI, 0 % t < oo , (4.1)

so that the first of (3.16) holds for any interval of unit length.

For the rest of (3.16) we assume that we have, for any T >w 0 ,

Re Q(t)~/ H'(t, T) , IH(t, T)I% A 2, T ~t ~ T+l, (4.2)

where H(t, T) is hermitian and continuously differentiable

in t , where (') denotes d/dt . We have then

Theorem 2. Under the above assumptions, if k is not in the

essential spectrum and y is a non-trivial solution of (1.3),

we have for some 72 > 0 either

e~ty(t) --> 0 , (4.3)

or else

e -2~t I Y2( -c ) d-~ -~oo (4.4) t

as t -9 oO

It follows from Theorem 1 that we have either (4.4) or else

e2~t I Y2(~)d~ -> 0 . (4.5)

We obtain (4.3) from (4.5) in view of (2.14).

In the case (1.1) , with real q(t) , the above extends

certain results of Putnam, Snol and others, referred to in i,

since we do not assume a pointwise bound on q, or Q .

The equation (1.3), with W(t) = I and a pointwise bound

on Q(t) has recently been considered by Rigler (15); other

related investigations are due to Kauffman (12,13).

Similar statements may be made regarding y'(t) . It may

be seen by means of (2.13) that (4.3) (or (4.5)) imply that

ea~t o. ( 4 . 6 )

It would not seem that the corresponding pointwise statement

for y'(t) follows from this. A deduction of this kind may

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

real q(t) ; 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 -~9

or else t+;

.

0,

-277t I I 2 )t2 e y(~) + y'(-c d~ -9~ . t

Simple conditions for the spectrum to contain ~0,Oo ).

Without at this point seeking the maximum generality, we

(4.7)

(4.8)

note that some criteria for this situation are almost immediate.

Theorem 3. Let (4.1) hold and, with W(t) continuously

differentiable, let

T-11olW'(t)Idt -9 0 (5.1)

as T --~ Oo . Let also

~ IQ(t)~ dt ~ 0 (5.2)

as T --> oo . Then every real 0 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(t) , as given by (1.11). We find that

E' = ~y*W'y * Y*'QY + Y*QY . (5,3)

It follows that

T-iIfE'E-lldt --~ 0

as 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

(5.4)

12

q(t) (not necessarily real-valued) is in LP(o, ~) for

some p with 1 ~ p < oo . A recent discussion of this'

situation (with real q) is due to Everitt ( 7 )3 spectra for

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

6. 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 is embodied in

Theorem 4. Let there be a sequence of intervals ~r' drD'

with

0 ~ c I <d I G 0 2 < .... (6.1)

and a positive, continuously differentiable function ?(t) /

such that, on the c r, d r ,

W(t) ~ ~2(t)l , (6.2)

~'(t)/~2(t) ~ 0 , as t-~oo , (6.3)

and

I• ( t)dt G

--~ Oo as r -->o~ (6.4)

Let R(t), S(t) be hermitian on the ~c r, drY, with R'(t)

continuously differentiable, S(t) continuous, and such that

on the ~c r, dr~ we have

Re Q(t) ~/ R ' ( t ) + S ( t ) ,

IR(t)l ~ K17(t) ,

and ~ I¢ I I t) dt <~- K2 ~ ~(t) dt

e - - '

(6.5)

(6.6)

(6.?)

13

where K I, K 2 are positive constants.

Let ~= ~(~ ) be such that if y(t)

solution of (1.3), and

F(t) = y*'y' + y*Wy ,

is any non-trivial

(6.8)

then for any ~ >0 there is an r I such that, for r >/r I ,

lln F(t") - in F(t')~ ~ (~ + @)lTdt, t', t" ~(Cr,dr).(6.9)

Then for any 6-> 0 we have

where

In (i + ~2D-I) ~ 2~-" , (6.10)

D -- lO 4 ~(~,÷ ÷ ~12)o ---2 ÷ ~ + 2K22 t. c 6 . n )

Here p is as in (1.8), and D 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. The quantity ~ is

similar in nature to ~ in (3.14), or to Lyapunov exponents

or to the general indices of Bohl. If ~ = 0 , the choice of

~--is arbitrary, and (6.10) shows that p = 0 , so that

is in the essential spectrum. If ~ 0, we choose ~-- so

as to obtain the best upper bound for ~ , at least so far

as order of magnitude is concerned.

We use G--to determine the (a m , bm). We may take it that

for some infinite sequence of k-values,

f 2ko--~ ~ 7dt ~ 2(k+l)O ~- , (6.12)

c~

We determine 2k intervals (Ckr, Ck,r+l), starting with

Ckl = c k , such that ~k~,~

7dr-- ~, r = 1 ..... 2k. (6.13)

ckw It then follows from (6.7) that at least k of these intervals

must satisfy

14

ISl~-ldt ~< 2K2G--. (6.14)

We specify that the (a m , b m) 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 (a m , bin), which are to be

numbered in ascending order. We shall have a m -9 oo as m -9 oo ,

since Cr-- ~ oo as r -9 ~o : this follows from (6,4) and the

continuity of y .

We next consider the lengths of the (am, bin). Let I

ly '(t)/72(t)l < ~ ' am %t ~< b m • (6.15)

Since ~a~dt = O-- , (6.16)

we have, over (a m , bm),

sup in 7 (t) - inf ln~ (t) ~o~ ,

and so

7(am) exp (-o--~) ~< 7(t) ~ ~(am) exp(6--(~ ). (6.17)

Hence, by (6.16),

(bin-am) ~ (am)eXp(-6-~) ~°-~<(bm-am ) V (am)eXp(~6)' (6.18)

We next calculate admissible values of A I, A 2 in (3,16)..

For A I we need, in view of (6.2), that

A I <~_ (bm-am)2~2(t) , a m <~t ~b m ,

and so, by (6.17-18), we may take

A I = o-2exp(

Passing to the remainder of (3.16), we take t

Hm(t) = R(t) + ~ S(-~)d-C,

so that, by (6.6), (6.14) and (6.16),

]Hm(t)) ~ R(t) + IO 2 (~)I~-I(~)S(Y~)IdxZ

~(am) exo(O--~) (K I + 2K2O-" ).

15

Hence

(bm-a m) IHm(t)l~<o-exp(2~--~)(K 1 + 2K2~--) := A 2 . (6.19)

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

Noting that ~ may have any positive value in (6.15) if m

is suitably large, by (6.3), we see that for large m we

may replace C by the value appearing in (6.11).

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

of (b m - a m ) with m . We claim that

I(bm÷k - am+k)/(b m - am)l 1/k -9 1 (6.20)

as m, k --~ o~ , subject to

Ck+ 1 ~ a m < b m <~ ... <bm+ k ~< dk+ I . (6.21)

Write now

sup I?'(t)/?2(t)l, Ck+ l< t~ dk+ 1 •

Using (6.17) with notational changes we have

lln (bm+ k - am+k ) - in (b m - am)l~ linT(am+ k ) - in ~(am)l+2o~

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

Suppose now that y is a non-trivial solution of (1.3).

We have trivially that

w m ~ (b m - a m ) max F(t) , (6.22)

where the maximum is over ~am, b~. For an inequality in the

opposite sense we denote by (o4 m, ~m ) the middle third of

(am, bin). We have ~ ¢~

I F(t)dt g w m + y*'y'dt . (6.23) 3

We now use ~2.13), which shows that

y*'y' dt ~ ClW m ,

16

where C 1 does not depend on m ; it is given, apart from a

numerical factor, by the expression in the braces in (2.13).

Hence, by (6.23),

3-1(bm - a m ) min F(t) ~ Wm(1 + C l) . (6.24)

It follows that, subject to (6.21),

lln Wm+ k - in Wml ~ sup in F(t) - inf in F(t) + in (3+3C l) +

+ lln (bm+k-am+k) - in (bm-am) ],

where the "sup" and "inf" are over (Ck+ l, dk+l). In view

of (6.20), it now follows from (3.14), (6.9) and (6.12) that

We now get (6.10) as a consequence of (3.17). This proves

the result of Theorem 4.

7. More on the case that ~0,o~) is in the spectrum.

We go back to the topic of ~ 5, to generalise the approach

there given. Whereas in 95 we needed that the solutions should

not grow or decay exponentially when considered over the

half-axis, we can now exploit the situation that this is the

case over a sequence of "large" intervals. As indicated after

the statement of Theorem 4, it is a question of imposing extra

hypotheses so as to ensure that ~= 0 in (6.9). The nature of

these extra hypotheses will depend on the stability argument

being used. With the technique indicated in (5.3) we have

Theorem 5. Let (6.1-4) hold and, with W(t) continuously

differen,tiable, let, as r ~ oo ,

~-21W' 1 dt = o ~ ~ ~ dtl ! (7.1)

I e~-ll Qldt = . o I !Jd ~. (7.2) Then every real ~ ~/0 is in the spectrum.

17

Suppose that ~ is real and positive. In (6.9) we may

equally use E(t) instead of F(t), and so we have (6.9) with

if

~ ' F . ~ - ~ l ~ t = o f ~ , d t I . (7.3)

This follows from (7.1-2) and (5.3) since E(t) as given by

(1.11) satisfies

E(t) ~ ly'(t)l 2 + ~ ~ 21y(t)} 2 (7.4)

v In particular, in the criterion (1.4-5) of Snol we may

weaken (1.5) to ~

(b n - an)-l !!q(t)Idt --~ 0 , (7.5)

and dispense with the requirement that q(t) be real-valuedl

it was not required in Theorem 5 that Q be hermitian.

In a second example, we work in terms of an integral of

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

take Q(t) hermitian.

Theorem 6. Let (6.1-4) held, and also (7.1). Defining R(t)

in the ~c r, dr~ by t

R(t) = ~ Q(x= )d~ , Cr~ t ~ d (7.6) ~ r '

where Q is hermitian, let

R(t), W-~(t)R(t)W~(t) = o ?(t) (7.6-7)

as t -~ oo . Then the spectrum contains £0,°0).

The proof is similar to that of the last theorem, using

this time the Lyapunov or energy function

El(t) = yl*y I + ~ y*Wy , Yl = y' - Ry • (7.8)

A simple calculation gives

E 1 ' = 2 Re (~ y*RWy - Yl*RYl - Yl*R2y) + ~ y*W'y . (7.9)

Hence

+ l,i + x +

18

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

E 1 may replace F in (6.9) when ~\~ 0, in view of (7.6).

In particular, this covers the case of (1.1) with real q

such that j iT+l, t max q(t)dt ~ 0 as T --~ ~o . (7,11)

O~ h<_ 1 T

For example, we can take q(t) = t sin t 3 . More generally, we

can require that (7.11) hold as T -~ oo through a sequence of

intervals whose length is unbounded.

As a further example, we take the case of (1.2) with

w continuously twice differentiable, and q continuous and,

for simplicity, real. For a non-trivial solution we form the

energy-type function

2 _i 2 A E2( t ) = y ' w 2 + ~ y W ~ + ½yy,w,w-3/2 , (7 .12)

for which

I y2qw ' w" 2 ) E2, = 2yy,qw-~ + ½ 3 /2 + ½yy , (w ,w-3 / ' • ( 7 . 1 3 )

Taking ~ real and positive, and assuming that

lw'w-3/ l < (7.14)

we deduce that

o 0 X-½1qrw-½ ÷ l w.w- l + By this means we obtain

Theorem 7. In a sequence of intervals Fcr , d~ , with L-

c~w ~ dt , (7.16)

let q w-1 -9 O, w'w-3/2--~ 0 and w"w-2-9 0 as t -9 =6 .

Then the spectrum includes ~,cO).

The proof follows the same lines, replacing F in (6.9)

by E 2 . The pointwise conditions imposed in the theorem are

easily generalised to integral conditions.

19

8. Order of magnitude of gaps in the essential spectrum.

In the hermitian case, to which we now confine attention,

t (~) gives the distance of ~ from the essential spectrum

and so, if ~ is real, the order of magnitude of p(~ ) as

-)Oo will be essentially the same as that of the function

~(~), the length of a gap with centre /~ in the essential

spectrum~ it is the latter function which has been investigated

by Eastham. In a series of papers (see (6)) he has extended

earlier results on the relation between this order of magnitude

and the continuity and similar properties of the coefficients,

The present method, as we have developed it for the second-order

case, seems to have similar power to the singular sequence

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

becomes a challenge to develop the stability techniques used

here so as to cover the remaining cases obtained by the

other method.

We consider the matrix case (1.3), with hermitian Q(t),

and consider the order of t(X) for large real ~ . A

natural choice of ~" in (6.10-11) is I/~( ~ )! here we

assume that ~(~ ) ~ O, since otherwise the choice of ~5- is

immaterial, and ~(k ) = O. We then have ~ 2 = O(D) ,

and so, explicitly,

Suppose now that, as ~ " ~ ,

{~) -- O{X½}, 1/~{~) -- O{X½).

We can t h e n s i m p l i f y ( 8 . 1 ) t o

e{X} _-0'{~,~ 1 j { x }? . {8..) Such a result will, of course, only be significant if

~ { X } = ocX½> . {8.5} We proceed to examine some such cases.

(8,2-3)

20

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

is in the spectrum, one obtains a variety of results according

to the choice of "Lyapunov function". Using E(t), as in (1.11),

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

Theorem 8. Let (6.1-4) hold. Let also (7.1°2) hold in the

modifed form that "small o" on the right is replaced by

°'big 0 ". Then, as ~ -~oo ,

(~) = 0 ( ~ ~) . (8.6)

Using ( 5 . 3 ) we now find that, in the ~r' dr]'

~,=-l= 0(~-21.,i)÷ O(,k-~7-11Q1), (8.7) where the last term is not significant, The result now follows

from (8.4).

If in (1.3) W(t) is constant, say W(t) = I, then the

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

#(~) = O(1) (8.8)

In particular, (8.6) holds for ~he scalar case (1.2)

if w(t) is bounded above and has a positive lower bound,

and has a continuous bounded derivative, and q(t) is real

and bounded. This is a result of Eastham ( 5 ), who considers

the operator associated with

(s(x))-l{ - (d/dx)(p(x)dy/dx)+ q(x)y~ = k y . (8.9)

In this scalar case, one may transform to the situation (1.2)

by means of the change of variable dt = dx/p(x); w(t) is

then given by s(x)p(x) . However we omit the details.

We next give results which fill in the range between (8.6)

and the weakest significant assertion, namely that

p ( ~ ) = o (1 ) . (8.1o) t

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

21

E3(t ) = y*'y' + ~Y*WlY , (8.11)

where r t+ ~" ~/~

Wl(t) : k½ I W(% )d~. (8.12)

t We now treat the case in which ~(t) is a positive constant

and in which W(t) is also bounded above. We write

Z ( k ) = lim sup max IW(t+s)-W(t)l. (8.13) t -~ 0<s<~ -~

Simplifying the situation somewhat, we have

Theorem 9. Let W(t) be continuous and bounded, and let

Q(t) be hermitian and bounded. Let W(t) also have a

positive-definite lower bound. Then, as ~ ~ 00 ,

0(~): O ( x ~ ( ~ ) + O. (8.1.)

For the proof we note that

E 3' = 2 Re {y*'(~ (WI-W) + Q)y~ + ~ Y*Wl'Y ,'

and so

We deduce that

'¢ _- 0

from which the result follows.

We get the case (8.10) if W(t) is uniformly continuous,

and a range of intermediate cases are given by making W(t)

satisfy a HGlder condition with a suitable exponent. In

particular, we get (8.6) if W(t) satisfies a uniform Lipschitz

condition, in addition to the hypotheses of Theorem 9. Of course

according to the present method we do not need to impose such

conditions for all t , but only on a sequence of intervals

of unbounded lengths.

The case (8.10) is among those considered by Eastham ( 5 ),

for the case (8.9). For results similar to (8.14) in the scalar

22

case reference may be made to Hartman and Putnam (i0), where

the oscillation method is used.

As is evident from Eastham's work, a reduction of the order

of ~(~ ) depends on more drastic assumptions regarding the

smoothness of the coefficients and, if the present method is

used, upon the use of a more developed Lyapunov function.

We illustrate this here by obtaining (8.8), the case of

bounded gaps, for the scalar case (1.2), with non-constant

w(tl). We use now the function E 2 given in (7.12), and

assume w(t) twice continuously differentiable, in a sequence

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

Theorem 10. In addition to (7.16) let w'w -3/2 be bounded on

the ~c r, dr~ , and let

I~¢~lqlw-½ + lw°'lw-3/21 dt =011~w~ dtl. (8.15) £ r ~" C~

Then I(~ ) = O (1 ) as XA --) 00 .

In particular, if w is bounded from zero and bounded

above, it is sufficient that q, w' and w" be bounded on a

sequence of intervals of unbounded lengths.

The result of the theorem follows from (7.15) together with

the previous arguments.

More elaborate Lyapunov functions than (7.12), involving

higher derivatives, have often been used to get, stability

theorems (13) . However it is not clear how to extend them

to the matrix case (1,3), or how to form them in general.

23

REFERENCES

I. F. V. Atkinson, Limit-n criteria of integral type,

Proc.Royal Soc. Edinb.(A), 73(1975), 167-198.

2. I. Brinck, Self-adjointness and spectra of Sturm-Liouville

operators, Math. Scand. 7(1959), 219-239.

3. N. Dunford and J. T. Schwartz, Linear Operators. Part II,

(Interscience, New York, 1963).

4. M. S. P. Eastham0 Gaps in the essential spectrum associated

with singular differential operators, Quart. Jour. Math.

(Oxford)(2), 18(1967), 155-168.

5. M. S. P. Eastham, Asymptotic estimates for the lengths of

the gaps in the essential spectrum of self-adjoint

differential operators, Proc. Yoral Soc. Edinb.(A),

74(1976), 239-252.

6. M. S. P. Eastham, Gaps in the essential spectrum of even-

order self-adjoint differential operators, Proc. London

Math. Soc.(3), 34(1977), 213-230.

7. W. N. Everitt, On the spectrum of a second-order linear

differential equation with a p-integrable coefficient,

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

8. I. M. Glazman, Direct methods of qualitative spectral

analysis of singular differential operators, (Israel

Program for Scientific Translations, Jerusalem, 1965).

9. S. G. Halvorsen, Counterexamples in the spectral theory of

singular Sturm-Liouville operators, Mathematies no. 9/74,

Matematisk Institutt, Universitet i Trondheim, Trondheim,

Norway.

10. P. Hartman and C. R. Putnam, The gaps in the essential

spectra of ~ave equations, Amer. Jour. Math. 72(1950),

849-862.

24

ll. R. M Kauffman, On the growth of solutions in the

oscillatory case, Proc. Amer. Math, Soc. 51(1975),49-54

12. R. M. Kauffman, Gaps in the essential spectrum for second

order systems, Proc. Amer. Math. Soc. 51(1975), 55-61.

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

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

193-196.

14. C. R° Putnam, On isolated eigenfunctions associated with

bounded potentials, Amer. Jour. Math. 82(1950), 135-147.

15. D. A. R. Rigler, On a strong limit-point condition and an

integral inequality associated with a symmetric matrix

differential expression, Proc. Royal Soc. Edinb. (A),

76(1976), 155-159 •

16. L. B. Zelenko, Spectrum of SchrGdinger's equation with a

complex pseudoperiodic potential, Parts I and II,

Diff. Urav. 12(1976), 806-814 and 1417-1426.

LAPLACE INTEGRALS IN SINGULAR DIFFERENTIAL AND

DIFFERENCE EQUATIONS

by B.L.J. Braaksma

0. INTRODUCTION

In this paper we consider singular differential equations

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

and difference equations

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

Here p is a positive integer, y £ ~n and f(x,y) 6 ~n, f(x,y) is an analytic function

of x and y in a set SX £(0; po ) where S = {x 6 ~ : Ixl > R, e < arg x < 8} and

p > O. 0

Assume

7 b (x) y~), where I = IN (0.3) f (x,y) ~6I

and

(0.4) -k

b (x) N [ b k x as x + ~ in S. k=0

dy We may transform (0.I) and (0.2) by x p = ~ to equations for ~ and y (~+i), but

then (0.4) is an expansion in fractional powers of ~. In general ~ will be a

singular point of the differential equation (0.i) of rank at most p. If D f(x,0) Y

I + 0(x -2) as x + ~ then ~ is also a singular poinb of the difference equation (0.2).

The construction of solutions of (0. I) and (0.2) near the singular point

often consists of two parts.

I. The construction of a formal series which formally satisfies (0. I) or (0.2)

if the formal series for y and the asymptotic series for b are substituted

in (0.3) and (0.1) or (0.2).

For example, in several cases there exists a formal solution of the form

26

-m (0.5) x X c x

m o

However, in general this formal series does not converge.

II.The proof that there exists an analytic solution which has the formal solution

as asymptotic expansion as x ÷ ~ in a certain region. We shall consider mainly

this analytic part.

First we consider the linear case of (0.I) and (0.2), where f(x,y) = A(x)y +

+ b(x). We assume that the n x n-matrix A(x) and the n-vector b(x) are representable

as Laplace integrals. We consider two classes of Laplace integrals A 1 and A 2 which

will be defined in sect. i. We will show that if A and b belong to A. and (0.5) is 3

a formal solution of (0.1) or (0.2), then there exists an analytic solution y(x)

which has (0.5) as asymptotic expansion and which is such that x-ly(x) is of

class A. with the same halfplanes of convergence as the Laplace integrals for A 3

and b (cf. sect. 2 and 3).

The class A 2 of Laplace integrals consists of functions which admit convergent

factorial series expansions. The problem whether there exists a factorial series

solution of (0.i) or (0.2) corresponding to a formal solution (0.5) is important

since if the factorial series solution exists it may be calculated directly from

the formal series (0.5).

In sect. 4 and 5 we consider the nonlinear case of (0.I) and (0.2). Now we

assume that f(x,y) is representable as a Laplace integral of a function ~(t,y)

which satisfies conditions similar to those for the classes A 1 and A2. Also

in this case we show that if there exists a formal solution (0.5) then there

exists a solution in the form of a Laplace integral which has (0.5) as asymptotic

expansion. Instead of (0.5) we may have formal solutions

-X k c k x , X k ÷ ~ as k ÷

o

of (0.i) or (0.2). For these formal solutions a result similar to that for (0.5)

holds.

Solutions of (0.i) and (0.2) in the form of Laplace integrals have been

studied by Poincar~, Birkhoff, Horn, Trjitzinski, Turrittin, Harris, Sibuya and

others. Following Horn ([8] - [12]) we transform the differential equation (0.i)

or difference equation (0.2) by means of

y(x) = Yo + S e -xpt w(t)dt

o

into a singular Volterra integral equation for w (here p = I in case of (0.2)).

We show that a solution of this integral equation exists in a suitable Banach

space of analytic functions with exponential bounds in a sector. This leads to a

27

solution of (0.i) or (0.2) with the desired properties. The r61e of singular

Volterra integral equations in asymptotics has been explained by Erd61yi [4].

In sect. 6 we give applications of the results in sect. 2-5. Here we show

when formal solutions of (0.i) and (0.2) exist to which correspond analytic

solutions in the sense mentioned above. Also an application to a reduction theorem

for linear equations is given.

Our results are related to the work of Horn [8] - [12], Trjitzinsky [17],

Malmquist [16], Turrittin [18], Harris and Sibuya [7] and Iwano [14], [15], cf.

aleo Wasow [24, ch. ll]. The linear case of (0.i) has been investigated by W.A. Har-

ris Jr. and myself in [2], where also functional differential equations of a certain

type are considered. The differential equation (0.I) where f(x,y) is a polynomial

in y has been considered in [i].

I. LAPLACE INTEGRALS AND FACTORIAL SERIES

We shall consider the differential and difference equations (0.I) and (0.2) in

cases where (0.3) holds and the coefficients b belong to a class of Laplace inte-

grals. We use two classes of Laplace integrals. They are defined as follows:

DEFINITION I. Let p be a positive integer, @I ~ @2' ~ ~ 0. Let S 1 = {t 6 ~ :

@i ~ arg t ~ 82 } inclusive the point 0. Then a I (@i' @2' g' P) is the set of

functions ~ such that

I i) ti-p ~ 6 C (SI, ~n) and, if @i < @2' then ~ is analytic in the interior

o S 1 o f S 1 .

ii) ~ (t) = O(1) exp (~lltl) as t ÷ ~ on S 1 for all ~i > ~" m

iii) ~ (t) ~ X tO m t ~ -i as t + 0 on SI, where ~m 6 ~n, m = I, 2 .... m=l

Let

(i.i) G 1 ~ Gi(~) = {x 6 ~ : B @ £ [@1,@2] such that Re (xPe i@) > g}.

Then AI(01, @2' ~' p) is the set of analytic functions f : G1 + ~n such that

i0 (1.2) ~(x) = fo + s~e e -xpt ~(t)dt~ f + L ~(x), x£Gi(~) ,

o p o

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 @=-arg ~. Let us agree that a

function is analytic on a closed set if it is continuous on the set and ana-

lytic in its interior.

28

DEFINITION 2. Let ~ 6 ~, m ~ 0, O = - arg m, H > 0. Let S 2 = $2(~) be the compo-

nent of {t E ¢ : Ii - e-~t I < I} that contains the ray arg t = @. Then a2(~,~) is

the set of functions 02: $2(~0) ÷ Cn such that:

i) 02 is analytic 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) > ~}. Then A2(m,~) is the set of analytic

functions f: G2(U) + Cn with the representation (1.2) where p = 1 such that

02 6 a 2 (~,~) and fo 6 ~n.

For short we will often denote the classes of Laplace integrals A defined

above by AI, A 2 or A 1 (H), A2(H) if we only want to stress the value of the

parameter ~. Moreover, we will use a similar definition for matrix functions.

It is well known (cf. Doetsch [3, p.45, 174] that f 6 AI(@I, @2' ~' p) implies

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

on any closed 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 (1.3) holds on closed subsectors of G I.

Conversely, if f is analytic on a closed sector G such that GIC G ° and (1.3) holds

on G, then f E A 1 (@I' 82' D' P) for some ~ ~ 0.

If f E A 2 (~, ~) then f is representable by a factorial series

m~fm+ I (1.4) f(x) = f + E , x E G2(~) ,

o m=0 ~ (~ + I) ... (~ + m)

where f 6 ~n if m 6 ~ (cf. Doetsch [3, p.221]. m

Conversely, if (1.4) holds, then f has a Laplace integral representation

(1.2) with p = I, @ = -arg ~ under somewhat weaker conditions on ~ than in

definition 2: 02(t) = 0(i) exp (~lltl) as t ÷ ~ on IIm ~t I ~ ~ - 6 for all HI > D~" o (~)

o < e < ~ and 02 is analytic in S 2

If f E A2(~, ~), then (1.3) with p = 1 holds as x ÷ ~ on any closed subsector

of G2: larg x - O I ~ ½7 - ~ (0 < e < ½z). Conversely, if f 6 A2(~, ~) and (1.3)

with p = i holds as x + ~ on larg x - 6 I < ½n - g, then we may construct the

factorial series (1.4) from the asymptotic series: we may expand each term in

(1.4) in an asymptotic power series, comparison with (1.3) now gives a recursion

formula for the fm+1" Alternatively we may write x -m as a factorial series;

substitution in (1.3) and comparison with (1.4) gives also a recursion formula for

f .For the explicit form of this formula cf.Wasow [23,p.330] In this way we sum m

asymptotic series for functions in A2(~,~) by factorial series. This is a useful

property since factorial series converge uniformly in half planes. This property

will be used in the following sections where we encounter formal power series solu-

29

tions which under certain conditions are asymptotic expansions of solutions in

A2(~,~) and consequently may be summed to any degree of approximation by facto-

rial series. If m > I, then S2(mw)C S 2 (~) and so A2(~,~) c A 2 (m~,~). Consequent--

ly factorial series (1.4) also are representable by factorial series (1.4) on G 2

with parameter me instead of ~ if m > I.

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

q91 ~ %92 6 aj if ~9 I, q02 6 aj .

2. THE LINEAR DIFFERENTIAL EQUATION

We now consider the differential equation (0. I) in the case that it is

linear and that it is a coupled system of a system with a singularity of the first

kind and a system with a singularity of the second kind.

To formulate this we partition n x n-matrices along the n I - th row and

column where 0 < n I < n:

w h e r e M j h i s a n n . x n h m a t r i x , n 2 = n - n 1 . A c o r r e s p o n d i n g p a r t i t i o n i n g o f

vectors f = 2 after the n I - th component will be used.

Now consider the system

(2.1) xl-P d__yy = A(x)y + b(x) dx

where p is a positive integer, and concerning 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 A2(~ , ~).

Then we have representations

(2.2) A(x) = A + Lp~(X) b(x) = b + L 8(x) o ' o p

and asymptotic expansions

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

in closed subsectors of 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 nonsingular in S in case j o n 2 j "

30

Then we have

THEOREM i. Suppose X o c x -m ~s a formal solution of (2.1). Then there exists m

an analytic solution y of (2.1) which belongs to AI(0 I, 0 2 , v, p)in case i and

to A2(~ , ~) in case 2 such that

-m (2.5) y(x) ~ X c x

m 0

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. In case 2 we may sum the formal solution ~ c x -m to a convergent

factorial series which satisfies (2.1) on Re (xe l@ > ~ (cf. sect. i).

N-I -m

PROOF. Let u = Z c x , a partial sum of the formal solution. Then m

0

xl-P d_uu = A(x)u + b(x) - c(x) dx

where c 6 A. and c 1(x) = 0(x -p-N ), c 2 (x) = 0(x -N) as x + ~ on closed subsectors 3

of G.. Hence with y - u=vwe get x I-p d v = A(x) v + c(x) as equation equivalent 3 dx

to (2.1).

I = 0, h = 0, ., So it is sufficient to prove the theorem in case b h ..

p + N - i, b 2 = 0, h = 0,I, ..., N-I for a sufficiently large integer N. We

assume this latter condition from now on or equivalently by (2.2) (cf. (1.3)

m__ 1 t p as t ~ 0 in 1 (2.6) B(t) ~ Z 8 m Sj, ~h = 0 if N < h < N + p - i.

m=N

We seek a solution y of (2.1) which is 0(x -N) as x ~ ~, and which belongs to

A.. If y = L w is of ckass A. then 3 P 3

(2.7) xl-P dxd--YY = Lp (-ptw) , A(x)y = Lp(A0w + ~ * w).

Hence (2.1) has a solution y = L w of class A. iff -ptw = A w + ~ * w + B an~[ P 3 o

w 6 a.(~). This equation for w is a singular Volterra integral equation. 3 1

l--

If t P v 6 C(S , %n) we define 3

(2.8) Tv = - (A + p t I) -I (~ * v). o

With

(2.9) ~ = - (A + p t I)-is o

31

the equation for w is equivalent to

(2.10) w = Tw + ~.

The assumptions on A 0 imply that

(2.11) (A ° + p t I)

and that

-i = diag {p-lt-iInl, (A~ 2 + p t In2)-l}

(2.12) (A 22 + p t I )-i and t(A + p t I) -I 0 n 2 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 in S. and 3 N

i-- _l/lit] (2.13) II vl~ = sup It p v(t)] e <

t6S. J

Here ~i is a fixed number, Pl > l/ where p is the parameter l/ in AI(81, 82, p, p)

or A2(~ , l/). It is clear that V N is a Banach space with norm If.If N. A similar

definition will be used for matrix-valued functions.

Since b £ Aj , it follows from (2.2) that 8(t) = 0(e l/lltl) as t + ~ in Sj.

Using (2.9), (2.6), (2.11) and (2.12) we deduce ~ 6 V N-

Next we show that T maps V N into V N. From the assumption A 6 Aj, (2.2) and

(2.4) we deduce that lh 6 Vp, 2h 6 Vl, h = 1,2. Since

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

- ~ 1- N+--L we see that t p (~ * v) I and t P (~ * v)

Moreover, if t 6 S. then 3

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

_N_ I

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

Hence, by (2.11)

(2.15) ]]{ (A + p t i ) - 1 ( ~ * v ) } l l l N o 1 (tl £1 £2 I t ~ IIp+ll Ip) tlvtlN

Similarly

I(~ * v)

and therefore

-- -I -- -I l/llti i 2(t) l < II all i e II v II N Itp * tp 1

82

< ll~lq livll Bc!,~) (2.16) I[{(A ° + p t I) -I (a • v)}211N _ 1 N p p

i

sup I t p (A 22 + - 1 p t I ) l- t6S o n 2

]

H e n c e w i t h ( 2 . 8 ) a n d ( 2 . 1 2 ) we s e e t h a t T m a p s V N i n t o V N a n d t h a t t h e r e

e x i s t s a c o n s t a n t K i n d e p e n d e n t o f N s u c h t h a t

[ITII <Krc~) {r(N+i)} -I -- p

Choosing N sufficiently large we see that T is a contraction on V N if o

N > N . Consequently there exists a unique solution of (2.10) in V N if N > N . -- o -- o

Now going backwards we easily verify that y = L w satisfies (2.1) and P

y = 0(x -N) as x + ~ in closed subsectors of G . ]

Hence, if N > N and Pl > Z there exists a unique solution y = c o + L w -- o p

of the original equation (2.1), without assuming (2.6), such that

1-! P t w is analytic on S. and

3 m N

N-I c -- -i -- -I w(t) = X m tp + 0(tP

m= 1 F (m/p)

P l l t l W ( t ) = O ( e ) a s t ÷ ~ i n S . .

3

) as t ÷ 0 in S. , 3

Now the uniqueness implies that w does not depend on N. Hence we have a unique

solution y 6 Aj (~i) such that (2.5) holds. By variation of ~i we see that this

solution y belongs to the class A. (~).~- 3

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

i and 2 are modified as follows: Assume

p-i p-i (2.17) A(x) = Z x-h ~(xP),-- b(x): X x-h b~(xP),-"

h=0 h=0

where ~, bh, h = 0,1 ..... p-l, are of class AI(81,82, ~, 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), there

exists an analytic solution y(x) = X~- 0 ~" x -h ~h(X p) where ~h 6 Aj (p) and (2.5)

holds as x + ~ in

(2.18)

where 81

- --2 - 82 + e _ 0),

= 82 = - arg ~ in case 2.

33

This may be shown using a rank reduction scheme of TURRITTIN [21]:

substitute

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

v(~) = (~o(~)

~T (~))T ' "''' Yp-I

..... ~p-i (~))T.

Then (2.1) is equivalent to

du (2.19) --= M(~)u + v(~),

d~

where M(~) ~ Z M ~-m, M - 1 0 m o p I AIo © 1 A 1 "'-

o • "-

Ao p-I ..... " .A I " A o o o

From (2.4) we may deduce that M ° has nlP rows of zeros and that 0 is eigenvalue

of Mo with multiplicity nlp. Hence Mo is similar to diag {0,M 22}O where M 22o is

nonsingular. So we may apply theorem I to (2.19) and the result follows. []

In case p = 0 in (2.1) we have a regular singular point in ~. This case

may be transformed to the case p = I by dividing both sides of the equation

by x:

dy _ x-i A(x)y + x-lb(x) (2.20) d-~ -

If ~(x) = x -I A(x), we have ~ = 0 and so we need not partition the matrices o

involved: we take n I = n, p = I and (2.4) is satisfied for (2.20). Our results

now give Laplace transforms related to formal power series solutions.

3. THE LINEAR DIFFERENCE EQUATION

Here we consider the equation

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

By means of the substitution

(3.2) z(x) = y(x I/p)

we transform (3.1) into

34

(3.3) y((x p + i) I/p) = A(x)y(x) + b(x).

We distinguish two cases. In case i we assume: A, b 6 AI(81, 82, p, p) and in

case 2 we assume: p = i, A, b 6 A2(w , p). We assume in case j:

(3.4)

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

if k 6 ~ k {0} then 2k~i~Sj;~ ~ Icos 8 I if 81 ~ @ < ~2;

A22 -t - e I is nonsingular on 8..

o n 2 3

Here S. and G. are defined in definition j of sect. i. Then we have ] ]

THEOREM 2. Suppose ~ c x -m is a formal solution of (3.3) and Re t is bounded o m

above on sj. Then there exists a function y E AI(8 I, 8 2 , u, P) in case i,

6 A2(~, p) in case 2 which satisfies (3.3) if (xP+l) I/p E Gj _ and such that (3.1) Y

with (3.2) is satisfied, and which satisfies (2.5) as x ÷ ~ on closed subsectors

of G in case j. The function with these properties is unique. ]

PROOF: The proof is quite similar to that of theorem i. The difference is that

instead of (2.7) we have

y((x p + i) I/p) = L (e -t w(t)) (x) , if (xP+l) I/p 6 G.. P ]

Hence the integral equation (2.10) with (2.8) and (2.9) now reads

(3.5) w(t) = (e -t I - A )-i (~ * w + 8)(t) . o

Instead of (2.11) and (2.12) we now have

(3.6) (e -t I - A )-I = diag { (e -t -I)-ii (e-tI - A22) -I} o n I ' n 2 o

and

(3.7) (e-tI - A22) -I and t(e-tI - A )-i n 2 o o

are uniformly bounded on S.. Here we use the fact that le-tl ÷ ~ as t ÷ ~ on S.. ] ]

with these alterations we may show that all steps in the proof of theorem i

with slight modifications remain valid, and theorem 2 follows. []

If Re t is bounded below on S. the assertion of theorem 2 does not remain

valid. In this case Re t ÷ ~ and e !t ÷ 0 as t + ~ on Sj. Hence t(e-tI - Ao)-i

is not bounded on S, (cf. (3.6)), and the proof of theorem 2 does not go through ]

in this case.

If A -I exists we may modify that proof for this case. First we may solve o

35

(3.5) in a neighbourhood of 0 in S.. Then the solution may be extended to a 3

global solution in S. (cf. sect. 4.3). We may estimate this solution by ]

majorizing the right hand side of (3.5) since (e-tI - Ao)-I is bounded in a

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 above, since this result

is a special case of theorem 6 in sect. 5.

However, a result corresponding to theorem 2 in the case that Re t is

bounded below on S. also may be obtained by transformation of (3.1). Let 3

~(x) = z(l-x). Then

~(x p + i) = A-I(x e ~i/p) ~ (x p) - A-l(x e ~i/p) b ix e ~i/p) ,

which is of the same type as (3.1). We now assume A -I, b 6 AI(8 I, 82' p' p) or

A2(~, p). Then it is easily seen using (1.2) that

-i ' A (x e~i/P), b(x e ~I/p) 6 A 1 (e I + ~, 82 + z, p, p) or A2(-~, 1.1).

Hence we deduce from theorem 2 :

THEOREM 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 RetiS ]

bounded below on 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

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 to the corollary of theorem 1 in sect. 2 we now have

COROLLARY: We make the same assumptions as in theorem 2 except that the cases 1

and 2 are modified as follows: Assume (2.17) where ~, hh, h = 0, 1 ..... p - I,

belong to Al(el, g2, u, i) in case 1 and to A2(~, ~) in case 2. Then, 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 NONLINEAR DIFFERENTIAL EQUATION.

We consider the differential equation

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

in the case that f(x, y) satisfies conditions similar to those in sect. 2,

theorem I. We again consider two cases j = i or 2 and use the same notation S. and 3

36

G as in definition j of sect. I. Assume 3

HYPOTHESIS H. (j = 1 or 2). Let Po ]

case j = 2. We have

> 0, p > 0, p a positive integer, p = I in

(4.2) f(x, Y) I= f°(Y) + {Lp~(y, t)} (x), if (x, y) 6 G.] x A (0; po ) ,

where fo(y) and t P qg(y, t) are analytic in ~ (0; p O ) x Sj, fo(0) = 0 and m

(4.3) qD(y, t) ~ Z %01n (y)t p as t + 0 in S m=l 3

uniformly on A (0; Do). Here ~m(y) is analytic in A (0; po ), m = I, 2, ....

Moreover, if ~I > Z then there exists a constant K depending on Pl such that

1

(4.4) I~(Y, t) l ! K It p I exp (~lltl) on ~ (0; DO) x S O . O

In the following we assume either hypothesis H I or hypothesis H 2.

= (\~i' "''' ~n ) 6 1 = ~n we denote I~I = ~i + "'" + ~n and

~%0 (y, t) = ~Ivl%0(Y' t) and similarly for ~ f (y). o

~Yl "'" ~nyn

If

Df will be the derivative of f • We use the partitioning of matrices and vectors o o

as in sect. 2. Our main result is:

THEOREM 4. Suppose j = I or 2 and in case j:

(4.5)

22 22 A = Dfo(0) = diag {O n , A ° } A ° + p t I is nonsingular on S. o ' 3

i II,~ 1 {~ %0 (0, t)} 1 = 0(t p )as t + 0 in S. if IvI ~ and an analytic solution yof (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 in several steps: in sect. 4.1 we give some lemmas

with estimates, in sect. 4.2 an integral equation equivalent to (4.1) will be

derived, in sect. 4.3 we show that a solution of this integral equation exists in

a neighbourhood of t = 0, in sect. 4.4 this solution will be extended to a

solution in S. and in sect. 4.5 we estimate the solution and we obtain the ]

solution of (4.1).

37

In section 4.6. we consider some generalizations of theorem 4.

4.1. SOME ESTIMATES

LEMMA 1. Let P, h and l be positive numbers. Thenthere exists a positive constant

K(n,l) independent of p such that

ml+h-I (4.6) ~ P < K(h,l) max (0 h-l , e p) .

m=0 F(ml+h) --

PROOF. The estimate is evident for p < i. If p > i, we use the Hankel-integral

for the gamma function and we get for the lefthand side of (4.6):

(0 +) (0 +) I ~ e s (~)ml+h-I ds I

2~i m~O I s s 2~i I e~S ( l - s - 1 ) - i s - h as .

In the last integral we choose as path of integration a loop enclosing the

negative axis and the points s = exp (2gni/1), g 6 ~. The residue in s = i gives

the main contribution to the integral as P + + ~- In this way we obtain (4.6)°

LEMMA 2. Let p be a positive number or p = ~ and sj(p) = s 5 ~ n~ (O;p),where

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

(4 .7) IZ( t ) i~ M I t l 1-1 exp ( ~ l l t l ) i f t 6 S . ( p ) , ]

where M > O, 1 > O, ~i ~ 0 are constants. Then

(4.8) IZ *~ (t) I MI~I FIUl (1) Itl I~11-1 ~l l t l < e if t £ S.(p),u 6 I, - r ( l ~ l l ) J

~ O, where z * v i s the convolution o f ~ l f a c t o r s z l , v 2 fac tors z 2 . . . . . . .

fac tors z . n n

and

pROOF: The proof easily follows by induction using (2.14).

LEMMA 3. Let z satisfy the conditions of lemma 2. Suppose dr6 %n if v 6 I and

there exist positive constants K I and Pl such that

- I~ I ~ ~, ,~+ o. (4.9) I%I <_ Ki~ 1 ,

Then d z *~ is uniformly convergent on compact subsets of Sj (po) , analytic v6I w~O in S~ (p) and

3

38

Itl} eUl (4.10) X I d~ z*~(t) I ~ KoKIk I sup {Ikltl I-I, kl Itl e i vEI

i f t 6 s j ( p ) , where k 1 = (p~l NnF(1) ) l /1 and Ko i s a cons tan t only depending on 1.

Moreover, if p = + ~, then

(4.11) L { X d z*V(t) } = X d (L z) v P 9EI w ~El w P

on {x E ~: Re (x p e i8) > Pl + kl} where the path of integration in the Laplace

integral is arg t = 8.

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

LEMMA 4. Let h be a positive number and t l-h ~ (t): sj (p) ÷ ~n be analytic in

S (p) if~ E I. Assume ]

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

where K 2 and P2 are positive numbers.

Then

X q * z *~ (t)

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

is analytic in S (p), ]

(4.13) [ I~9 * Z*~ (t) I -- < K'O K2 k21-h max ([k2tl h-l, elk2tl) e 71tl ~EI

if t 6 Sj(p), where ~ = max (PI' P-)' k. = (Mn r(i)) z z Pl

pending only on h and i. In case p = ~ we have

I/i and K' is a constant de- c

(4.14) Lp(~EiX ~ * z *~) (x) = ~EIX (Lpq) (x) (LpZ)V(x) ,

on {x 6 ~: Re (xPe i@) > ~ + k2} , where the path of integration in the Laplace

integral is arg t = e.

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

K2 IVl ft u21t-TITI~II-I z (Mr (i)) (4.15) ID9 * *W (t) l < I (t-T) h-I e - r(l~ll) ~ o

PIITI dT I -- ~ (MF(1)) I~l jltl h+]~ll-1 e < - - F(I~ll) ~ 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 y , <0 (y, t) = [ By(t) Y , ~6I ~6I v+o

if y 6 A (0;P), t 6 S~ and o ]

1 (4.17) b - SVfo (0) 1 ~v , ~(t) = m--~ q) (o,t)if ~ 6 I, t 6 Sj .

From Cauchy's formula for derivatives of analytic functions and hypothesis H. we 3

deduce

1 -i ~lltl

-l~i I t~ 1 e , (4.18) Ib I ~ KoPo[~l , l~(t) l ~ KoP °

where Ko is the maximum of K (cf. (4.4)) and maxl fo(Y) lon ~ (0;Po)"

Let N 6 IN, where N will be chosen later on sufficiently large.

Define m

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

i 1

Then u N = LpZ N and uN6J''3 Moreover fUN(X) I < Po if Ixl is sufficiently large.Let

(4.19) x I-p d d-x UN - f(x, UN(X)) = gN(x)-

-m From the assumption that Z 1CmX is a formal solution of (4.1) and hypothesis

H. we may deduce that J

(4.20) gN(x) = 0(x-P-N), g~(x) = 0(x -N) as x + ~ in G.. ]

From (4.19) and (4.17) we may infer with lemmas 3 and 4

(4.21) gN = Lp YN' - YN (t) = PtZN(t) + ~6ZI {~* z*VN + bY zN*m} (t).

These lemmas also imply YN 6 aj(~ 2) for some ~2 > ~ " From (4.20) we deduce

N N

2 ( 4 . 2 2 ) ~ ( t ) = 0 ( t P ) , Y N ( t ) = O.t p { ) a s t + 0 i n S . .

]

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

40

(4.23) xl-P dxdV = f(x, uN(x) + v) - f(x, UN(X)) - gN(x).

We show that (4.23) has a solution v = L w 6 J(v') for some U' > ~2 iff P 3

w E a.(~') and ]

(4.24) -ptw(t) = AoW(t) + H(ZN, w) (t) - XN(t)

Here

(4.25) H(z, w) = 7 b {(z + w) *~ - z*~ } + E B * { (z + w) *~ -z .9} . ~EI ~6I I~I>2 ~+0

First we remark that H(z, w) exists in S~ if z, w 6 aj(~') and that H(z, w) Ea=3 ~'') ]

for some ~" > z' on account of lemmas 3 and 4. Moreover, these lemmas imply that

Lp(AoW + H(ZN, w)) (x) = f(x, ~N(X) + v(x)) - f(x, U N(X)) on Gj( ~" ). Hence

(4.23) and (4.24) are equivalent.

Next we rearrange the terms

(4.26)

where

H(z, w) = ~(z, .) * w + I {B (Z,.) * W *~ + b w'W},

vEI ~ I~I>2

(4.27)

e(z,.) = DqO(0,.) + r

vEI

{ID3m f (0)Z*9 + i *~ ~! O 9--~ D~9~(% ")* z } ,

~ (z,t) = 8 (t) + E (v+O) *O z*O} O {b +O z + 8 +g ~ (t) .

We prove this for z = z N and w 6 aj(p'). To this end we use the estimate (4.7) • , > p, 1

for Z = z N and for z = w wlth ~, replaced by ~ , 1 = --and a suitable ~+o J~I +Iol P

M, and the estimate ( ~ ) ~ n . Then lemmas 3 and 4 imply that the series

in (4.27) converge uniformly and absolutely on compact sets in S. and that ]

-i eP31t I l~(z N, t) l < MIItP I

(4.28) 1

-i e~31t I 189(Z N, t) I ~ M 1 (~)IWIItP I

for some ~3 > ~ and a constant M 1 independent of ~. A second application of

lemma 4 shows that the series in (4.26) are uniformly absolutely convergent on

compact ~ets in Sj, and so (4.26) follows from (4.25) if z = ZN, w 6 aj(pl).

41

4.3. LOCAL SOLUTION OF THE INTEGRAL EQUATION

We first solve (4.24) in a Banach sp~ce VN(a) of functions w: Sj(e) ÷ ~n

where S.(£) = S. n ~ (0,£) such that tl-p w is analytic in S.(c) and 3 3 3

N

(4.29) I IWlIN = sup {It P w(t) I: t 6S (£)}. 3

Here g will be chosen later on, a > 0.

We rewrite (4.24) as w = Tw, where

= + ptI) -I H(ZN, w) + ~N (4.30) Tw -(A °

-I (4,31) ~N = (Ao + ptI) YN "

Using (4.22) and (4.5) we deduce ~N 6 V N, so in particular ~N 6 VN(¢).

Choose M 2 > ICll {F(p-l)} -I +i, where c I is the first coefficient in the

formal solution of (4.1). Then i

(4.32) IzN(t)I < (M2-1) i tp i o n S <6 ) -- 3 O

for some sufficiently small positive £ depending on N. We consider H(z, w) o

for w 6 V N(£) and z 6 V(6) where i

(4.33) V(£) = {s 6 Vl(e) : Iz(t) I <_ M 2 It p I on Sj(¢)].

In particular z N + w ° 6V(a) if w ° 6 VN(£) and £ is sufficiently small, because

of (4.32).

We first estimate a and B defined by (4.27) on S (£), if z 6 V(E) and 3

0 < £ <__ £i" The estimates (4.18) , the assumption (4.5) and lemmas 3 and 4

imply that there exists a constant M 3 independent of z, t,N and ~J such that

i

I s(z, t) l<_ M31 tP !, I ~lh(z, t) l < M 3 if h = 1,2 ( 4 . 3 4 ) i -i

18~(Z, t) I <__ M 3 (Q~) I~I It p [, "~6I,

if z 6 V(£), t 6 Sj(a) and 0 < £ ~ e I.

Now choose N > 16M3, N > p. Then we have analogous to (2.15) N -- -i

I{(Ao + ptI)-I e(z,.) * w(t)}ll _< 81 [[WI[N I tp I

if w 6 VN(g) , 0 < e <_ el, t 6 Sj(6), z 6 V(g). Analogous to (2.16) we have under

the same conditions

42

N

I{(Ao+ ptI) -I ~(z,.) * w(t)}21 < K' I I wl I N [tl p

where K' is a constant independent of z, w and t. Hence there exists E2,

0 < s2 ~ E1 such that

i (4.35) If (A O + P tI)-[ ~(z,.) * W 1 IN < ~ l lwl IN ,

if w ~ VN(£), z 6 V(s), 0 < ~ _< E 2 .

Let L = 211~NI I N , where the norm is the norm in VN(E 2) (cf. (4.31)), and

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

From (4.26), (4.34) and lamina 4 we deduce 2N

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

, , < 2L, 0 < e < e 2. if t6S.(s) z 6 V(e) w 6 VN(E), IlW[IN _ 3

Here M 4 is a constant independent of z, w, t , g. Choose E so small that 0 < e < ~2'

2N

i sup It P (Ao + ptI)-ll M4 < 4 t£S. (g) ]

and that z N + w 6 V(E) if w 6 VN(E) , j lwl JN < 2L. Let D(S) = {w 6 VN(E):

J lwJ 'iN < L}. Then we deduce from (4.35) - (4.37) that

_ 1 _ (4.38) II (Ao + ptI)-I H(ZN + W, W2 Wl)IIN < : ii w 2 WllIN,

if wl, w 26 D(E). With w I = 0 this implies that T maps D(e) into D(£). If

Wl, w 26 D(e), then H(ZN, w 2) - H(ZN, w I) = H(z N + w I, w 2 - w I) in view of

(4.25). Hence (4.25) and (4.38) imply that T is a contraction on D(e). Conse-

quently there exists a unique solution of w = Tw, so of (4.24), in D(E).

4.3. EXTENSION OF THE SOLUTION OF THE INTEGRAL EQUATION

Suppose the solution w of (4.24) is known on S.(p) for some p > 0 (cf.4.2):.

Choose to 6 S.(p)3 with 21P < jt ol < P" Let sT3 = s.(p)] N (s.-t3 o ) and+ 3 = S~+t3 o"

Here S + t denotes the set S translated over t . Then S.(p) and S~ are convex o o 3 ]

sets. +

We transform the integral equation (4.24) on S. using the following ]

decomposition of u . v for scalar functions u, v continuous on S.(p)G S~ and ] 3

analytic in its interior:

43

(4.39) (u ~v) (to+tl) = {U(to+ .)~ v} (tl) + {V(to+ .)• u} (t I) + R(u,v)(tl),

where

t (4.40) R(u,v) (tl) = S o v(t + t I - T)U(T) d~, t I E Sj,

t 1 o

and the paths of integration [0, t l ] a n d [ t l , t o ] b e l o n g t o S j ( p ) . H e r e

R(U, v) = R(V, u) and R(u, v) only depends on the values of u and v on [to, ti].

Using induction we may show

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

k-I + E { v *(k-l-j) * R(v, v~J)} (tl),

j=l

if k > 2 and v ~° • R = R, v ~I ~R = v ~R. From this formula we may deduce that

for an n-vector function z whose components satisfy the assumptions above and

k 6 I, I kl > 2 we have

n (4.41) ~(z, tl)m zmk(t + tl) - I k {z.(to+ .)~ z~(k-ej ) } (t i) =

o ~=~ ] ] ] I

n kl-i z~(k_(j+l)el ) m R(Zl' Zl ) (tl) + = E E ~J

i=i j=l

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

l=l

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) then we have from (4.8)

and (4.40)

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

~k 1 * k l _ [ *k 1 k i + . . . . k 1 IR(z I , Zl_ 1 *--- *z I ) (tl)l 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)I I _< ~]-~° , t I C Sj .

Now we transform the operator T into an operator ~ on $7. Let w be the solution

] n of (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 , ( ' ~ - e , )

* z(t I) = ~(ZN,.)* z(t I) + [ [ ~j {8 (ZN,.) * w 3 V61 j:l v

I~I>_2 * (~-e)

+ b w ] } • zj (t I) ,

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

+ E {Sv(ZN, .) , Rv (w,.) + ~ (z N, to+ .) * w *v + v6I

I~I>_2

+ R(Sv(z N, -), W *v) + b v R (w, .) }(tl).

Using lemmas 2 and 4, (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 (t o + .) is a

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

z = ~ z has a unique solution in S7 which is analytic in (sT) O and continuous 3 3

on S~, this solution is a continuation of w(t + .). Denoting this continuation J o

also by w(t + .) we see that w = T w on S.(O) US< because 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. whzch is continuous o ]

on S. and analytic in S.. 3 ]

4.5. EXPONENTIAL ESTIMATE FOR THE SOLUTION

NOW we estimate the solution w of (4.24) on S,. Let 3

(4.44) g(p) : sup {lw(t) l: t 6 S., ItI: p}, if p > 0 . 3

We rewrite (4.24) in the form (4.30) and use estimates for (A + ptI) o

-i from

45

(2.12) for H(ZN, w) from (4.26), (4.28) and (4.18), and for ~N from (4.31) and

the fact that YN 6 Gj(~2) (cf.sect.4.2). Then we see that there exists positive

constants M and d such that for @ > 1 we have

(4.45) g(p) < (Tlg) (p) ,

where

1 {eP3 p ~ ~ (p~- - 1 ePBP ) ( ' r i g ) (p) = M + 2 d m *m d m g~m( g (p) + X :~ p } }

m:2 m=1

By choosing M > sup {g(p): p £ [0,1]}we get (4.45) for p > 0.

F o l l o w i n g W a l t e r [ 2 3 , p . 1 7 ] we f i r s t s o l v e v = T t v . I f u = L l v , t h e n

1

u ( x ) = M ( x - P 3 ) - I + t,~ 5- dmum(x) + M I" (1) ( x - p 3) P X dmum(x) m=2 P m:l

This equation has a unique solution u in a neighbourhood V of ~ which is analytic

in x I/p, positive for x > 0, x 6 V and

1 1

u(x) = Mx -I + 0(x P) as x p ~

Let V contain the halfplane Re x ~ P4' where P4 > ~3" Then

1 ; 4 ~ i ~ v(p) = (Lllu) (p) = M + 2~ ePX {u (x) -Mx-l }dx'

P4-i ~

if p > 0. It follows that v is real-valued, v(0) = M (cf.[3, p.174]) and v(p) =

0(exp p4p) as p + +~. In particular we have g(0) < v(0).

Suppose there exists Po > 0 such that 0 < g(p) < v(p) if 0 < p < Po and

g(po ) = V(Po). Then (4.45) implies

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

which gives a contradiction. Hence g < v on ]R and so +

(4.46) lw(t) I ~ K O exp (P41tl), if t 6 Sj

for some constant K . o

C o n s e q u e n t l y LpW e x i s t s o n Gj a n d i s a s o l u t i o n o f ( 4 . 2 3 ) . T h e r e f o r e y = UN+

+ L w is a solution of (4.1). In the same way as in the proof of theorem i we may P

show y 6Jj(P4) and (2.5) as x ÷ ~ on G . This completes the proof of theorem 4. ]

46

4.6 A GENERALIZATION

In several cases (4.1) has formal solutions of the form

-k.K (4.47) y(x) = Z d k x

Ikl=1

(ko, < ) where k = ..., k ) 6 ~g+1 (g a positive integer), K = (i, <i, ..., g g

Re <j > 0 if j = i , ..., g and k.< = ko+ klKl + ... + kg<g (cf. sect. 6,

application V). In these cases we have the following generalization of theorem 4.

THEOREM 5. Assume hypothesis H I . Let

(4.48) A m = F(~) Dqgm(0) , bvo = by' bvm _- l__u! F(~)agp q)m(0), m = I, 2 ....

and assume

A l l = 0 , A 12 = 0 , b 1 = 0 i f m = 0 . . . . . p - 1 a n d w 6 I , I~1 + 1 , m m ~m

(4.49)

A 21 = 0, A 22 + ptI is nonsingular on S I. o o

If (4.1) has a formal solution (4.47) then there exists a real number" ~' > ~ and

an analytic solution y = L w of(4.1) on GI(~') (cf. (l.l))such that P

oo -k. K

(4.50) y(x) N y d k x

Ikl=*

as x + ~ on closed subsectors of GI(V'). This solution is unique.

PROOF. The proof is similar to that of theorem 4. Let the set of numbers k. K

with Ikl> i be arranged in order of increasing magnitude of their real parts to

the sequence 11, 12, .... Hence 0 < Re I i < Re 12 < ... and Re A m ÷ ~ as m + ~.

Then (4.47) may be rewritten as y(x) = Z c x -Am . Let 1 m t

N-I -i N-I 1 m -i

(4.51) UN(X) = Y c x m, UN =LpZN, z N = Z c {F(--m)} -I t p i m i m p

where N is chosen in such a way that Re IN_ I < Re A N. In general u N ~ A I. With

(4.19) we deduce

2 -I giN(X) = O(x -p-IN), gN(x) = 0(x N)

and (4.21). Instead of (4.22) we now have

h h l TiN(t)= 0(t p N) 2(t ) = 0(tP N ) as t + 0 in S I

, T N

with y = u N + v, v = LpW we deduce (4.23) - (4.27). Because of (4.51) we may

47

estimate e and 8~ in (4.27) with lemmas 3 and 4 with 1 = Ii/p. The result is

(4.28) with I/p replaced by Ii/p.

In sect. 4.3 we adapt the definition of VN(e) by replacing N/p by IN/p

(cf. (4.29)). In (4.32) - (4.34) and (4.45) we now have to replace i/p by

ii/p and in (4.37) N/p by IN/p. With such modifications the reasoning in sect.

4.2 - 4.5 remains valid and the prood of theorem 5 is completed.

REMARK. Cases where a formal solution of (4.1) also contains logarithmic terms like

in Iwano [15] may be treated using solutions y = L w where the expansion of w near P

the origin contains logarithmic terms.

5. THE NONLINEAR DIFFERENCE EQUATION

We now consider the difference equation

(5.1) z(~+l) = f( I/p, z(~))

or equivalently with (3.2) :

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

where f satisfies hypothesis H. (j=l or 2) as in sect. 4. We also assume 3

(5.3) A = Df (0) = diag {Inl, A 22} o o o '

(5.4) A 22 - e-tI is nonsingular on S., o n 2 3

(5,5) 2k~i ~ S. if k 6 ~ \{0} p + cos 8 > 0 if 81 < @ < e2 3 , _

22 (5.6) A is nonsingular if Re t is bounded below on S..

o 3

Then we have

THEOREM 6. Assume hypothesis H., (5.3) - (5.6), with j=l or2. Assume 3 1-19t

~0(O,t)} 1 = 0(t P ) as t ÷ 0 in S. if 191 ~ and an analytic solution y of (5.2) such that y 6 A 1 (e l, e2,v', p) in

case I and y 6 A2(~o, ~') in case 2 and such that (2.5) holds on closed

subsectors of G. This solution is unique. 3

PROOF. The proof is a modification of that of theorem 4. This modification is

the same as used in the proof of theorem 2: the lefthand side of (4.24) now

48

reads e-twit) and in (4.30), (4.31) etc. we replace (A + ptl) -I by (A -e-tI) -I 91 o

In view of (3.6), (5.4) and (5.6) the matrix (A -e-tI) is bounded on S.~ A(0;I) o 3

and the matrices in (3.7) are bounded on S. N A(0;I) and even on S~ if 3 3

Re t is bounded above on S.. Using these modifications the proof of theorem 4 3

goes through. D

REMARK. Theorem 6 is an extension of a result of Harris a~id Sibuya [7].

Analogous to theorem 5 we have the following generalization of theorem 6.

THEOREM 7. Assume hypothesis HI, (5.3) - (5.6) with j=l and in the notation of

(4.48):

(5.8) = b I : o i f m=o . . . . . p - 1 and ~ ~ ~ , b t+ I~ A 11 0, A 12 = 0,if m=l,..., p-l; um m m

If (5.2) has a formal solution (4.47) 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 proof is a modification of that of theorem 5, with analogous modifica-

tions as in the proof of theorem 6.

6. APPLICATIONS

In this section we first give sufficient conditions in order that formal

series solutions of (0.I) and (0.2) exist, so that the previous theorems may be

applied. Finally we deduce a reduction theorem for linear differential equations.

I. A formal non-trivial solution ~ c x -m of (2.1) with b ~ 0 exists if (2.4) is o m

satisfied, A 12 = 0, A II is singular and A 11 + mI is nonsingular for m = I, 2 .... P P i P nl

Now we may choose for c an eigenvector corresponding to the eigenvalue 0 of A II o p

and e 2 = 0. o

Such a formal solution exists for (3.3),and so for (3.1), with b -= 0 if (3.4)

is satisfied, A 12 = 0, A 11 is singular and A II + m I is nonsingular for P P P P n 1

m = i, 2, ...

The difference in the conditions for (2.1) and (3.3) stems from the formal

relations:

(6.1)

dy x l - P d ~ = X - mc x - m - p ,

m 1

y ( ( x P + l ) I / p ) - y ( x ) = X c x - m - p ~ ~ + X x - P g } . i m P g=l g+l

Formal solutions of the type considered above exist for the nonhomogeneous

A II equations (2.1) and (3.3) under the same conditions except that now also has P

49

to be nonsingular; then c differs from c in the previous case. A detailed o o

treatment of formal solutions of (2. i) and (3.3) has been given by ~irrittin [19 ].

II. Results for formal solutions ~ c x %-m analogous to theorems 1-3 may be obtained o m

by a transformation of the equations (2.1) and (3.3) via the substitution

y(x) = x%~(x). The equation for ~ is of the same type as that for y and has the

formal solution ~- c x -m o m

The applications I and II of theorem i contain the results of Turrittin [18]

concerning factorial series as solutions of (2.1) with b -= 0. However, here the

solutions have the same halfplane of convergence as the coefficients, whereas

Turrittin gets a smaller halfplane of convergence for the factorial series which

represents the solution (cf.[2]).

III. A matrix solution of (2.1) and (3.3) with b ~ 0 may be obtained as follows.

Assume A 12 = 0 and there is no pair of eigenvalues of A 11 which differ by a P P

positive integer in the case of (2.1), whereas in the case of (3.3) we assume

11 41 the same for p Ap . If the assumptions concerning A in theorems 1,2 or 3 are

satisfied and b ~ 0 we may construct an n x nl-matrix solution U(x)x ~ of (2.1)

and U(x)x of (3.3) such that U £ J. and U(x) ÷ as x ÷ ~ in closed 3

subsectors of G.. We give the proof for the difference equation (3.3). Substitute 3 pA~l

Y(x) = U(x)x in (3.3). Then we get

_A 11

U((xP+I) I/p) = A(x) U(x) (l+x -p) P

The righthand side defines a linear transformation T in the space V of n x n l-

matrices U. Partitioning these matrices after the nlth row we get

ii -A

(TU)I(x) = {All(x) Ul(x) + Al2(x) U2(x)} (l+x -p) P ,

hence

(TU)l(x) = ul(x) + x-P(A~IuI(x) - UI(X)A ll)p + 0(x-P-l).

ii ii t Here the linear transformation U 1 b N U. - U,N has as eigenvalues he

Ii p i i p differences of the eigenvalues of A . Hence p times this transformation has

P eigenvalue 0 with eigenvector I and no other integer is eigenvalue. Also

n 1 (TU)2(x) ~ A 22 U 2 (x) as x + ~ in G . Hence all conditions of application I are

o 3 satisfied and the result for U follows. Similarly, the differential equation

(2.1) may be treated (cf.[2]).

IV. We now give sufficient conditions such that there exists a formal solution

c x -m of (4.1) in case hypothesis H and (4.5) are satisfied. From (4.2) and 1 m 3 (4.48) we deduce the formal expansion

50

f(x, y) : (2 A x-m)y + [ ( [ b x -m) y~ + [ b x -m 0 m ~EI m=0 vm m=l om

I~I>S

From this we may deduce that there exists a formal solution of (4.1) if n 1

or if n. ~ I and

i ) A l l l h a s n o e i g e n v a l u e w h i c h i s a n e g a t i v e i n t e g e r a n d P

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

=0

or

ii) A II has eigenvalue -i, but no other negative integer is eigenvalue of A II , P P

(6.2) holds and

A 12 (A22)-Ib 2 = b 1 p o ol op+l

In these cases we may apply theorem 4.

An analogous result holds for the difference equation (5.2): instead of Ii

(4.5) we now assume (5.3)-(5.6) and the conditions i) or ii) above with A ii P

replaced by p A P

V. A formal solution (4.47) of (4.1) assumed in theorem 5 exists if hypothesis

H~ and (4.5) are satisfied and ]

are eigenvalues of -A II where Re< • > 0, j= I, ..., g; a) ~ i' "''' <g P ]

b) k + kl 2 or k I + ... + k = i k > 0; k 1 + . . + k g _ g ' o

c) there exists at most one positive integer which is eigenvalue of Ail - ; if P

1 is such an eigenvalue then 1 6 {~i <g}' bl = b 2 = 0 for ' ' "''' op+h 0, oh

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

AI2 (A22)-i b 2 = b I . p o ol op+l

The number of formal solutions (4.47) depends on the dimensions of the nullspaces

of A II + K.I , j=l, ..., g. We refer to Hukuhara [13] who computed all formal P 3 n 1

solutions of systems (0.i). Theorem 5 is related to results of Iwano [14].

A formal solution (4.47) of (5.2) assumed in theorem 7 exists under the same ii

conditions as above except that (4.5) is replaced by (5.3)-(5.6) and A is P

replaced by p A II P

VI. We may apply IV to obtain a block diagonalization for linear systems

(6.3) xl-P d_yy = C(x)y dx

analogous to Wasow [24, theorem 12.2] and Malmquist [16, sect. 3 ].

51

Theorem 8. Suppose C 6 A(U), C(x) ~ E C x -m, C = diag {C 11 C 22 ) o ] Q m 1 o o

Let X I .. ~ be the eigenvalues of C 1 are h . ~ those of C 22 ' " ' r o r+l' "" ' n o "

Assume Z !p (lg - I h) ~ S 9 if g _< r < h.

Then there exists a transformation y : T(x)z which takes (6.3) into

(6.4) xl-p dz _ ~(x)z, ~(x) = diag {~ll(x) C~92(x)} dx

such that T and ~ 6 A (~' ) for some ~ ' > ~ and ~Ii = cll , ~22 = C22 3 o o o 0

If CI2(x) , C 21 (x) = 0(x -N) , then T(x) = I + 0(x -N).

PROOF . First we substitute y = Q(x)y, where

o x/to fix t Then (6.3) is transformed into

(6.5) xl-P d w = D(x)w, dx

where Dl2(x) ~ 0 iff

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

Now C II 12 12 22 o Q -Q Co defines a linear transformation in the linear space of 12

r x (n-r)-matrices Q with eigenvalues hg- hh, g = I, .... r; h = r + I, ..., n.

Hence we may use application IV with n I = 0 and a solution of (6.6) exists in

A.(~') for some ~i > Z" In a similar way we may transform (6.6) to (6.4). [] ]

A special case of theorem 8 with j = 2 has been given by Turrittin[22].

Theorem 8 with j = i corresponds to theorem 12.2 in Wasow [24] with a different

sector. Results of this type may be used to reduce linear differential and

difference equations to canonical forms, cf. Malmquist [16], Turrittin [18] ,[20].

REFERENCES

[i] BRAAKSMA, B.L.J., Laplace integrals, factorial series and singular diffe-

rential equations, Proc. Bicentennial congress of the Wiskundig

Genootschap, Amsterdam 1978.

[2] BRAAKSMA, B.L.J. & W.A. Harris, Jr., Laplace integrals andfactorial series

in singular differential systems. To appear in Applicable Mathematics.

[3] DOETSCH, G., Hendbuch der Laplace Transfor~ationj Band II. Birkh~user Verlag,

Basel, 1955.

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, 1964, 211-229.

[5] HARRIS, Jr., W.A. & Y. SIBUYA, Note on linear difference equations, Bull.

Amer. Math. Soc., 70 (1964) 123-127.

[6] HARRIS, Jr. W.A. & Y. SIBUYA, Asymptotic solutions of systems of nonlinear

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 (1966) 120-135.

[8] HORN, J., Integration linearer Differentialgleichungen durch Laplacesche

Integrale und Fakultdtenreihen. Jahresber. Deutsch. Math.Ver., 24

(1915) 309-329; 25 (1917) 74-83.

[9] HORN, J., Laplacesche Integrale als L~sungen yon Funktionalgleichungen,

J. reine angew. Math., 146 (1916) 95-115.

[i0] HORN, J., 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 (1924) 85-95.

[13] HUKUHARA, M., 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., (4) 19 (1940) 35-44.

[14] IWANO, M., 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., (4) 94 (1972) 109-160.

[16] MALMQUIST, 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~termi-

nation, 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] TURRITTIN, H.L., The formal theory of systems of irregular homogeneous

linear difference and differential equations, Bol. Soc.Math. Mexicana

(1960) 255-264.

[20] TURRITTIN, H.L., A canonical form for a system of linear difference

equations, Ann. Mat. Pura Appl., 58 (1962) 335-357.

[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, 1970.

[24] WASOW, W., 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 reg-

ularity of the equation (c.f.~7]). In fact it is only in very

exceptional circumstances 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 by means

of locally defined integral representations. Until quite recently

both of these approaches have been confined to the case of elliptic

equations.

The generalization of the Schwarz reflection principle for

harmonic functions to the case of elliptic equations in two inde-

pendent variables satisfying a first order boundary condition along

a plane boundary was established by Lewy in his seminal address to

the American Mathematical Society in 1954 ([20]). In this address

Lewy considered the elliptic equation

+ u + a(x,y)u x + b(x,y)Uy + c(x,y)u = 0 (I.i) Uxx yy

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) = f(x) • (1.2)

Then under the assumption that u(x,y) e C2(D) A ~(D 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 through-

out 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 complex variables z

and z* for z e DV o~ 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 line segment lying in

y < 0. 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 dimen-

sional 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 Huygen'sprinciple 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 exten-

sively 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 +k -- u = 0 (1.4)

xx yy y y

where k is a real parameter ([8~). Erd~lyi's result was to show that

if u(x,y) is a regular solution of (1.4) in a region containing the

singular line y = 0 and u(x,O) is a real valued analytic function in

a y-convex domain D (i.e. if (x,y) e D then so is (x,ty) for -l<t<l)

then, provided k # -1, -3 ..... u(x,y) is a regular solution of (1.4)

/ in D. For generalizations of this result of Erdelyl the reader is

referred to Gilbert's book (~). The method of integral operators

outlined above has a variety of applications, among them being the

58

analytic continuation of the solution to Cauchy's problem for elliptic

equations which arises in connection with certain inverse problems in

fluid mechanics (c.f. [iI], [14]). More recently the author used such

an approach to establish a relationship between the domain of regul-

arity of axially symmetric solutions of the Helmholtz equation defined

in exterior domains and the indicator diagram of the far field pattern

([2]). In contrast to the rather extensive investigations into the pro-

blem of continuing solutions of elliptic equations, until recently

relatively little work has been done in connection with the corresponding

problems for parabolic equations. One reason for this is of course the

fact that solutions of parabolic equations do not enjoy the same reg-

ularity properties as solutions of elliptic ~quations, i.e., in general

solutions of parabolic equations with analytic coefficients are not

analytic functions of their independent variables. Until about ten

years ago there were (to the author's knowledge) only two rather isolated

results on the global continuation of solutions to parabolic equations,

both of them concerned with the heat equation in one space variable.

The first of these was the fact that solutions of the one dimensional

heat equation defined in a rectangle and satisfying homogeneous Dirichlet

or Neumann data on a vertical side of the rectange could be reflected

as a solution of the heat equation into the mirror image of the rec-

tangle ([25]). On the other hand it was shown by Widder in [26] that

if h(x,t) is a solution of the one dimensional heat equation

h = h (1.5) xx t

which is analytic in x and t for Ix[ < x Itl < t , then h(x,t) can o' o

be continued as a solution of the heat equation into the entire strip

59

< to, and expressed there as a uniformly convergent Ixl < =, Itl

series of heat polynomials

h(x,t) = Z a h (x,t) (1.6) nn

n=O

where the heat polynomials are defined by

/~ x n-2ktk h (x,t) = ~]: E (1.7) n k=O (n-2k) ~k:

This result is noteworthy since it is one of the few cases where it

can be stated that eyery sufficiently regular solution of a partial

differential equation defined in a given (real) domain has an auto-

matic continuation into a larger (real) domain regardless of what

the boundary data is. Note that such behaviour is only true for

analytic solutions of the heat equation, and not in general for

classical solutions which are infinitely differentiable, but not

analytic, in the time variable. Hence in the development of the ref-

lection and continuation properties of solutions to parabolic equations

with analytic coefficients one can expect a different behaviour

depending on whether or not the solutions are analytic in the time

variable. In this connection it is worthwhile to note that if u(~,t)

is a solution of a linear parabolic equation with analytic coefficients

defined in a cylinder~.x(O,T) in ~R n+l, and if u(~,t) assumes analytic

Dirichlet data on the analytic boundary ~.C~x(O,T), then u(~,t) is an

analytic function of its independent variables in-/~ x(O,T)(c.f. ~0]).

In the following sections we shall survey, with an outline of

proofs, some recent results on the reflection and continuation of sol-

utions to linear parabolic equations with analytic coefficients. We

shall restrict ourselves to parabolic equations of second order, although

60

corresponding results should also be valid for certain higher order

equations. The theory we shall present is far from complete. In

particular we have omitted questions concerning removable singul-

arities as well as the problem of backward continuation of solutions

to parabolic equations. Some aspects of this last topic will be

discussed in a survey paper to be presented at the conference on

"Inverse and Improperly Posed Problems in Differential Equations"

to be held next year in Halle.

II. Parabolic ERuations in One Space Variable.

The basic idea behind Lewy's reflection principle for elliptic

equations was to develop an integral operator which allowed the

general reflection problem to be reduced to one in which the Schwarz

reflection principle for analytic functions could be applied. This

is also the main idea behind the method for developing reflection

principles for parabolic equations, where in this case the basic

theorem employed is the reflection principle for solutions of the

heat equation. The operators employed in this analysis are no longer

based on the idea of a complex Riemann function as in Lewy's work but

instead on a generalization of the transformation operators for

ordinary differential equations developed by Gelfand, Levitan and

Marcenko in their investigations of the inverse scattering problem in

quantum mechanics (c.f. [7]). To introduce these generalized trans-

formation operators we first consider solutions of the parabolic equation

Uxx + a(x,t) Ux + b(x,t)u = u t (2.1)

defined in a domain D of the form D = {(x,t): Sl(t)<x<s2(t)10<t<to }

and make the assumption that the coefficients of (2.1) are analytic

61

Eunctions of x and t in Du~ ~ D* where o is the arc x = Sl(t) and

3* is the reflection of D across ~ defined by D* = {(x,t): 2Sl(t) -

s2(t) < x < Sl(t) , 0 < t < to }" Assuming that x = Sl(t) is analytic

and making the change of variables

= x - sl(t) (2.2)

T = t

changes (2.1) into an equation of the same form but now defined

in a domain D to the right of the t axis with lateral boundary

x = O. A further change of variables of the form fx

u(x,t) = v(x,t) exp {- 12 I a(s,t)ds} (2.3) J 0

reduces (2.1) to an equation of the same form except that a(x,t)

is now equal to zero. Hence without loss of generality we can

consider equations of the form

u + q(x,t)u = u (2.4) xx t

where q(x,t) is analytic in D u ~ ~ D* and D = {(x,t):O<x<s2(t),

O<t<to} , ~ = {(x,t): x=O}, D* = {x,t): - s2(t)<x<O, O<t<to}.

Assume now that u(x,t) is a classical solution of (2.4) defined

in ~ ~ ~ ~ ~*. Then u(x,t) can be represented in the form ([3])

u(x,t) = (I + T)h

= h(x,t) + (s,x,t)h(s,t)ds

where h(x,t) is a solution of the heat equation

(2.5)

hxx = h t (2.6)

in ~ ~ ~* and E(s,x,t) is a solution of the initial value

problem

62

E - E + q(x,t)E = E xx ss t

fx E(x,x,t) = - ~ q(s,t)ds (2.7) O

E(-x,x,t) = O .

A solution of (2.7) can be constructed by iteration and is anal-

ytic for -s2(t)<x<s2(t ) -s2(t)<s<s2(t ) O<t<t . The fact that ' O

every solution of (2.4) defined in Duo~D* can be represented

in the form (2.5) follows from the fact that ~ is a Volterra

operator and hence I + T is invertible. The reflection principle

for parabolic equations is obtained by a judicious application of

various modifications of the operator (2.5). This will be illus-

trated in the proof of the following theorem:

Theorem 2.1: Let u(x,t) be a classical solution of (2.1) in D,

continuously differentiable in D k7 ~, and satisfying

~(t)u(sl(t),t) + 8(t)Ux(Sl(t),t) + y(t)ut(sl(t),t) = f(t)

(2.8)

on o, where Sl(t), ~(t), B(t), y(t) and f(t) are real analytic

on (O,to). If, for t E (O,to), the non zero vector ](t) =

(~(t), y(t)) is never tangent to x = Sl(t) (or always tangent to

x = Sl(t)) and never parallel to the x axis (or always parallel

to the x axis), then u(x,t) can be uniquely continued as a sol-

ution of (2.1) into D~ o~)D*.

Proof: By the above discussion we can reduce the problem to the

case when a(x,t) = O(i.e. equation (2.4)), Sl(t) = O, and assume

that B(t) = 0 or B(t) # O for t ~ (O,to). Furthermore, by solving

an appropriate non-characteristic Cauchy problem for (2.4) (to be

discussed at the end of this section) we can assume that f(t) = O.

63

Hence without loss of generality we can consider the problem of

continuing solutions of (2.4) subject to (2.8) with Sl(t ) = 0 and

f(t) = O, where u(x,t) is defined in D and is continuously differ-

entiable in D ~ o. We shall only consider the special case when

6(t) # 0 and y(t) # 0 for t c (O,t o) and refer the reader to [5]

for full details. In this special case by making a preliminary

change of variables of the form

U(X,t) = v(x,t) exp {- ~(~)dT} (2.9) 0

we can assume without loss of generality that ~(t) = 0 and the

boundary condition on o is

Ux(O,t ) + q(t)ut(O,t) = 0 (2.10)

Y(t)16(t ) where q(t) = . We can now show that in D, u(x,t) can be

represented in the form

XK(1) (i) u(x,t) = h(1)(x,t) + h(2)(x,t) + (s,x,t)h (s,t)ds

~0 (2.11)

+ {XK(2)(s,x,t)h(2)(s,t)ds

JO

where K(1)(s,x,t) is the solution of

K (I) _ K (I) + q(x,t)K (I) = K (I) xx ss t

i 'x

K(1)(x,x,t) = - ~ q(s,t)ds 0

K(1)(O,x,t) = O, s

K(2)(s,x,t) is the solution of

K (2) - K (2) [q(x,t) q (t)] K(2) xx ss + q~)'J

K(2) (x,x,t) = _ ½ ix J 0

K(2)(O,x,t) = O,

= K (2) t

(2.12)

Eq(s,t ) q (t) ~ds (2.13) q(t)

and h(2)(x,t) = -n(t)h~l)(x,t) where h(1)(x,t) is a solution of (2.6)

in D satisfying h(1)(O,t) = O. K(1)(s,x,t) and K(2)(s,x,t) can be X

constructed by iteration and are analytic for -s2(t)<x<s2(t),

-s2(t)<s<s2(t), O<t<to. The reflection principle now follows from

(2.11) and the fact that h(1)(x,t) can be continued into D ~ ~ ~*

by the reflection principle for the heat equation. The uniqueness

of the continuation follows from Holmgren's uniqueness theorem.

From the above analysis it is see that for the original problem the

domain of dependence associated with a point in X<Sl(t) is a one

dimensional line segment lying in X>Sl(t).

It would be desirable to know if the restriction on the dir-

ection of the vector ~(t) can be removed.

As a corollary to Theorem (2.1) we have the following version

of Runge's theorem for solutions of (2.1) defined in a domain

D = {(x,t): Sl(t)<x<s2(t), O<t<t } where Sl(t) and s2(t) are anal- O

ytic for O<t<to([3]):

Corollary: Let u(x,t) e C2(D) ~ C°(~) be a solution of (2.1) in

D and assume that the coefficients of (2.1) are analytic for -~<x<=,

O~t@t o. Then for every c>O there exists a solution Ul(X,t) of (2.1)

in -~<x<~, O<t<to, such that

~Xlu(x,t ) - Ul(X,t) l <

Proof: There exists a solution Ul(X,t) s C2(D)~ CI(~) satisfying

analytic boundary data on x = Sl(t ) and x = s2(t) such that the

above inequality is valid. By reflecting Ul(X,t) repeatedly across

the arcs x = Sl(t) and x = s2(t ) it is seen that Ul(X,t) can be

continued as a solution of (2.1) into the infinite strip -~<x<~,

O~t~t o •

65

We now turn to the case when the solution of (2.1) is known

to be analytic in some neighbourhood of the origin and make the

assumption that the coefficients of (2.1) are analytic for -=<x< ~,

-t <t<t . In this case we have that u(0,t) and Ux(O,t) are anal- o o

ytic functions of t for -t <t<t for some constant to, and if (2.1) o o

is reduced to the canonical form (2.4) and u(x,t) is represented

in the form (2.5) we have that h(O,t) = u(O,t) and hx(O,t) =

Ux(0,t) are analytic functions of t for -t <t<t . Hence by the o o

Cauchy-Kowalewski theorem and Holmgren's uniqueness theorem h(x,t)

is analytic in a neighbourhood of the t axis for -t <t<t , By o o

Widder's theorem referred to in the Introduction h(x,t) is analytic

in the strip -=<x<=, -to<t<to, and hence from (2.5) and the anal-

yticity of the kernel E(s,x,t) so is u(x,t).

Theorem 2.2: Let the coefficients of (2.1) be analytic in the

strip -~<x<=, -to<t<to, and let u(x,t) be any solution of (2.1)

that is analytic for -Xo<X<Xo, -to<t<to, for some positive con-

stant x . Then u(x,t) can be analytically continued into the strip O

-~<x<~ -to<t<t o.

We close this section be giving a simple derivation of Widder'

result on the analytic continuation of analytic solutions of the

heat equation which was used to prove Theorem 2.2 Suppose h(x,t)

is an analytic solution of the heat equation for -Xo<X<Xo, -to<t<t o.

Then locally h(x,t) can be represented in the form (~5])

= i h(x,t) - 2hi J E (X,t-T)h(O,T)dT X

It-el =

- 2~ii ~ E(x,t_T)hx(O,r)d ~

Jt-T I = 6

(2.14)

66

where

E(x,t) =

oo E x2j+l (-I) J ~'

j=O (2j+l) : t j+l (2.15)

Since E(x,t) is an entire function of x we immediately have the

result that h(x,t) is analytic in the strip -~<x<~, -t <t<t . If O O

we further assume that h(O,t) and h (O,t) are analytic for Itl<t ° X

and r e p r e s e n t t h e s e J u n c t i o n s by t h e i r T a y l o r s e r i e s

oo

h(O,t) = E b t n , Itl < t n= O n o

oo

hx(O,t ) = )2 Cn tn Itl < t O

n=O

(2.16)

we have from (2.14) and termwise integration that for -~<x< ~, -to<t<t o,

h(x,t) = E a h (x,t) (2.17) n= 0 n n

where hn(X,t) are the heat polynomials defined in (1.7) and a2m=bm ,

a2m+l=Cm for m = O,1,2, ....

Returning to the proof of Theorem 2.1 we note that from the

above results a special solution of (2.4) satisfying (2.8) (for

Sl(t ) = O) can be obtained in the form u(x,t) = (~ + ~)h(x,t) where

h(x,t) is given by (2.14) with h(O,t) and hx(O,t) chosen appropriately.

Ill.Parabolic Equations in Two SEace Variable @.

In contrast to the case of parabolic equations in one space

variable, the results on the global continuation of solutions to

parabolic equations in two space variables are rather limited. Even

in the case of analytic solutions of parabolic equations in two

space variables with analytic coefficients it is not yet known

whether or not such solutions can be reflected across a plane

boundary on which they assume homogeneous Dirichlet data. However

67

it has been shown by Hill that the domain of dependence associated

with an analytic solution of a parabolic equation in two space

variables at a point on one side of a plane on which the solution

vanishes is a one dimensional line segment on the other side (El6]).

This suggests the possibility of establishing a reflection prin-

ciple for parabolic equations in two space variables, at least in

in the case of analytic solutions. However this has yet to be done.

The main problem seems to be to establish the domain of regularity

in the complex domain of the normal derivative of the solution

evaluated along the plane on which the solution vanishes. This

suggests the problem of determining the domain of regularity in

the complex domain of analytic solutions of parabolic equations,

given either their domain of real analyticity or the domain of reg-

ularity of the Cauchy data in the complex domain. As will be seen,

modest results such as these can be applied to derive a Runge

approximation property for parabolic equations as well as a global

representation of the solution to certain non-characteristic Cauchy

problems arising in the theory of heat conduction. Although we

shall derive these basic results in this section, we shall post-

pone their application to problems associated with the heat equation

until Section IV.

We consider analytic solutions of parabolic equations of the

form

[u~ + u + a(x,y,t)u x + b(x,y,t)Uy L = Uxx YY

--0

+ c(x,y,t)u - d(x,y,t)u t

(3.1)

and for the sake of simplicity make the assumption that the coeffi-

cients of (3.1) are entire functions of their independent complex

68

variables.

the space of two complex variables into itself defined by

z = x + iy

z* = x - iy

we can rewrite (3.1) in its complex form as

By making the nonsingular change of variables mapping

(3.2)

(3.3) i i I i

where A = [(a + ib), B = ~(a - ib), C = [ c, D = ~ d. The basic

tool used in the investigation of analytic solutions of (3.1) (or

(3.3)) is the Riemann function for (3.3) first introduced by Hill

in [16]. This is defined to be the unique solution R(z,z*,t;~,~*,T)

of the (complex) adjoint equation

My] -- V-zz , a(AV) 3z ~z*

satisfying the initial data

R(z,~*,t;~,~*,T) = --!-I exp { B(q,~*,t)d~} t-~

(3.5)

R(¢,z*,t;C,¢*,T) = ~ exp { do}

along the planes z -- ¢ = ~ + iQ, z* = ~* = $ - in. The Riemann

function can be constructed by iterative methods ([3], [16]) and

3(B~/) + C_~r+_ ~ (D'V') = 0

(3.4)

(under the assumption that the coefficients of (3.1) are entire) is

an entire function of its six independent complex variables except

for an essential singularity at t = T. In the case of the heat

equation

u + u = u t xx yy

the Riemann function is given by

(3.6)

I (z-C) (z* -¢* ) R(z z*,t;¢,¢*,T) = ~ exp { } .

• 4 (t-T) (3.7)

69

Now let u(x,y,t) be an analytic solution of (3.1) defined in

a cylinder Dx(O,T) where D is a bounded simply connected domain.

Assume that D contains the origin and that d(x,y,t) > 0 in Dx(O,T).

By standard compactness arguments we can conclude that if ~ C D, O

6 o > O, then u(x,y~t) is analytic in some thin neighbourhood in

C 3 of the product domain box ~o,T-~o3. We now use Stokes theorem

applied to u and R log r (where R is the Riemann function and

r = /(x-~) 2 + (y-n) 2 to represent u(x,y,t) in terms of the Riemann

function, where the domain of integration is the torus D x O

with ~L= {t:It-T [ = ~}. This yields the result that

4~ 2

~D x~h- D x ~L o o (3.8)

where~ is the adjoint operator to (3.1) and

H[u,v 3 = {(VUx - UVx + auv)dydt - (VUy UVy + bul)dXdtv (duv)dxdy}.

(3.9)

Since ~ [R log r] is an entire function of its independent complex

variables except for an essential singularity at t = r and

H[u, R log r~ is analytic for (x,y,t) ¢ ~Do x~ , ~ ¢ Do, ~ ~ Do,

¢ ~o' T-6~, we have the following theorem (Note that subject

to the above restrictions D and 6 are arbitrary): O O

Theorem 3.1: Let u(x,y,t) be an analytic solution of (3.1) defined

in D x (O,T). Then'U- (z,z*,t) = u(x,y,t) is analytic in D x D* x (O,T)

where D = {z:z ~ D}, D* = {z*: z* e D}.

Theorem 3.1 is analogous to the result that every regular solution of

a linear ordinary differential equation with entire coefficients

defined on a finite interval can be extended to an entire function

70

of its independent complex variable. For partial d~ffe~emtial e~fnations

singularities can of course occur even though the coefficients are

entire. However Theorem 3.1 shows that in the case of analytic solutions

of parabolic equations in two space variables the location of the sin-

gularities in the complex domain is determined in a simple way by the

location of the singularities on the boundary of the domain of def-

inition of the solution in the real domain. For elliptic equations

in two independent variables analogous results have been found by

Bergman ([I]), Lewy (E20]) and Vekua ([24]).

Using Theorem 3.1 we can now represent?~(z,z*,t) in D x D* x (O,T)

in terms of the Riemann function and the Goursat data on the character-

istic hyperplanes z = O and z* = O in a manner almost identical to the

use of the classical Riemann function to solve the Goursat problem for

hyperbolic equations in two independent variables. Since this Goursat

data is defined in a product domain we can apply Runge's theorem for

several complex variables to approximate this data on compactsub-

sets by polynomials and hence, since the Riemann function is entire

except at t = T, approximate-~f(z,z*,t) on compact subsets by an

entire solution of (3.1). If we now note that if d(x,y,t) > O every

classical solution of a parabolic equation with analytic coefficients

defined in a cylindrical domain with analytic boundary can be approx-

imated by a solution having analytic boundary data, and hence from

the Introduction is an analytic solution of the parabolic equation,

we can now deduce the following version of Runge's theorem ([3]):

Corollary: Let u(x,y,t) be a classical solution of (3.1) in D x (O,T)

where d(x,y,t) > O in D x (O,T). Then for every compact subset D cD o

and positive constants ~ and e there exists an entire solution Ul(X,y,t)

of (.3.1) such that

71

max lu(x,y,t) -Ul(X,y,t)I < e .

o

We now turn our attention to the problem of determining the

domain of regularity of solutions to non-characteristic Cauchy pro-

blems for parabolic equations with data prescribed along an analytic

surface. Such problems arise if inverse methods are used to study

free boundary problems in the theory of heat conduction (c.f. Section

IV) and a local solution can always be found by appealing to the Cauchy-

Kowalewskl theorem. However in addition to being impractical for com-

putational purposes such an approach does not provide us with the

required global solution to the Cauchy problem under investigation,

and hence we are lead to the problem of the analytic continuation of

solutions to non-characteristic Cauchy problems for parabolic equations.

We shall accomplish this through the use of the Riemann function in

connection with contour integration techniques and the calculus of

residues fn the space of several complex variables.

We assume that u(x,y,t) is an analytic solution of (3.1) in a

domain containing a portion of the non-characteristic analytic surface

S along which u(x,y,t) assumes prescribed analytic Cauchy data. Let

the intersection of the plane t = constant with the surface S be a

one dimensional curve ~(t). Suppose ~(t) is described by the equation

F(x,y,t) = O. Then since S is analytic we can write

F.Z+Z* z-z* t) = O (3.10) £ 2 ' . . . . 2i '

where z and z* are defined 5y (3.2) and this is the equation for ~(t)

in (z,z*) space. We now choose C(t) to be an analytic curve lying on

this complex extension of ~(t) and for (~n,r) not on S let G(t) be a

72

cell whose boundary consists of C(t) and line segments lying on the

characteristic planes z = ~ = ~+i~ and z* = ~ = t-in respectively

which join the point (~,~) to C(t) at the points P and Q respectively.

If we now use Stokes theorem to integrate RL[I~r]--~rM[R] (where R is

the Riemann function and L and M are defined by (3.3) and (3.4) res-

pectively) over the torus {(z,z*,t):(z,z*) e G(t),It-T I = 5} we have

-~r(~,~,T) = ~ .... ~ [R(P,t)~ (P,t) + R(Q,t)7-F(Q,t)]dt 4~i

It-~1 =~ (3.II) J

2~i

[t-T I=6 C(t) - (BR~-+ ½ R T_)- z - ½ Rz-O-)dzdg

where we have suppressed the dependence of the Riemann function on

the point (~,~,T) and an expression of the form 7jr(P,t) is a

function of three independent variables, i.e.7_~(P,t) =~r(~l,~2,t)

where ( 1, 2) are the Cartesian co-ordinates of the point P in C 2.

For (~, ~,~) sufficiently near the initial surface S and for

sufficiently small, (3.11) gives an integral representation of the

solution to the non-characteristic Cauchy problem for (3.1) with

analytic data prescribed on S. Equation (3.11) can now he used to

obtain a global solution by deforming the region of integration,

provided a knowledge of the domain of regularity of the Cauchy data

and analytic function F(x,y,t) is known. In particular such a pro-

cedure yields results on the analytic continuation of solutions to

parabolic equations along characteristic planes in terms of the

domain of regularity of the Cauchy data and the domain of regularity

of the function describing the non-characteristic surface along which

73

the Cauchy data is prescribed. For example suppose for each fixed

t, t e [O,TJ, z = $(~,t) maps the unit disc conformally onto a

domain D t and let the analytic surface S be described by

im ~-l(z,t) = 0 (3.12)

Assume that ¢(z,t) and ¢-l(z,t) depend analytically on the para-

metec t. Then setting ~(~,t) = ¢(~,t) we have that the equation

for C(t) is given by @-l(z,t) = ~-l(z*,t). Hence if we assume

that the Cauchy data is analytic in D t for each t we have from

(3.11) the following result (c.f.~]):

Theorem 3.2: Let u(x,y,t) be an analytic solution of (3.1) which

assumes analytic Cauchy data on the surface (3.12) where @(z,t)

conformally maps the unit disc onto the domain D t. If the Cauchy

data for u(x,y,t) is analytic in D t then, for each fixed t,

7J(z,z*,t) = u(x,y,t) is an analytic function of z and z* in

O t x Dr* where O t" = {ze:z* e Ot).

Theorem 3.2 is a generalization of the corresponding result

obtained by Henrici for elliptic equations in two independent

variables ([lhJ). For partial progress on extending the results

of this section to parabolic equations in three space variables

we refer the reader to [23S.

IV. Applications to Problems in Heat Conduction.

In this last section we shall apply the results of the

previous two sections to derive an explicit analytic representation

of the solution to the inverse Stefan problem and to construct

some complete families of solutions to the heat equation which are

suitable for constructing approximate solutions to the standard

initial-boundary value problems arising in the theory of heat

74

conduction (c.f.[3]). We first consider the Stefan problem. This

is a free boundary problem for the heat equation and we are inter-

ested in the inverse problem where the free boundary is assumed to

be known a priori. Such an approach allows one to construct a

variety of special solutions from which a qualitative idea can be

obtained concerning the shape of the free boundary as a function

of the initial-boundary data. In certain physical situations, e.g.,

the growing of crystals~ the inverse problem is in fact the actual

problem that needs to be solved. The simplest example of the type

of problem we have in mind is the single phase Stefan problem for

the heat equation in one space variable, which can be mathematically

formulated as follows: to find u(x,t) and s(t) such that

u = ut; O<x<s(t), t>O xx

u(s(t),t) = O; t>O (2.z)

U x ( S ( t ) , t ) = - s ( t ) ; t>O

u(O,t) : @ ( t ) ; 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 par-

ticular 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) = 12wil t-~l=6 E(x-s(T),t-x)s(T)dT (4,2)

Computing the residue in (4.2) gives

u(x,t) = ~ I S n n=l ~--St 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, = constant , (4.4) n=l

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

+ u = ut; ¢(x,y,t)<O Uxx yy

u I = y(x,y,t) so x

u I = o ~=0

(4.5)

S u I _ i

¢=0

S¢

St

¢=O

where u(x,y,t) is the temperature in the water, ¢(x,y,t) = O is the

interphase boundary, D is a region originally filled with ice,

y(x,y,t) ~ 0 is the temperature applied to the boundary SD of D,

is the normal with respect to the space variables that points into

76

the water region $(x,y,t) < O, and V denotes the gradient with respect

to the space variables. In this case from the analysis of Section !II

we see that in order to solve the inverse Stefan problem it is neces-

sary to restrict the function #(x,y,t) to be more than just analytic,

and we assume ¢(x,y,t) is given by an equation of the form (3.12)

where D t D D for t ~ [O,T]. In this case we have from (3.11) and

(3.7) that Z/(~,~,~) = u(~,n,T) is given by

1 ! i I exp [(z-~)(z*-~q i ~¢ ,dzidt 7 J (~, ~,T) : ~ t -~ [ ~ h ( t i T ' ) ~ J IV$1 St

I -~I :a c ( ) (~.6)

where C(t) is a curve lying on the surface $-l(z,t) = ¢ -l(z*,t)

with endpoints on the characteristic hyperplanes z =~ and z* = ~.

Computing the residue in (h.6) now gives

i ~ i ~n ~ (z-~)n(z*-~) n 3¢ 7 j ( ~ , L + ) : + z )a { j -- Id+l} •

n:0 hn(n! ~T n twl ~ c(+) (~.7)

Equation (h.7) gives the generalization to two dimensions of the

series solution (h.3) for the one dimensional inverse Stefan pro-

blem. We note that the integral in (h.7) is pure imaginary.

We now turn our attention to the construction of complete

families of solutions for the heat equation. By completeness we

shall mean completeness on compact subsets of a given domain D

with respect to the maximum norm. We first consider a solution

h(x.t) of the one dimensional heat equation (2.6) defined in a

domain D of the form D = {(x.t):sl(t)<x<s2(t) , 0<t<t O} where sl(t )

and s2(t) are continuous functions. If %cD then we can assume

without loss of generality that D has an analytic boundary and o

hence from the Corollary to Theorem 2.1 we have that for every

77

E ~ 0 there exists a rectangle R P o

heat equation in R such that

and a solution hl(X,t) of the

max lh(x,t) -hl(X,t) I <

o

(~.8)

Since it is a relatively easy matter to show that the heat poly-

nomials h (x,t) defined by (1.7) are complete for solutions of the n

heat equation defined in a rectangle, we can now conclude that the

heat polynomials are also complete for solutions of the heat

equation defined in a domain of the form described above.

We next want to decide under what conditions on the separation

constants k do the solutions n +

hn(x,t) = exp (+_ i~nX - ~2t)n (4.9)

form a complete family of solutions to the heat equation in D.

From the above results it suffices to show that every heat poly-

nomial can be approximated by a linear combination of the functions

(4.9), subject to certain restrictions on the constants ~ . From n

the representation (2.1h) we see that it suffices to show that the 2

set {e -~nt} is complete for analytic functions defined in an ellipse

containing [O,to]. The type of restriction necessary is indicated

by the following theorem based on the theory of entire functions

(B@ ,p.219):

Theorem h.l:

the limit

If {~n } is a sequence of complex numbers for which

lim n d = ~ > 0

n'~ ~n

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 --- > 0 . (4.10)

n~ ~2 n

Theorem 4.2: Let D = {(x,t):sl(t)<x<s2(t), O<t<t o} where sl(t) and

s2(t) are continuous functions. Then the following sets are complete

for solutions of the heat equation defined in D:

[n/~ xn-2ktk I) h n(x,t) = [~]'. Z ........ ; n = 0,1,2 .....

k=O (n-2k)!k!

+ 2) h~(x,t) = exp (+ ilnX - k2t)~ lim n > O.

-- n n ~-~ ~2 n

We now conclude this section by using the results of Section III

to derive a result analogous to Theorem h.2 for the heat equation in

two space variables defined in a cylindrical domain D x [O,T] where

D is a bounded simply connected domain. From the proof of the Corollary

to Theorem 3.i we see that it suffices to obtain a set of solutions

to (3.6) assuming data on the complex hyperplanes z = 0 and z* = 0

that is complete for product domains in the space of two complex

variables. It is easily verified that one such complete set is given by

+ +ine m r2k+n tm-k U~,m(r,e,t) = e- Z (h.ll)

k=O 4kk~(m-k)!(n+k)!

where (r,e) are polar co-ordinates, since on the above mentioned

+ (z,z*,t) = u ~ (r,e,t) characteristic hyperplanes we have that~/~, m n,m

satisfies

79

znt TM + (z ,O,t) . . . . . . n,m m'n'

V n,m ( O ' z * ' t ) : z*ntm

m'n'

(h .12 )

It follows from the uniqueness theorem for Cauchy's problem for

the heat equation that another complete set for (3.6) defined in

D x (O,T) is given by

hn,m(X,y,t) = hn(x,t) hm(Y,t) (h.13)

where hn(x,t) is the heat polynomial defined by (1.7). On the

other hand if we separate variables for (3.6) in polar coordinates

we find that

+ h-- (r,e,t) = Jm(Anr)exp(+im8 - X~t) (h.lh) n ,m

where J (z) denotes Bessel's function, is a solution of the heat m

equation satisfying the complex Goursat data

z exp(- ~ t)

H~n+'m(Z'O't) -- 2 m m:

z*mexp(- ln2t)

n,m(O,z*,t . . . . . .. -. H ) = 2 m m' ( h .15 )

Hn+--m(Z,z*,t) = h +- ( r , B , t ) n ~ m m

and hence from Theorem h.l we can conclude that (h.lh) is another

complete set of solutions for the heat equation in D x (O,T) pro-

vided (~.10) is valid.

Theorem h.3: Let D be a bounded simply connected domain in the

plane. Then the following sets are complete for solutions of the

heat equation defined in D x (0,T):

l) hn,m(X,y,t) = hn(x,t)hm(Y,t) ; n,m = 0,1,2,...

2) h+,m(X,y,t) = Jm(Xn r) exp(+_ ires - X2t)n ;n'm= O,i .... limn~<o X- >-Vn O.

n

80

Theorem h.3 can also be extended to higher dimensions ([6~).

For other methods of proving Runge's theorem for the heat equation

see [18~ and [22].

1.

2.

.

.

.

.

.

.

.

References

S. Bergman, Integral Operators in the Theor~ofLinear Partial

Differential Equations, Springer-Verlag, Berlin, 1969.

D. Colton, Partial Differential Equations in the Complex Domain,

Pitman Publishing, London, 1976.

D. Colton, Solution of Bound ar ~ Value Problems by the Method of

Integral Operatgrs , Pitman Publishing, London, 1976.

D. Colton, A reflection principle for solutions to the Helmholtz

equation and an application to the inverse scattering problem,

Glas~ow Math ,. J. 18(1977), 125-130.

D. Colton, On reflection principles for parabolic equations in

one space variable, Proc. Edin. Math. Soc., to appear.

D. Colton and W. Watzlawek, Complete families of solutions to

the heat equation and generalized heat equation in R n,

J. Diff. Eqns~ 25(1977), 96-107.

V. De Alfaro and T. Regge, Potential Scattering, North-Holland

Publishing Company, Amsterdam, 1965.

J °

A. Erdelyl, Singularities of generalized axially symmetric

potentials, Comm.: " Pure Appl. Math. 9(1956), hO3-hlh.

V. Filippenko, On the reflection of harmonic functions and

of solutions of the wave equation, Pacific J. Math. lh (196h),

883-893.

81

i0. A. Friedman, Part%al Differential Equations, Holt, Rinehart

and Winston, New York, 1969.

ii. P. Garabedian, An example of axially symmetric flow with a free

surface, in Studies in Mathematics and Mechanics Presented to

Richard yon Mises, Academic Press, New York, 195h, 149-159.

12. P. Garabedian, Partial differential equations with more than

two independent variables in the complex domain, J. Math. Mech.

9 (1960), 2hi-271.

13. R.P. Gilbert, Function Theoretic Methods in Partial Differential

Equations, Academic Press, New York, 1969.

P. Henrici, A survey of I. N. Vekua's theory of elliptic partial

differential equations with analytic coefficients, Z.An~ew.Mat h.

Physics 8(1957), 169-203.

15. C. D. Hill, Parabolic equations in one space variable and the

non-characteristic Cauchy problem, Comm. Pure Appl. Math. 20

(1967), 619-633.

16. C. D. Hill, A method for the construction of reflection laws

for a parabolic equation, Trans. Amer. Math. Soc. 133 (1968),

357-372.

17. F. John, Continuation and reflection of solutions of partial

differential equations, Bull. Amer. Math. Soc. 63 (1957),

327-3hh.

18. B. F. Jones, Jr., An approximation theorem of Runge type for

the heat equation, Proc. Amer. Math. Soc. 52 (1975), 289-292.

19. B. Levin, Distribution of Zeros of Entire Functions, American

Mathematical Society, Providence, 1964.

82

20. H. Lewy, On the reflection laws of second order differential

equations in two independent variables, Bull. Amer. Math. Soc.

65 (1959), 37-58.

21. H. Levy, On the extension of harmonic functions in three vari-

ables, J. Math. Mech.lh (1965), 925-927.

22. E. Magenes, Sull' equazione del calore: teoremi di unicit'a e

teoremi di completezza connessi col metodo di integrazione di

M. Picone, Rend. Sem. Mat. Univ. Padova 21 (1952), I, 99-123,

II, 136-170.

23. M. Stecher, Integral operators and the noncharacteristic Cauchy

problem for parabolic equations, SIAM J. Math. Anal. 5 (1975),

796-811.

I. N. Vekua, New Methods for Solvin~ Elliptic Equat%ons, John

Wiley, New York, 1967.

25. D. Widder, Th 9 Heat Equation, Academic Press, New York, 1975.

26. D. Widder, Analytic solutions of the heat equation, Duke Math. J.

29 (1962), h97-503.

Legendre polynomials and singular differential operators

W N Everitt

Introduction The Legendre polynomials can be defined in a

number of different ways which we review here briefly before

discussing the connection with differential operators. For convenience

let (-I,|) be the open interval of the real line, and let

N = {0,1,2,. .... } be the set of non-negative integers. o

All the definitions given below are inter-connected, as may be

seen in the standard accounts of the Legendre functions given in [2]

by Erdglyi et al, and [9] by Whittaker and Watson.

(i) The Legendre differential equation

For our purposes this is written in the form

(here ' ~ d/dx)

(I - x2)y"(x) - 2xy'(x) + n(n + l)y(x) = O (x ~ (-l,l)) (I.I)

which is derived from the Laplace or the wave equation, considered in

polar co-ordinates. This equation has singular points at ±l where

the leading coefficient vanishes. This equation has a non-trivial

solution which is bounded on (-I ,I) if and only if n £ No; the

corresponding solution is P (-), i.e. the Legendre polynomial of order n. n

See [2, chapter III] and [9 sections 15.13 and 15.14]

(ii) The Poisson generating function

This definition takes the form

I = ~ Pn(x)hn (x ¢ (-I ,I)) /(I - 2xh + h 2) n=°

valid for all h with I~ < I. See [2, section 3.6.2] and [9, section 15.1]

(iii) The Rodrigues formula

This definition has the form

Pn(x) 1 d n - ( x 2 - 1 ) n ( x e ( - 1 , 1 ) , n ~ N o ) ; n v 2 n . d x n

see [2, section 3.6.2] and [9, section 15.|I]

84

(iv) The Gram-Schmidt ortho$onalization

The Legendre polynomials may be defined by the Gram-

Schmidt process applied to the set {x n : x ~ (-l,l), n E N } o

in the integrable-square inner-product space L2(-1,I). See Akhiezer

and Glazman [1, section 8], [2, sections 10.1 and IO. I0] and

[9, section ll,6].

This definition leads to the fundamental or thogonal

property of the Legendre polynomials 1

f Pm(X)Pn(X)dx = (n + ~)6mn (m,n E N o ) (1.2)

-l

where ~ is the Kronecker delta function. mn

Whichever definition is adopted it is important

subsequently to prove that the set of Legendre polynomials {P (.):n e N o } n

is complete, equivalently closed, in L2(-1,1); see [I, sections 8 and 9]

and [2, section I0.2]. Once the orthogomal property of the polynomials

is known the completeness may be obtained by classical means, as for

the completeness of the trigonometrical functions, using the Weierstrass

polynomial approximation theorem for continuous functions on compact

intervals; for details see [] , section 11] and [2, section 10.2].

Our interest in this paper is to discuss the definition

and completeness of the Legendre polynomials in L2(-I,I) from the

viewpoint of the Titchmarsh - Weyl theory of singular differential

operators. This theory is concerned with the differential equation

- (py')' + qy = %wy on (a,b) (1.3)

where p, q and w are real-valued coefficients on the interval (a,b),

and % is a complex-valued parameter.

If in (1.3) the coefficient w is non-negative on (a,b)

then this is a so-called risht definite problem and is studied in the

integrable-square space ~(a,b), i.e. the collection of those functions

f f or which b

l w(x) If (x) 1 2dx < ~ (1.4)

a

If in (1.3) it should happen that, whether w is of one

sign or not, both p and q are non-negative then the problem is called

85

left-definite and is studied in the space of those functions f for which b

f {p(x) If'(x)I2 + q(x) If(x)12}dx < ~ (1.5) a

For both the right- and respectively left-definite cases

the differential equation is classified as either limit-point or

limit-circle, at an end-point a or b, according as to whether not all or

all solutions of (1.3) are in the spaces (1.4) respectively (1.5),

in the neighbourhood of the endpoint in question. For reference to

the classification of the differential equation (1.3) in the right-

definite case see Naimark [4, section 18.1] and Titchmarsh

[8, sections 2.1 and 2.19]; and in the left-definite case see Pleijel

[6, sections I . 4 and 5].

For the purposes of the study of the Legendre equation (1.1)

in this paper we write the equation in the form

- ((I - x2)y'(x)) ' + ¼y(x) = % y(x) (x ~(-l,l)), (1.6)

so that in comparison with the standard form (3.3) we have a = -I, b = | 2

and p(x) = I - x q(x) = ¼ w(x) = | (x c (-1,3)).

Thus we may study the Legendre equation as right-definite in the

space L2(-I,I), and left-definite in the space I

I {(I - x2llf'(x) I 2 + ¼1f(x) 12}dx < -I

which, for convenience in this paper, we denote by H2(-;,I).

The original study of Legendre's differential equation in

the right-definite case in L2(-l,l) is due to Titchmarsh; see [8, sections

4.3 to 4.7]. It should he noted that the Legendre equation (1.6) is

limit-circle at both end-points ~I in the right-definite case; for

details see [8, section 4.5] and [I, II, appendix II, section 9, II].

Here it is emphasized that the analysis of Titchmarsh is essentially

'classical' with no reference to operator theoretic concepts. This

work was followed by the studies of Naimark [4] and Glazman as given in

[3, appendix II, section 9]; in particular Glazman characterized the

elements in the domain of the differential operator giving rise to the

Legendre polynomials.

The first study of Legendre's differential equation in the

left-definite case is due to Pleijel; see [5] and [6] in which may be

86

2.

found references to earlier results of Pleijel and the work of his

school at Uppsala. In particular we owe to Pleijel the observation

that the Legendre equation (1.6) is limit-point at both end-points ±I

in the left-definite case; see [6, page 398].

Our purpose in this paper is to study the right- and

left-definite problems for the Legendre differential equation with

the methods of Titchmarsh [8] in mind. We link the Titchmarsh method

with operator-theoretic results in the Hilbert function spaces L2(-|,])

and H2(-I,I).

The paper is in six sections; after this introduction

the second section considers essential properties of the Legendre

differential equation; the third and fourth sections are given over to r#mgrks.

a study of the right- and left-definite cases; five and six to certain/

Notations R - real field; C - complex field; L - Lebesgue integration;

ACIo c local absolute continuity; if D is a set of elements f then

'(f £ D)' is to be read 'the set of all f • D'; N O - the set of all

non-negative integers.

The Legendre differential equation

In this section we consider certain essential properties

of Legendre's differential equation required for consideration of the

right-definite and left-definite cases.

As before we write the equation in the form

~I - x2)y'(x~ ' + ¼y(x) = % y(x) (x • (-I,])). (1.6)

For convenience let

p(x) = J - x 2 (x ~ (-l,I)). (2.1)

The standard form of the Legendre equation is given in (1.I)

but for the purpose of considering both right- and left-definite problems

the form (1.6) is to be preferred. A detailed discussion of the classical

properties of the Legendre equation (I.I) is to be found in Erd~lyi

et al [2, volume I, chapter III].

The account of the Legendre equation in Titchmarsh

[8, chapter IV] is based on the Liouville normal form of (1.6) i.e. ~ 2

- y"(x) - ¼ sec x. y(x) = ~ y(x) (x • (-½~,½n~ .

However, as is evident from the results in [8, section 4.5],

this differential equation does not of itself enjoy the property of having

87

polynomials solutions. Here we adapt the analysis of Titchmarsh to

apply similar methods to the equation (1.6) which does have the

Legendre polynomials directly as solutions.

We write s = /%, i.e. s 2 = ~, and determine the /(.)

function by requiring

O -< arg /% <~ when O -< arg ~ < 2~

Following the analysis in [8, section 4.5]

of (1.6) above may be determined by -I

COS X

= [ cos st dt (x £ (-l,l) ~ ¢ C) Y(X, ~) -I (cos t - x) I/2 J

-COS x (2.3)

(2.2)

two solutions

1 ½~ + sin-lx)

Z(x, ~) = cos st dt

sin-lx, J (cos t + x) I/2 (x e (-l,l) % E C) +

where the positive square root is implied. In (2.3) the inverse -I

trigonometrical functions are determined by requiring cos x to decrease

from ~ to O, and sin-lx to increase from -In to ½~ as, in both cases,

x increases from-l to I.

To show that the Y and Z, as defined by (2.3), are

solutions of (1.6) we follow [8, section 4.5] and write the integrals

as contour integrals

Y(x, ~) = ½ [ cos sz

d z G (cos z - x ) | /2 J

(2.4)

f cos sg Z(x, ~) = ~ J F (cos z + x) I/2 d z

(2.5)

where now the sign convention of [8, lemma 4.4] is taken to hold for

the square root terms in the complex-valued integrands. In the integral

for Y the integrand is made single-valued by cutting the z-plane from -1 -I

- cos x to cos x; similarly for Z the cut is from

- (½~ + sinlx) to (~ + sin-lx). Here the contours G and F can be

taken as circles with centre the origin of the z-plane and of radius

r and p respectively, where cos-lx < r < ~ and ~ + sin-lx < p <~.

88

We then follow the method of [8, section 4.5] by

differentiating under the integral sign and integrating by parts, in

order to show that Y and Z are solutions of the differential equation (1.6)

for all x c (-1,I) and all % ~ C .

From (2.4) and (2.5) and the properties of the inverse

trigonometrical functions it may be shown that

Y(- x,~) = Z(x,~) (x ~ (-1,1) ~ ~ c).

The initial values of Y and Z at 0 may be calculated

on using [8, lemma 4.4]; we find, for all % E C,

21/2~I/2

Y(O,%) = r(~ + Is) r(~ - is) = Z(O,%) (2.6)

25/2~;/2 Y'(O,%) = - r(¼ + ½s) r(¼ - ½s) = -Z'(O,%). (2.7)

We note from these results that the solutions Y and Z

are linearly independent except when Y(O,%) = 0 or Y'(O,%) = O, i.e.

when % = (n + ½)2 for n = 0,I,2, .... In fact the Wronskian

(Y(x,%) Z'(x,%) - Y'(x,%) Z(x,%)~ (2.8) p(x)

is independent of x and has the value, on taking x = 0 and recalling

that p(O) = 1~

Y(O ~) z' (o,~) - Y'(O,~) z(o,~) = 2z(o,~) z'(o,~)

= 8~ cos ~s (2.9)

on following the analysis in [8, section 4.5].

The asymptotic forms of the solutions Y and Z in the

neighbourhood of the singular end-points ±I can be calculated; we give

some details for the point | and there are similar calculations for -l.

For the solution Y we find, for all % E C ,

Y(x,k) ~ ~ /2 Y'(x,X) ~ (% - ¼)~//2 (x + 1-). (2.]0)

These results follow from the integral representation for Y given by

(2.4); as x + I- the cut in the z-plane within the contour G tends to the

single point at the origin of the plane and we obtain

lim Y(x,X) = [ co~ s z dz (~ C). £

x ÷ I- JG (cos z - l) I12 ~

80

Notwithstanding the square root term the integrand is regular within

and on G except for a simple pole at the origin. Using the calculus of

residues yields

lim Y(x,%) = 7/2 (~ e C). x+l-

Similarly

lim Y'(x,~) = ~ [ COS SX

x + l - JG ( c o s z - 1) 3 /2

also valid for all % e C.

For the solution Z we find

Z(x,%) ~ 2/2. cos ~s.

dz = (~-¼)~//2

In ('/(l-x) (x + 1-)

( 2 . 1 1 )

Z' (x,%) ~ 2/2. cos ~s. (I - x) -I (x + I-)

both valid for all % c C except for the set of points

{(n + i)2 : n = O, I, 2 .... }. The proof of the first of these results

follows from the analysis given in [8, section 4.5]; the second result

then follows from the constant value of the Wronskian of Y and Z as

given by (2.8) and (2.9).

The asymptotic forms of Y and Z at -| follow from the

above results at I and the relationship Y(-x,%) = Z(x,%) (x e (-1,1)).

It follows from these asymptotic results and the linear

independence of the solutions Y and Z, except for certain exceptional

values of %, that the Legendre equation (1.6) has a non-trivial solution

which is bounded on the internal (-1,1) if and only if % lies in the

set of points {(n + ½)2 : n = O, I, 2 .... }. For all other values of %

any non-trivial solution of the equation is unbounded at 1 or -I or

at both points.

From Y and Z we now form solutions O and ~ of the

Legendre equation (1.6) which satisfy the following initial conditions

0 ( 0 , ~ . ) = 1 9 ' ( 0 , ~ ) = 0 ( p ( O , ~ ) = 0 ~ ' ( O , ~ k ) = | . ( 2 . 1 2 )

In fact we have

O(x,~,) = Y(x,%) + Z ( x , k ) ¢ ( x , % ) = Y ( x , ~ ) - Z(x,~.) (x E ( - 1 , 1 ) ) ( 2 . 1 3 ) 2Y(O,~) '~ 2Y' (0,%)

for all ~ ~ C (but with care needed at the set {(n + ½)2 : n = 0, I,2,...}).

90

From the general properties of the differential

equation (1.6) we know that the Wronskian p(@~' - 0'~) is constant

on (-I,I) and so

(1 - x 2 ) ( e ( x , ~ ) ~ ' ( x , k ) - @ ' ( x , ~ ) ~ ( x , ~ ) = 1 (x e ( - l , l ) , k ~ C)

The asymptotic forms of 0 and ~ follow from the earlier

results for Y and Z given by (2,10) and (2.11).

e(x,X) ~ r(¼ ÷ ~s)r(i ~ iS)

2~ 1/2 ] e ' ( x , ~ ) ~ r ' ( ¼ ÷ ½ s ) r d - ~ s ) " 1 - x

*(x,k) ~ - 2F(~ ÷ ½s)r('% -~;) in

I/2 1

~'(x,l) ~ - 2F(% +' ~s')'F(% ½s) " 1 - x

There are similar results at -I. Note again that not

all these results are valid at the set {(n + 3) 2 : n = 0,1,2 .... } of

the k-plane.

With these results established it is now possible to

look at the classification of the Legendre differential equation (1.6)

in the right-definite, and left-definite cases, as introduced in

section ; above.

In the risht-definite case; from the asymptotic results

above for 0 and ~ it is clear that although these solutions are unbounded,

in general, in the neighbourhood of the end-points ±1, both @ and

are in L2(O,I) and L2(-I,0) for all % e C; thus the equation is

limit-circle at both ! and -I in this case. (This result may also be

established independently on considering the solutions of (1.6) in

the special ease i = ¼, i.e. 1 and In~l + x)/(l- x 9 (x e ( - 1 . 1 ~ : both

these solutlons are in L (0,I) and L2(-I,O) and so from a general result

[8, section 2.|9] the equation is limit-circle at 1 and -I.)

In the left-definite case; here the spaces concerned

are, see section ! above, H2(O,I) and H2(-l,O); the asymptotic results

for the solutions 0 and ~ and their derivatives, as given above, show

that for all non-real ~ e C neither @ nor ~ is in H2(0,I) or H2(-;,O)

and this implies that the equation is limit-point at 1 and -I in this

case. (In the left-definite case greater care has to be taken in looking

For x ~ I- these are

(2.14)

(2.15)

(2.16)

(2.17)

3.

9~

at the nature of solutions at real values of %, in order to determine

the classification of the equation; we do not discuss this point

here but see the works of Pleijel [5] and [6], and references therein.)

The risht-definite case. In this section we consider the right-definite

case for the Legendre equation on the interval (-1,1). This section

is dependent in part on the original analysis of Titchmarsh in [7]

and later in [8, chapter II, section 4.4].

Consider the Legendre differential equation in the

form (1.6) on the interval (-1,1)

i.e. - ((l-x2)y'(x)) ' + ¼y(x) = %y(x) Cx £ (-1,1)).

For solutions of this equation in the neighbourhood of the singular

point l, the general theory in [8, chapter 2] gives the existence of

the Titchmarsh-Weyl m-coefficient; the analytic function m(') : C ÷ C

and determines a p~rticular solution ~ of the equation in the form

~(x,%) = e(x,%) + m(%)~(x,%) x £ (-;,I) % £ C R , (3.;)

where e and ~ are the solutions determined by (2.13). As the differential

equation is limit-circle at both ±I, the m-coefficient is not unique

and, in order to determine the differential operator associated with

the Legendre polynomials, we have to make a suitable choice from the

family of m-coefficients belonging to the end-point I. We do this by

following the limit process which determines m(') from the 1-functions;

for the general theory see [8, section 2.;].

With the solutions 0 and ~ determined from (2.;2) and

with % ~ C R, let ~X be a solution of (1.6), here X• (0,1), given by

~x(X,%) = e(x,%) + l(%,X,B)~(x,%) (x • (-1,1) .

The function i is chosen so that ~X satisfies the following boundary

condition at X

@X(E,%) cos B + P(X)~x'(X,%)sin B = 0

for some B • (-½~,½~] ; thus, for all ~ • C~R,

I(%,X,B) = e(X,%) cos ~ + p(X)e'(X,%) sin - ~(X,%) cos B YP(X)~'(X,%) sin

Now let X + ;- and choose ~ as a function of X so that 1 tends to a

limit m(.), where m(-) : C + C and is regular on c~R. In this the

92

Legendre case we can see how this is done explicitly by writing I in

the form l(l,x,B) =

_ O(X~I){In((I-X)-;) } -| + p(X)Ol(X,l ) tan ~(X).{in((l-X)-l)} -I (3.2)

~(X,X){In((I-X) LI)}'I '+ p(X),'(X,X) tan B(X).{In((I-X) -l)}-I

and then choose B(X) (X e (O,l) so that for some y e (-~,~r] we have

tan B(X).{In((I-X) -I) }-| + tan y as X ÷ I-. (Note that we can evaluate

the limits of all the remaining terms in (3.2) from the asymptotic

formulae (2.14 to 17) . In our case, guided by [8, section 4.5], we

take y = 0 and so choose ~(X) = 0 (Xe (O,l)); it then follows that

m(l) = lira I(I,X,0) = lira {O(X'X)/~(X,I)} X -~ 1 X -~ 1

lira { Y(X,I) + Z(X,I) . 2Y'(O,I) } =X + I .... 2V(0~,I) Y(XTi7 r ZT(X,~)

y'(o,k) 4F(% + ½s)r(%- Is) (3.3) = Y(o,------7) = - r( ~, + ½s)r(~ - ~s)

where we recall, see (2.2), that s = /A.

From (3.3) we then find that the resulting solution 4(.,%)

has the properties, see (3.1),

(i) 4(x,l) = Y(x,I)/Y(0,%) (x ~ (-I,]) I e ckR)

(ii) 4(',1) e L2(-I,I) (I e CNR)

• .. lira (xXl)x + ] 4(x,l) = Y(I,I)/Y(0,1)

= ?r-I/2r(% + ~s)r(% - ½s)

. , lira 4'(x,l) = (l - ¼)F(% + ~s)F(% - ½s)/(2~I) ivJ x ÷ ]

(v) 4(.,i) and 4'(',I) are unbounded in the neighbourhood of -I

(1 c CxR) .

A similar analysis holds for the singular end-point -l;

there is a solution X(.,I) of (1.6) which has the form

X(X,1) = O(x,%) + n ( t ) ¢ ( x , t ) (x e ( - 1 , I ) t E C~R)

where

n(1) = + Z'(O,I)/Z(0,1) = - m(1) (% e C',R) (3.4)

such that X(.,I) has the properties

(i) X(X,1) = Z(x,1)/Z(0,%) (x E (-l,l) I ~ CxR)

(ii) X(',I) ~ L2(-I,I) (I £ CNR)

(iii) lim X(X,l) = ~-I x->-I /2F(% + ½s)r(%- ½s)

83

(iv)

(v)

l im X'(X,%) = - ( l - ¼)1`(% + ~s)r (% - ½ s ) / ( 2 ~ ) x + -1

X ( ' , l ) and X' ( . , l ) a r e unbounded i n t h e n e i g h b o u r h o o d of 1

(X • C\R).

The Green's function for this choice of m(-) and n(.)

(-1 < x < g < 1)

(-1 < g < x < 1 )

is given by

_ ~ ! x , ~) ~ (~, ~) G(x, ¢ ; X) ffi {p W(~,~)}(%)

=- X(~) ~ (x,~) {p W(~,X) } (%)

where, from the form of ~ and X,

{p w(~,×)}(X) = p(x) (~(x,X)×'(x,k) - ~'(x,~)×(x,l) (x • (-I,I)

= n(~) - m(~) (~ • C~R).

From the general theory of differential equations of the

form (1.6) with two singular end-polnts, it is known, see F8, section 2.18],

that the eigenvalues of the equation, with the particular choice of

and X above, are given by the zeros and poles of n(') - m(-). In this

case, from (3.4),

n(%) - m(X) = - 2m(%)

= 8 I'(% + ½s)F(% - ½s) (3.5)

r(~ + Is)r(¼ - ½s)

and from this a calculation shows (recall s = ~; see (2.2)) that the

eigenvalues are given by

%n = (n + ½)2 (n • No). (3.6)

Anticipating the definition of the operator T below, let Po(T) denote

the set of eigenvalues given by (3.6).

Following the analysis in [8, section 4.53 it may be shown

that the eigenfunetlons {~n(. ) : n ~ N o } corresponding to the eigenvalues

P~(T) are given by

~n(X) = (n + ~)I/2en(x ) (x ~ (-I,I) n • No) (3.7)

where {P (-) ; n ~ N } are the Legendre polynomials, see [2, sections 3.6.2 n o ~ and 10.6] and [9, section 15.1].

Here we leave the classical study of the differential

equation of Legendre and turn to the study of the associated differential

94

operator in this, the right-definite, case.

Let the symmetric differential expression M be defined by

M[f](x) = -((I - x2)f'(x)) ' + ¼f(x) (x ~ (-1,1))

for any f : (-I,I) -~ C with both f and f' E ACloc(-1,1). For any f and g

with these properties, integration by parts shows that

B

I {~ M[f] - fM[g]} = [fg](')l~ (3.8) J

for all compact intervals [~,B] c (-l,l), where

[fg](x) = p(x)(f(x)g'(x) - f'(x)g(x)) (x c (-l,l)). (3.9)

We define a differential operator T, later shown to be

self-adjoint in the Hilbert function space L2(-l,l), as follows:

firstly define the linear manifold A c L2(-I,I) by

f ~ A if (i) f : ( - l , l ) -~ C and f ~ L2(-I,I)

(ii) f and f' c ACloc(-l,l)

(iii) M[f] ~ L2(-I,I) ;

secondly define the domain D(T) c A by f e D(T) if

(iv) f ¢ A and for some I c C\P ~ (T)

(a) lim If(x) ~ (x,l)] = 0 (b) lim If(x) X(x,I)] = O ; x - + 1 x ÷ - I

thirdly define the operator T by

Tf = M[f] (f e D(T)).

Note that the limits in (iv)(a) and (b) exist and are finite in view of

the Green's formula (3.8), and since f, ~ and X all satisfy conditions

(i) and (iii) for A. The genesis of (iv)(a) and (b) as the correct general

form of boundary condition at a singular end-point may be seen in

[8, section 2.7], and receives its full development in Naimark

[4, section 18]. Note also that ~ and X themselves satisfy the boundary

conditions (iv)(a) and (b) respectively, and that once ~ and X are

determined, as above, then D(T) is independent of the choice of I in (iv)(a)

and (b); these results follow essentially from the important result given

in [8, lemma 2.3].

95

Similar analysis also shows that for all f, g E D(T)

(but not for all f, g e A )

lim If g](x) = O lim If g](x) = 0 x÷l x'+-I

and then from Green's formula (3.8) it follows that

(3.|0)

(Tf,g) = (f,Tg) (f,g E D(T)) (3.11)

where (-,.) is the inner-produc~ in L2(-I,|).

If Co(-l,l) represents all infinitely differentiable

functions with compact support in (-],l) then clearly Co(-l,;) c D(T);

hence D(T) is dense in L2(-|,I). From this result and (3.11) it follows

that T is symmetric in L2(-],I).

If also we define ~ : (-1,1) x C\R x L2(-I,]) ÷ C by, see

[8, section 2.6] I' (x,l;f) = G(x,~;l)f(~)d~ (3.|la)

-I

then ~,%;f) e D(T) and

MIni = % ~ + f on (-I,I); (3.11b)

from this result it may be shown that

(T i il)D(T) = L2(-I,;).

Thus T is a self-adjoint (unbounded) differential operator in the space

L2(-l,l).

We now give a number of different but equivalent descriptions

of the elements of the domain D(T), and, in particular, give a number of

alternative forms of the boundary conditions (iv)(a) and (b) which reduce

A to D(T). For an alternative account of these boundary conditions see

[|, appendix II, section 9, example II].

We first show that if f £ D(T) then lim f exist and are both ±l

finite. We have from [8, lem~na 2.9], but with our sign convention, for

f e D(T) and % ~ C\Po(T)

f(x) = ~ (x,l;M[f]) - ~ $ (x,l;f) (x £ (-I,1));

it is essential for this result to hold that f satisfies the boundary

condition (iv)(a) and (b).

Next we prove the general result that if g ~ L2(-1,I) and

E C\Po(T) then lim ~ (.,~;g) both exist and are finite; for ±I

(3.12)

96

fx 11 (n(X) - re(X)) ~ (x,X;g) = ~(x,%) X(~,X)f(~)d~ + X(X,%) @(~,X)f(~)d~ -I x

and

IX (x,X) x*(~'X)f I< "'h-x' • d~.xlg(~)l 2d

= O(ln ( ( l -x ) - l ) . (l-x) 1/2)

= o(1) (x÷l)

on using the results in (2.10 and ll). Also

lim I x ~2 I~IX(~,X)g x-~l ~(x,X) _iX(~,X)g(~)d~ =Y-~-,X) (~)d~

which is finite. There is a similar result if we take the limit at -I.

Thus it now follows from (3.12) that if f £ D(T) then lira

f both exist and are finite; if f(±l) is defined by these limits then ±I

f c C[-I,I] for all f ~ D(T).

Now suppose that f e A and lim f exist and are both finite; ±

then we state that lim + pf' = O. To see this we can suppose, without loss

of generality, that f is real-valued on (-I,I); we have M[f] E L2(O,I),

i.e. M[f] ~ L(O,I) and (pf')' ~ L(O,I); hence for some real k

lim I 1 1 Pf' = (pf')(O) + (pf')' = k; O

now if k ~ 0 then we can take k > 0 and obtain

f(x) ~ f(O) + ~k I~p-I

for x close to I, i.e. lim f = = which is a contradiction; hence k = 0 1

and lim lim 1 pf' = O; similarly -I pf' = O.

lim Now suppose f e ~ and ±I Pf' = O; then

If(x) l ~ If(o) l + If'l = If(o) l + Ipf'l 0

If(o) l + K in ((I- x) -I) (x ([o,I))

for some positive real number K; this last result, and again using lim 1 pf' = O, together with the known properties of ~, proves that

lim [f X (-,l)] = O; lira if ~ (. ,l)] = O for any % ( C\Po(T); similarly -I I

(3.13)

hence f E D(T).

97

If now f E D(T) then, from the results overleaf,

lira pf,~ = O; hence from +

{plf' + ¼Ill 2} = pf,.? + .f (3.14) -X

L 2 |/2f, it follows that pl/2f, e (-I,I). Conversely let f E A and p ¢ L2 _i,I~; t ~

then M[f] e L(o,I) and as in (3.13) lim pf, = k (say); if k ~ 0 then +

for x close to I it follows that, for some k > 0, o

iXp i x. ]f,[2 zk O 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

lim [f~] = lim [fx] = 0 f ~ D(T) if f £ A and either (~) 1 -I

lim or (8) +-I f exist and are finite

lim or (y) _+ pf' = 0

or (8) pl/2f, e L2(-I,I)

lim [fl] = lim [fl] = 0 or (Z) ; -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} = M[f]'f (f c D(T)) -I -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)) (3.15)

with equality if and only if f is constant over (-I,I). This is a special

98

4.

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 the

Legendre polynomials {Pn(') : n • No}; clearly Pn(') • D(T) (n • No)

and, in particular, 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 operators,

see [ 1, chapter VII now yields the completeness of the set of Legendre

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. The normalized

eigenvectors 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.

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

= {pf'g' + ¼fg} (4.1) (f'g)H -I

and norm llfllH; 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 in the right-definite theory. We again use

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 I, neither solution @(',~) or ~(.,l) is in

H2(O,I) for any ~ e C\R. However there exists a unique coefficient m(')

(we use the tilda notation to distinguish the left-definite case) which is

analytic (regular, holomorphic) in C\R and such that the solution

= @ + m ~ ~ H2(o,I). Now for )~ c C\R there is only one linearly

independent solution of the equation (1.6) which lies in H2(o,I); from

the asymptotic results (2.10 and 11), for the solutions Y(',%) and Z(',%),

it is clear that this solution in H2(O,]) must be Y(',%). Thus

~(.,~.) = 0(-,%) + m(%)~(.,%) = k(~)Y(-,%) on [o,])

with k(-) to be determined. If we differentiate this result and evaluate

both sides at O we find that 1 = k(l)Y(o,%) and ~(%) = k(l)Y'(o,l), i.e.

m(l) = Y'(o,I)/Y(o,I). (4.2)

Similarly at the end-point -I the solution in H2(-I,o) 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) (4.4)

X(',%) = Z(.,%)/Z(o,%) (= X(.,%) of section 3).

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)

i 2 but now defined for x ¢ (-|,l), % ~ C\{(n + 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) (x £ (-I,1)) (4.8)

for all f ¢ H2(-l,l). 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) ~(.;f) ~ C[-],|] (f £ H 2)

(ii) lira ~, = +l p (';f)g O (f,g E H 2)

Proof (i) 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.

H 2 (ii) We note that if g E then

Ig(x) l = Ig(°) + I-< Ig(°)l + [Jo op]g,12 i/2

i.e. g(x) = O({In((l - x)-l)} I/2) (x -~ I).

Hence, from (4.6), (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 tha t the ope ra to r A i s bounded on H2; in f a c t

IIAflIH = ll~(';f)][ H -<~llfl[H (f c H2). (4.9)

We have, fo r a l l x e ( o , l )

fXxf (~) I x (~;f)d ~ = M[~(~;f)] ~ (~;f)d -- --X

p~'(-;f)~(-;f) x x I x = + {p(~)l~' (~;f112+¼1~(~;f) 12}d~. -- --X

Now l e t x -~ 1 to g i v e , on us ing the lemma above,

,[~(.;f),[2 = If I f(~)~(~;f)d~ <{I I 'f(~)'2d~I j l'(~;f)12d~I I/2 -I -I I

< ~o

since f £ L2(-I,I) (recall H 2 c L 2) and ~ (.;f) ~ C[-I.I],

i.e. ~ (';f) c H2(f ~ H2). Also

if: 4f: Hence

I IAfl[H < 411fIIN (f c H 2)

and from this (4.9) follows as required.

I n t e g r a t i o n by p a r t s g i v e s f l p~,g I f l

= {p~flg, + ¼~g} = + M[~]g (Af 'g)H - I -1 -1

= I I M[~]g (on using the lemma) -I f'

= f g (f.g ~ H 2) (on using (4.7)) -1 f'

= f M[~(-;g)] = (f,Ag) -1

on r e v e r s i n g the argument; thus A i s symmetric . S ince A i s bounded i t

follows that A is self-adjoint.

102

Suppose now Af = O, i.e. ~(x;f) = 0

then from (4.7)

0 = M[~(x;f)] = f(x) (x E (-1,I)),

i.e. f = 0 in H 2. Thus A -! exists.

by

and

(x ~ (-!,l));

Now define an operator S : D(S) ¢ H 2 ÷ H 2

D(S) = {Af : f e H 2}

-! S f = A f (f c D(S)) (4.10)

A standard theorem in Hilbert space theory, see [1, section 4l, corollary

to theorem 1], implies that S is self-adjoint (bounded or unbounded) in

H 2. In fact S must be unbounded since, from the properties of ~, it follows

that if f e D(S) then f' e ACloc(-l,] ) and so D(S) is strictly contained

in H 2, even though the closure D(S) = H2; it may be seen that if S is

bounded then D(S) = H 2 and this gives a contradiction.

We conclude that S is an unbounded self-adjoint operator

in H2(-1,1); we now show that S has a simple discrete spectrum with

O(S) = P~(S) = {(n + ½)2 : n ¢ No}; we note that this spectrum is identical

with Po(T) of the operator T introduced in (3.6).

Suppose that % is an eigen~lue of S, i.e. for some

eigenvector f ~ O, Sf = Xf; then X ~ O, since S -! = A exists, and

Af = A-!f; hence ~(.;f) = X-!f or, on using (4.7), f = M[~(-;f)] = ~-|M[f].

Thus f is a non-trival solution of the Legendre equation with the property

that f ~ H2(-I,I); from the properties of the Legendre equation given in

section 2 this can happen only when the solutions Y and Z are linearly

dependent, i.e. when % c Po(S) and then the eigenvector f is linearly

dependent on the corresponding Legendre polynomial from the set

{Pn(') : n e No}. Conversely every point in the set Po(S) is an eigenvalue

with corresponding eigenvector in {Pn : n £ N o }.

For all real numbers D @ Po(S) we can show that the range of

S-ZI is H2(-I,I), i.e. {(S - B I)f : f ¢ D(S)} = H2(-l,l); this follows

from the properties of the resolvent function ~ as defined in (4.5).

Hence all these points are in the resolvent set of S, and so O(S) = Po(S).

The spectral theorem for self-adjoint operators in Hilbert

space, see [I, chapter VII, now implies that the Legendre polynomials form

5.

103

a complete, orthogonal set in H2(-],I). The derived complete orthonormal

set in this space is then, say, {~n : n ¢ No} where ~n = (n + ½)-;/2P n

(n E No).

From this last result it may be shown that the Legendre

polynomials are dense in those vectors of L2(-I,;) which are also in

H2(II,I); however this set is d~nse itself in L2(-I,1) and so we obtain,

indirectly, yet another proof~he completeness of the Legendre polynomials

in L2(-I, I).

Remarks on the operators T and S. It is of some interest to compare the

operators T and S as defined in sections 3 and 4 respectively.

T is an unbounded self-adjoint, differential operator in

L2(-I,I) with a simple, discrete spectrum {(n + ~)2 : n E N o } and

corresponding eigenvectors {Pn : n e No}.

S is an unbounded self-adjoint operator in H2(-I,I) with

the same simple, discrete spectrum and eigenvectors; we may hesitate to

call S a differential operator for the reasons given below.

The operator T, and its domain D(T), is defined directly

in terms of the Legendre differential expression M; also we are able to

give alternative and simplified descriptions of the elements of D(T), as

detailed in section 3.

The situation for the operator S is different; we defined S

as the inverse A -I of a bounded, symmetric operator in H2(-I,I). Whilst

we can say something about the elements of the domain D(S), in particular

as sets of complex-valued functions on [-1,1] we have D(S) c D(T)

(see the alternative definition T below), it does not seem possible to

characterize the operator S directly in terms of the Legendre differential

expression M. The definition of S in section 4, i.e. S = A -I, depends

upon a general theorem in Hilbert space theory (see the result quoted

in section 4 from [I]) which provides for the existence of S but, due to

the generality of the theorem, cannot give a constructive definition in

general. Thus S appears as a differential operator only in an indirect

sense in comparison with the operator T.

It is of interest to note that we could have defined the

operator T in the same way as S is defined in section 4. With the resolvent

function # defined by (3.|]a) let the operator B be defined on L2(-I,|) by

(Bf)(x) = ¢(x,o;f) (x ~ (-l,l))

104

6.

L 2 for all f E (-I,I); compare with (4.8). With arguments entirely similar

to those used in section 4 we may prove that B is a symmetric, bounded

operator on L2(-I,I) into L2(-I,;), and that the inverse B -I exists;

also that the operator T, as already defined in section 3, satisfies -1

T=B

Note that in the sense H2(-I,I) c L2(-I,I), we have

A = B on H2(-;,]). However, the inverse A -I has to be determined in

H2(-I,I) and B-; in the space L2(-I,I) so that there is no identification

of S with T, even on D(S).

It happens then that in the right-definite case, we are -I

able to give an explicit characterization of the inverse B = T, a

characterization which is not possible in the left-definite case.

These remarks should also be read in the context of the

general theory developed by Pleijel for both the right- and left- definite

cases; see, in particular, the illuminating results given in the concluding

remarks in [6, section 8].

H alf-ranse Lesendre series. In the theory of Fourier series the two

collections of functions {sin n x : x ~ [o,~], n E N o } and

{cos n x : x E [o,~], n e N o } give separate orthogonal sets in L2(o,~),

both of which are complete in this space; these are termed half-range

Fourier series. The two collections conbined and extended to the interval

[-~,~] give an orthogonal set which is complete in L2(-~,~).

The same phenomena is seen in Legendre series. The two

collections of polynomials {P2n+l (') : n e N o } and {P2n (.) : n e N o }

give two separate, complete orthogonal sets in L2(O,I).

The associated self-adjoint, differential operators are

given respectively, say, by T o and T 1 where

D(T o) = {f:[o,|) ÷ C : f and fl E ACloc[O,l)

f and M[f] E L2(O,I)

f(o) = 0 and lim [f~] = O} I

and Tof = M[f] (f e D(To))

(in the notation of section 3); and D(TI) and T; are given by replacing

the boundary condition at the regular end-point O by f4(o) = O. It may

105

be seen that the spectrum (simple and discrete) of T is given by o

{(2n + ½)2 : n E No}; similarly for T! but with spectrum

3 2 {(2 n + ~-) : n £ No}.

There are similar half-range Legendre expansions in the

left-definite case.

106

References

I. N.I. Akhiezer and I.M. Glazman. Theory of linear operators in

Hilbert space: I and II (Ungar, New York, 1961; translated from

the Russian edition).

2. A. Erd~lyi et al. Higher transcendental functions: I and II

(McGraw-Hill, New York, 1953).

3. W.N. Everitt. Some remarks on a differential expression with an

indefinite weight function. S~ectral theory and asymptotics of

differential equations. 13-28. Mathematical Studies 13,

E.M. de Jager (ed), (North-Holland, Amsterdam, 1974).

4. M.A. Naimark. Linear differential operators: II (Ungar, New York, 1968;

translated from the Russian).

5. A. Pleijel. On Legendre's polynomials New developments in differential

equations 175-180. Mathematical Studies 21, W. Eckha~s (ed),

(North-Holland, Amsterdam, 1976).

6. ~. Pleijel. On the boundary condition for the Legendre polynomials.

Annales Academiae Scientiarum Fennicae; Series A.I. Mathematica 2,

1976, 397-408.

7. E.C. Titchmarsh. On expensi~s in eigenfunctions II. Quart. J. Math.

(Oxford) ii, 1940, 129-140.

8. E.C. Titchmarsh. Eigenfunction expansions associated with second-order

differential equations; I (Clarendon Press, Oxford, 1962).

9. E.T. Whittaker and G.N. Watson. A course of modern analysis (University

Press, Cambridge, 1927).

I0. F.V. Atkinson, W.N. Everitt and K.S. Ong. On the m-coefficient of

Weyl for a differential equation with an indefinite weight function.

Proc. London Math. Soc. (3) 29 (1974), 368-384.

SINGULARITIES OF 3-DIMENSIONAL POTENTIAL FUNCTIONS

AT THE VERTICES .AND AT THE EDGES OF THE BOUNDARY

Gaetano Fichera

Icl raw, tory of Arthur Erd@lyi

In this paper we shall extend to more general surfaces the results

which have been proved in [ i ] in the particular case of a cube.

Let E be the boundary of a bounded domain A of the cartesian

space X ~ of the point P .:: (×,v,z). Let E be the set of the points ex-

terior to A . We consider ~ as a conducting surface and suppose r_

smooth enough so that the potential function u of E does exist. The

potential function u is continuous in E : R u E and satisfies the con-

ditions

~ z ~ :- + + = 0 in K x a ~ y ~ ~ z ~

= t o n £ , l i r a ~ ( P ) : O .

L e t 0 b e t h e o r i g i n o f X 3 a n d 5 t h e u n i t s p h e r e I P - 0 I = t . L e t

5* be a domain of 5 which does not coincide with 5 If [k is a posi-

tive number, we shall denote by H R the cone

H R - I P ; o I P - O I ~ R , P - O ~ S f o r P~O }, I P - O I

I f w e d e n o t e b y ~ R t h e u n i t b a l i I P - O f ~- }% w e s u p p o s e t h a t

n B~ = Ft R f o r 0 < R ~ - R o •

We shall assume that 9 5 ~ is a connected set formed by a finite

collection of arcs of great circles, two of them meeting, eventually,

only in one end point. Let oJ I , .... '~I be the vertices of 9S ~ and let

~h (h = ~,...,q ) be the size of the angle of ~S" in ~h , measured

on the side of 5 ~ We denote by I h the segment of X ~ defined bythe

c o n d i t i o n s o - ~ I P - O I -~ R o ~ P - O = t P - O t ( ~ h - O ) .

Let L~ be the Laplace-Beltrami operator on 5 In the domain 5"

of S we consider the eigenvalue problem

L ~ f ) , ~ : 0 i n S ( 1 )

~r ~ 0 o n 9 5 ~

108

The following results can be assumed to be known:

i) The eigenvalues of the problem [considered in the space ~,(5 ~)

of functions which have a gradient belonging to La(S ~) ] are real posi-

tive and form an ascending sequence which tends to +

2) For every eigenvalue the algebraic multiplicity is equal tothe

geometrical multiplicity and this multiplicity is finite.

3) The eigenfunctions of the problem (i) are continuous in 5~

4) The smallest eigenvalue has multiplicity 1 and the correspond-

ing eigenfunction is always positive (or always negative) in 5"

Let ~4 H R be the part of the boundary ~ H R of H~ defined by

the conditions

P-0 , - ~ OS for P ~ o -

I P - O l

Let ~ be the unit normal in a regular point ~ of ~ H R pointing

towards E ° We shall consider the density of the electric charge in

field

edges of ~ 1 4 R-

THEOREM I.

(2)

where

~ C Q ) : - _ L ~___! ~ .

The following theorem describes the behaviour of the electric

grad u and of the electric density ~ near the vertex and the

For P ( H ~ - (i~ u ..... ~ i~)

I g r a d ,,,~ ~P h, t

, ~: IP -O I , o~ = - -

2 a< :

(o < R < R~ ) we have

P - O J

I P - O I

r _ ,

: Ira- ~0 k) if ~h > ~

~ :h (~ ) : I ~ - ~h i l o g S i f k~h : ~ ,

: Im - u~ h I if ~h 4

(~) The symbol ~ is the Landau capital 0

109

The asymptotic formula (2) cannot be improved in the sense that

the symbol G9 in it cannot be replaced by o (z) if x tends to an ar-

bitrarily fixed point of I u ..- Iq in a suitable neighborhood of 0 .

If Q is a point of the plane 0 m h ~I,+~ ( ~ = ~" ~ ; ~9+~ = ~ )

and ~ ~ ~ FI~ ( I h u [h+,) ( I~ = i ~ ) we have

( 3 ) 3(0) = 0 [ ~ : ~ : ~ ( ~ ; ~ : , , ~ , ~ ' ~ ) ] '-~,~ ~ ~ : , ) '

of

(4)

where

(5)

in the right hand side of (5), which are needed in order to get

are permitted because of lemma I of [ ~ ] The same lemma permits

apply the Green formula

Hence

PROOF. For 0 < ~ ~ R o , a) ¢ 5" we have

The convergence of the series must be understood in the sense

L' (5') We have

' ~ ' U 2 ~ U ~ h U o ~ 2 U - : - - + + - ~ .

and integrating over 5 ~ we have Multiplying both sides by ~r k

Zrk L,(oUdo)

"Uk(~ ) = f U ( t c , a') V (CO)dCO . g~, k

The differentiations with respect to ~ under the integral sign

(4) ,

to

_ _ Z dU K _ ~ U. o a = U. ~- __ _ _ =

UK( ~): c k? + %( :

and therefore

whe re c and K

c are constan~ and -k

2 k

(~) The symbol ~ is the Landau small o

110

S i n c e U ( q ~ )

a n d f o r m a l ] y

is continuous as ~ -) 0 , we must have

(6) g r a d U(qco) : i c g r a d ~ zrk(co ) k:~ k

: Since ~or every C~ ~notion ~ we have I grad #I ~ ~, {, I gra%~

we get

~ ~ o(~. I IZ 20( -i 2 (7 ) I g r a d ? u~(~)l : I% ~ ~ + ~ ~ I grad ~ I

M a p p i n g t h e d o m a i n 5~ o n t o a b o u n d e d p l a n a r d o m a i n by a s t e r e o - (~)

graphic projection and applying lemma II of [I ] , we have

(8 ) l ~ k ( ~ ) l ~_ c 1~ ,

q

(9) grad "G K I z C s u p I kk~ ~ I hilt ~t (co) ~ (2 ~2 IS ~h(c°)

where C is a constant only depending on ~.

Set r,

~h(~ ) : l~-cohl log

2

I~ - coal

if Ph ( ~ "

From (8), (9) we deduce q

(io) Ivk(~)l -~ F ~ ~ ~h (~) k

where ~ is a constant only depending on ~

Let ~ be a constant such that for co ~ 5 ~

q q

h:~ h=¢

£

If we set 6 = ( rues ~)2 then

(12)

i

(f ) .............. ~ tU(R ~)i~d~ _z R~ ~

(3) See,in particular, (1.7),(1.8),(1.9),(i. I0),(I.II) of [I].

111

From (7), (9), (i0), (ii), (12) we deduce

z -- C~ z

It is known from the asymptotic theory of eigenvalues (see [2] -± ±

p.442) that lim k ~ ~k ~ ~0. It follows that lim k ~ -- q~ ~ 0 .

Therefore, there exist two positive constants po and p~ such that for

a n y k ~ I 3 P° ~.

We have for 0 4 ~ ~ R

Jim k i k s { -~q )P°J~ : lira exp ~-~ ( { logk + Po l o g i ) = O.

J~ Ro

which proves the total convergence of the series

in the interval [0, R ]. Hence the development of

is justified.

We have in H R - (i u...u |~ ) , assuming ~ ~ ~ ,

I grad ~ I 4_ 6 ~ II rh(~) E ~z~ (d ¢~'+C 4 - - h.:t k:~

grad~ given by (6)

If the constant M is such that for 0 ~ ~ ~ ~R ~

we have in H a- (I~u ,,. u i~ )

~.~ --

I grad ut I < M 6 ~ 11 ~ (~°) - h:~ h '

i.e. the estimate (2).

(13)

~_ M ,

Suppose that there exists a point P?E i I u-. • u ]q such that

[grad -]p :o[~"~.-li ~(~)] (forP-'P~) , I~ ~-01 < ~

We can exclude the case P~: 0 . In fact in the development (6)

we have c~ < 0 since ~ >o and U< 0 in ~ ~ . On the other hand

the estimate (13) when P~O implies ¢~ = 0 .

Hence ~j~ O. If (13) holds when P-, P~ , then there

exists a ~ > 0 such that

112

a-~ q (lap l[grad~I~l ,_ 8~ ]7 r (co)

h : { h

for _P~ H R - ( I ~ u . . . ~ I ~ ) , I P - @ l < 8~, We may s u p p o s e o < c~£ _~ ~.. F o r s u c h _P we have

o o

Igrad L~I >. i c , ] lg rad~ ~ ( o o ) l - ~ ICi~l I g r a d ~ ( a ) ) I

± e( -~ 2. 20( - .2 ,2 ]

(14) ~ [c,l la~ %(c~)I + ~ l g r a d ~Y (co [ - -

_ o__~ ?°,=h (~),,r-:z : x , , ( ~ g r * ~ ) < ~. "

Let M~ be a positive constant such that for 0 ~ ~ e

(~5) ~ ~ ~ ~" + )'

Hence for (13)',(14),(15)

]] ~(~) _> Ic, l~ ]grad oa~r~(a))[- M t 6 (~ g "~i (co) I,.:~ h h.:~

[ l C , ] g r a d ~ ( c o ) I - V1 ~(2a) 11 =~,(~)]. h:l

The re f ore q

] grad I.tr4(co)l z Ic,I .

Since P~ ~ 0 and ~(a))>O in S ~, the last estimate contradicts

lemma III of [if.

If Q = ~w is a point of the plane 0 cohogh~ ~ , then for ~e~H~

we have

~ ( ~ ) . . . ~h.,(~)~t,~(~) -.. V9 (~) "- L

where L is a positive constant. Since Ic~(G)I -~ I[ grad"]QIwe obtain

the estimate (3).

THEOREM II. A necessary and sufficient condition for a~ C*(HR}

( 0 < P, 4 R o) is that HR ° should be convex.

PROOF. From the convexity of HRo we deduce that ~h < ~ (h={.

..,~) and that ~ is contained in a hemisphere. Hence 11 is greater

than the smallest eigenvalue of the problem (i) for the hemisphere ,

which can be easily computed and equals 2. From the inequality ~i > 2

113

we deduce ~ > ~ . Hence from the proof of Theorem I we see that grad

is continuous in FI R and vanishes on Hg ~ i h ( h ~ ~ , " " ' ' @ )

The necessity follows from the fact that the estimate (2) cannot

be improved.

From now on we shall assume that for at least one value of h we

have Mh > ~ ( HRo is not convex).

THEOREM III. Let p be the smallest of the numbers

3 2 1"1 2 IX~ ) J , • . j

t h a t a r e p o s i t i v e . Then I g r a d u l e L P ( H ~ ) ( 0 ~ 1~ ~ Ro ) f o r any p

s u c h t h a t 4 a p ~ ~ .

PROOF. We have f rom (2) t h a t } g r a d ~ l c: L P(Ft R) i f t h e f o l l o w i n g

integral is finite:

FI h~ R

On the other hand this integral is finite if and only if the

following integrals are finite:

o

where the second integral must be considered for any h such that

~h > ~T . From this remark the proof follows.

THEOREM IV. Let p be the smallest of the numbers

that are positive. Then the electric density ~(Q) belongs to LP(g4H~)

( 0 < R ~ Ro ) for any p such that ~ 4 p ~ p .

PROOF. Denote by ~h the planar sector which is the intersection

of H R with the plane O~0hC0h, .We have ~(Q)E LP(~ H R) if the follow-

ing integrals are finite:

fZh~ (a'~) P (16) l't::h(r~)Z'h~ ( c u ) J P d Y'~

for any h such that at least one of the following inequalities holds:

h > ~ ' ~h#~ > ~ "

Assume in the plane 0 ~0 h ~oh+ ~ a polar coordinate system with )

pole 0 and polar axis 0 co h . The integrals (16) are finite if and

only if for all the h such that ~h > ~ and for 0 ~ ~ 4 2'E we ha~e

114

0 0

From this the proof follows.

Theorem IV improves a result~obtainable from a theory developed

in [3] , which states that ~ e Lz(94 H R )

The theory developed in this paper reduces the description of the

behaviour of the electric field and of the electric density near 0 and

i, , .. , iq to the computation of the constant ~ ,i.e. the lowest

eigenvalue ~ of the problem (i) o In the case that A is the cubic

domain: o< x < ~ , O < y ~ i , o < z ~ i , the following rigorous

numerical results have been obtained for ~ ( [4], [ i] ) :

0.62153 < ~ ~ 0 . 6 8 0 2 5

which lead to the fo l lowing bounds f o r ~ :

0 . 4 3 3 5 < ~ < 0 . 4 6 4 6 .

T h e s e b o u n d s t o g e t h e r w i t h t h e o r e m s I I I a n d IV l e a d t o t h e f o l l o w -

i n g function theoretic results for the electric field and the electric

density of a cube 6000

where £ is an arbitrary number such that D ~ a < I.

R e f e r e n c e s

[ 1 ] G. FICHERA, Asymptotic behaviour of the electric field and density

of the electric charge in the neighborhood of singular points of a

conducting surface, Uspekhi Mat. Nauk,30:3,1975,pp. lO5-124; English

translation: Russian Math. Surveys,30:3,1975,pp.107-127; Ital.trans-

lation:Rend, del Seminario mat.dell'Univ, e del Politec. di Torino,

32,1973-74,pp. II~-143.

[ 2 ] R. COURANT-D. HILBERT, Methods of Mathematical Physics,vol. I; Inter-

science Publ. New York, 1953.

[ 3 ] M.A.SNEIDER, Sulla capacit& elettrostatica di una superficie chiu-

s_aa, Mem. Acc. Naz. Lincei,X,3,1970,pp.99-215.

[ 4 ] G. FICHERA-M.A. SNEIDER, Distribution de la charge ~lectrique dans

le voisinage des sommets et des ar~tes d'un cube, C.R.Acad. Sci.

Paris,278 A, 1974,pp.1303-1306.

SINGULAR PERTURBATIONS OF ELLIPTIC BOUNDARY VALUE PROBLEMS

P. Habets

. Introduction

A major contribution to singular perturbations of linear elliptic

boundary value problems (BVP) is due to W. Eckhaus and E.M. de Jager

[5]. An extension to nonlinear equations has been worked out by

A. van Harten [11][12] (see these references for a complete

bibliography). The present paper extends these results using monotone

methods and differential inequalities,

As a first step, we consider the nonlinear elliptic BVP in a

domain ~ C ~2

eL2u + Liu = f(x,u) in (1.1)

u = g(x) on ~

where E > O, u E ~, x E ~2, Li is a linear i th order differential

operator; f(x,u) and g(x) are given functions and f is increasing

with respect to u, Using monotone iteration [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 = g(x) on F

with F G 3~. Convergence is of order E except in an arbitrary small

neighbourhood of 3~ \ F, where exponential boundary layer can appear,

and in an arbitrary 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 (1.2). However, in such cases,

we give no informations on the behaviour of the solutions of (I.I) in

a neighbourhood of these free boundary layers. Our work is very much

in the spirit of A. van Harten [II] chapter 6. In [II] a solution of

(I.I) is supposed 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 and the limiting behaviour using monotone iteration.

As a by-product this also gives an explicit scheme to cumpute the solution.

Further our method is not restricted 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 of a known

theorem on elliptic BVP (see H. Amann ['l] where other BVP are considered).

In a second part, we allow nonlinearities with gradient dependance.

More precisely, we consider the BVP

EL2u = f(t,U,Ux,EUy) in

u = g on ~

Using a theorem of M. Nagumo [9] based on differential inequalities we

extend the existence and convergence results of the first part. More general

BVP could be investigated using H. Amann [2]. The main drawback of this

approach is that it is restricked 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 as describing a beam with pin-ends. This extends

to high order equations 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 • ~ ul(x) ~ u2(x)

is an ordered Banach space []]

If ~ e C(~), B • C(~), ~ ~ B, let us define [~,8] = {x ~ C(~) ~< x< $~.

An operator T : [~ B] ~ C(~) is said increasing if u| < u 2 implies

T(u l ) ~ T(u 2)

1t7

2.2. PROPOSITION [I] Suppose that : I : [a,B] ~ C(~) is an inereas~ng map

which is co.act and such that

a ~Ta

then the iteration schemes

TB~<6 ,

and

u_~ = T~_ 1 Uk = TUk-I

converge to fixed points ~ and u of T such that

<u_k~ ~ = T(~) < ~ = T(~) < u k ~ B.

3. An elliptic boundary value Problem

3. I. Let ~ C fl~2 be a bounded domain whose boundary 3~ is a I- dimensional

C 2+v manifold for some v @ (o,I). Consider the differential operators

L 2 = - all(X,y)DxDx - 2a 12(X,y)DxDy - a22(x'Y) DyDy

L 1 = - p(x,y)D x + q(x,y)

where the following assumptions are satisfied :

(H-i) aij , p, q E cV(~);

(H-ii) for some ~ > 0 and all (x,y) E ~, ~ E ~2,

Z 2 lSjaij(x,Y)~i~j ~ Kol~ i ;

(H-iii) V(x,y) • ~, q(x,y) i> 0.

Consider the linear BVP

~L2u + LlU = f in (3.1)

u = g on 3~

where E > 0. It is well known [l] that under assumptions (H-i) to

(H-iii), one can define the solution operators

K : cV(~) ~ c2+V(~), f ~ Kf

R : ~+v~)~ c2+V(~), g ~ Rg

where Kf denotes the solution of the BVP (3.1) with g m 0 and Rg

the solution of (3.1) with f m O. The solution of (3.1) reads then

u = Kf + Rg.

118

3.2. Let f : ~ x ~ ~ ~ and g : ~ ~ ~ be such that

(H-iv) f E cV(~x~), fu E C(Hx~), g E c2+V($~);

(H-v) V ( x , y ) e ~, u t > u 2 implies

f(x,y,u I) > f(x,y,u2) .

In the following we consider the non]inear BVP

gL2u + Llu = f(x,y,u) in H ( 3 . 2 )

u = g ( x , y ) on ~

which can be s o l v e d u s i n g the f o l l o w i n g Lemma ( s e e H. Amann [ 1 ] ) .

3.3. LEMMA If assumptions (H-i) to (H-v) are satisfied, the BVP (3.2) is

equivalent to the fixed point equation

u = Kf(.,u) + Rg = lu ,

in C(~l) and the map.

I : c ( ~ ) - ~ c ° ( ~ ) , o ~ [ 0 , 2 [ ,

is increasing and completely continuous.

3.4. We are now ready to prove the main tool of this section.

THEOREM Suppose there exist functions ~i C C2(~), i = 1,2,..,k and

Bj E C2(~), j = 1,2,.,,1 such that for any i and j, ai ~ Bj,

£L2a i + LI~ i ~ f(.,a i) , £L26 j + LIB j ~ f(.,~j) in H, (3.3)

l j

Then if assumptions (H-i) to (H-v) are satisfied the iteration

schemes

u O = a = max a. (resp = B = min B ) i i j j

Uk+ I = Tu k = Kf(.,u k) + Rg

converge in C2(~) to fixed points ~ (resp. u) of I i.e. to solutions of

(3.2) such that

119

Proo~ Let u = Kf(.,~ i) + Rg. Then

(sL 2 + LI)(~ i - u) = eL2~ i + LI~ i - f(.,~i ) < O, in ~,

~. - u = ~. - g ~ O, on ~, I i

and from the maximum principle [lO]

a. - T~. = ~. - u ~ O i i l

Further ~ = max ~. > ~.. Hence from Lemma 3.3 l i

Ta > Ta. ~ a . l i

This proves T~ ~ = max a i. Similarly one proves TB ~ @ = min @j.

From Lemma 3.3 and Proposition 2.2, everything follows, except that

we only know u k ~ u in C(~). To prove the convergence in C2(~), notice that

from Schauder's estimate

I~Ic2+~ < K(i f(.,Uk)l C N + llukll)

the sequence [u k] is bounded in C 2+~ (~) and by Arzela-Ascoli's theorem

there exists a subsequence [u.,] converging in C2(~). At last any 2 K

converging subsequence ukC~ v converges in C(~) i.e. v = u. This implies C 2

u k ~ u.

3.5. Consider the reduced problem

L|u = f(x,y,u) in (3.4)

u = g(x,y) on F

with F = {(x,y) C $~ I 3 h > O : [x - h, x [× {y} C ~ }.

Direct integration of simple examples shows right away that the solution

u O of (3.4) has in general singularities at points on horizontal lines

tangent to the boundary of ~. Let D C ~ he such that u O ~ C2(~ \ D)

E

fig. ]. example of set D = {A,D,E} U

120

Choose ~ > 0 as small as we wish and define ~o @ C2(~) such that

u°(x,y) = u°(x,y) if (x,y) E ~ \B(D,~).

3.6. Consider next a set

= l(x,y) : a < y < b , ~ (y) < x < ~÷(y) }

such that ~o n ~ = {(x,y) I a < y ~ b, x = ~+(y)

@_, ~+ e C2([a,b]).

C ~ \ B(D,6)

and

(3.s)

&

fig. 2. example of a set ~o C

Define O E C~(~) such that

p(y) = ] if y ~ [a,b],

p(y) = 0 if y e [a + 6, b - 6],

p(y) @ [0,1]

and y E C2(~) such that

y(y) = ~_(y) if y @ [a,b] .

&

Consider the function

= ~o - Ae-%](x - Y(Y)) - cB(e%2(d - x) _ 1) - Cp(y)(d - x) (3.6)

where A,B,C,%] and 4 2 are positive constants and d > sup {x I (x,y) @ ~} + 1.

121

and C such that

Let us choose A large enough so that y E [a,b], (x,y) E ~,

x # ~+(y) implies

~(x,y) < uO(x,y) - A ~ g(x,y)

y ~ [a,b] , (x,y) @ ~ implies

a(x,y) < ~°(x,y) - C < g.

For such a choice of A and C

Let I be an interval large enough such that for any (x,y) E

~0 (x,y) ± A* ± l ± C*(d - c) @ l

with A* = sup {I: ° - g I : (x,y) E ~ , y E [a,b]} ,

C* = sup {I: O- g I / (d - x) , (x,y) E ~ , y ~ [a,b]} and

c = inf {xl (x,y)E~ }. In order to prove cL2~ + L1a < f(.,a) we shall

assume :

(H-vi) for some ~ > 0 and all (x,y) E ~ , u E I

p(x,y) k ~ , q(x,y) - fu(X,y,u) k 0

One computes

am = u~ + %iAe -%l(x - Y) + %2EBe %2(d - x) + CO

ay = ~ - hlY'Ae -%l(x - Y) - C0'(d - x)

~xx = USx %~ Ae-%l(x - Y)- h~gBeX2(d - x)

~xy = U~y + %~Y' Ae-Xl(x - Y) + CO'

~yy = U~y - (%lY" + %~X'2)Ae -%l(x - X) - CO"(H - x)

e2a < e2u° + K[(%~ + hl)Ae -%l(x - Y) + %~gBe %2(d - x) + C]

with K larger than :

sup (all - 2a12 Y' + a22 ~'2) , sup (a22X") , sup all ,

sup (-2a120' + a220"(d - x)).

Next, one has

gL2~ + LI~ - f(x,y,~) < gL2:°

+ £K[(%~ + %l)Ae -%l(x - Y) + ~2 g~e~2 ~ %2(d - x) + C]

_ P [:~ + %lAe-Xl(X - y) + %2~BeX2(d - x) + CO]

- f(x,y,:0) + q:O

_ [q _ fu(X,y,e)][Ae-Xl(X - X) + CB(e%2(d - x) _ I) + Co(d - x)]

122

with 0 E [a(x,y) , uO(x,y)]. Using assumptions (H-vi) one gets

~L2~ + LI~ - f(x,y,~) < EL2u° + L1uO - f(x,y,u °)

+ gK [(%~ + %l)Ae -%l(x - Y) + %~gBe ~2(d ~ x)

- ~ [%iAe-%l(x - y) + %2EBe%2(d - x) + CO ]

+ gB(e%2(d - x) ~ I).

+ c]

Let us choose %1 and %2 such that

P - ~K)% I = 0

EK%2 - P%2 + ! = 0

= ! + O(£). It follows i.e. %1 = ~ - 1 and %2 ]~

EL2cz + LI~ - f(x,y,~) ~< gL2u° + LlUO - f(x,y,u O)

v_~i. -~I (x - y) + EKC - 2 Ae - ~Cp - gB < 0

since if y ~ ]a,b[

gL2e + LI(~ - f(x,y,~) ~< gL2u° + L1u° - f(x,y,u O) - (~ - gK)C < O

for g <~ ~/2K and C large enough ,

if (x,y) e ~o

~L2~ + L1a - f(x,y,~) ~< sL2u° + gKC - CB < O

for B large enough, and if y @ [a,h] , x ~< ~_(y)

sL2(~ + LI~ - f(x,y,~) _< gL2u° + L1u° - f(x,y,u O) + EKC - ~2 A < O

for A large enough.

In a similar way, one proves that for appropriate choice of the

constants, the function

B = ~o + Ae-%l( x - ~) + ~B(e-%2( d - e) - I) + Cp(y)(d - x) (3.7)

is such that

gL2B + LI~ - f(x,y,B) ~ O in ~ ,

B ~ g on ~

123

3.7. Theorem 3.4 and the construction 3.6 prove the following theorem,

THEOREM Let ~>o be fixed and f~o = f~l U "'" U~k be such that each

~. is defined as in (3.5) i.e. i

~i = {(x,y) : a i < y < b i , ~_i(y) < x < ~)+i(y)} C ~ \B(D,~),i= l,..,k,

with ~-i ' ~+i e C2([ai,bi])

Assume assumptions (H-i) to (Hvi) are satisfied.

Then the BVP (3.2) has a solution u such that for ~ small

enough and any i

where

u(x,y) = u°(x,y) + O(e -%I(x-@-i (y)) + e) in ~. 1

o < %1 = 0(½) Further u can be computed using the iteration scheme

u O = max ai (res~ min ~i ) , u k = Kf(.,Uk_ l) + Rg

with ei defined by (3.6) (resp. B i defined by(3.7~.

(3.8)

3.8. Notice that assumptions (H-iv) and (H-vi) can be weakened by considering

only points (x,y) E ~ and u e [a(x,y),$(x,y)]

4. Nonlinearities with gradient dependence

4.1. If one gives up the computing procedure (3.8), one can consider a more

general BVP.

Let ~ C ~2 be a bounded domain such that for each p E ~,

there exists a cone K with vertex p and a neighbourhood 0 of p such

that (K ~ O) lies outside of ~.

Consider next the differential operator

L 2 = - al](x,y)DxD x - 2a|2(x,y)DxDy - a22(x,y)DyDy

together with the following assumptions :

(A-i) for some C and all (x,y),(x',y') e ~ , i,j = 1,2,

l aij(x,Y) - aij(x',y')l ~ C H(x,y) - (x',y')ll;

(A-ii) for some K 0 >°0 and all (x,y) e ~ , ~ e ~2

i~jaij(x,Y)~i~j ~ KO~ ~

124

At last, let f : ~ x ~3 ~ and g : 3q ~ ~ be such that

(A-iii) f E C~(~ x R 3) , g • C(~) for some ~ e ]O,I[

(A-iv) there exists a function c : ~+ ~ ~+ = [0,~[ such that

If(x,y,u,v,w)I ~ c(p)(l + v 2 + w 2)

for every p > O , (x,y) E ~ , u • [-p;+p] , (v,w) • ~2

4.2. Consider the BVP

eL2u = f(x,y,U,Ux,gUy) in (4.1)

u = g(x,y) on ~

with c > O. The following result is then a consequence of theorem 6 in

M. Nagumo [9].

THEOREM Suppose there exist functions ~i @ C2(~) ' i = 1,2,,..

k and ~j • C2(~) , j = 1,2,...1 such that for any i and j, a i ~i ~j

En2a i ~ f(.,~i,aix,g~iy) , ~L2B j ~ f(.,Bj,Bjx,SBjy) in

a. ~ g B. ~ g on ~ l 3

Then if assumptions (A-i) to (A-iv) are satisfied there exists a

solution u of (4.1) such that

max a . ~ u < min B- i i j J

4.3. As in 3.5 we assume there exists a set D c ~ such that the reduced problem

f(x,y,U,Ux,O) = o in

u = g(x,y) on r

where F = {(x,y) E~43 h > o : Ix - h,x[×{y} C ~}, has a solution

u°(x,y) • C2(~ \ D). We choose ~ > o as small as we wish and define

~o • C2(~) such that

u°(x,y) = u°(x,y) if (x,y) • ~ \ B(D,~)

Using the notations and assumptions of 3.6 we consider a set ~o and

a function

a = ~o _ Ae-~1(x - y(y)) _ ~B(e%2(d - x) - I) - Cp(y)(d - x)

125

Choosing A and C large enough

~<g on 3fl .

and (~,~x,e~y). Using assumption (A-v) and if %1 > O and

chosen such that

Let I be an interval defined as in 3.6 and let us replace

assumption H-vi by

(A-v) for some ~ > o , Q > o and all (x,y) • ~ , u • I , (v,w) ~ ~2

- Q ~ fu < 0 , fv ~ ~ ' ~fwI ~ Q ,

One computes then

EL ~ - f(x,y,~,~x,~y) < ~L2u°

+ gK[(%~ + %1)Ae -kl(x - Y) + A2g~e-2 ~ %2(d - x) + C]

- f(x,y,~O,~Ox,O )

+ fu(X,Y,Sl,e2,@3)[Ae-%l(X - y) + cB(e%2(d - x) - I) + Cp(d - x)]

_ fv(X,Y,el,e2,83)[%iAe-%l(X - y) + %2sBe%2(d - x) + CO]

+ Sfw(X,Y,el,62,e3)[%iY'Ae -%l(x - Y) + Cp'(d - x) - ~Oy]

with (el,e2,e3) a point on the line segment joining the points (~o, ~o,o)

%2 > O are

one has

gK% 1 - (~ - gK - ~Q sup :y'i) = 0

EK%~ - ~%2 + l = 0

gL2a - f(x,y,a,ax,Cay ) < EL2~° - f(x,y,~O,~Ox,O)

- ~%IAe -%|(x - Y) - EB - pO___CC2

At last, using the type of argument used in 3.6, one proves

EL2~ - f(x,y,~,~x,E~y) <0 0

Similarly, for an appropriate choice of the constants it can be

shown that the function

6 = ~O + Ae-%l(X - y(y)) + EB(e%2(d - x) _ I) + Cp(y)(d - x)

is such that

~L26 - f(x,y,6,6x,e6y) ~ 0 in ~ ,

6 ~ g on 3~

126

4.4. Theorem 4.2 and the construction 4.3 prove the following theorem,

THEOREM Let ~ > 0 be fixed and ~o = ~l U "'" U ~k be such that each

~. is defined as in (3.5) i.e. l

~'1 = {(x,y) : a i < y < b i , ~_i(y) < x < ~+i(y)} C ~ \ B(D,~)

with ~-i and ~+i E C 2.

Assume assumptions (A-i) to (A-v) are satisfied

Then the BVP (4.2) has a solution u such that for g small

enough and any i

u(x,y) = uO(x,y) + O(e-%l (x'~-i (y)) + ~) in ~i '

where o < ~1 = O ( I / E ) .

5. The beam string problem

5.1. One interest of Proposition 2.2 is that it can be applied to higher

order problems. As an example consider an elastic beam with pin ends

described by the BVP

s2uiV - p(t)u" = f(t,u)

u"(o) = u"(1) = o (5.1)

u (o) = u (2) = o

with s > 0 , u E ~ , t ff [0, I] and let p :[0,I] ÷ ~ ,

f : [0,2] x R + ~ be such that :

(i) p is continuous and for some p > o and all

t @ [0,I] , p(t) ~ V 2 ;

(ii) f and fu are continuous and for all (t,u) e [0, I] x ~ ,

fu(t,u) > O.

5.2. To solve the reduced problem

- p(t)u" = f(t,u) (5.2)

u(0) = u(1) = o,

let us define the operator

K o : C([0, I]) ~ C([O,I]) , f ~ Kof

where K o denotes the solution of the linear BVP

127

- p(t)u" = f(t)

u(O) = u(1) = o .

From the maximum principle [I0] , K c is increasing. Next we introduce

the Nemitskii operator

V : u -~ Fu = f(.,u) . (5.3)

It is then obvious that solutions of the BVP (5.2) are the fixed points

of the increasing operator

To = KoF.

Their existence can be obtained as follows

5.3. PROPOSITION Suppose there exist functions s o E C2([0,I]) and

So E C2([0, I]) s u c h t h a t

- p(t)a~ < f(t,~o) , - p(t)S~ ~ f(t,S O) , t e ]0,I[,

%(o) <o , So(1) <o , So(O) ~o , So(1) >o ,

then the iteration schemes

~o = % Us = Bo

and

~k = To~k-I ~ = rouk-I

converge to solutions u and u of the reduced problem (5.2) such that

% < $ < _ u O , To~o(1) - as(1) >-- 0 and the

maximum principle imply To~ O > ct O. Similarly, one proves ToS o ~< S O

and Proposition 2.2 applies. U

5.4. To study the BVP (5.1), let us introduce the operator

Kg : C([O,I]) ~ C([O,I]) , f~K~f

where KEfdenotes the solution of the linear BVP

e2uiV - p(t)u" = f(t)

u " ( o ) = u " ( 1 ) = u ( O ) = u ( 1 ) = o .

128

By applying twice the maximum principle, it is easy to show that

f(t) i> O implies u" < O and next u ~> 0, i.e. the operator K E is

increasing.

Hence solutions of the BVP (5.1) are the fixed points of the

increasing operator

Tg = K~F.

Once again, Arzela-Ascoli's theorem imply T E is completely continuous and

as in 5.3, we can prove the following proposition.

5.5. PROPOSITION Suppose there exist functions s E C4([O,I]) and

E C4([O,I]) such that

g2siv - p(t)S" < f(t,s) , E2B iv - p(t)B" ~> f(t.$) , O < t < 1 ,

W'(o) i> o , W'(1) i> o , B"(o) ~< o , S"(1) ~< o ,

s ( 0 ) < 0 , s ( 1 ) ~< 0 , B ( 0 ) ~> 0 , IB ( 1 ) > 0 .

Then there exists at least one solution 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 ~ f(t,B o) , t e [O,1] ,

So(O) <o , %(1) ~<o , Bo(O) /> o , Bo(l) >/ o .

Then, for ~ small enough there exists at least one solution u~

of the BVP (5.1) such that

a o ~ O ,

~"(I) = ~o(I) + ~2Ae-~/c +~2B >~ 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)~" 2 iv + D4Ae-~ ~(]~t) - f(t,~) = E ~O + ~4Be

,, Dt p 2Be_D(~-t ) - p(t)~ O - p~2Ae-~- _

- f(t,~o) - S2fu(t,~ O + ~(~ - ~O))[Ae ~ + B e - -

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).

~(1-t) c -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 of the maximum

principle to singular perturbation problems, SlAM Review 15 (1973),

43-88.

[5] W. Eckhaus, E.M. de Jager, Asymptotic solutions of singular perturbation

problems for linear differential equations of elliptic type, Arch. Rat.

Mech. Anal. 23 (1966), 26-86.

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 2 nd 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

(1)

where s

I sy" + f(y,t,e)y' + g(y,t,g) = 0,

y(0), y(1) prescribed

OJt_< 1

is a small positive parameter (cf., e.g., Amundson (1974), Sethna and

Balachandra (1976), and Cohen (1977)). Scalar problems of this form are analyzed

quite thoroughly in the forthcoming memoir, Howes (1978). An enlightening case

history of such analyses was given by Erd~lyi (1975), and important early work

includes that of Coddington and Levinson (1952) and Wasow (1956).

For simplicity, let us assume that f and g are infinitely differentiable

in y and t and that they possess asymptotic power series expansions in e as

E ÷ O. We'll first consider the vector problem under the assumption that the

reduced problem

(2) f(uR, t,0)u ~ + g(uR, t,O) = 0, UR(1) = y(1)

is stable throughout 0 < t 0

there (i.e., -f is a strictly stable matrix having eigenvalues with negative

Supported in part by the National Science Foundation under Grant Number MCS 76- 05979 and by the Office of Naval Research under Contract Number N00014-76-C-0326.

132

real parts). We first realize that u R cannot generally represent the solution

to (i) near t = 0 because we cannot expect to have UR(0) = y(0). Instead, we

must expect boundary layer behavior to occur near t = 0, providing the required

nonuniform convergence from y(0) to UR(0) as ~ ÷ 0. For a (very) small

"boundary layer jump" fly(0) - UR(0)il or for a constant f(y,0,0), no extra

hypotheses are needed. More generally, however, we must require an additional

"boundary layer stability" assumption, namely that the inner product

(4) T I ~ f(uR(0)+s, 0, 0)ds > 0 0

remains positive for ~ + UR(0) along all paths connecting UR(0) and y(O)

with 0 < il~li ~ fly(0) - UR(0)Jl. (Here, T represents the transpose and llzli =

~zTz.) We note that if f(z,0,0) is the gradient VF(z - UR(0)) , (4) is

equivalent to the condition that

~T(F(~) - F(0)) > 0

since the integral is then path-independent. Indeed, (4) directly generalizes the

(minimal) hypotheses used by Howes for the scalar problem and it is weaker than

the common assumption that f(y,0,0) > 0 for all y.

Pictorially, the boundary layer stability assumption must hold within the

circle shown

y(1) = UR(1)

Figure 1

The results of Howes and others suggest that under such hypotheses, (i) will

have a solution y(t,c) of the form

(5) y(t,~) = U(t,e) + H(T,e)

where the outer solution

(6)

U has an asymptotic expansion

U(t,e) ~ Z U.(t)s j j=0 J

133

providing the asymptotic solution for t > 0, while the boundary layer correction

has an expansion

(7) ~I(T,E)

0o

Z K. (T)c j

j=0 J

whose terms all tend to zero as the stretched variable

(8) T = t/E

tends to infinity. We would expect this solution to be unique. Under weaker

smoothness assumptions on f and g, we'd have to limit the expansions to finite

order approximations. For the scalar problem, Howes doesn't actually obtain

higher order terms or complete boundary layer behavior, but they can easily be

generated. Applying his results to the boundary value problem for the remainder

terms, however, shows the asymptotic validity of the expansions so obtained.

The outer expansion (6) must provide the asymptotic solution to (i) for

t > 0, since ~ is then asymptotically negligible. Thus, the terms U. can be 3

successively obtained by equating coefficients in the terminal value problem

(9) f(U(t,e),t,c)U'(t,e) + g(U(t,e),t,e) = -cU"(t,~), U(I,B) = y(1).

Evaluating at c = 0, then, shows that U 0 must satisfy the reduced problem

(i0) f(U0(t),t,0)U&(t) + g(U0(t),t,0) = 0, U0(1) = y(1)

(which has a unique solution UR(t) under (2) and (3)). Succeeding terms Uj,

j > 0, will satisfy linear problems of the form

(ll) f(U0(t),t,0)U](t) + fy(U0(t),t,0)Uj(t)U&(t)

+ gy(Uo(t),t,O)Uj(t) = hj_l(t) , U.(1)] = 0

where hi_ I is known in terms of t, U0(t), ..., Uj_l(t). The stability assump-

tion (3) implies that (ii) is a nonsingular initial value problem, so it also has

a unique solution throughout 0 < t < i. Thus, there is no difficulty in generat-

ing the outer expansion U(t,e) with U(t,0) = uR(t).

The boundary layer correction H must necessarily be a decaying solution of

the nonlinear initial value problem

I__ d~ = (12) d2~ + f(U(~T,~) + ~(T,e), ~T e) ~ -e[(f(U(~T ~) + ~(T,e) eT, ~) - dT2 ' , ,

134

(12)

dU - f(U(ET,e),ez,e)) ~ (e~

- g(U(eT,e),er,e) ] ,

17(0,~) = y(0) - u(0,e).

, c ) + g(U(~'~,e) + n ( z , ~ ) , ~'v, e)

T > 0

Thus, the leading term ~0 must satisfy the nonlinear problem

(13) d2~ 0 d17 0

dr 2 + f(Uo(O) + ~O(T), O, O) d--~ = O, n0(0) : y(0) - u0(0)

while later terms must satisfy linear problems

(14) d2~j dN.

J + f(U 0(0) + 17 0(T) 0, 0) dr

dT2

d~ 0 + fy(Uo(O) + nO(Z), O, O)Hj(z) ~ = kj_l(T), ~j(0) : -Uj (0)

where kj_ I is a linear combination of preceding terms

dN~/dT, % < J, with coefficients that are functions of

decaying solution of (13) must satisfy

~ and their derivatives

T and HO(T). The

d~ 0 fT d~ 0 dY + f(U0(0) + H0' 0, 0) d-~- d% = 0

and, thereby, the initial value problem

dn 0 f~0 (T)

d~ : -J0 (15) f(U0(0) + w, 0, O)dw, ~ i 0, 10(0) = y(O) - UR(0),

T Multiplying by ~0' the boundary layer stability condition (4) implies that

~0(z) 1 d ]l~0(T)l]2 = "H~(T) I f(U0(0) + z, 0, 0)dz < 0 (16) 2 dT 0

for nonzero values of H0(T) satisfying [I~0(T)}] ! fly(0) - UR(0)H = IIH0(0)[I.

Thus, our boundary layer stability implies that ~0(r)~ will decrease monotonic-

ally as T increases until we reach the rest point H0(T) = 0 of (15) at T = ~.

Ultimately, ~0(T) will become so small that (3) (for t = 0) implies that the

eigenvalues of f(U0(0) + H0(T), 0, 0) will thereafter have real parts greater

than some K > 0 and (15) then implies that

(17) H0 (T) = 0(e-<T),

135

i.e., Y0 is exponentially decaying as T + =. Although we can seldom explicitly

integrate the nonlinear system (15), we can approximate its solution arbitrarily

closely by using a successive approximations procedure on (15) (cf. Erd~lyi

(1964)). Knowing ~0' we next integrate (14) for j = 1 and then proceed term-

wise. Rearranging (14) and integrating, we obtain

where

dH. 3 + f(U0(0) + ~0(T), 0 0)Hj + ~.(T) = 0

dT ' 3

f T

£j(T) = {fy(U0(0) + N0(r), 0, 0)[~j co dK 0

-d~T . (r)H.(r)]3 + kj_l(r)}dr

is known whenever

equation

(18)

where P(~)

dK 0 (r) ~ (r)

~. and dH0/dT commute. Thus, ~. satisfies the integral 3 3

IT H.(T) = P(T)U.(0) - P(T)P-l(r)%j(r)dr 3 3 0

is the exponentially decaying fundamental matrix for the linear system

d~ d-~ + f(U0(0) + K' 0, 0)K = 0, T > 0, H(0) = I.

In general, (18) must also be solved via successive approximations, though it

directly provides the solution of (14) when the commutator [~j, dH0/dT] ~ 0.

We note that the boundary layer jump lIH0(0)II = Hy(0) - UR(0)II is limited by

the minimum value of II~II > 0 such that the inner product

(19) ~T I ~ f(uR(0 ) + z, 0, 0)dz = 0. 0

The jump can, in practice, be quite large. It involves no restriction, for exam-

ple, if f(z,0,0) > 0 for all z since then (19) cannot ever hold for a ~ # 0,

It gives precise limits to the jumps for certain scalar problems (Howes (1977)

reconsiders an example of 0'Malley (1974)).

We could also consider the reduced problem

(20) f(uL,t,0)u L + g(uL,t,0) = 0, UL(0) = y(0).

Then the stability condition (3) and the boundary layer stability condition (4)

would be replaced by

136

(21) f(uL(t),t,0) < 0

for 0 < t < 1 and the assumption that

(22) eT fe f(uL(1) + z, I, 0)dz < 0 0

for all 8 + UL(1) on paths between UL(1) and y(1) satisfying 0 < IlOlf

IIy(1) - UL(1)il. Nonuniform convergence of the solution to (i) would then take

place near t = i, depending on the stretched variable

o = (i- t)/E,

and the limiting solution on 0 j t < 1 would be uL(t). If f were nonsingular

with eigenvalues having both positive and negative real parts along an appropriate

solution of the reduced system, we must expect boundary layer behavior near each

endpoint (cf. Harris (1973) and Ferguson (1975) for discussions of problems where

f (y,t,0) E 0). Y

2. Problems with boundary layers at both endpoints

Let us now consider the "twin" boundary layer problem

( (23) ~ ~Y" + g(y,t,e) = 0, 0 < t < 1

y(0), y(1) prescribed

under the assumption that g is infinitely differentiable in the region

interest and that the reduced system

of

(24) g(u,t,0) = 0

has a smooth solution U0(t ) throughout 0 < t < 1 which satisfies the stability

assumption

(25) gy(U0(t),t,0) < 0

there, i.e., gy is a stable matrix when evaluated along (U0(t),t,0) , 0 < t < i.

Motivation for this assumption is obvious if one considers the linear scalar prob-

lems with cy"± y = 0, while generalized stability assumptions are sometimes

appropriate and necessary (cf., e.g., Howes (1978) or consider the scalar problem

with g = y2q+l). With (25), one can hope that a solution to (23) exists which

converges to U0(t ) within (0,i). Since we won't generally have either

137

U0(0) = y(0) or U0(1) = y(1), we must expect "twin" endpoint boundary layers

(i.e., regions of nonuniform convergence of thickness 0(~-e) near both t = 0

and t = i). Our previous experience (cf. Fife (1973, 1976), Yarmish (1975),

O'Malley (1976), and Howes (1978)) suggests that we must add "boundary layer

stability" assumptions. These generally limit the size of the boundary layer

jumps [[y(0) - U0(0)I; and ily(1) - U0(1)[[. They'll certainly be guaranteed if

gy remains stable throughout the boundary layer regions (cf. Kelley (1978)).

Indeed, for small boundary layer jumps, the stability assumption (25) is suffi-

cient. Under appropriate assumptions, then, we can expect to obtain an asymptotic

solution to (23) in the form

(26) y(t,c) = U(t,e) + 9(p,/~) + ~(o,/~)

~

where U, 9, and Q all have power series expansions in their second variables

and the terms of the left boundary layer correction ~ tend to zero as the

stretched variable

(27) p = t/~TEe

tends to infinity while the right boundary layer correction Q + 0 as

(28) o = (i - t)//~e

becomes unbounded.

The outer expansion

(29) U(t,E)

should therefore satisfy

oo

Z U. (t)g j j=0 j

(30) eU" + g(U,t,s) = 0, 0 < t 1 j j-i ' --

11 where C j_ 1 is known termwise (e.g., C O = -U0).

implies that the systems (31) are all nonsingular.

cients are simply and uniquely obtained termwise.

The stability condition (25)

Therefore successive coeffi-

(Different roots U 0 of (24)

138

would, of course, result in different sequences of perturbation terms U., j > 0, J

under appropriate stability assumptions.)

According to the Borel-Ritt Theorem (cf. Wasow (1965)), there is a (non-

unique) function U(t,s), holomorphic in c, having the outer expansion (29).

If we set

(32) y(t,~) = U(t,e) + z(t,g),

we convert the problem (23) into the two-point problem

(33) cz" = h(z,t,~), 0 < t < i, z(0,~) = y(0) - U(0,s).

Here

satisfies

~

h(z,t,e) = -~U"(t,s) - g(U(t,£) + z, t, S)

(34) h(0,t,c) = 0(~ N) for every integer N > 0

since EU" + g(U,t,e) = 0(eN). In particular, the reduced system

h(z,t,0) = 0

corresponding to the transformed problem (33) has the (not necessarily unique)

trivial solution and the outer expansion :for (33) is also trivial. Henceforth,

then, we shall deal with (33) and, corresponding to (26), we shall seek an asymp-

totic solution of the form

(35) z(t,c) = v(p,~) + w(u,~)

providing the needed boundary layer decay to zero within 0 < t < i. Our smooth-

ness assumptions will be required in a domain

~EI,6 = {(z,t,e): 0 _< llz - U0(t)ll ida(t), 0 < t < i, 0 < ~ < ~ I}

where s I is a small positive number and, for any 8 > 0, we define

i llz(0) - U0(0)II + 6, 0 < t <

d6(t) = 6, ~ < t < 1 - 8

139

l ilz(1) - U0(1)li + 6, i- ~ < t < i.

We shall determine the asymptotic behavior of z by first determining that

of [[zll =~zTz. Here IizI[ satisfies the scalar problem

(36) ellzll" = [hT(z,t,s)z + e(llz'll 2 - (llzll2)']/llzll,

0 < t < i, where llz(0,a)II and llz(l,c)i[ are prescribed.

This follows via simple calculations, namely

d iizii2 = 211zlllIzii' = 2(z')Tz dt

and

d 2 -- lizll 2 = 21izllilzli" + 2(llzli')2 dt 2

= 2(z")Tz + 211z'il 2

imply the differential equation for llzll. Further,

(37) llz'112 > (llzll')2

since the Cauchy-Schwarz inequality ((z')Tz) 2 < llz'li211zll 2 implies that llz'[i 2 >

((z')Tz/llzII) 2 = (lizI[') 2. Thus, with a loss whenever z and z' 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) hT(z,t,s)z _> ~(llzll,t,e)]Izn

where

140

(40)

and

(41)

f

¢(O,t,c) ~ O, ¢(O,t,O) = O, ~ (O,t,O) > 0

~(s,0,0)ds > 0 whenever 0 < ~ ~ Ilz(0,0)ll if z(0,0) # 0

whenever 0 < ~ ~ Itz(l,O)li if z(l,O) ¢ O.

and

I ~ ~(s,l,0)ds > 0 0

Existence of such a function $ will constitute our stability hypotheses. Spe-

~n (0,t,O) > 0 implies the stability of the trivial solution of the cifically,

reduced system within (0,I) while (41) implies boundary layer stability at both

endpoints. Hypotheses (39)-(40) imply that

where

(42) 0 < ilz(t,~)il <_ m(t,e)

m(t,c) satisfies the scalar two-point problem

(43) cm" = ¢(m,t,e), 0 < t < I, m(O,c) = iiz(O,c)ll, m(1,g) = i~z(l,c)II.

The bounds (42) follow from the elementary theory of differential inequalities

since zero is a lower solution for lizil and m is an upper solution (cf. Nagumo

(1937), Dorr, Parter, and Shampine (1973), and Howes (1976)). Further, ~(0,t,0)

= 0 and ~n (0,t,0) > 0 imply that the zero solution of the reduced problem

~(m,t,0) = 0 corresponding to (43) is stable and, according to Howes (1978), (41)

is the appropriate hypothesis for the needed boundary layer stability of this

solution. Indeed, the solution of (43) satisfies

(44) m(t,g) = rO(P) + So(O) + O(~-c)

where r 0 is the decaying solution of the boundary layer problem

d2r 0 (45) = ¢(r0,0,0)

dp2 P > 0, r0(0 ) = lJz(0,0)il = fly(0) - Uo(0)ii

while s O is the decaying solution of

d2s0 (46) = ¢(so,l,O), ~ > O,

do2 s0(0 ) = ;Iz(l,0)li = fly(1) - U0(1)ll.

141

The solutions to (45) and (46) are easily shown to exist and be unique. Multiply-

ing (45) by dr0/dP, for example, and integrating from p to infinity implies

that

(dr0) ir0 = ~(s,0,0)ds > 0

d-7- 0

(by (41)). Thus, r 0 satisfies the initial value problem

dr° S °(p) (47) dp 2 ~(s,O,O)ds, ro(O ) = fly(O) - Uo(O)II.

0

Hence, r0(P) will decrease monotonically to zero as p increases, reaching the

rest point r 0 = 0 at p = ~. Since ~(s,0,0)~n (0,0,0)s for s small,

~_i (0,0,0) > 0 implies that the decay of r 0 to zero is exponential as p ~ ~ Sn (When r0(0) = 0, we have r0(P) E 0 since there is no need for a boundary

layer correction.) Continuing by solving linear problems, we could obtain an

asymptotic solution of (43) in the form

(48) m(t,s) = r(p,~) + s(o,~).

In terms of the original problem (23), our stability hypothesis (39) becomes the

inequality

gT(u0(0) + z, t, e)z _< ~(ilzl),t,E)llzll

where ~ satisfies (40) and (41). The expansion (44) corresponds to the expected

expansion (26) for an asymptotic solution for the vector problem (23).

Now, we return to the vector boundary value problem (33) and its asymptotic

solution in the form (35). Near t = 0, w and its derivatives should be asymp~

totically negligible (o being infinite), so (33) and (35) imply that the initial

boundary layer correction v should be a decaying solution of the nonlinear

initial value problem

(49) v = h(v,~p,e), p > O, v(O,~) = z(O,e). pp

Thus, it is natural to seek an expansion

(50) v(p,/T) E v. (p)s j/2 j=0 J

by substitution into (49). The leading term v 0 must then satisfy the nonlinear

142

problem

d2v0 (51) ...... h(v0, 0,0)

do2

Later terms v., j > I, must satisfy the linear problems J

d2v.

dP 2~ = hz(V0,0,0)v j + dj_l(p), p _> 0

(52) v.j(0) = 0, j odd; v.j (0) = -Uj/2(0), j even

p ! 0, v0(0) = y(0) - U0(0), v 0 ÷ 0 as O * ~.

v. -+ 0 as p ÷~ 3

where dj_ 1 will be determined successively as an exponentially decaying vector.

Since (51) and our hypothesis (39) imply that

llv011p0 ~ ~(llv011,0,O), @ ~ 0, llv0(O)ll = r0(0),

we are guaranteed a decaying solution v0(P) such that

(53) 0 ~ llv0(P)ll j r0(0), P ~ 0

(and v0(P) ~ 0 if U0(0) = y(0)). No explicit solution v 0 can be provided,

though an approximate solution can be obtained as usual. Introducing the matrix

= hz(0,0,0) > 0

(whose eigenvalues have strictly positive real parts by our stability assumption

(25)), variation of parameters can be used to express the solution of (52) in the

form

(54)

f -- (r)dr + 2 0 J p

e~(P-S)Fj (s) ds]

where F.(p)3 = [hz(V0(P)'0'O) - ~]v (p)3 + dj_l(p). This provides the exact solu-

tion to (52) whenever h(v0,0,0) is linear. Otherwise, the linear integral

equation (54) must also be solved by successive approximations. In analogous

fashion, we could generate the terms of the terminal boundary layer correction

w(o,~) of (35). Thus, we've formally obtained (35), which we expect is a locally

unique asymptotic solution.

143

We note that the assumptions on ~ automatically hold if gy(y,t,0) or

-hz(z,t,0) are everywhere stable. Thus, if we take

h (v,0,0) - ¥I > 0 z

(i.e., positive definite) for some real y > 0

llz(0,0)ll, the mean value theorem implies that

and all v satisfying 0 ! livJi

hT(z,0,0)z = zTh (~,0,0)z > yilzll 2 g

for some "intermediate" point ~. Thus, taking ~(n,0,0) = yn, both ~ (0,0,0)

> 0 and /0 ~(s,0,0)ds > 0 for 0 < D ~ Jlz(0,0)ll hold.

We could also extend our discussion to systems of the form

sx" = F(x,x',t,s)

with SF/~x' small. Thus, Kelley (1978) considered problems where S F _

1 ~F (~F)T ~x 2s ~x' Sx' > 0, just as Erd~lyi (1968) considered scalar problems somewhat

more nonlinear than semilinear.

3. Examples

a. A problem with an initial boundary layer

Let us consider the vector equation

gy" + f(y,t,c)y' + g(y,t,~) = 0, 0 < t < 1

where

y = lyll f (yll Y2 1 Y2

In order to have a limiting solution

the reduced problem

, and g = -

YI+ i ).

Y2 + I

u R of the two-point problem which satisfies

f(uR,t,0)u ~ + g(uR,t,0) = 0, UR(1) = y(1)

we must require u R to be stable in 0 < t < ], i.e.,

-f(uR(t),t,0) < 0

must be a stable matrix, and we must also require boundary layer stability at

144

t = 0, i.e., we ask that

T f~ f(uR(0) + z, 0, 0)dz > 0

0

for all $ such that 0 < lJCJl ! fly(0) - UR(0)ll.

More specifically, the reduced problem has the solution

where C = URl(0) = -i + Yl(1)

requires the matrix

to be stable throughout

UR(t) = ( t + D t + C

and D = UR2(0) = -i + Y2(1). Stability of u R

-t - C

-I

0 < t < i.

-i

-t - D

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 w21 l(dwll 0 1 + D dw 2

0,

i.e.,

2 ~I 3 + 2C~ + 4~i~ 2 + ~ + 2D~ 2 > 0

= satisfying 0 < ll~ll = lJy(0) - UR(0)ll = (Yl(0) - C) 2 + for all ~ ¢2

(Y2(0) - D)2. Our initial values y(0) are thereby restricted to a circle about

(C,D) with radius less than the least norm II~II of the nontrivial zeros of the

cubic polynomial. Setting ~2 = t~l' such a ~ 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, t = =, is 2D.) This calculus problem,

then, determines an upper bound for lly(0) - UR(O)~.

For C = D = 2, i.e., y(1) = (~), we'd obtain the minimum value 3.390

for d(t) corresponding to tmi n = -0.291. Thus, we're guaranteed that the

limiting solution of our two-point problem is provided by UR(t) if y(0) lies

in the circle of radius 3.390 about (~). This is presumably a conservative

estimate for the "domain of attraction" of the reduced solution uR(t). We expect

that boundary layer stability need only hold for ~ + UR(0) on the actual trajec-

tory joining y(O) and uR(0). Finally, we observe that this example is quite

analogous to the simplest cases occurring in the analysis of solutions of the

scalar problem ey" + yy' - y = 0 (cf. Cole (1968), Howes (1978), and elsewhere).

b. A problem with twin boundary layers at the endpoints

Consider the vector problem

gz" = h(z,t,~), 0 < t < i

where

z =

Zl

z 2

and h =

3

-z I + z 2 - z 2

Here U 0 = 0 is a stable solution of the reduced problem

the Jacobian matrix

hz(0,t,0) = ( -II ii )

h(U0,t,0) = 0 since

has the unstable eigenvalues i i i. Boundary layer stability involves the deter-

mination of a scalar function ~ such that

hT(z,t,e)z _> ~(lJzll,t,s)IEz]l.

Here

4 Since z I +

2 2 4 4 hT(z,t,e)z = (z I + z 2) - (z I + z 2) _> IIzJl2(l - 11z~2).

4 2 2 2 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 ° 1 0 ~(s,i,0)ds = ~ n2(l - n2/2) > 0 for 0 < n < /2, i = 0 or i.

Our preceding results, then, guarantee the existence of an asymptotic solution to

the two-point problem which converges to the limiting solution U 0 = 0 within

(0,i) provided the boundary values satisfy

llz(0,0)l[ < ~ and llz(l,0)il < ~.

Indeed, we then have

0 i l~z(t,a)lJ ~ m(t,¢)

where m satisfies the scalar problem

gin" = %(m,t,¢), 0 < t < i, m(i,¢) = llz(i,¢)ll < /2, i = 0 and i.

The asymptotic behavior of m follows from the scalar results of Howes (1978)

and others.

c. A problem with internal transition layers

We now consider the very special problem

¢y" + f(y,t,c)y' + g(y,t,s) = 0, 0 < t < i

where

y = , f(y,t, ~) =

Y2 0

and g = gl (yl,Y2)

-Y2

f2 (yl,Y2, t, ~)

Y2

and

- =

eY2 + Y2Y2 - Y2 0

ey~ + fl(Yl,t,c)yl + [f2(Yl,Y2,t,E)y~ + gl(yl,Y2,t,e)] = O.

This system decouples into the two nonlinear scalar equations

147

If Y2(1) > Y2(0) + 1 and -Y2(1) - 1 < Y2(0) < 1 - Y2(1), it follows from

Howes (1978) that the limiting solution for Y2 will satisfy the reduced problem

1 UL( ~ - i) = 0, UL(0) = Y2(O) 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*

Y2 U

uR(t) = t + Y2(1) - I, t* < t ~ i.

Thus, the limiting solution is generally discontinuous at t* and its derivative

(which is asymptotically one elsewhere) becomes unbounded there. Indeed, Y2

increases monotonically near t* from UL(t*) to uR(t* ) = -UL(t* ). For other

relations between the boundary values Y2(O) and Y2(1), other limiting possi-

bilities occur (cf., e.g., Howes).

One must generally expect the transition layer at t* in Y2 to generate a

corresponding discontinuity there in YI" To simplify our discussion, however,

let's assume that f2(Yl,Y2,t,0) = 0 and attempt to apply Howes' scalar theory

to the equation for YI" Thus, consider the reduced problems

fl(VL,t,0)vL + gl(VL,U,t,0) = 0, 0 < t < i, VL(0) = Yl(0)

and

fl(VR, t,0)v~ + gl(VR,U,t,O) = O, 0 < t < i,

The limiting solution for Yl will be provided by vR(t)

dition

vR(1) = Yl(1).

if the stability con-

holds throughout

for q between

that the limiting solution is

fl(VR(t),t,O) > 0

0 < t < 1 and the boundary layer stability assumption

VR(0) r (vR(0) - Yl(0)) J fl (s'0'0)ds > 0

q

vR(O) and (including) Yl(O). Similar conditions would imply

VL(t) on 0 < t 0 on t R < t < i

fl(VL(t),t,0) < 0 on 0 < t < t L

with t R < tL, we can expect Yl to have a limiting solution

/

J VL(t), 0 ~ t < t

Yl V

VR(t), t < t ~ 1

as c + 0 provided we can find a t in (tR, t L) such that

J(t) = O, J'(t) ~ 0

for

VR(t) r

J(t) = J fl(s,t,0)ds VL(t)

(cf. Howes (1978)). Pictorially, we will have limiting solutions

shown in Figures 2 and 3.

Y2 and

Y2

Y2 (0)

Y2

u R

i .............. !

(i)

t

Figure 2

Yl as

yl 149

v R

A

t t* 1

b t

Figure 3

' has a jump at t*, corresponding to a Note that Y2 has a jump at t and Y2

Haber-Levinson crossing (cf. Howes (1978)). Much more complicated possibilities

remain to be studied.

Acknowledgment

We wish to thank Warren Ferguson for his interest in this work and for cal-

culating 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.

8.

9.

i0.

ii.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

A. Erd~lyi, "Approximate solutions of a nonlinear boundary value problem," Arch. Rational Mech. Anal. 29 (1968), 1-17.

A. Erd~lyi, "A case history in singular perturbations," International Con- ference on Differential Equations (H. A. Antosiewicz, editor), Academic Press, New York, 1975, 266-286.

W. E. Ferguson, Jr., A Singularly Perturbed Linear Two-Point Boundary Value Problem, Ph.D. Dissertation, California Institute of Technology, Pasadena, 1975.

P. C. Fife, "Semilinear elliptic boundary value problems with small parameters," Arch. Rational Mech. Anal. 52 (1973), 205-232.

P. C. Fife, "Boundary and interior transition layer phenomena for pairs of second-order differential equations," J. Math. Anal. A~. 54 (1976), 497- 521.

W. A. Harris, Jr., "Singularly perturbed boundary value problems revisited," Lecture Notes in Math. 312 (Springer-Verlag), 1973, 54-64.

F. A. Howes, "Singular perturbations and differential inequalities," Memoirs Amer° Math. Soc. 168 (1976).

F. A. Howes, "An improved boundary layer estimate for a singularly perturbed initial value problem," unpublished manuscript, 1977.

F. A. Howes, "Boundary and interior layer interactions in nonlinear singular perturbation theory," Memoirs Amer. Math. Soc.

F. A. Howes, "Modified Haber-Levinson crossings," Trans. Amer. Math. Soc.

W. G. Kelley, "A nonlinear singular perturbation problem for second order systems," SIAM J. Math. Anal.

M. Nagumo, "Uber die Differentialgleichung y" = f(x,y,y')," Proc. Phys. Math. Soc. Japan 19 (1937), 861-866.

R. E. O'Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.

R. E. O'Malley, Jr., "Phase-plane solutions to some singular perturbation problems," J. Math. Anal. Appl. 54 (1976), 449-466.

P. R. Sethna and M. B. Balachandra, "On nonlinear gyroscopic systems," ~echanics Today 3 (1976), 191-242.

W. Wasow, "Singular perturbation of boundary value problems for nonlinear differential equations of the second order," Co_~. Pure Appl. Math_. 9 (1956), 93-113.

W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Wiley- Interscience, New York, 1965 (Reprinted: Kreiger, Huntington, 1976).

J. Yarmish, "Newton's method techniques for singular perturbations," SI~ J. Math. Anal. 6 (1975), 661-680.

HIGHER ORDER NECESSARY CONDITIONS IN OPTIMAL CONTROL THEORY

H.W.Knobloch

1.1ntroduction

The lecture is intended to give a survey on some recent research in

non-linear systems theory. Here the notion of "system" refers to

a dynamical law given by an ordinary differential equation

(1.1) x = f(X,U) = d/dt,

plus a constraint on u of the form u~U, U being an arbitrary set

in u-space (control region). The state variable x=(x 1,...,xn) T

and the control variable u=(ul,...,um) T are finite dimensional

column vectors. We admit specializations of the control variable

by piecewise C~-functions of t which assume values in U (ad-

missible control functions). A pair (u(-),x(-)) where u(') is

an admissible control function and x(.) solution of the differen-

tial eq. x = f(x,u(t)) is called a solution of (1.1).

Attention will be focused on two topics: (i) Sufficiency tests for

local controllability along a given arbitrary solution of (1.1) .

(ii) characterization of special solutions of (1.1),the so called

singular extrema!s. We present a unified approach to both problems

which has two advantages compared with other relevant work in this

area. It provides more accurate results and requires rather ele-

mentary tools. The background work consists, roughly speaking,

essentially in a careful analysis of the way in which the solutions

152

of the differential eq. (1.1) depend upon the choice of the control

function. This analysis will be carried out in detail in a forth-

coming paper.

2. Local controllability and cones of attainability.

We consider a fixed solution (u(.),x(-)) of (1.1) (reference solu-

tion) on some interval [0,t I] . The system is said to be locally

controllable along the reference solution at the terminal point

x I = x(t I ) if it can be steered within time t I into any point of

a full neighborhood of x I along admissible trajectories initia-

ting from x o = x(0). This concept is the local equivalent of the

notion of complete controllability in linear systems theory. The

most common access to sufficiency tests for local controllability

of nonlinear systems uses cones of attainability. These are convex

approximations to the set ~ of all points attainable at time t I

from x ° . The usage of such cones in connection with separation ar-

guments is a familiar tool in optimal control theory, e.g. the

standard proof of the Pontryagin Maximum Principle involves the

construction of a cone of attainability. It was felt however since

long that the Pontryagin cone does not yield the best possible con-

vex approximation and attempts have been made to find suitable ex-

tensions. The one which has received most attention so far is the

recent work of Krener (c.f. [2]) which leads to a statement of a

"High Order Maximal Principle". To begin with we shortly outline a

construction method for a cone of attainability (i.e. a derived

cone for d at x(t I ) in the sense of Hestenes) which contains

both the Pontryagin cone ond the cones used in the work of Krener.

The procedure consists essentially of two steps. SteD I. We pick a

fixed intermediate point (u(~),x(~)) of the reference solution q7

and associate with it a certain non-empty subset II = II~ of the

153

state space. II can be described roughly as follows (for a more de-

tailed description 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 k and

which is such that u(t,k)=u(t) (= reference control) except a neigh-

borhood of ~ which shrinks to zero for ~0. Let then x(',k) be

the solution of the differential eq. (1.1) (with u=u(t,~)) which

has for t=0 the same initial value as the reference trajectory. Take

then the asymptotic expansion at k=0 of x(t,k)-x(~).

Step 2. Each set lit, 0 ~ t ~ tl, is transferred to the terminal

point x I by means of the linear mapping induced by the solutions

of the variational equation. 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 K is a derived cone for ~at x(t I) (in the sense of

Hestenes, cf. [3]). In particular, if K = ~n then x(t 1) is in-

terior to ~.

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 be a subset of the R n and let x(tl) be a

regular point of M. If there exists a neighborhood of x(tl) which

does not contain a point of ~which is relative interior to M, then

K and the tangent cone T to M at x(tl) are separable (in the

R n) . This implies the existence of a non-trivial adjoint state

variable y(-) such that the following relations hold

y(t)Tp ~ 0 for all PE ~t and all tE[0,t I] , (2.1)

Y(tl)Tk- ~ 0 for all k~T .

Note that this result allows applications to optimality problems

with terminal constraints both in equality and inequality form, e.g.

to problems where one is seeking for a Pareto-optimum.

3. Singular extremals. General remarks.

From now on it is assumed that the reference control u(') assumes

values in the interior of U. If in addition the reference solution

is optimal (with respect to some performance criterion) one speaks

of a singular extrema!. Since in case of a singular extremal the

application of the Pontryagin Maximum Principle frequently turns

out to be of little use, there is particular interest in those ele-

ments of K which are not contained in the Pontryagin cone. The

necessary conditions which arise from those elements are then called

"higher order". In the lecture all known second order conditions

will be touched upon, in fact they are contained in the two basic

results which will be discussed in some detail. The second depicts

a linear space which is contained completely in II t. It gives rise

to equality type necessary conditions, special cases of which are

widely used in applications and mostly attributed to Robbins and

Goh. The first result is an inequality type relation which is

known in the literature as generalized Clebsch-Legendre condition.

In case of a multivariable control we obtain a conclusive result

155

which seems not to be known so far (cf. related work by Kelley, Kopp,

Moyer, Jaeobson, Krener and others).

The theory of second order necessary conditions is not just a

straightforward application of convex approximation theory, but

also relies heavily on some non-trivial algebraic facts about non-

linear systems. The formalism which brings out these facts is of

interest in its own right and can be developed out of two ideas

which are in complete agreement with our general line, namely to

treat control systems strictly from the differential equations

viewpoint. Firstly we introduce an analogue of the Kalman controlla-

bility matrix into the non-linear theory. Let us consider the Hamil-

tonian system for the state and adjoint state variable, that is the

system which arises from the Hamiltonian H(x,y,u) =yT. f(x,u):

( 3 . 1 ) x = f ( x , u ) y = - f x ( X , u ) T y

L e t u s a s s u m e f o r s i m p l i c i t y t h a t u i s s c a l a r . T a k e t h e f o r m a l

v-th time derivative of the scalar function

(~H/~u)(x,y,u) = yT-fu(X,U) with respect to the system (3.1). I t

to see that it can be represented in the form yT'b is easy

where b is a n-dimensional vector having as components certain / %

functions of x,u and further independent variables ~,~,...u k~j

In case of a linear system (i.e. a system of the form x = Ax + bu)

b just coincides with the v-th column AVb of the Kalman matrix.

It now turns out that in the general non-linear situation the se-

quence of the b -provided they are defined as above, namely as

functions of the independent variables ~,~,... - play the key

role not only for the first order necessary conditions (which is

not surprising) but also for the understanding of the second order

conditions.

The second idea - which so far seems to represent the real novelty

of our approach - is to study systematically invariance properties

with respect to the substitution

156

(3.2) u ~ u(x,v)

In other words, we consider along with the given system all those

which arise by making the control depend upon the state. It turns

out that the quantities and relations elaborated by our approach

exhibit a rather transparent behaviour with respect to the substi-

tution (3.2). With respect to the sets Ilt this is somehow ob-

vious from the explanation given in Section 2. The application of

invariamce properties is our main technical tool. It allows - in

contrast 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 many independent variables

ui, i=0,1,... , each u i being a m-dimensional vector. The symbol

U will be used in order to denote the sequence lUo,Ul,... I ;

vector-valued functions of x and finitely many of the variables

u i will be denoted by g(x, U ) for shortness. It will always

tacitly be assumed that these functions are infinitely often diffe-

rentiable with respect to all variables. If g is a n-dimensional

column-vector, then the Lie-bracket [f,g] = gxf-fxg is well defi-

ned, where f=f(X,Uo) is the right hand side of the given differen-

tial equation (1.1) (with u o instead of u). Hence

(4.1) r(g) = [f,g] + ~ (~g/~ui)'ui. I i=O

is also well defined. The mapping g ~ F(g) represents a linear

operator acting on the set of all n-dimensional vectors g=g(x, U ).

It is then easy to see that in the case m=l the vectors b which

we introduced in the last section can be obtained from bo=fu(X,U o)

by applying the operator F v and writing u,u,~ etc. instead of

u o , Ul,U 2 ....

157

We now turn to the case of a multivariable control (m > 1). The role

of the b is then played by a sequence of matrices By = (B~, .... B~),

where the n-dimensional column vectors B ~ are recursively defined

as follows (the u-th component of u is denoted by u (~) henceforth)

(4.2) ~o = (~fl~u~)(X,Uo) ~ = rv(~ o) , v=1,2, .... '

The elements of ~ are C ~° functions of x, U = IUo,Ul,.-.l and

they are polynomials in the components of u i for 1 ~ i ~ v • As

in the case m=l the B also can be obtained with the help of the

Hamiltonian function H(x,y,u) = yT.f(x,u) and its Jacobian matrix

Hu(X,y,Uo) = yTfu(X,Uo) = (yTf 1 .... ,yTf m) u u

Indeed, if H u is differentiated v-times with respect to t and

if differentiation is carried out according to the rules

(4.3) x = f(x,u o) , y= -fx(X,uo)Ty , ui = ui+1' i=0,I ....

then the result can be ~itten in the form yTB (x,u) , i.e. we have

(4.4) dV yT. dt ~ Hu(x,y,Uo) = By(x, U ) .

This remark leads to a further interpretation of the matrices B v

Let us take the B along a given reference solution u(.),x(') of

(1.1), i.e. let us consider

(4.5) By(t) = Bv(x(t), 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 according to the rule

(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 order necessary conditions":

If u('), x(') is a singular extremal and y(') an adjoint state-

vector (i.e. a non-trivial solution of (4.7) for which the Pontrya-

gin Maximum Principle holds) then y(t) ist orthogonal to the

158

columns of By(t), v=0,1 .... These conditions are also a consequence

of the following more general result. Here x('), u(') need not to

optimal, however the assumption u(t)Eint U has to be made

Theorem 4.1 ~t contains the linear space spanned by the columns

of the matrices Bv(t),v=0,1, .... This space will be denoted by

$(t) henceforth.

There are good reasons for introducing the B not along given

solutions, i.e. via the relation (4.6), (4.7) but introducing them

instead formally as functions of x and U(cf. (4.3),(4.4)). This

enables one to express not only the first order, but also all the

second order conditions in terms of the B as we are going to

demonstrate in the next two sections. An important step in this direc-

tion is the following result which is of interest in its own right.

Theorem 4.2 The following relation holds identically in x, U for

all ~,v ~ O, p,o = 1,...,m ~+I

m=O

The importance of this result lies in the fact that it links two

different operations which can be performed with the B v - namely

differentiation with respect to the control variable u o and the

forming of Lie-brackets out of its columns (which involves differen-

tiation with respect to the state variable x only).

A proof of Theorem 4.2 which is based on the invariance principles

mentioned in connection with (3.2) will be given in the forthcoming

paper.

159

5. The generalized Clebsch-Legendre condition.

Given a reference solution u(.), x(') on some interval I and let

us assume that u(t)EintU for all tEI. We denote as before by ~(t)

the linear subspace of R n which is spanned by the columns of the

matrices B (t) (cf. (4.5)).

Theorem 5.1. Let there be given an integer ~ ~ O and real numbers

~1,...,g m such that the following conditions are satisfied

m i ( ~.E ¢(t)for all tel if ~< ~ ,

Ci~j(bB~/~UoJ))(x(t)' ~ ( t ) ) L ' ~ ~ t ) for some tel i f ~ = , i,0=1

then ~ is even and

m ~ ( -1)~ /~ ~i~ ~ ( bBi/bu~ ) ) (x( t ) , U (t))E

i,j=l

Assume now that u('),x(-) is a singular extremal and that a multi-

plier rule of the form (2.1) holds with some non-trivial adjoint

state vector y(-). It follows then from Theorem 4.1 that y(t) is

orthogonal to ~(t) for every tel. If ~(t) in addition happens

to have the maximal rank n-1 (i.e. if the multiplier y(') is

determined up to a positive scalar constant by means of the first

order conditions) then conversely~t) consists of all vectors which

are orthogonal to y(t). The statement of the theorem can then be

given this simple form.

Corollary I. If the dimension of ~(t) is constant and equal to

n-1 on I then for each choice of the m-tuple ~I' .... '~m the first

non-vanishing among the numbers m

(5.1) ~ ~i~Y(t)T(bB~/bU~J))(x(t), U(t)) ,~=0,1,2, . . . . i,j=1

carries an even subscript ~ (which may depend upon ~1,...~m ) and

has the sign (-I) v/2

160

As one observes from (4.3) and (4.4) the coefficient of Ci~j can

also be interpreted as the quantity

(5.2) ~ # d~ ~ H(x,y,u O)) : : hi'J(x,y,~ U )

°

taken "along" the singular extremal, i.e. taken for x=x(t),y=y(t),

U = U(t). Hence the corollary can be rephrased in a way which is

more close to what is known as generalized Clebsch-Legendre condition

in the literature. Note however that the standard version of this

condition in case of a multivariable controlconGerns the quadratic

form (in indeterminates zl,z2,...Zm)

i zizoh~'J(x(t)'y(t)' U(t))

i,0

as such, whereas our corollary yields the analogous statement for

each i n d i v i d u a i value of this quadratic form.

We state a further result which is an immediate consequence of Theo-

rem 5.1. In the following corollary the reference solution is not

required to be optimal and the rank of ~(t) need not to be maxi-

mal.

Corollary. 2. Let ~ >- 0 be an integer such that

(5.3) bB~/bu(J))(x(t), U(t))+ (bBJ/bu(i))(x(t), ~(t))E ~(t)

for i,j=l,...,m, all tel and v=0,...,~-1 Assume furthermore

that (5.3) is n o t true for all tel, and all i,j if V=~

Then ~ is an even number and we have

(-1)~/2~bB~/bu(J))(x(t), U(t))+ (bB~/b/u(i))(x(t), U(t))~T t

for all tel , and i,j=i,...,m .

161

6. Higher order equality-type conditions.

These are conditions of the form y(t)Ta = 0 and they arise,

according to the multiplier rule (2.1), from elements a of the A n

which are such that A aE I~ t . All elements of the linear subspace

~(t) have this property, but there may be more, as we are going

to show in the next theorem. We assume again that the reference

solution u(.),x(') satisfies the condition u(t)Eint(U) for all

tEI; but otherwise it can be arbitrary. In particular it need not

to be optimal.

Theorem 6.1. Let ~ _> 0 be an integer such that the following ele-

ments belong to the space ~(t), for every tEI:

(i) (bB~/bUo(J))(x(t), U(t)) for v ~ ~, i,j=l ..... m

for i,j = I .... m.

Conclusion: The elements

(6.1) ~ (~B~/bu~J))(x(t), U(t))

belong to "If t for every tEI and i,j=1,...,m.

There is an important special case of Theorem 6.1 which is known

(or rather its consequences for singular extremals are known).

Corollary 1. Let f(x,u) be a linear function in u

Then

+ (~B~/~Uo(J))(x(t)' U(t))E ~t

for i,j=l,...,m and all tEI

Proof. If f is linear in u, then Bo = fu is independent from

162

Bi, {j) U and hence b o/bU o is zero identically in x, • . It follows

then from Corollary 2 to Theorem 5.1 that the hypotheses of Theorem

6.1 are all satisfied if one takes ~=I .

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 con-

tains the linear space generated by the union of ~(t) and the ele-

ments (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 y(t),~(t) is necessarily

the maximal linear subspace of II t if it has dimension n-1. Hence

the elements (6.1) actually belong to ~(t) 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 ~It , for

every tEI. Assume furthermore that (5.3) holds for all tEI, for

i,j=1,...,m and for v=O,...,a ,~ 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. Since

the standard techniques of sensitivity analysis are based on the

general properties of derived cones only, they can be applied to

the cone K which was introduced in Sec. 2 - the same cone from

which we have deduced all necessary conditions discussed in this lec-

ture. Thereby one arrives on sensitivity results which take the

163

higher order variational effects into account. This leads to an in-

crease of accuracy.

Sensitivity analysis in general is concerned with estimates for the

changes which the value function (i.e. the optimal value of the

performance criterion) undergoes if the data of the control problem

are changed. We confine ourself to a sketchy outline of a typical

result and its extension. A more detailed account can be found in

[5] where also an illustrative example is discussed.

Let us consider an optimal control problem where the terminal mani-

fold consists ef a single point x 1. We now change x I in a certain

direction p, that is we replace x I by x1+kp, k being a positive

paramter. Let us assume that for each sufficiently small k~O the

value function V(k) is well defined and that a right-hand side

derivative V' of V(k) at k=O exists. It is then known that V'

can be estimated from above in terms of the values which a certain

linear functional assumes on the (suitably normalized) set of all

those multipliers y for which the statement of the Pontryagin

Maximum Principle holds true. The estimate of V' now remains valid

if we let y vary instead on the set of those multipliers which

satisfy the first of the conditions (2.1). That this indeed means

a restriction of y to a subset of the original set and therefore

also a lowering of the bound for V' follows simply from the fact

that the Pontryagin cone is contained in the convex cone K

R e f e r e n c e s .

[I] H.W.KNOBLOCH, Local controllability in nonlinear systems,

Dynamical Systems, A.R.Bednarek and L. Cesari eds., Academic

Press 1977, pp. 157-174.

[2] A.J.KRENER, The high order maximal principle and its application

to singular extremals, SIAM J. Control Optimization 15 (1977)

pp. 256-293.

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

Ins t i tu t Meth~matique Universit~ de Louvain

B-134B Louvain-la-Neuve Belgium

I . INTRODUCTION

Much work has been devoted in recent years to the so lvab i l i t y of nonlinear equations

of 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 assu~tion. 5ee for example the mono-

graphs[ 8 ] , [ 1 1 ] a n d [ 1 3 ] . B a s i c f o r t h i s s t u d y i s t h e r e d u c t i o n o f e q u a t i o n ( 1 . 1 ) t o

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 is the associated generalized inverse of L and J : Im Q -~ ker L is an isomorphism.

The compactness assumption on N generally implies that (JQ + KpQ)N is a compact mapping

on some bounded subset of X and, P being by def in i t ion of f i n i t e rank, (1.2) is a f ixed

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 (1.3), then P is no more compact

and is at best non-expansive, which makes the study of the fixed point problem (1.2)

very d i f f i c u l t even fo r (JQ + KF~)N compact (see e . g . ~ ] , chapter 13).

In a recent paper, Br~zis and Nirenberg [4] 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 ~given 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 and with the

growth restrictions replaced by a condition of Leray-Schauder's type on the boundary

of some set. This existence theorem is then applied in Section 5 to abstract problems

of Landesman-Lazer type, the first one corresponding to a result of Cesari and Kannan

[9] when dim ker L <two , and the second one being essentially a version of the

result of Br~zis-Nirenberg. In Section 6, the main theorem of Section 4 is applied

to a second order periodic boundary value problem of the form

- x " = f ( t , x )

in a H i l b e r t space, and the existence r e s u l t which i s obtained i s s f i r s t but s t i l l

partial answer to the question raised in 60] about the solvability of this periodic

problem for differential equations in infinite dimensional spaces. In Section 7,

we consider equation (1.1) in a reflexive Banach space X and obtain a Leray-Schauder's

type theorem when the assumptions of monotonicity and compactness are replaced by

some assumptions of strong continuity.

Another problem in differential equations and leading to equations of the type

(1.1) with an infinite dimensional kernel is the periodic boundary value problem

for first order ordinary differential equations of the form

x' = f(t,x)

in an infinite dimensional Banach space. Such a problem was considered by Browder

in [5] and Z6]and will not be treated here. The interested reader can consult the

recent paper ~9 lwhere some of the results of Browder are generalized, leading in

particular to existence theorems of the Landesman-Lazer's type for this situation.

167

2. SOME RESULTS ON H.AMMERST.EIN E~UA.TI.ONS IN HILBERT SPACES

Let H be a real Hilbert space with inner product ( , ) and corresponding norm ~.~ •

DEFINITION 2.1. A pair (M,N) of mappings from H into H is said to be Hammerstein-

compatible (shortly h-compat.~ble) with constants a an___.dd b if the following conditions

hold :

(i)

(ii)

(iii)

0 ~ b ~ a .

N is demi-continuous, i.e. if x ~x in H, then N(x )--~N(x) . n n

(V u ~ H ) ( V v ~ H ) : a|Mu - My| 2 ~ (Mu - Mv,u - v) •

( i v ) (V u ~ H ) ( V v ~ H ) : - b { u - vl 2 _~ (Nu - Nv,u - v) .

This class of pairs of mappings is related to the unique solvability of the

abstract Hammerstein equation

( 2 . 1 ) x + MNx = f

for every f ~ H, by the following results, which generalizes and completes a theorem

of De Figueiredo and Gupta [10] .

PROPOS3TION 2.1. Let (M,N) be. a.. pair .of h-compatible mappinqs from H t__ooH, with

constants a an__.d.d b • Then.~ for every f ~ H, equ..ation (2.1) has a unique sol.u.tion.

If, moreover ,

( 2 . 2 ) M(O) = 0 ,

the unique s.o..lution x o f (2.1) s a t i s f i e s the foll.pwinq estimate.

(2.3) Ix - f | ~ (a - b ) - I I N f l .

Pro pf. We only sketch the proof, details of which will be given in a subsequent

paper. Equation (2.1) is trivially equivalent to equation

y + MN(y + f ) = 0 ,

and hence to

(2.4)

i f T : H .-~ 2 H

O ~ T y ,

is def ined by

Ty = - M-1(-y) + N( f + y) .

By the assumption on M and N and basic results of the theory of maximal monotone

operators (see for example [ I] ), it is easy to show that T is maximal monotone

and strongly monotone, so that (2.4), and hence (2.1) has a unique solution. To

prove (2 .3 ) , l e t us not ice t ha t

- b l f - x~ 2 ~ ( f - x, Nf - Nx) = (MNx, Nf - Nx) = -(MNx,Nx) + (MNx,Nf)

- a~MNx~ 2 + (MNx,Nf) ~ - ~ I x - f {2 + ~x - f ~ N f ~ ,

and the result follows.

168

Under some more assumptions, one can obtain results about the continuous

dependence of the solution of equation (2.1) with respect to M, N and f .

The formulation of the following Proposition 2.2 is modelled after a theorem

of Brezis and Browder (~2] , Proposition 5) and the proof will be given elsewhere.

PROPOSITION 2.2. Let (M,N) .be a pair of h-cqmpatible m appinqs, f~ H, (Mn,Nn)n~N.

be a sequence of pairs of h-compatible mappinqs with .c.onstants a and b , and (fn)n~_N .

be e sequence of elements of H which converqes to f. AssuMe that the followinq

conditions hold :

(I} for each bounded subset 5 C H, U N (5) is bounded . n

n ~ N*

(2) For each u ~ H, M Nu- -~ MNu i f n- -P=o . n

(3) For each uC¢. H, N u - -~Nu if n---~=~. n

(4) For each n~ N*, Mn(Oi = 0

Then, if u = ( I + MN)- I f , u = ( I + M N ) - I f , (n E N*) ,

n n n n

one has

u --~u if n-.~=o. n

If we call a mapping F : H --PH bounded when it takes bounded subsets of H into

bounded subsets of H, we have the following immediate consequence of Proposition 2.2.

COROLLARY 2.1. Let (M,N) be a pair of h-compatible mapp.~nos, with M(O) = D sndN :

H--~H bounded. Then (I + MN) -I H --~H is continuous.

If we now call a mapping F : H --~H closed when F{S) is closed for every closed

subset S of H, we deduce at once from Corollary 2.1 the following

COROLLARY 2.2. Under the es@umptions of Cuorollary 2.1, I + MN : H--p H i~s

a cl,psed mappinq.

3. A CONTINUATION THEOREM OF LER.A..Y.-SCHAUDER' ~ TYPE

Let still H be a real Hilbert space with inner product ( , ) and corresponding

norm I.| , let I = [0,1] and, if E C_H and T : E x I -~H , (x,~) w-pT(x,~),

let us denote, for each ~Q I, by T~ the mapping T~ : E --m H, x ~T(x,~) .

Let now ~. C H be an open bounded set, with closure cl~ and boundary fr ~'~ ,

and let us consider the mappings M, N : H x I --~H and C : cl~. x I --~ H .

The following result is a continuation theorem of Leray-ichauder's type for mappings

which are not necessarily compact perturbations of the identity. Recall that F : E--P

H is called ~ compact on E if F is continuous on E and F(E) is relatively compact.

169

THEOREM 3.1. Assume that the mappinqs M, N and C satisfy the followinq conditions.

I. There exist real numbers a enid b such that, for each ~_ I, the pair (M , N~)

is h-compatible with constants a an_..~d b , and M~(O) = 0 for each ~I.

2. N : H x I -~ H is bounded . and, for each x ~_H, the mappinq

p ~. N(x, p)

is continuous.

3. For .each x~ H and each V~ I, the mappinq

~ ~-~ M(N(x,y ) , ~ )

is compact on c l U • Moreover, i t follows from condition (4) that

(O,O) ~ U .

Now, by Corollary 2.2, T~ is a closed mapping for each ~E I, and hence

cI(TF(~L)) c T (cl ~ ) .

Therefore, T~ being moreover one-to-one, one has

f r T (-~) = c l (TB(~) ) k in t (TM(~)) = cI(TM(3"L)) ~ T (~L) C

C T ~ ( c l ~ ) ~- T~.(J'L) = T ( f r - ~ ) ,

which, together with assumption (6), implies that, for every ( y , ~ ) ~ f r U, one has

y + K(y ,~ ) ~ 0 .

by

Proof. Let us define, for each ~ I, the mapping T~: H --~H by

T M = I + M p N p .

It follows therefore from the assumptions (I) to (3), Proposition 2.1 and Proposition

2.2 thst the mapping

is defined, continuous and bounded on H x I . Consequently, the set U defined by

U = { ( T B x , ~ ) : ( ~ , ~ ) ~ ~ . x I ]

is an open bounded subset of H × I , and, using assumption (5), the mapping K defined

is continuous.

4. 0 ~ (I + MoNo)(D~) .

5. C : cl~ x I ~ H is compsct and C O = 0 .

for every ( x , ~ ) ~ f r ~ . x I .

Then, for each ~I, equation

X + MpN~x + C x = 0

has at least one solution x ~ ~L .

170

As, by assumption (5),

K 0 = 0 ,

all the conditions are satisfied to apply the usual Leray-Schauder's continuation

theorem [14] to the family of equations

(3.2) y + K ( y , ~ ) = 0 , ~ 6 I ,

implying that, for each ~ E I, equation (3.2) has at least one solution y such that

( y , p ) ~ U,

and hence, l e t t i n g

y = Tpx , x E ~ C ,

which i s possible by d e f i n i t i o n o f U, we see tha t x

the proof is complete.

satisfie equation (3.1) and

REMARK. Theorem 3.1 is distinct from but in the spirit of generalized Leray-Schau-

der's type continuation theorems due to Browder [ 73 and Br@zis and Browder [2].

4. A CONTINUATION THEORE M FOR SOME NONLINEAR PERTURBATIONS OF LINEAR MAPPINGS WIT H

AN INFINITE DIMENSIONAL KERNEL

Let still H be a real Hilbert space with inner product ( ~ ) and corresponding

I'~ We shall be interested in existence results for equations in H of the norm

form

(4.1)

where

Lx = Nx

L is a linear and N a not necessarily linear mapping from H to H satisfying

some regularity assumptions we shall describe now. Following the line of recent

work by Mawhin [16], Br~zis and Haraux ~] and Br~zis and Nirenberg ~] , and in

contrast with most of the literature devoted to equations of type (4.1) (see for

example the su rveys tB ] , D I ] ,D3] ) we shall not assume tha t ker L is of f i n i t e

dimension.

More precisely, let L : dom L~ H -~PH be a closed linear operator with

dense domain dom L and closed range Im L such that

(4.2) Im L = (ker L) "L

Consequently, using the closed mapping theorem, the restriction of L to dom L ~ Im L

is a one-to-one linear mapping onto Im L with a continuous inverse

K : Im L--P dom L ~ Im L .

We shall denote by P the orthogonal projector in H onto the (closed) subspace ker L ,

so that condition (4.2) is equivalent to

Im L = ker P .

Let now N : H--~ H be a bounded mapping which is moreover demi-continuous on H,

i.e. such that Nx --~ Nx for every sequence (Xn) n which converges to x in H. n ~ N*

171

Recall that a mapping F : H-~H is called monotone if, for every xE H and every y ~ H,

ore has

(Fx - Fy,x - y) ~ 0 .

THEOREM 4.1. Let us assume that the mappinqs L, N, K, P defined abpve verify the

fol.lowin.q F onditions :

I. I - P - N : H--~ H is monotone.

2. Ther# exists an open bounded set ~).C H such that 0 ~ ~ and

i. K(I - P)N is compact on cl ~- .

ii. Lx + (I - ~)Px - ~Nx ~ 0 for all (x,~) ~ (don L t3 fr ~. x ]0,I~ .

Then, equation (4.1) has at least one solution

Proof. First step. Let us first show that, for every ~]0,I~ , the equation

( 4 . 3 ) Lx = - ( 1 - ) . )Px + ~ N x

has at least one solution x ~ dom L (~ ~L By a result of~53 (see also't33 ),

equation (4.3) is equivalent to the equation in H

(4.4) x +%P(-P - N)x -%K(I - P)Nx = O

and hence, by assumption (2.ii), one has

(4.5) x +~P(-P - N)x - ~ K ( I - P)Nx ~ 0

for every (x,~) ~ fr ~. x ]0,1[ , because the solutions of (4.4) are necessarily

in dom L . Let ~ ~0,I~ be fixed; then, one immediately obtains , using the

orthogonal character of P and assumption (I), for every x G H and y E H, and each ~(I,

(%(Px - Py),x - y) =~|Px - py|2 = X-1]%px _ %py]2 ,

(F(-Px - Nx + Py ÷ Ny),x - y) ~ -~Ix - y l 2 ~ - I× - y|2 ,

so that the pa i r (~P,~-P-N)) is h-compatible with constants ~-I and I f o r every ~ 6 I ,

P(O) = O, the mapping (x, ~ ) ~-~ ~(-Px - Nx) bounded, the mapping ~ - F(-Px-Nx)

continuous fo r every x ~ I , the mapping ~ ~-~P(y( -Px-Nx) ) constant and hence cont i -

nuous f o r every (x, ~ ) ~ H x I , and I + ~pP(-P - N) reduces to I fo r ~ = 0 .

Now the mapping (~, ~ ) ~ K ( I - P)Nx is c lear ly compact on c l J~L x I , vanishes fo r

~ - : 0, ands, by ( 4 , 5 ) ,

x + ~.'.XP(-P - N )x - f f " ~ K ( I - P)Nx #~ 0

for avery (x,~)~ fr ~ x I, because, when ~= 0 (the only case not covered by (4.5)},

this is implied par the condition 0 ~-~ . The result follows therefore from Theorem 3.1.

Second step. ~ being bounded, it follows from the first step, the compactness of

K(I . P)N on cl~ ~ . and the properties of L that one can find sequences (~ ) n n~N* '

(~n)n~.N . , auch that )~n--.~ 1 if n--~oo, X h ~ ~ 0 ~ , n ~ - N ~ ,

(4.6) Lx = - ( I - Xn)PX n +~ Wx , n ~-N*, n n n

172

and, if we write

Yn = ( I - P ) x , z = Px , n n n

such that, for some y~ H and z E H,

(4.7) yn~y , Zn---~ z and LXn = Ly n ~ Ly if n--~.

Now, from the monotonicity of I - P - N, we deduce, for every n ~_N* and v ~_-H, that

(Yn - NXn - ( I - P - N ) v , x - v ) ~ O , n ~ _ N * , n

and hence, using (4.6),

(4 .8 ) (Yn - A n 1(1 - )~n)zn - 'Xn lLYn - ( I - P - N)v, Xn - v) ~' 0 , n ~ N*.

Using (4.7), we can go to the limit in (4.8), which gives

(y - Ly - {I - P - N)v, x - v) ~ 0

for every v ~ H. Since I - P - N is cZearly maximal monotone, this implies

and hence

y - Ly = ( I - P - N ) x = y - Nx ,

Lx = Ly = Nx

and the proof is complete.

REMARK 4.1. It is immediately checked that -L has the same properties as L, with

the same orthogonal projector P. Consequently, the assumption I - P - N is

monotone can be replaced by

I'. I - P + N : H ---~H is monotone

if (2.ii) is replaced simultanuously by

2.ii'. Lx - (I - ~)Px -%Nx ~ 0 for all (x, ~)~ (dam L~ fr~L) x ]O,i[ .

Of course, the monotonicity of I - P - N (resp. I - P + N) is implied by the

monotonicity of -N (resp. N), as it is easily checked because I - P is monotone.

5. APPLICATION TO SOM E ABSTRACT LANDESMAN-LAZER PROBLEMS

We shall show in this section how Theorem 4.1 allows simple proofs for abstract

Landesman-Lazer problems, i.e. results about the range of L - N when N satisfies

some growth restrictions. The first theorem is modelled after a result of Cesari

and Kannan Z 9] for the case where ker L is finite-dimensional. This assumption

is suppressed here at @e exp.nse of a monotonicity condition on I - P - N or I - P + N.

COROLLARY 5.1, Let us assqme that the mappinqs L, N, K, P verify the conditions

listed at the beoinninq of Section 4 an d that the followinq assumptions hold;

a. I - P - N : H--~H is mop£tone.

b,' K(I - P)N : H--~H is completel.y continuous, i.e. continuous and such that

K(I - P)N(B)~s relatively compact for every bounded subset B of H.

c. There exists r~ 0 such that, for all x~ H,

i K ( I - e ) N x ~ r .

173

d. There exists R > 0 such that I for all x E H for which

IPxt = R and |(I - P)x~ ~ r ,

one has

(Nx,Px) ~ 0 .

Then t equation (4.1) has a t leas t one so lu t ion .

Proof. We shall apply Theorem 4.1 with the open bounded subset~ of H defined by

Q- = ~ x E H : I P x | ~ R and | ( I - P ) x ~ < r } ,

so that

with

fr ~'L = 51 U 5 2 ,

51 = ~ x G H : I P x l = R and l ( I - P)xl_~r~ ,

S 2 = {X ~ H : IPxl ~- R and l ( I - P)xI= r } •

By applying P and I - P to equation (4.4), we see that, for each ~]0,I[, equation

(4.3) is equivalent to the system

(5.1) ( I - P)x = ~ K ( I - P)Nx

(5.2) - ( I - ~)Px + ~PNx = 0 ,

and hence every possible solution of (4.3) is necessarily such that

(5.3) l(I - P)xI = ~ K ( I - P ) N x ~ Xr < r ,

i.e. such that x ~ 5 2. Now, (5.2) implies that, for every ~]0,I~ ,

- ( I - X) IPxl 2 + k(Nx,Px) - 0 ,

because, by the or thogonal i ty o f P, (PNx,Px) = (Nx,Px) = (PNx,x). Therefore, by

assumption (d) and (5.3), one has x ~ 5 I. Thus, assumption(2) of Theorem 4.1 is

satisfied with our choice of-~ and the result follows.

REMARK 5.1. By using Remark 4.1, one easily obtains that, in Corollary 5.1,

conditions Ca) and the inequality in condition (d) can be simultanuously and respectively

replaced by

a'. I - P + N : H~H is monotone.

d'. (Nx,Px) ~ 0

with the same conclusion f o r the equation (4 .1) .

We shall now deduce in a very simple way, from Theorem 4.1, a recent result of Br~zis

and Nirenberg[4] . Assume that L : dom LC H --~H satisfies the conditions listed at

the beginning of Section 4. Then, since, for every x E dom L,

l ( I - P)xI = ~KLx ~ ~; IK I |Lx | ,

one has 2

(Lx,x) = (Lx,(I - P)x) ~ -$Lx| l(I - P)x~ ~ - IKllLx|

for every x ~ dom L. Let us denote by @ the largest positive constant such that

174

(5.4) (Lx,x) ~ - ~-I--|Lx~2 ,

for all x~ dom L. Let now B : H-~H be a bounded demi-continuous mapping. We have

the following result.

COROLLARY 5.2. Assume that the mappinqs L ands satisfy the above conditions and

that the followinq conditions hold;

a. I - P + B : H~ H ~s monotone.

b. K(I - P)B : H--~ H ..i..s...complete.l.E..continuous.

c. There exists ~ , with O f ~ ~, such th..a.t,, for all x~H and y ~ H,

(5.5) (Bx - By,x) >~ ~-I~Bx%2 - c(y)

where c(y) depends only on y.

Then,

i n t ( Im L + cony Im B) ~ Im(L + B) .

Proof. Let

f ~ i n t ( Im L + cony Im B)

and let us apply Theorem 4.1 with N = f - B. By the assumptions of regularity made

on L and B, it suffices to prove that the set of solutions of the family of equations

(5.6) Lx + (I - X ) P x +kBx = %f

is a priori bounded independently of ~]0,I[ , because then condition (2) of Theorem

4.1 will be satisfied with ~ an open ball of center 0 and radius sufficiently large.

For all h ~ H of sufficiently small norm, we can write n

f + h = Lv + ~ - t . Bw , I l

i=I n

where v ~ don L, w i ~-H, t i ~ 0 (1~i~_n) and ~ t i = I . i=1

(5.6) that n

I t fo l lows then from

~-1~Bxl 2 - ~ t . c (w . ) + ( h , x ) ~ L v | l ( I - P)x{ + (~@)-IILx~2 . i i i=I

But, by (5.6) and (5.1) with N = f - B , one gets eas i ly

~Lx~(~( ~x~ + If~ ) , ~(I - P)x |6 XIK~(~BxI + |f~) ,

which, together with (5.7), gives, for all h E H of sufficiently small norm,

( h , x ) ~ ( 8 - I ~'- l ) [Bx~2 + c ' (h) IBx~ + c"(h)

and hence

(5.7)

-~ ~ l L v | | ( I - P)x~ + @ - l l L x l 2 ,

(1 - ~ ) P x + ~ ( B x - ~ t . B w , ) * ~ h = ~ L v - Lx . l z i=1

Taking the inner product of this equality with x and using (5.4) and (5.5), we obtain

(1 - X ) l P x l z + A ~ t i ( ~ r - l l B x l 2 - c(w.))~. ÷ X ( h , x ) ~ X(Lv,x) + e - l l L x I 2 -~ i=1

175

where c' and c" only depend from h. As @-I ~ -I , this implies that

( h , x ) ~ c" ' (h )

for some c"t(h), and hence, by the Banach-Steinhaus theorem, there will exist R > 0

such that

I x l ~ R , which achieves the proof.

REMARK 5.2. More general versions have been given by Br~zis and Nirenberg in [4]

and they could be treated similarly. We have restricted to this one for simplicity.

Let us also notice that Br~zis and Nirenberg assume that B is monotone instead of

I - P + B .

COROLLARY 5.3. If r in Corollary 5.2, condition (a) is replace db~

a'. B : H-~H is monotone and B is onto,

~hen L + B is onto.

Proof. By Corollary 5.2,

int(Im L + cony Im B) = H ~ Im (L + B) ~ H .

In particular, by the theory of maximal monotone operators, B will be onto if

~Bxl--*~ if Ix~---~,w~ .

We shall refer to [4] for an interesting application of Corollary 5.3 to the

existence of generalized solutions in L 2, 2Tt-periodic in t and x, of the nonlinear

wave equation

=

utt - Uxx f(t,x,u) .

A generalization of some of the results of [17]is obtained which replaces some

Lipschitz condition in u for f by a monotonicity assumption.

6. AN APPLICATION TO A SECOND ORDER PERIODIC BOUNDARY VALUE PROBLEM. IN A HILBERT SPACE

In this Section we shall apply Theorem 4.1 to the second order periodic boundary

value problem

(6 .1)

(6.2)

where x' = dx/dt

- x " ( t ) = f ( t , x ( t ) ) , t ~ I = [0 ,1 ] ,

x ( O ) - x ( 1 ) = x ' ( O ) - x ' ( 1 ) = 0 ,

and f : I x HI--~ HI, with H I a real Hilbert space with inner

product ( , )I and corresponding norm |'~ I " One could consider similarly the

Neumann boundary conditions

x'(O) = o = x ' ( 1 )

which share with the periodic ones the feature of furnishing a non-invertible linear

part in the abstract formulation of (6.1-2). When the boundary conditions make this

linear part invertible and f is completely continuous, rather general results for

the solvability of the corresponding boundary value are available, even for H I a

176

Banach space (see e.g.[20] ). But, in contrast with the situation where H I is finite-

dimensional, the general case with periodic or Neumann boundary cnnditions is much

more difficult and ~es not seem to have been explored. Hence, even the very special

results we obtain here may be of interest.

= L2(I,HI) with the inner product Let H

= ~ (u(s),v(s))ds (u,v) J I

and the corresponding norm ~.~ , end let us define N on H by

( N x ) C s ) = f ( s , x ( s ) ) a . e . on I

for x ~ H. To avoid lenghty technical discussions, which will be given in a

subsequent paper, we shall directly make our regularity assumptions on N instead of on

the original mapping f . Let us assume that :

I. N maps H into itself in a completely continuous way.

2 . ( = 1 r > O ) ( V x ~ H ) : INx I ~ r .

3. N is monotone, i.e.

(f(s,x(s)) - f(s,y(s)),x(s) - y(s))ds ~ 0 I

for all x~ H and y~H .

Let us now define dom L ~ H by

dom L = {xEH : x is absolutely continuous in I together with x', x"~= H and

x(O) - x ( 1 ) = x ' ( O ) - x ' ( 1 ) = o ~ ,

so that dom L is a dense subspace of H~ let 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 : ~I x(s)ds = 0 } = (ker L) "L

ker L = Im P with P : H--~H the orthogonal projector onto ker L defined by

= ~ x ( s ) d s . Px I

THEOREM 6.1. Assume that the condit~pns above hold for N and that there exists

R~ 0 such that~ for a.e. t E1 and all x~ H I with ~x|1~ R, one has

(6.3) ( f ( t , , x ) , x ) I ~ (r/21¢) 2 .

Th.en ~ problem (6.1-2) has at least a (~aretheodor.v) so lu t ion.

.Proof. We shal l apply the var iant of Theorem 4.1 mentioned in Remark 4.1 to the

equivalent abstract equation in dom L ~ H

177

Lx = Nx .

C lear ly , the sssumptions we have made imply that I - P + N i s monotone and K(I - P)N

completely continuous (one shall notice that for H I infinite-dimensional, K is a

continuous but not s compact mapping). It suffices therefore to show that the

possible solutions of the family of equations

(6.4) Lx = (I - A)Px + $~Nx , X E ] O , I [ ,

are a p r i o r i bounded. By applying I - P and P to both members o f (6.4) , we obtain

Lx = ~ ( I - P)Nx ,

O = (I - %)Px + APNx , (6.5)

and hence,

I x " l = I L x I ( I N x | ( r .

Letting 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 Iv(t)| _~ ItrI((21~)23 I/2) = r' • I

t~.I

Therefore, if lu~ = |u | 1 ~ R + r', one has, for all t~ I,

I x ( t ) ~ l ~ l U l l - max l v ( t ) l 1 ~, R , t ~ I

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 ) - ~ ( N x , v ) .

Consequently, using (6.G) and (6.7)~ one gets

0 >~ (1 - ~ ) | u 1 2 + A ( r / 2 ~ ) 2 - ~ ( r / ( 2 I t ) 2) = (1 - % ) l u l 2 ~ (1 - ~)(R + r ' ) 2 ,

a c o n t r a d i c t i o n . The re fo re

l u l 1 < R + r '

and hence, by (6.6) ,

Ix l 1 ~ lul 1 + Iv~ 1 < R + r' + ( 2 T c ) - 2 r ,

and the proof is complete.

178

7. A CONTINUATION THEOREM FOR SOME NONLINEAR PERTURBATION OF LINEAR MAPPINGS IN

REFLEXIVE BANACH SPACES

Let X be a real reflexive Banach space, Z a real normed space,

L : dom L~ X --~ Z

a linear mapping such that there exist continuous projectors P : X-~X, Q : Z-~Z

for which

ker L = Im P, Im L = ker Q ~ ~P|=~

and such that there exists a linear homeomorphism J : Im Q--~ker L . Let us

denote by Kp,q : Z --~ dom L~ ker P the linear mapping Kp(I - Q) where Kp : Im L

dom L ~ ker P is the inverse of the one-to-one and onto restriction of L to

dom L ~ ker P . Let I~LC- X be a bounded open convex subset with O~L, and

let N : cl~L ~ Z be a (not necessarily linear) mapping such that the mapping

(dq + KpQ)N : cl~l-C X --~ X p

is stronqly continuous on cl~- , i.e. such that

(JQ + Kpq)N(x n) ~ (JQ + Kpq)N(x)

for every sequence ( x ) N* in cl3~- such that n n ~

x ~ x i f n---~ . n

Let us reca l l that the strong cont inuity on c l ~ implies the compactness on c l ~ . ,

for X a reflexive ~anach space, but the converse is not true (see e.g. L 12~ ).

THEOREM 7.1. Assume that L and N satisf~ the conditions above and that

(7.1) Lx ~ -(I - ~)J-Ipx + ~Nx

for every (x, %) ~ (dom L ~ fr &"L) x ]0,1~ . Then equation

(7.2) Lx = Nx

has at least one solution.

Proof. As shown in [153 (see also [133 ), equation

( 7 . 3 ) Lx = - ( 1 - ~ ) J - 1 p x + ~Nx

i s e q u i v a l e n t t o t h e e q u a t i o n

x - Px = (JQ + K p ~ ) ( - ( 1 - ~ ) J - 1 p x + ~ N x ) ,

i.e. to the equation

x - ~Px = X(JQ + Kp,Q)NX ,

in cl~ . Let us fix ~]0,I[ and consider the family of equations in cl~ ,

(7.4) x - ~Px = ~(JQ + Kp,Q)Nx , ~ ~ I = [0, I~ .

For each ~ ~ I, the mapping I -~P is a linear homeomorphism of X onto itself

with

( I -~XP) - I = (I - ~ ) - I P + ( I - P) ,

and hence the family of equations (7.4) is equivalent to the family of equations

179

x = (I - h ~ ) - 1 ~ J Q N x + ~KpQNx = TA(x ,~) , ~ i ,

in c l~rL . By our assumptions and the r e f l e x i v i t y o f X, T~ : cl~'Z x I ~ X is

a compact mapping, and by (7.1) with ~ replaced by ~ and the equivalences of

equations described above,

x - T~(x, /~) ~ 0

for every (x,~) E fr~.x I, the validity for ~= 0 being a consequence of the

condition O ~ . Therefore, using the Leray-5chauder's continuation theorem,

T (.,I) has at least a solution in ~ t and hence equation (7.3) has at least one

solution in dom L A~ , and that for every ~]D,I[ . Let now (~n)n ~ N* be e

sequence in ]0,I[ which converges to I and (Xn) n~N. a sequence in dom L ~ such

that

Therefore,

(7.5)

Lx = - ( I - X )J-Ipx + ~ Nx , n E N * . n n n n n

Xn " ~nnPx = ~ n (JQ + KpQ)NXn , n ~ N* .

I f we w r i t e x = Yn + z w i t h Yn = PXn ' n ~ N* , t h e n ( Y n ) n ~ N . and ( Z n ) n ~ N . a r e n n

bounded and, hence, the compactness of (JQ + KpQ)N and the reflexivity of X imply

that, going is necessary to a subsequence, one has

PXn = Yn .~b y ~ ker L , (JQ + KpQ)NXn --~. z EX,

so that, by (7.5),

z - - ~ z E k e r P i f n - ~ ¢ ~ . n

Consequently,cl~being weakly closed,

x ~ y + Z = x i f n--~o~ , and x ~ c l ~ , n

and then, by the strong cont inu i ty o f (JQ + Kp~)N, one gets

(JQ + KpQ)N(x n) --~(JQ + KpQ)N(y + z) •

The uniqueness of the limit implies that

z = (JQ + KpQ)N(y + z ) ,

i . e .

x - Px = (JQ + KpQ)Nx J

which is equivalent to (7.2) as noticed above. Thus the proof is complete.

REMARK 7.1. It is easy to check that Theorem 7.1 can be used instead of Theorem 4.1

to prove the existence of a solution for the periodic boundary value problem (6.1-2) is

assumptions I and 3 on N are replaced by the assumption

I'. N maps H into itself and is strongly continuous on every bounded subset of H ,

the proof being entirely similar to that of Theorem 6.1. It is an open problem to us

to know if the result still holds if one replaces in (I') the strong continuity by the

complete continuity without the monotonicity condition used in 5ection 6.

180

REFERENCES

I. H. BREZIS, "Op~rateurs maximaux monotones et semi-groupes de contractions dens les espaces de Hilbert", Mathematics Studies 5, North-Holland, Amsterdam, 1973.

2. H. BREZIS and F.E. BROWDER, Nonlinear integral equations and systems of Hammerstein type, Advances in Math. IB (1975) 115-147.

3. H. BREZI5 et A. HARAUX, Image d'une somme d'op~rateurs monotones et applications, Israel d. Math. 23 (1976) 165-IB6.

4. H. BREZI5 and L. NIRENBERG, Characterizations of the ranges of some nonlinear operators and applications to boundary value problems, Ann. S puola Norm. Sup. Pisa, to appear.

5. F.E. BROWDER, Existence of periodic solutions for nonlinear equations of evolution, P~Rc- Nat. Ace d. 5ci. U.S.A. 53 (1965) 1100-1103.

6. F.E. BROWDER, Periodic solutions of nonlinear equations of evolution in infinite dimensional spaces, in "Lectures in Differential Equations", vol. I, A.K. Aziz ed., Van Nostrand, New York, 1969, 71-96.

7. F.E. BROWDER, "Nonlinear Operators and Nonlinear Equations of Evolution in Benach Spaces", Proc. S~mp. Pure Math., vol. XVII, part 2, Amer. Math. Soc., Providence, R.I., 1976.

8. L. CESARI, Functional analysis, nonlinear differential equations and the alternative method, in "Nonlinear Functional Analysis and uifferential Equations", L. Cesari, R. Kannan and J. 5chuur ed., M. Dekker, New York, 1976, 1-197.

9. L. CESARI and R. KANNAN, An abstract existence theorem at resonance, Proc. Amer. Math. Soc. 63 (lg77) 221-225.

10. D.G. BE FIGUEIREDO and C.P. GUPTA, Non-linear integral equations of Hammerstein type with indefinite linear kernel in a Hilbert space, Indaq. Math. 34 (1972} 335-344.

11. 5. FUCIK, "Ranges of Nonlinear Operators", 5 volumes, Universites Carolina Pragensis, Prague, 1977.

12. S. FUCIK, d. NECA5, J. SOUCEK, Vl. 50UCEK, "Spectral Analysis for Nonlinear Operators", Lecture Notes in Math. n ° 346, Springer, Berlin, 1973.

13. R.E. GAINES and J. MAWHIN, "Coincidence Degree and Nonlinear Differential Equations", Lecture Notes in Math. n ~ 568, Springer, Berlin, 1977.

14. J. LERAY et J. 5CHAUDER, Topologie et @quations fonctionmelles, Ann. Sci. Ec. Norm. ,5,~. 51 (1934) 45-7B.

15. J. MAWHIN, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces, ~. .D i f fe ren t ia / .E~uations 12 (1972) 610-636.

16. J. MAWHIN t Contractive mappinge and periodically perturbed conservative systems, Arch. Math. (Brno) 12 (1976) 67-73.

17. J. MAWHIN, Solutions p~riodiques d'~quations aux d@riv~es partielles hyperboliques non lin~aires, in "M~langes Th. Vogel", B. Rybak, P. Janssens et M. dessel ed., Presses Univ. de Bruxelles, Bruxelles, 1978, 301-315.

18. J. MAWHIN, Landesman-Lazer's type problems for nonlinear equations, Confe~ t Sam. ~t. Univ. B ari n= 147, 1977.

181

19. J. MAWHIN and M. WI|LEM, Periodic solutions of nonlinear differential equations in Hilbert spaces~ in "Proceed. Equsdiff 78",Firenze 1978, to appear.

20. K. 5CHMITT and R. THOMPSON, Boundary value problems for infinite systems of second- order differential equations, J. Diffe!en~al Equations 18 (1975) 277-295.

SOME CLASSES OF INTEGRAL AND INTEGRO-DIFFERENTIAL

EQUATIONS OF CONVOLUTIONAL TYPE ~)'

E. Meister (TH Darmstadt

Abstract .

S ta r t i ng wi th c lass ica l convo lu t iona l in tegra l equations on ~ n , t r a n s l a t i o n - i n v a r i a n t operators and t h e i r symbol representa t ion according to H~rmander are introduced. The var ious genera l i za t i ons concerning the domains G o f ~ n t e g r a t i o n lead to Wiener-Hopf i n teg ra l and i n t e g r o - d i f f e r e n t i a l equations on ~+ and on cones. Compound in tegra l and i n t e g r o - d i f f e r e n t i a l equations of the p r inc ipa l value and L1-kernel type are discussed on ~n using resu l ts by Rakovsh~ik. Simonenko's theory of local type operators permits us to i nves t i ga te genera l ized t~ans la t ion i n v a r i a n t operators. Wiener-Hopf in tegra l equations wi th s t rong ly s ingu la r kernels correspond to equations wi th piecewise continuous symbols in both va r i ab l es . Convolut ional equations on quadrants and wedges are studied v ia the theorey of operators of b i - l o c a l type.

O. Notat ion. In the sequel the f o l l ow ing abbrev ia t ing terminology is used:

]R n : = {x : x = (x I . . . . . Xn), x EIR} : n - dim. Euclidean space with

n (0 . I ) <x,y> : = Z xvy v : sca lar product

~=1

(0.2) Ixl : = <x,x> I /2 : length of vector x

(0.3) IR n • = ]R nu{~} : one-po in t -compac t i f i ca t ion

(0.4) ~ • = iRnu ~r : ray -compac t i f i ca t ion by adding one ideal X i n f i n i t e element to each d i r e c t i o n e : = - ~ on

~n : n-sphere = {xE]R n : Ix l = I }

and completing the topologies by the usual way

" = (~1 . . . . . Un)eIN n : mu l t i - i ndex

n

Ipl : = z G Cmo ~=1

(o.5)

~1 ~n (0;6) D ~ " = D l . . . ' D n where

(0.7) D : = i ~ ; ~ = I , . . . . n

*)Extended vers ion of a General Lecture at the Dundee Conference on D i f f e r e n t i a l Equations, 31st March 1978.

183

(0.8)

(0.9)

(0.10)

(0 .11)

(Q.12)

Pl ~n : = ~t " " ' ~ n ' ~ : dual v a r i a b l e to

: measurable subsets

: c h a r a c t e r i s t i c f u n c t i o n o f se t E

= ) O(X) - i and G = ~R n put l l f i I p : I I f l l LP(IR n )

~P G, Ec~ n

XE(X) LP(o;G) : = space of equ iva lence c lasses of f u n c t i o n s f such t h a t

II f I l " = IS P ( X ) ' I f ( x ) I p d x ] l / p < = , l~p<~ , or LP(p;G) G

i I f l l • = ess sup p ( x ) . I f ( x ) I < ~ , r e s p e c t i v e l y , L=(o;G) xEG

f o r a measurable we igh t f u n c t i o n p (x ) ~ 0

( i n case o f

: space of k - t imes c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s

on G;kc~ o u{~}

: space of f u n c t i o n s fcck(G) such t h a t f is c o n t i n u o u s l y

ex tendable to G : = GU~G w i t h

ck(G)

ck(~)

II f l l ck (~ )

C~(IR n

• = max sup I D~ f (x ) l < ~ " kE~ O~l~ l~k x ~ ' o

• = { fEck( IR n ) : l im D'#f(x) : 0 f o r O<l~l~<k} ; kE~ o I x I ~

tr • = {mEC~(IR n ) : sup ( l+ I x l 2 )k /21D '~m(x ) l < = f o r k E ~ o , ~ e ~ n}

(0 .13)

(0 .14)

(0 .15)

(0 .16)

F

(Fm)(~) =8(~) : =

w i t h the i nve rse

( F - I ~ ) ( x )

F is de f ined on

<F f , m>

s ince F maps

(FD~m)(C)

Schwar tz 'sspace o f r a p i d l y decreas ing f u n c t i o n s

: the space of tempered d i s t r i b u t i o n s dual to

: n -d im. F o u r i e r t r a n s f o r m a t i o n de f ined by

i ei<X ~ n f '~>m(x)dx , ~E~ n

- a t ~ l e a s t on ~ -

i f e - i<x ,~>~(C)d~ 2J2~-~ n ~n

~' by

• - < f ,F - lm> f o r f E ~ ' and mE~,

b i j e c t i v e l y onto i t s e l f , w i t h p rope r t y

= ~ (Fm) (C ) = ~ ( ~ ) f o r mE~ and

184

(0.17)

(0.18)

<DUf,~> = ( - l ) lh I< f ,DU~> fo r fE ~'

GM • = {~EC~(~n) : lDU~(x) l~ Ipu(x) l f o r a l l xE~ n}

where p~ is a polynomial depending on ~ and ~ :

"space of m u l t i p l i e r Functions of J~ and ~ ' "

3E , ~:: Banach spaces wi th norms I I ' I I z and I I ' i i ~

m(m,9)

~(~,t~) : subspace of a l l compact operators

~(~,~) : set of Fredholm-Noether operators

ac ~(Z,~) i f f ( i ) ~ ( A ) : = A(~C) is closed in

( i i ) ~(A) : = dim ~(A) = dim ker A <

( i i i ) ~(a) : = dim ~/R(A)= dim coker A <

~(A) = ind A : = ~(A) -~(A) : " index of A".

In case of ~ = JC one wr i tes ~(~) etc.

: Banach space of a l l l inear and bounded operators

A : ~ ÷ I ~

~(~,~)

1. In t roduct ion.

De f i n i t i on 1.1: AI ,A2E~(~,~) are ca l led "equiva lent" i f f AI-A2E~(~,V)

Or: = {BC~(~,~) : B~A} gives the quotient-space with norm

( 1 . 1 ) l lc~l : = i n f II A+VII

There is the well-known theorem by ATKINSON (1951) [2 ] :

and

Theorem 1.1 : Let ~ , ~ be B-spaces then AE~(~,~) is Fredholm-Noether, i . e~

E~(~,~) , i f f there ex i s t a " l e f t - r e g u l a r i z e r " B~E~(~,~) ~ a " r i g h t - r e g u l a r i z e r "

B r E ~ , ~ , V IE~Z ) , and V 2 E ~ ) such that

(1.2) B~A = IZ + V 1 and AB r = I~ + V 2

are Fredholm-Riesz operators.

t85

Remark 1.1 : This has been generalized to pairs of Fr~chet spaces ~, l~ and

linear, densely defined, closed operators A (cf, eg, GOHBERG & KRE[N (1957) [42]

or PRUSSDORF (1974) [71]. There you may find the properties of Fredholm-Noether operators listed.

Let kELl(~n) , ~ELp (~n) , 1~p~ , then

- k (x-y)~(y)dy ex is ts a, e~ on ~n (1.3)

and

II kll II ~lIp holds ( 1 . 4 ) Ii k~mllp I" "

The Fourier t ransformat ion may be continued from ~ onto Lp ( ~ n ) , 1~p~2 , such

I i = I . F is then un i ta ry on L2(~n) that FE~(LP(~ n), LP'(~n)) where ~ ÷ ~,

The "convolutional integral equation of the second kind"

(1.5) (Am)(x) : = re(x) - (k*m)(x) = f(x)ELP(m n) is algebraized by F to

(1.6) [1-~(~)]~(~) ~({)EL p' : (m n )

1<~<2

(c f . e@. TITCHMARSH [ 101]).Asolution ex is ts - then uniquely - i f f the "symbol of A":

(1.7) ~a(~) : = Z-k({ ) # 0 on ~ n

I t is given by

; (~) : ~(~) + ........ k(~) .~(~)

(1.8) 1-C(~) A

: [1 + ~(~)] . f (~ )

where 1 + ~(C) is an element of the "Wiener-algebra" ~ : = { ÷ FLI(~n) the Wiener-Levi theorem.

On the other hand let

due to

(1.9) (p(D)@)(x) : S a .(D~m)(x) : f ( x ) e ~'

be a d i f f e r e n t i a l equation of order m wi th constant coe f f i c i en t s . Applying the

F-transformat ion th is y ie lds

(1.10) p(~).~(C) = ~(~)E ~ ' .

This leads to the "problem of d i v i s i on " which asks fo r condit ions under which

( I . I i ) ~(~) = ~ E 3 ~'

186

gives the transform of a so lu t ion. Taking f = ~ c 9 ' one is led to the question

of existence of the "fundamental so lut ions" EE9 ' such that

~(~) = [p(~)]- I E~,.

Convolutional in tegra l equations and d i f f e r e n t i a l equations with constant coe f f i -

c ients are a l l special types of the fo l lowing operators:

De f in i t i on 1.2a) Let 3E be a B-space of funct ions or d i s t r i bu t i ons on R n and

l e t T h : ~ + 3 E be defined by

(1.12) (Thm)(x) : = m(x-h) fo r mc~ and hER n

then ~h is cal led a " t rans la t ion operator on ~ ". I f fC3E ~, the dual o f 3 E , !

then Th : 3(1' ÷ 3( ~ is defined by

(1.13)

on

< ~ f,m> : = <f , ~_hm> for f c ~ ' and me~ .

b) A e ~ ( 3 ( , ~ ) - or at least l i near , densely defined, and closed

- is cal led " t rans la t ion invar ian t " i f

(1.14) (AThm)(x) = (ThAm)(x) fo r mc3E and hC~ n .

In th is case one wri tes AC~(JE,~). H~RMANDER [51] proved in 1960

Theorem 1.2 : Let ~ c 3 E , ~ c ~ be B-spaces of funct ions on ~n _ or the i r

duals - where ~ is dense in the topologies of ~ and ~ , resp., AE ~(~,~).

Then there ex is ts a uniquely defined ~A C ~' such that

(1.15) Al~m = F - I OA.Fm for a l l ~ ~ .

~A is cal led the "symbol of A".

I f JL~E,~) denotes the set of a l l s~T~ibols to A E ~ ) - also cal led ' ~ u l t i -

p l i e r funct ions" - then i t is known that [51] J~(k2(~ n ) ) = L~(~ n)

~ (LP(~ n) c L~(~ n)

~V~(LP(m n) , Lq(mn)) = {0} i f l ~ q < p ~ .

Extending to Frech~t-spaces one has

or fo r i t s dual ~' :

Now, e. g. on

(1.16) (am)(x) = (F-loA.Fm)(x) = f ( x ) E k 2 ( ~ )

is found via F- t ransformat ion:

(1.17) ~a(~). ~(~) = ~(~) e L 2 ( ~ ) .

I t ex is ts f o r every f (uniquely) i f f A

and hence [OA(E)] - Ic L~(R n ) , given by

(1.18) ~(~) = ~ 1 ( ~ ) . ~ ( 5 ) , thus

(1.19) m(x) = ( F - l ~ A Z ( ~ ) ' F f ) ( x ) , i . e.

A -I : F-IGA1.F e ~(L~ n))

and C ~ ( L 2 ( ~ n ) ) too .

1~p<2 th is argument works only i f Remark 1.2 : For

187

L2(• n) , the so lu t ion to the t rans la t i on i nva r ian t equation

is " e l l i p t i c " , i . e. in f I~A(~) l > 0 ~c~ n

OAIEj{(LP(LRn)) which is

the case fo r s u f f i c i e n t l y smooth

Examples:

(1.20)

i t s symbol is

(1.21) OH(~) = - i - s i g n

OA(~) (c f . e.g. MICHLIN (1965) [61 ] ! ) .

I . The " H i l b e r t t ransformat ion" in one dimension is given by

= _ ~ m(y)dy (Hm)(x) : 1 S y _ x f o r mc LP(m) , l<p<~ , IT = c o

(c f . e. g. TITCHMARSH [101 ] , p. 120, ( 5 .18 ) ) .

(1.22)

where

2. The "Calder6n-Zygmund-Michl in-operator (CMO)" given by

(M~)(x) : = ( a l + A f / . ) ( x ) : = a.~(x) + p.v. f f(~I) ~R n Ix-Yl n ~(y)dy

ae~ , f(O) eLq(zn) , l<q<~ , S f (e )de = O. I ts symbol equals Sn

f(~-~), (1.23) ~M(~) = a + (p.V.Fx,,c Ixl n )(~)

(c f . e.g. MICHLIN [61] , p. 100).

Modern theory of convolut ional i n t eg ra l , i n t e g r o - d i f f e r e n t i a l , a n d pseudo-d i f fe r -

en t ia l equations aims in to three main d i rec t i ons :

188

A. To admit more ~eneral domains G of i n t eg ra t i on instead of ~n , e g .

( ! ) ~n : = {xE~n : Xn ~ O} kELI (~ n ) : +

Wiener-Hopf in tegra l equations (WHIEs),

( i i ) Fc~ n cones def ined by smooth (n-2)-d imensional manifolds on s n and other

smooth s e m i - i n f i n i t e domains c~ n ,

( i i i ) q u a d r a n t s l i k e ~2 : = {xE~2 Xl > O} or wedges W(~) : : {xE~ 3 , x l + i x 2 = r ~ , ++ ' ~X 2 =

0 g ~ , X3ER} or cones w i th edges: polyhedral domains.

A l l o f these are special cases of

D e f i n i t i o n 1.3 :

a B-space, and

given by

"General Wiener-Hopf operators (WH0s)" : Let AE~(]E) , 3£

P = P2E~(3E) a l i n e a r , continuous pro jec tor on 3C . Then i t is

(1.24) (Tp(A)~)(x) : = (PAlm(p)~)(x)

Remark 1.3 : In the "c lass ica l cases" mentioned above we got ~_= LP(~R n) ,

l<p< = , and P = ×G" the "space p ro j ec to r " f o r Gc]R n and A E~((3E)

B. To admit " va r iab le kernel f unc t i ons " , e. g. the "general ized L l - convo lu t i on

in tegra l equat ions"

(1.25) (A~)(x) : = a (x )~ (x ) - ~ k ( x , x - y )~ ( y )dy = f ( x )ELP(~ n) IR n

or - even more -

D e f i n i t i o n 1.4 : Equations w i th "genera l ized t r a n s l a t i o n i n v a r i a n t operators (GTI~) "

(1.26) (A~)(x) : = ( g i l x ~A(X,~) .Fy~ ~ + ~ ) ( x

= s .OA.(~ ) , aj being continuous c o e f f i c i e n t s on ~n where~,,e, g. OA(X,~) j = l a j ( x ) O

or R n ' ~A being m u l t i p l i e r symbols, and V being a completely cont inuous

operator on ~ .

Combinations of A and B lead to important classes of s ingu la r i n teg ra l

equations on manifolds ~(~ , e. g.

( j ) 30 = ~ c ~ , a Ljapounov-curve: "c lass ica l Cauchy-type s ingu la r i n teg ra l

equat ions"

(1.27) (K~)(x) : = a (x )~ (x ) + ~ i f k ( x , y ) ~ ( y ) d y = f ( x ) x - y

spaces ~ = C~(r) , mE~ , O<~<I ; or = LP(p;F), l<p<~ , p ( x )~O in var ious w i th 1 - ' o

p and p P E L I ( r ) .

189

( j j ) CMOs or singular integral equations with "Giraud kernels" with a(x) and

f (x ,e ) instead of ac-{ or f (o) eLq(En) in eq. (1.22), respect ively.

C. To admit d i f f e r e n t i a l operators which means to study " in tegro -d i f fe ren t ia l

equations of convolutional or pr incipal value type" l i ke

m

(1.28) S (a (x) l+b (x)H+K)D~m(x) = f (x ) ~ = 0

in Sobolev-spaces Wm'P(IR) or Wm'P(IR+) or even

(1.29) ~ (A D ~# )(x) + (Vm)(x) = f ( x ) e ~ = LP(8) , GclR n

where the K are one-dimensional generalized Ll-convolutions (cf . eq. (1.25)), n the A~ = F-I OA(X,~). F , ue]li ° , are a rb i t ra ry generalized t ranslat ion i n -

var iant operators and V : Wm'P(G) ÷ LP(G) a compact operator, respect ively.

A l l these operators, of course, are special versions of "pseudo-dif ferent ial

operators", a term introduced by KOHN & NIRENBERG in 1965 [ 5 2 ] , and, since then,

in standard use as L : 3£ ÷ I~ defined by

(1.30) (Lm)(x) : = (F -10L(X,~)Fm)(x ) fo r mE~

where OL(X,~ ) may be expanded asymptot ical ly with respect to ~ into a series

of functions ~ _k(X,~) homogeneous with respect to C of degree m-k. Here

o _ k ( x , t ~ ) = t ~-k ~ k (X,~) for t > 0 , kE]N o, and

co

( i .31) °L(X'~) ~k=oE ~m-k (x,c) for I~I ÷ ~

We shall desist from entering into the discussion of the resul ts on pseudo-

d i f f e ren t i a l equations since there are many excel lent survey a r t i c l es (cf . e. g.

FRIEDRICHS [36], SEELEY [83], CORDES [12], ESKIN [31], KUMANO-GO[57]). Here we shall

keep close to the l ines of SIMONENKO's theory started in (1964, '65)[91,92 ]

and shall report mainly on resul ts of our research groups.

Acknowledgement. This paper has been prepared partly during the time when the

author was a visiting professor at the Depar~ent of Mathematics, University of

Regina, Saskatchewan. He wants to thank the Head of the Department, Prof. E. Lo Koh,

for arranging ideal wo~ki~ conditions and for the great hospitality there. The

author wishes also to thank Dr. F.-O. Speck, TH Darmstadt, for his valuable

criticism and substantial remarks during the preparation of the manuscript.

190

2.,~,,,Integral- and in tegrg-d i f ferent ia l equat!ons of the Wiener-Hopf type

WIENER & HOPF studied in 1931 [ 107 ! certa in types of homogeneous convolutional

integral equations on the ha l f - l i ne R+ in connection with problems of rad ia t ive

t ransfer . They developed the funct ion- theoret ic method, a f te r applying the Fourier

transformation. This is now cal led the "Wiener-Hopf-technique". KREYN in 1958154]

completed the classical theory of WHIEs in LP-spaces

(2.1) (W~(x) : = ~(x) - I _ ~ k(x-y)~(y)dy = f (x ) E ~+ 0

where kELI (~) and fE3(÷= LP(~+) , 1~ps= , or certain closed subspaces of

L~(~+) such as Co(~+) for instance, are given and me~.+ is sought.

GOHBERG & KRE~N [43 ] extended these funct ional -analy t ic invest igat ions to systems

of WHIEs.

Making use of the convolution theorem of the F-transformation (1%p~2) one

arr ives at the image equation to eq. (2.1)

(2.2) [ i -~(~) ] ~+(~) ^- • - h (~) = fP'*(~)e L P ' ( ~ )

where

(2.3) ~÷(~) : = (FP+m)(C) : : (F×~m)(~)

and A

(2.4) h-(~) : = (FP_h)(c) : = (Fx N -h)(~)

are one-sided F-transforms which may be continued holomorphically into the upper,

H +, and lower, H-, complex hal f -plane, respect ively. Here we put

[ - - ~ f k(x-y)m(y)dy for x < 0 o

(2.5) h(x) : :

0 for x > 0 .

Equation (2.2) actual ly denotes a "Riemann boundary value problem for the l ine"

which is a special form of

(2.6) ¢+(t) = G( t ) -¢ - ( t ) + g( t ) , tEF ,

F being a smooth contour c C,G(t), g( t ) given data and ¢±(t) the unknown

boundary values of a (sect ional ly) holomorphic function ~(z) vanishing for z-~.

In solving such problems (cf . e. g. the books by MUSHKHELISHVILI (1953) [62 ] or

GAKHOV (1966) [37]) one of the crucial steps is the poss ib i l i t y of " fac tor iza t ion"

191

^ -1 of G(t) - or in our example of [1 -k (~) ] - in to

(2.7) G(t) = a - ( t ) - f t - P ÷ 1 < A+(t) ~t-p_ z

D ± wi th funct ions Am(z) holomorphic in , the i n t e r i o r and e x t e r i o r domains of F

. D ± in ~ , being bounded and # 0 on D ± : = D~F p+~ are chosen a r b i t r a r y and

denotes the "winding number of G along F "-

(2.8) 1 [ a r g G ( t ) ] < : = ~ r "

This makes sense only f o r G(t) # 0 under ce r ta in smoothness assumptions - at leas t A

one has to know a b i t more than GELS(?). In case of 1 - k ( ~ ) E ~ ) , the one-dimension~

Wiener-algebra, f a c t o r i z a t i o n is always possib le fo r " e l l i p t i c " convo lu t ion o p e r a t o r ,

i . e . Ow(~ ) : = I -~(~) # 0 on ~E. The r e s u l t by KRE~N[34] is given in the f ~ l w i n g

Theorem 2.1 : Let k E L I ( ~ ) , f E X ~ a s above) be given. Then the Wiener-Hopf-operator

(WHO) W = P+(I-k~)P+ wi th P+ : = x~+- is

(2.9) i n f IOw(~)I = i n f I i - ~ (~ ) I > 0 .

I f t h i s cond i t ion holds then

( i ) m(W) = max(O,<) , 8(W) = max(O,-<) and

( i i ) f o r < > 0 there ex is ts a base {mol . . . . . mo<}

~ok EL l (~+ )nCo(~+ ) and

(2.10)

( i i i ) f o r

(2.11)

where

E ~(3C~+) (Fredholm-Noether) i f f

ind W = u(W) = K

of ker W

mo,k+l(X) =

mo,j(+O) =

%,<(+0) =

d mok(X~., , k = 1 . . . . . <- i

0 , j = 1 . . . . . <-1

< < 0 the orthogonal r e s o l v a b i l i t y cond i t ions are

/ f (x ) .~ok(X)dx = 0 fo r k = 1 . . . . . I<l o

(2.12)

(iv)

(2.13)

where we have the reso lvent equation

such tha t

(W"~ok)(x) : = ~ok (x) - 7 k(y-X)~ok(Y)dY = 0 0

f o r < ~ 0 a so lu t i on to the inhomogeneous WHIE is expressed by means of

the reso lvent kernel

minh(X ) = f ( x ) + I ~ ( x , y ) f ( y )dy o

192

oo

(2.14) y (x ,y ) : Y l ( x - y )+ - {2 (Y -X )+~ l ( x - t )Y2 (Y - t ) d t 0

with uniquely defined Ll(IR)-functions 5,1,y 2 via the fac tor iza t ion of

A - i K (2.15) l - k (~ ) : ( I + ( F P + Y I ) ( ~ ) ) ( ~ ) . ( I+(FP+~2)(-~)) .

Remarks: 2.1 : The reasoning which leads to theorem 2.1 has been car r ied over to n , the case of ~ + , n m2; instead of ~+ by GOLDENSTEIN & GOHBERG in 1960 [48]

where the f a c t o r i z a t i o n appl ies to the n-th F-var iab le Cn in ~. In th is case A

i [ a rg ( l_k (~ l , " ,~n))] is always 0 fo r an e l l i p t i c WHO since due to K : : T ~ " "

~n =-~

the Riemann-Lebesgue l e n a , ~(~i . . . . . ~n)ECo ( ~ n ) impl ies that K depends

cont inuously on ~' : = (~I . . . . . ~n-I ) and tends to 0 fo r l~'I ÷ ~ •

2.2 : Concerning the group B of genera l iza t ions {AHBAGJAN in 1968 [7 ] ] and

RAKOVSH~IK in 1963 [ 75 ] t r ea ted "var iab le kernel WHOs"

(2.16) (Wm)(x) = m(x) - f k(x ,x-y)m(y)dy = f ( x ) E3E+ n IR+

co

where k(x,.)ELI(IR n) and k(x, t ) = z a i (x )k i ( t ) with the ai(x ) i=1 sa t i s fy ing the conditions:

and k i ( t )

(2.17) a i ( x )eC( IR n) such that sup l a . ( x ) I < ~ fo r a l l X # ~ n 1

oo

(2.18) kieL1(IR ) such tha t = I Ik i II 1 < i= i

A

(2.19) i n f I i - k ( - , ~ ) I > 0 EIR n

in ~AHBAGJAN's paper, whi le RAKOVSH~IK[75 ] assumes ~o

(2.20) m(x)~+ I k(x ,x-y)m(y)dy

to be a bounded operator on X = LP(IR), ISp<~ , be compact from

fo r any f i n i t e

(2.21)

(2.22)

[ a,b] c]R ,

(2.23)

lira k ( x , t ) =; k+( t ) CLI ( ]R) X-~_+~

I k ( x , t ) - k+(t) l ~_+ ( t ) -n+ (x ) f o r large

and ~+( t )ELI ( IR) , lim n+(x) = 0 +X_~+oo -

A

i n f _ _II-N+(C)] > 0 i

~elR

+x >0

ielN

LP(N) ÷ LP(a,b)

193

Here A oo

__

1-k+(~) -~

The proofs are worked out by insert ing terms, e. g.

(2.25) (W~)(x) = ~(x) i~ la i (~) .~nk i (x-y)m(y)dy

+

- s [ a i ( x ) - a i ( ~ ) ] . I k i (x-y)~(y)dy i= l Nn

+

= (l-W=lm(x) + (wlm)(X) = f ( x ) E3E~+

where I-W ~ c~(3E+) and even boundedly inver t ib le for ~ = 0 ( i . e . always' for n~2!) W 1 and 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 ke that of the f i r s t term.

2.3 : WHOs, par t i cu la r l y for ~+ , have been studied also for the whole scale of

Sobolev-Slobodezki-spaces ws'P(~+) and w~'P(~+), the functions of

ws'P(R) : = { f ~ ' : F-I(1+I~I2) s/2 FfcLP(~)} ; l~p<~ , sE~, also called "spaces

of Bessel potent ia ls" , and the subspace of them having supports, supp f , in ~+ while

W s f of fEwS'P(~) to ~+ Due to the 'P(~+) denotes the res t r i c t ions P+

Sobolev embedding theorem i t is well-known (cf . e.g. TALENTI (1973)[100], p. 28)

that Wo' s,2,~+)^wS,2(o ~ - ) = {0} for s~-1 /2 but span { ~ , 6 ' , . . . , ~ ( n ) } for , s , 2 , ~ .s ,2 , s < -1/2 and the largest integer n < - s - i / 2 while w o ~ + ) m w o (~_) = wS'2(~)

for Is~ < 1/2 and = { mEwS'2(~) : m(k)(o) = O, k = 0 . . . . . n - I } for

n-1/2 < s < n + 1/2, nE~. The defect pair (~(W),B(W)) of the one-dimensional

WHO then depends on the number n being related to n- I /2 < s < n+1/2 (see for the

deta i ls the paper by TALENTI, pgs. 63 - 71!)

2.4 : The theory of WHOs on the h a l f - l i n e has been generalized to operators of the

f i r s t kind where the symbol is given by C(¢) vanishing at i n f i n i t y and to more

general "non-e l l i p t i c or non-normal WHOs" whose symbols may be wr i t ten in the form

~-i < N ~-~. n. = ~ (~-TT ~) J (2.26) aw(C) (c - i ) -n~ G'(C)(~-~-~) G+(C)'j= 1

with ~jER being d i s t i nc t , n , NE]~ ° ; n jE~. Considering the deta i ls to these

questions being studied on a functional analy t ic basis since about 1965 by

SAMKO [78] and mainly by PR~SSDORF (1965, '67, '69) [68~69~70 ] look at PR~SSDORF's

book (1974)[71]! TALENTI ( loc. c i t ) is involved mainly in WHOs of the f i r s t kind

(pgs. 71 - 77) which play a dominant role in the study of mixed boundary value

problems in ~+2 (cf . e.g. PEETRE (1963)[64] , SHAMIR (1962, '63) [84,85 ] and

194

for the higher dimensional case: ESKIN's book (1973)[31]! )

2.5 : In qui te a s im i l a r way " two-part composite convolut ional equations" and t h e i r

ad jo in ts the "dual in tegra l equations" may be t reated v ia the F-transformat ion and

the Riemann boundary value problem:

0

(2.27) (W2m)(x) : = ~ (x ) '~ (x ) - ~ k l (X-y )~(y )dy - ~ k2(x-y)m(y)dy = f ( x ) , x ~ " ~ 0

where u(x) = u I fo r x < 0 and u(x) = u 2 fo r x > O, respec t i ve ly .

(2.28) (.~)(x) ={ Ul '# (x ) " 7 kI(Y-X)~(y)dy : g_(x) , x < 0

~2"~(x) - 7 k2(Y-X)~(y)dy = g+(×) , x > 0 .

A good survey on th is subjects wi th app l ica t ions to other dual in tegra l equations is

provided by the book by ZABREYKO et a l . (1975)[108] in Chap. V I I I , §§ 4 - 6.

We are now going to review the resu l ts on i n t e g r o - d i f f e r e n t i a l eqs. of the Wiener-

Hopf type, i . e. some kinds of general WHOs act ing between su i tab le Sobolev spaces,

v i z .

m (2.29) (Lm)(x) : = S {a (x)D~m(x)+b (x)~ c (x)k (x-y)D~m(y)dy} = f ( x ) , x > O.

BANCURI in 1969 [3] t reated only the case of constant coe f f i c ien ts a , b , c

assuming b c - 1 or - O , k ELI(]IR), fEL I (~ ) given and ~ , l ( m ) ~ ~ • sought, ' q)~'O L =~+ , ~(m-I )~ +0 He i . e . m . . . . . m(m)ELI(]R+) and m(+O) . . . ) = O. appl ied the F- t rans fo r -

mation and ar r ived at the Riemann boundary value problem (BVP) a f te r some manipulations:

m ^ ^ ^ ^ (2.30) ~, [a u + ku(~)] ~u m+(~) _ h-(~) = f + ( ~ ) E F L l ( ~ + ) .

~=0

Set

~+(~) : = am.(i+~)m~+(~) (2.31) ^_ ^

(~1 : = h - ( O A

(2.32) G(~) : = 1÷~(~) : ={~=om am " ~ + ~ m j a +k (~) ~ - i and

A

(2.33) g(~) : = f÷ (~) , g(~) where

^ m oo

(2.34) h-(~) : = (Fx] R (x) . z f k (x-y)D~m(y)dy)(~) !J=O 0

we get the equiva lent BVP A+ A

(2.35) ~ (¢) = G(~)~-(~) + g(~) in ~]~(]R)

195

which is e l l i p t i c i f f G(~) # 0 on JR, GERLACH (1969) [38] admits a l l the Kre~'n spaces and the subspaces

)[~m) : = {q)e)E+ : D u m ~ fo r u = 0 . . . . . m} or ]E[(+m) : = {mE~(+m): (Djm)(+O)= O;

j = 0 . . . . . m- l } . F i r s t he t rea ts the constant c o e f f i c i e n t s ' case too and reorders

by wr i t i ng D = ( D - i l ) + i l

m

(2.36) (Lm)(x) = P+ ~ (a , l+kp , l~ ) (D- i l )Um(x) = f ( x ) c ~ + p=o

Now, in t roducing ~(x) : = (D - i l ) -mm(x ) = (Gmm)(x) = (gm*m)(x) where

GmC~C(~+,3E~ m)) he gets

(2.37) (kgm~)(x) = (W+Vm)~(x) = f (x)c3E+

as a perturbed WHIE on ~+ where

m-1 (2.38) W = P+ + P+(km,l, + ~ Gm_ . [a ,1 -1+kp, l * ] )P+

~=o

is a c lass ica l WHO, since ( . . . ) is a L l ( ] R ) - k e r n e l convolut ion and V m is compact,

ac tua l l y having f i n i t e rank, given by

m-1 v

(2.39) (Vm~)(x) : = ~ [(D-i l )Vgm~](+O). E G~- j+ IX]R+'K j , I ( x ) l " v=o j=o

Then LGmE~(J~ F) i f f WE~'(~c+) which is the case, according theorem 2 . 1 ~ i f f the

symbol ~w(~) # 0 on ~ . Now, th is is given by

m ^ Ow(~) = OA(~)'gm(~ ) = E (a +k (~) )~ '¢ [2 /~(~- i ) ] -m

p=o

Then Gerlach t reats the case of continuous coe f f i c i en t s ap(x) ,b (x ) , cp (x ) using a

simpler version of RAKOVSHCIK's compactness theorem (1963)[75] of convolut ions

mu l t i p l i ed by func t ions : Put m ,Bp,yu fo r the l im i t i ng values fo r a p , b p , c as

x + ~ and rearrange eq. (2.29) to read

m (2.40) (km)(x) = P+ ~ (~ l+~p~ .k~*)(D~m)(x) ( : : ~)m(x))

p=o

m + P+ ~] [ (a ( x ) - ~ ) I + (b (x ) -6p)k * c ( y ) ' I +

p=o

+ ~ . k ~ ( c ~ ( y ) - ~ ) • l ] (DUo)(x) .

Then ~ ) is a WHIDO of the type above and the las t term of eq. (2.40) being

compact from 3(~(m) in to ~F.. The same argument appl ies to the subspaces ~ )

which coincide with W m o ' P ( ~ + ) f o r l<p<~.

196

GERLACH's resu l t s may be summarized in

Theorem 2.2 : Let a ,b , c ~ C ( ~ ) , - - - - k f~Ll(~)_ f o r u= 0,1 . . . . . m wi th

on ~ + , ~ : = l im a~(x) ; 6 similarly___ def ined. Then the WHIDO L

) ' Y ~ ( ~ ) , ~ o + ) i f f i t s "main symbol" eq. (2.29) is x++~e~+,~+) or E

m ^ (2.41) ~ ) : = z [a +8 ~ "k (~) ] -~ ~ # 0 on N~

~=0 ~ P p

I f th is is t rue one has the re la t ions

am(X ) ~ 0 in

(2.42 a) 0 s e(Lo) ~ ~(L) s ~(Lo) + m

(2.42 b) 0 ~ 6(L) g 6(Lo) where L o : = Ll~(m ) ~ 0 + and

(2.43 a) ind L = v(L) = v ( ~ = < + m/2

(2.43 b) ind L o = v(Lo) = v ( ~ = K - m/2

with the winding-number

(2.44) <(L) = ~(Lo) = ~ L a r g Ow(~)]~=_~

The spectrum of L is given by

(2.45) s(L) : = {zE¢ : z = q L ( ~ ) , ~ e ~ } u { z c C : ( ~ ( A - z l ) , 6 ( A - z l ) ) # (0 ,0 ) } .

Remarks: 2.6 : The theory of WHIDEs has been general ized from ~+ to R n+. I f

one takes (n- l ) -d imensional F-transformat ion with respect to x ' : = (x I . . . . . Xn_ I )

one ar r i ves at an equation in the remaining va r iab le x n > 0 but containing now

~' : = (¢1 . . . . . ¢n- i ) as parameters. So i t is qu i te natural to inves t iga te

equations

(2.46) P+A(e~Dn)m+(Xn) = f(Xn) , x n > 0

where 8' : = ~ ' / I ~ ' I has been f ixed and P+ denotes the r e s t r i c t i o n operator

to ~+ . Here A(8',Dn) is defined as a pseudodi f ferent ia l operator (PDO) by

_i ̂ (2.47) A(e',Dn)~+(Xn) : = (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 , '71 , '72) [72 ,73 ,74 ] , BOUTET DE MONVEL (1969, '71) [5 ,6 ] , DIKANSKI[

(1971,'73) [17,1~ and others!) .

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 n problem for ~+ , 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 rad ia t ive energy t ransfer . GRABMOLLER (1976, 1977)

[49,50] discussed such in tegro -d i f fe ren t ia l equations on ~+ of f i r s t order with

a l inear , closed operator - A generating an analyt ic semi-group involved

(2.48) m'(t)+c(Am)(t) + 7 ho(t-s)(Am)(s)ds + ~7 h l ( t -s)m' (s)ds = f ( t )E~.~ O O

where ho ,h lEL l (~ ) are scalar-valued funct ions, ~ + a re f lex ive B-space and

denotes the strong der ivat ive , 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

solut ion as t ÷ ~.

3. Compound integral and in tegro -d i f fe ren t ia l equations of the princ!pal value

and Ll-kernel type on ~n

MICHLIN [ 60 ]introduced in 1948 the notion of the symbol fo r singular Cauchy-type

integrals along curves rc £ and also fo r operators on ~ n , nm2 . He and mainly

CALDERON & ZYGMUND studied 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

t rans lat ion appeared in 1965 [61]. A more recent account, also on in tegro -d i f fe re~

t ia l equations with CMOs as coef f i c ien ts , may be found in Chap. IX of the book by

ZABREYKO et al . ( loc. c i t . ) . AGRANOVI~ (1965) treated equations of the fol lowing

type in his extensive survey a r t i c l e [1 ] :

(3.1) (Am)(x) : = s (MuDUm)(x) + (T~ (x ) : f (x ) l~l~m

as an operator A : wm+~'2(~ n) ÷ W~'2(~ n ) , or ~n replaced by

compact manifold ~O . The symbols

n ~+ or a smooth

(3.2) o M (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]N o and q ~ such that , by

Sobolev's embedding theorem, they form an algebra of continuous functions on

]Rn>~ E n which are homogeneous of degree zero in ~ . He shows that to every such

function o(x,~) there corresponds a character is t ic 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 'S compact for a compact manifold ~9 instead of ]R n or JR+ by Rel l ich c r i t e r ion .

198

AGRANOVI~ proves a couple of theorems which give necessary and su f f i c ien t condi-

t ions for 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 -pr io r i estimates (theorem 12.1). He obtains then the

well-known properties for e l l i p t i c operators, such as regu la 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 IR n or IR n he does not give regular izers in the sense of theorem i . I above. + SEELEY[82] investigated at the same time (1965) singular in tegro-d i f fe ren t ia l

operators on vector bundles of smooth manifolds ~P and tensor-products of such.

We are not going to enter into th is detai led material but jus t want to give

two d i f fe ren t approaches: one rely ing 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]. In 1973 DONIG[ 19] treated the case of ~R

(n=i) with the singular in tegro-d i f fe ren t ia l operator (SIDO) m

(3.3) (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<p<~, with coef f ic ients

am(X ), bm(X ), Cm(X)CC(]R ) and a (x)-a (=) etc. CLo(]R ) = {~L=(]R): mes {Ix~i>n : I~(x) l > E} +O,n + ~ , for a l l c > O} and where fEWs-m'P(~R) is

given. He applies the RAKOVSH~IK technique (1963)[75] by defining the symbol of

A through

A (3.4) OA(X,~ ) : = [am(X)_ibm(x)sig n ~ + Cm(~)km(~) ] ~m +

m-1 + Z ~a (~) - ib (~)sign ~ + c#(~)C ( ~ ) ] ~ p=O )~ ~ I]

on ]RxIR , where which is called " e l l i p t i c " i f f ~A(X,~) # 0

Using the Bessel potent ial operators

: = ~ u { - ~ } u { + = } .

jm : = F-1(1+I~I2)-m/2F he gets the re lat ions

(3.5) Dmj m : ( i l l ) m (l+rmw) with a rmcLl(]R).

Then he treats the case of constant coef f ic ients f i r s t . A is mul t ip l ied from the

r ight-hand side by jm _ s imi la r to GERLACH's approach - leading to the equation

m (3.6) (A dm~)(x) = (B'm~)(x) + ~ ~ r ~( where p

lJ=o

(3.7) B" : = I + b H + k ~ ) " { (iH)m for ~ = m

( a u ~ I for O~ u-<m-I

and some r~ELI(1R). He succeeded in cons t ruc t ing a bounded inverse operator

CmC~(LP(IRn)), l<p<~, in the form of

(3.8) C m = Mml(iH) m + hl,m~ + h2,m~H

199

where M m : = am.l + bm.H, H is the Hi lber t transform and hl , m , h2, m E L I ( ~ )

ex is t due to the Wiener-Levi theorem. He gives an e x p l i c i t formula, though a ^

complicated, for f inding hl,m(~ ) and h2,m(~ ). The existence of C m then implies

the a-pr io r i estimate

(3.9) ¥ li~ II m,p ~ II Aml] o,p

with a y > 0 , that means "strong coerciveness". In the general case the constant

coe f f i c ien t operator with ~m : ~m (=) ' Bm : = bm(=) is s p l i t o f f and is inverted

with C m as the inverse (A .J m) on LP(~) . The product with the perturbed

terms, and par t i cu la r l y the lower order ones, gives a compact operator on LP(L~)

(Rakovsh~ik's resu l t from 1963!). This leads to the coerciveness inequal i ty:

(3.10) Y'IImllm,p ~ IiAmtlo,p + IIV~llo,p

with a certain VE~(Wm'P(R) , LP(IR)). This one implies that A~'(Wm'P(IR),

LP(IR)) having an ind A = v(am(x)+bm(X)H ) which may be calculated from the a_iblx~l ~ winding number K : = [arg ~ ~j . In case the data are smoother, e. g.

- oo

a m , bm~S(~R)~sE~ , av,bvccs- l (~ ) and a (s) b(S)CLo(]R) one obtains for any

fEwS'P(]R) smooth solutions mEWm+s'P(IR). Af ter discussing the dual operator - - c

on W m,p (IR), 1/p' + l /p = 1,

m v

(3.11) (A*~)(X) : : S (-1)~DV{au(x). l - H b ( . ) . l +kv~}~ (x ) p=O

where ki(x ) : = kv(-x ) i t is possible to show that A and A* may be extended

continuously - for su f f i c i en t l y smooth coef f ic ients a (x) , bv(x ) - from Wm'P(~)

to J)Lp(N) and from w-m'P' (N) to j ) ~ p , ( ~ ) .

DONIG in 1976 [ 21]extended his reasoning in order to include the n-dimensional

version of the singular in tegro-d i f fe ren t ia l eq. (3,3) to the case of coef f ic ients

M + K where the M are CMOs and the K of L l ( ~ n ) . kernel type, both being

x-dependent. In contrast to AGRANOVI~ in (1965)[ 1 ]he succeeded in arr iv ing at an

exp l i c i t r ight and l e f t regular izer in the case of e l l i p t i c operators A . I t is

impossible to give a l l deta i ls here, but his main resu l t is formulated in the

fol lowing

Theorem 3.1 : Let n ~2, l<p,q<= such that q ~ p-P1

I n- i for i<o<_2 q .-

go (q) : = n_@_ fo r q~2 and

> ~o(q).

200

Furthermore assume tha t a (x)EC(~ n) , f (x ,o)EC(~ n , Wg°'q(sn)) wi th

S f~(x ,~)d~ = 0 fo r a l l xEIR n and k (x,t)EC(IRn, L I ( N n ) ) , I~I < m.

Let the symbol of f ( x , ~ )

(3.12) (Am)(x) : = lu!~ -~m~ {a~(x)(D~m)(x) + p.v, mn~ u Ix Ix[yl(Dum)(y)dy-yl'' +

+ f k~(x,x-y)(D~m)(y)dy} = f(x)ELP(~R n) IR n

be def ined by

(3.13) f (x, - ~

aA(X,~) : = S {a (x)+ p.v.F ( (~) + I~ I ~m Y~+~ iYl n (FY'+~k~(x 'Y)) ( ~ ) }~

and i t s "p r inc ipa l symbol" by

(3.14) oAo(x,~) : = i~ l=m{a~(x)+(P 'V 'Fy~ f~(x,T~- F)

]y [n ) ( ~ ) } ~

Then put t ing l im aA(X,~ ) = : OA(~) and assuming [a~(~ ) ] ' l . (1+ l~12)m/2 to be a

Four ier m u l t i p l i e r symbol on LP(~ n ) the fo l low ing statements are equ iva lent :

I . A is e l l i p t i c on ~n i e

(3.15) i n f I ( x ,~ ) l > 0 and x C~ n aA° ~dR n

(3.16) i n f lOA(¢) I > 0 or ~E~ n

2. A is coerc ive on Wm'P(~R n ) , i . e .

there ex i s t s a T > 0 and a compact semi-norm l . ] such that

(3.17) x'l[~llm, p <_ IIA~LIp + L~I fo r a l l eEWm'P(IR n ) ,

o r

(3.18) 3. AE~(Wm'P(IR n ) , LP(~ n ) )

where ind A = ind ( s M • R u) = 0 and the

the "Riesz-operators"

(3.19) Rj : = F - I T ~ F , j = l . . . . . n .

1. ~n R ~ = R 1 . . . ' R n are powers of

201

I f one of the condit ions holds then a two-sided regu la r i ze r

by

(3.20) B = B~.M -M °

where the M =, M ° are CMOs. M °

symbol

(3.21) M ~ : = z M ~. R ~

and

- i ~ 12)-m/2)-iF (3.22) B ® : = F (OA(~). ( l+I~

is a bounded regu la r i ze r to

aAo(x,~)- l~I-m being homogeneous of degree 0 , lui=m

where f (@) : = l im f (x,o) IxF~ ~

B may be calculated

M "R u with

Remarks : 3.1 : DONIG (1977)[ 22] appl ied his method, in case of n = 1 , to give

condit ions fo r the Fredholm property of a transmission problem fo r sec t i ona l l y

holomorphic funct ions ~(z) , z c C - i ~ , which have i n t e g r o - d i f f e r e n t i a l transmission

condit ions on the common boundary i ~ . On the other hand (1974) [20 ]

he solved the Cauchy-problem

(3.23 a) ~ + A(t)~ = f ( x , t ) E C ° ( [ o , ~ ] , L 2 ( N n ) )nCz( (o ,~ ] , k2 (N n ) )

(3.23 b) m(x,+o) = mo(X)cL2(Rn)

with e l l i p t i c operators A as before of the compound Ll -kernel and CMO-type, now

even depending on t but with r e g u l a r i t y condit ions such as the usual ones for

parabol ic evolut ion equations. The pr inc ipa l symbol is assumed to be "uni formly

s t rongly e l l i p t i c " :

f ( x , t , 4 1 ) ) > 12m (3.24) Re(- l ) m z {a ( x , t ) + (p ( ~ ) } ~ cl~

lul =2m ~ .V .Fy~ ly ln =

for a l l x c ~ n and tE [o,~] .

The resu l ts are qu i te s im i l a r to those in FRIEDMAN's book (1969)[ 35 ]

4. Simonenko's theory fgs. general ized t rans la t i on i nva r ian t operators

We want to give now a short out l ine of SIMONENKOs approach which he establ ished in

1964,'65 [91,92] and which was appl ied by RABINOVI~ (1969-'72)~72,73,74] and SPECK

(1974-'77)~96,98] in the theory of GTIOs. For abbreviat ion we shal l w r i te A ~ B

i f f A-BE~T(~L,I~) fo r A,BG~(3(,~) where ~ , ~ are Banach spaces. ATKINSONs

theorem then may be formulated in the fo l lowing way ( ~ = ~ fo r s i m p l i c i t y ) :

Ac~(3E) i f f there are RL, RrE~(~ ) such that R~A~ARr~-I. In what fo l lows we

shal l mainly be in terested in the spaces ~ = Wm'P(IRn), mcIN o , l<p<~ , and the i r

202

' 1 1 duals W -m'p (JR n ) , ~ + ~, : I . So, for the fol lowing de f in i t ions and resul ts , l e t

us assume ~c~_c~' being dense in the topology of the bigger space. Let the

t ranslat ion operator ThE~(~ ) and the mul t ip l icat ion operator m(x). l with a C~(]R ~ ) - f u n c t i o n be a bounded operator on ]E , too. ]R n may stand for ~n or

]R n (see sec. 0).

Def in i t ion 4.1 : ( i ) AE~(3E) is said to be "of local type (with respect to ]Rn) ''

i f f m1.A~2.1~O for a l l ml,~2~C~(]R n) with ~im2 = 0 . We then wr i te AE.A,(L~]. x o

( i i ) A,BE~(~E) are said to be " loca l l y equivalent at Xo~]R n , wr i t ten as__ A~B ,

i f f for a l l ~ > o there exists a neighborhood %(Xo) and an ~IzEC=(]R n) with

mlL(x) m 1 on ~ (Xo) such that

(4.z)

(4.2)

( i i i ) AC£(~)

i f f there exists a neighborhood ~(xo) , and a pair of operators

such that

in f II ~- I (A-B) -V l I~ (z ) < ~ and V~0

inf II < too. VN) (A-B) ~z" I-V 1 [ZL~) c ,

is said to be " loca l l y Fredholm at Xoe-R -if ' ' , wr i t ten as

RL,x o

A E ~xO(~)

and Rr,xo

(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

a l l ~CC~(R n) and ( i " ) X E . A xG.I-~O for a l l E ,Gc~ n with EnG = ~.

For th is 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 fol lowing theorems.

Theorem 4.1 : Let AEy~(X). Then A E ~ ( ~ ) i f f AE~×o(~_ ) for a l l XoEIRn.

Xo Xo~ A~x ° Theorem 4.2 : Let A,B c~(~) and A-~B for a f ixed . Then (3~) i f f

B c ~Xo(~" Some of SIMONENK0's addit ional resul ts had been generalized by SPECK (1974)[96] ,viz.

Theorem 4.3 : Let A E~0E), where ~_= Lp ( ~ n ) , lz:Jp<~ or .3E= Co(~n ). Then

AE~Xo for an i n f i n i t e remote point, x oE~I~-R n i f f A is (continuously) in-

ver t ib le .

Theorem 4.4 : Let X= LP(~ n) , l<p<~ , and

203

(4.4) (Am)(x) = am(x) + p.v. [ i n- m(y)dy + [ k(x-y)m(y)dy Rn Ix-y ~ n

a compound CM- and Ll-convolut ion operator, (cf . eq. (1.22)) ; then AE~Xo for a

f i n i t e point x oE~n i f f al + Af/ . is inver t ib le .

Remark 4.2 : These resul ts c~rry over to more general s i tuat ions as e. g~ ~n

a) Z : [ # ~ { n ) , 1<p<~ , A = r ~ ( ~ ) . P , c be ing ~ - s t a b i l i z e d , i . e . c = o + %, O being homogeneous of degree zero and c o tending to zero at i n f i n i t y , or

b) 3C_= L2(R n) , ~ being an arb i t ra ry L ' - funct ion (cf. SPECK [97] ! )~ Thus we are

led to the following general izat ion of Def in i t ion 1.4.

Def in i t ion 4.2 : ( i ) A~(3E) is cal led a "generalized t rans lat ion invar iant

operator (with respect to ~-T~),, i f f for any XoE-~ there is an AxoC~(~) NA (3~) ~ch that AX~°A

Xo -I then(ii) I f A is a GTIO "enveloping" the family {Ax}xE-~n where A x ~ = F CAx(~)-F,

(4.5) o(x,g) : = a A (g) , (x,~) E~n ~n ,, is cal led a "presymbol of A X

Remark 4.3 : Denoting the presymbol space by ~ and the subideal of a l l functions

corresponding to compact GTIOs A by ~o we see, that the map T defined by

( 4 . 6 ) A ~ o ~ ~ + ~o

is a homomorphism of the space of GTIOs onto the quotient space ~/~'o with kernel

O~, the compact operators (the sequence {o} + d~ ÷ GTIOs +~/I~" o ~ {o} is "exact") . I T Thus de f in i t i on ( i i ) makes sense.

The main resul ts in th is context may be summarized in the

Theorem 4.5 : (SIMONENKO (1964) - SPECK (1974-77)): X o

I . Let A'~A x E ~ ) N A(~) for a l l XoC~ "~. Then the fol lowing conditions

equivalent, o

( i ) Ac~(~_)

( i i ) AxoE~Xo(~( ) for a l l XocNn;

I I . For p = 2 at the same time holds

( i i i ) sup ess in f l~(x,~) - Co(X,~)i > O. ~o e ~o (x '~)e-~Trxmn

Thus we get

( iv) e s s i n f Lo(x,~)I > o ( x , ~ ) e ~ n ~ n

asa su f f i c ien t condit ion, while a necessary one in any case is given by

are

204

(v) ess in f l { ( x ,~ ) [ > 0 ( x , ~ ) e ( ~ - ] R n ) x IR n

where x ~ a(x , . ) has to be continuous from ~'T~_ ~Rn into L=(~n).

I I I . For p ~ 2 , A being of type (4.4), in formula ( iv) we have to add

(v i ) ~l(xo,~ ) is a ~ -Four ie r mu l t i p l i e r symbol for a l l XoE-~-11-~ n

( th is leads to more complicated condit ions).

Applying th is to the compound pole-dependent Calder~n-Zygmund-Michlin and L 1-

integral operators on ~ = L2(]R n) given by x-y

f (x , + f k(x (4.5) (Am)(x) = a(x)m(x) + p.v. I in m(y)dy ,x-y)m(y)dy ]R n Ix-Y ]Rn

one gets the Fredholm c r i te r ion by the symbol conditions

(4.6) in f laA(~,~)i ~ 0 ~e~ n

(4.7) min in f I o (~)I > 0 , xE~ n ~EZ n a (x ) I+A f ( x , . ) / .

since the symbols of the CMOs are homogeneous of degree zero.

These are exact ly the same conditions l i ke 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 for a rb i t ra ry (non-stabi l ized)

GTIOs in the case of p = 2. Due to the kind of the characterizat ion theorem for

Ac~" x at f i n i t e points x o there occur rest-classes of functions in the de f in i -

t ion °of the symbol.

Now, a l l the considerations above may immediately be generalized to integro-

d i f f e ren t i a l equations

= z (ApDUm)(x) = f (x)E (4.8) (Am)(x) l ~ l ~

where the coef f ic ients A are generalized t ranslat ion invar iant operators

according to de f in i t i on 4.2 ( i ) and ~ c ~ m ) c ~' be d is t r ibu t ion spaces such that

D~mc~ for ~E~ o , 0 ~ lul ~ m. F i rs t we notice that the concept of th is section

can be completely transferred to the case of A : ~ ~ operating between d i f ferent

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 fac t here

is that the js are not only t ranslat ion invar iant operators from ~ ( t ) = wt,p(~n)

onto ~ ( t+s ) = wt+s,p(~n) for t,sEIR, l<p<~ but also of local type with

respect to ~n , cf. RABINOVI~ (1972)[74 ]and SPECK (1974)[96]. So we can state the

205

Theorem 4.6 : Let AE ~(wS'P(IR n ) , wt'P(]R n)), s,tG]R ; l<p<= . Then

( i ) A is a generalized t ranslat ion invar iant operator form ws'P(]R n) to wt'P(]R n)

i f f ~ : = j - t A j s is one on LP(~ n) (with respect to ]Rn!)

( i i ) AE~(wS'P(]Rn), wt'P(IRn)) i f f ANc~(LP(]Rn)),

I t turns out that "generalized convolutional in tegro-d i f fe ren t ia l operators"

l ike in eq. (4.8) are generalized t ranslat ion invar iant operators from Wm'P(]R n)

into LP(]Rn), l<p<=, with respect to ]~n So i t is easy to reformulate the Fred-

holm conditions for such operators.

Theorem 4.7 : (cf . RABINOVI~ [72] , p. 87, Th. 4.1; SPECK [96] , p. 89)

A D u be a generalized convolutional in tegro-d i f fe ren t ia l operator. Let A : = i ~ l ~

I t is c~:'(Wm,p(~n), LP(~n) ) , l < p ~ , i f f A x jmE~x(LP(~n) l for a l l xE~ n

is the family of operators ~ A xDUc~NA(Wm'P(~ n ) , LP(~ n) ) where {A x } . xem n I~ I <-m

(operator with "frozen coe f f i c ien ts " ) .

I t is necessary that A= be b i jec t i ve and i t is su f f i c i en t that A x be b i jec t i ve

for a l l xE~ n. This holds for

(4.8) in# I #~rn~A (x,~) '~U(l+l¢I2)-m/21 > 0 ×c~ n ~ ~cR n

i f p = 2 in which case the pseudo-inverse of AJ m can be constructed as an

enveloping operator to the local inverses (AxJm)-IE#NA(L2(~n)). For p # 2

local existence of these operators must be assured (cf . theorem 4.6).

the

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 fol lowing one

Def in i t ion 4.3 : Let ×, Y be Hausdorff-spaces being homeomorphic by ~ : X ÷ Y

and le t (T~u)(x) : = (u,~)(x) be a non-distort ing transformation for al l ueLP(Y) ,

l<p<~ , f ixed. I f Yo = ~(Xo) and ~L(Xo)C× and ~(Yo)CY are homeomorphic neigh- borhoods then AeA(LP(x;~)) and B~A(LP(Y;~)) are cal led " loca l l y quasi-equivalent

with respect to the pair Xo' Yo"' wr i t ten as AXe9 ° Y~°B i f f

(4.9) T _IP A Yo P~B P~r

in the sense of de f in i t i on 4.1 ( i i ) with space projectors P~ = ×~.I and P~= 7~I,

respect ively, u and n denote complete ~-addi t ive, e - f i n i t e measure5 on X and Y

respect ively.

This notion allows us to t reat semi- in f in i te smoothly bounded regions Gc~ n i f

One has the corresponding resul ts for half-spaces ~n n >2

206

De f i n i t i on 4.4 : Let ~ = LP(]Rn), l_~p<~ , and NEj, j = 1 . . . . . N, be a f i n i t e set

of d i s j o i n t measurable sets in ~R n such that u E. = ~ n I f A j , j = 1 . . . . N ,

are general ized t rans la t i on invar ian t operatorsJo I J ~ (wi th respect to ~I~) then

N (4.10) W N : = z AjPE.

j = l j

is ca l led a "component GTIO" or an "N-part composite WHO". I f ~A- denotes the

presymbol of A j , j = 1 . . . . . N, then the "presymbol of WN" is def~lned by

(4.11) ~WN(X,~) : = OA ( x , ~ ) j fo r (x,~)CEj~<IR n.

Theorem 4.8 : (SIMONENKO (1967)[94] , p. 1322) :

LetW N be a component GTIO as above where Ej = Fj are smooth conical sets, Aj are

GTIOs of L l - t ype , and IR~ = ~ n Then W N ~ i f f

(4.12) min in f IOWN(X,~) ! > 0 j = l . . . . . N (x,~)cAj

where

(4.13)

In th is case the index of

Aj : = [ ( ~ N } ~ j ) ~ n ] u [ F j ~ { ~ } ] , j = l . . . . . N.

W N equals zero.

Remarks: 4.4 : This theorem can be general ized in var ious ways - espec ia l l y fo r

p = 2 - by the fo l lowing more general assumpt ions: l ) .Let Ajx , the l o c a l l y quasi- co

equiva lent t rans la t ion invar ian t operators in xcIR n have symbols ~Jx = ~J x+ ~jx

where ' f jx are " r e l a t i v e l y th in " L ~ - funct ions instead of being EFLiCCo(]Rn),

i . e.

(4.14) sup ~ l~ jx(~) Id~ = o ( r n) fo r r÷ ~ , l~oi~2r I~- o Isr

(c f . SPECK [98]. 2).Take fo r Ej measurable sets which are only asympto t ica l ly

smooth cones at i n f i n i t y , i . e.

(4.15) mes(Ej-Fj)n{~cIR n: I~O-~l ~ I } ÷ 0 fo r l~oI ÷

where the Fj are smooth cones, c f . SPECK [98] , too, 3). Let r j be piece-wise smooth cones, cf . ~!EISTE~ & SPECK [58] . 4).Assume that l o c a l l y at every 9oint xE~ n

more than two sets ~ j are allowed to i n te rsec t , c f . HEISTER & SPECK [59],

4.5 : SIMONENKO (1964)[ 91] general izes to composite WHOs wi th space pro jec tors

pEjE~(L2 ( ~ n ) ) where the Ej are domains c~ n bounded by smooth Ljapounov

manifolds of f i n i t e area and the operators Aj are s ingular having homogeneous

207

symbols (or symbol matrices) of (posi t ive) order 0 being continuous on ~n ~ He

proves that the symbol ~w(X,~) : = ~A(X,~) for (x,~) cE jx~ n and the i r l im i t ing

values as x ÷ x oC~Ej from both sides has to be zero and a certain index-con-

d i t ion to be f u l f i l l e d in order that W N be Fredhlom-Noether. This can be done

by reduction to a two-part composite problem loca l l y at every boundary point x o

since the quasi-equivalence is established by the local mapping of ~Ej onto

~ = ~ n - l . This idea of local coordinate mapping is inherent to a l l e l l i p t i c

boundary value problems, for smoothly bounded domains. SIMONENKO's method has been

applied by RABINOVI~ in (1969)[72,§ 5] to boundary value problems for generalized

convolutional in tegro -d i f fe ren t ia l equations - and systems of such - in semi- inf i -

n i te domains GC~ n with smooth boundaries behaving l i ke a cone for large distance

or ig in . Again the non-vanishing of the symbol on A G := [#N~R~]U[G~{~} ] from the

is necessary and su f f i c ien t for

(4.18) PGA(X,D)PG m + TPGm = fcHS-~(G)

to be Fredholm-Noether. Here PG denotes the res t r i c t i on operator on s+C s+c,2 °s+c H (G) : = W (G) and TE~(H (G), Hs-z(G) where ~s+~(G) is the closure

of C~(G) with respect to II.llHS+~(G)-norm.

For the case of G = R n, or a bounded domain, a thorough discussion of the

Fredholm property, n par t icu lar with boundary conditions or potent ials carried V

on ~G , and the related conjugate problem has been performed by DIKANSKII(1971,'73)

[17,18]. RABINOVI~ (1972)[74] then studied pseudo-dif ferent ial operators on classes

of noncompact manifolds with boundary conditions. Quite recently CORDES (1977)

stQdied C*-algebras of e l l i p t i c boundary problems [10]. Short ly before he derived

a global regular izer - or even parametrix - to pseudo-dif ferential operators on

R n (1976)[14].

208

5. Wiener-Hopf type integral equations with strongly singular kernels

So far we have been interested in convolutional equations having smooth symbols on

G~R~admittingc.l~x for the x-dependence of the factors or kernels more general domains n _ { 0 } but x varying in the or having symbols being continuous on IR~

whole of ]~n We want to look now into equations combining stronly singular ker- J •

nels, of the Cauchy pr inc ip le value or Calderon-Zygmund-M1chlin type, with piece-

wise constant coef f ic ients on ]R n. The simplest case arises from the well-known x " a i r f o i l equation"

+I (5,1) _1 f q~(y)dy

-1 'y - X = (m(-1,1)m)(x) = f (x ) , XE(-1,1)

involving the " f i n i t e Hi lber t transformation Hr- , . j l I~"" BETZ in (1920) [,4 ] gave

an inversion formula for su f f i c i en t l y smooth fEC#oc( ( -1 ,1) )NLI ( ( - I , I ) ) 0<),<1, by

(5.2) m(x) = - - - - ~ _ 1 "

with a rb i t ra ry CE~ (or ~). Now, th is is completely in contrast to H 2 = -I

acting on LP(~ ) , l<p<~ , or C~oc(~) fol lowing from eq. (1.21). I f one wants

to solve eq. (5.1) above by means of the Carleman-Vekua funct ion- theoret ic method

introducing #(z) 1 +} my_~d z = , Z

g'~3- _1 give for r = IR:

(5.3)

and

~+(x - G - ( x ) =

~+(x + ®-(x) : ( 5 . 4 )

[ -1,1] then the Sochozki-Plemelj formula

{ ~ ( x )

0

T(H(-1,1)m)(x)

+i __1 f < y )dY Ti L1 y - x

for xE( - l ,1 )

for .xEIR-[-1,1]

for xE(-1,1)

f o r xcIR-[ -1,1]

So eq. (5.1) is equivalent to the Riemann boundary value problem with a piecewise continuous factor only:

@+(x) = G(x) .~ ' (x) + g(x) , XE~-{ -1 ,1} ( 5 . 5 )

where

(5.6) G(x)

and

=I - I for x E ( - I , 1 )

l +I for xeIR - [ -1,1 ]

209

1 f(x) for xe(-1,1) T (5.7) g(x) :

0 for xE~- [ -1 ,1 ]

So the jumps of G at x = ±i must be the reason for the non-uniqueness of the

solut ion given by eq. (5.2). This f i n i t e H-transformation, and the semi - in f in i te

along ~+ as wel l , have attracted the at tent ion of many mathematicians up to now

concerning one- and multidimensional integral equations involving them or the i r

multidimensional counterparts.

While the Russian school around MUSHKHELISHVILI (cf . his book (1953)[6211 has

mainly looked into Hoelder space solutions by the classical method TRICOMI in 1951

[102], NICKEL in 1951 [63], and SUHNGEN in 1954 [95] applied the LP-theory, in

the l a t t e r case by a transformation which diagonalizes the f i n i t e H-transformation.

KOPPELMAN & PINCUS in 1959 [53] and J. SCHWARTZ in 1962 [81] derived spectral

representations for H(_I , I ) on L 2 ( ( - I , I ) ) . WIDOM in 1960 [106] and SHAMIR in

1964 [86] studied singular integral equations and systems of them, respect ively,

on measurable subsets Ec~ and on ~+ or [ - I , + i ] , respect ively. They treated the

LP(E) - ws'P(~+) - , s ~ O, l<p<~, cases, respect ively, but were and apparently not

aware of SUHNGEN's work before.

CORDES & HERMAN in several papers, (1966) [15], (1969) [11] introduced Banach-

algebraic techniques, the Gelfand theory, to singular integral operators in L2(~+)

of purely Cauchy-type and of compound ones with addit ional Ll-kernels. IndependenCy

of the l a t t e r paper KREMER in his thesis (1969) [55] treated the equation

(5.8) (W,)(x) : = a(x)~(x) + . y - x + c ( x ) . f k(x-y)~(y)dy o o

= f(x)EL2(N+)

where a,b,cEC(~--T) , kELI(~). The studies were continued, for c(x) z O, to

LP(~+) with the symbol calculus by GERLACH & KREMER in (1972,'73)[39,40 ] •

Following quite a dif ferent line GOHBERG & KRUPNIK started in (1968) [44] to

study Cauchy-type singular integral equations along curves rc{ with piecewise

continuous coefficients anew and introduced 2 x 2-matrix symbols for scalar equati~ons

in 1970 [45]. The whole theory, even under much weaker conditions on the coefficients

a( t ) , b(t), tEr , in the equation

(5.9) ( K ~ ( t ) : = a ( t )~ ( t )+b( t ) . (Sr~) (F ) = f ( t ) , tErc~ ,

has been displayed in a l l deta i ls in GOHBERG & KRUPNIK's book (1973) [47]. DUDU~AVA

applied the i r theory to Wiener-Hopf type equations and compound equations with

symbols having weak smoothness assumptions in a number of papers in 1973,'74,'75,'76

[24,25,26,29] .

210

SHAMIR in 1966,'67 [87,88] studied systems of singular integral equations in

n ) , ~UBIN in 1971 [90] and ESKIN in 1967,'73 [104,30] applied the method of LP( ~+

mat r ix - fac tor iza t ion with respect to the n-th Fouriertransform variable ~n in

order to solve general boundary value problems for pseudo-di f ferent ia l operators

being homogeneous of degree ~ + iB in ~n with addit ional potent ials and/or +

boundary conditions on ~ = R 'n-1 added• In his book (1973) ~SKIN is also

concerned with mixed-type boundary conditions on ~, n-I so that boundary

pseudo-di f ferent ia l operators with piecewise continuous coef f ic ients on ~ , n - I

ar ise.

A main part in the technique is played by the "Mell in transformation" and the

"Mell in convolution". This enters in a natural way by studying certain operator-

algebras acting on LP(R+).

To show th is we are going to fo l low the l ines of KREMER (1969) whose work

strongly para l le ls CORDES'(1969). In order to build up the algebras he introduces for ~EL 2(IR+) :

(5.10) S : = SIR +

(5.11) T : = T]R +

(5.12) K k : = P+k~P+

(5.13) C k

defined by ( S ~ + ~ ) ( x ) : = ~Ti ! ~ '

defined by (T~+m)(x) : = i i y +xmlY)d~

• _ 1 ! k(x-y)~(y)dy defined by (Kkm)(x)

• _ 1 7 k(x+y)m(y)dy defined by (Ckm)(x) 2 ~ o

Then the basis for the whole theory are the compactness re lat ions given by the

fol lowing [55], p. 4 & 11:

Lemma 5.1 : Let a(x)cC(~+)

(5.14) S 2 = I + T 2

( 5 . 15 ) [ S , T ] ~ O , i . e.

(5.16) [al,S]~-O , [a l ,T] ~ 0

(5.17) a.K k --- 0 in case of

(5.18) C k ~ 0

(5.19) [S,Kk]~O

(5.20) [T,Kk]~-O

(5.21) Kkl.Kk2-Kkl.k 2 0

and S,T,Kk,C k with kcL l (~) as above• Then

C~(L2 ( ~+ ))

lim a(x) = 0

211

Then KREMER studies, as does CORDES (loc. c i t . ) , the algebra (~, generated by

S and T which is commutative, modulo the compact operators, with an ident i ty .

Def in i t ion 5.1 : I f mcL2(~+) then

(5.22) ~ ( t ) : = l . i .m. 1 }/~x - ( z+ i t ) / 2 m(x)dx ~ ~ 0 2d~-

1,11 is cal led the "Mell in transform of ~ (at Re s = ~) .

I t has the property

(5.23) (S~o)(t) = tanh - ~'(t) and

(5.24) (Tm)(t) = - i • ~'~t) cosh ~ t

2

(cf. e.g. CORDES [11], p. 897!) .

The algebra ~ I proves to be isometric isomorphic to the funct ion algebra C(~)

( ~ = ~ in th is context ) where ~( t )EC(~) corresponds to K EC~ 1 with

( K ~ ) ( t ) = ~ ( t ) . ~ ' ( t ) . ~1 is cal led the "generalized Mel l in convolutional algebra"

since i t contains an algebra O~ of integral operators of a "quot ient convolution"

type

( 5 2 5 ) : dy , x > o

o which transforms mELl(x- i /2; ~+) into i t s e l f when the kernel qcL l (x - I /2 ; ~+

(cf. e.g. CORDES [11] , p. 897/88!),The general Mel l in 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 mu l t ip l ie rs 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

aa l (X, t ) = a(x) on 0 ~ x ~ ~ , t = ±~

~Kq(X,t) = ~( t ) on -~ ~ 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 ~ (LP(~+ ) ) , l<p<~, by GERLACH & KREMER in

1972,'73 [39,40] where only

212

t i _ ~ ) ] } ~ P ) ( x , t ) = tanh {~ [~ + i ( ~

changes.

Now, i f we apply these resu l ts we get

Theorem 5.1 : (CORDES (1966), KREMER (1969)). Let a,bEC(R +) be given.

Then

(5.27) (Kom)(x) : : a(x)~(x) + b(x)(S~)(x)

defines a Fredholm-Noether operator on L2(R+)

(5.28)

on sE

(5.29)

i f f

~Ko(X,t ) = a(x) + b ( x ) . t a n h ~ 0

where E : = ~ + ~ t or , equ iva len t l y , i f f

a2(x) - b2(x) ~ 0 on ~+ and

alOl+b(O) (5.30 a) -~ < arg a 0 -b(O) < ~

(5.30 b) -~ < arg a ~ < ~

and

The index is given by

(5.31) ind K o = v(Ko) = ~ fDE darg ~Ko(X,t).

In order to s imp l i f y the wr i t i ng in what fo l lows l e t us assume

be Fredholm-Noether and the coe f f i c i en ts to be normalized s. th.

on ~. . . I f we def ine K I : = a l l +b lS where a I = a(a2-b2) - I

b I = -b(a2-b2) - I = -b then K I is Fredholm-Noether a l i ke wi th

i (5.32) ind K 1 = - ind K o = - ~ f darg DE ~Ko(X't)

and the two products

Fredholm-Riesz due to

K o = al + bS to a2(x)-b2(x) = 1

= a and

KoK 1 and KIK o have zero indices but are in general not

(5.33) KIK o = I + blbT 2 + V 1

(5.34) KoK 1 = I + bblT2 + V 2

KREMER's idea ( th. 23, p. 51) [55] is to construct a two-sided regu la r i ze r U to

L = I + blbT 2 by inver t ing the symbol OL(X,t).

He then invest igates the Wiener-Hopf-algebra 6( 2 being generated by a l l oper-

ators K- k , I and the compact ones on L2(~+ ) and being closed under the oper-

ator norm of ~ ( L 2 ( ~ + ) ) . ~2 = ~ 2 / ~ i s commutative semi-simple with an i d e n t i t y , A

i t s space of maximal ideals being homeomorphic to ~ and ~2 is isomorphic to

213

C(~) . The Gelfand theory then allows us very easi ly to characterize the Fredholm

operators, that are the inver t ib le elements in ~2 ' of the type I + K k on L2(R+), A

by the condit ion i + k(~) # 0 on ~ as we know already.

To t reat the compound integral equation on L2(~+) the algebra ~N. being gener~

ating by OC I and 02 is constructed. Again (X /~ is commutative and semi-simple with

an ident i ty . The space T~C(~J of maximal ideals is homeomorphic to a compact subset

of J~, (cf . KREMER [55] (ths. 19,20) where

~ : = ~ u{ ( t ,~) : t ¢ ~ , ~ =0}

: = ( m - - ~ )

Then the compound integral equation with constant coef f ic ients

(5.35) Km e (I +aKk+~SKk)m(x ) = f(x)CL2(IR+)

may be treated. There holds the

Theorem 5.2 : (KREMER, th. 24, p. 54). Let

I f the funct ion G(~) : = I + (~+B'sign ~)~(~) @ 0 on ~

then ( i ) KE~(L2(~+))

(5.36) ( i i ) ind K = v(K) = - ~ [arg G(~)] =

1 I_S)Kk 2 5.37) ( i i i ) R : = I + ½(I+S)Kkl+ 7(

is a two-sided regular izer to K where k 1, k2ELI(~)

A ^

(5.38) 1 + k1,2(~) = [1 + (~±B)k(C)] -1

m,BEC , kEL I (~ ) , Then

and

Ke ~c ~,

~(o) = o ,

are defined from

as elements of the Wiener-algebra i ~ ( ~ ) ,

Now, simi lar to the argument by GERLACH [38] (cf . p. 27/28!) one is ready to dis-

cuss the general case of continuous coef f ic ients a,b,c on R+:

(5.39) (W~)(x) : = (a(x)l+b(x)S+c(X)Kk)~(x) = f (x )EL2(~+)

making use of Rakovsh~ik's resu l t of (1963). So KREMER arr ives at his main resu l t , v iz.

Theorem 5._33 i ( [55] , th. 26, p. 62). Let a,b,cEC(~'--~), k e L l ( ~ ) , al + b.S

Fredholm-Noether operator on L2(~+) . Assume a2(x) - b2(x) = i on T+~

: = c(~)a(~), B = -c(~)b(~). Let

be a

(5.40) i +(~±B)k(~) # 0 on and

214

(5.41)

wh~re

(15.42)

Then W is

1 + k (0 ) {~ -s . tanh ~t T - ~p( t ) + B tanh p ( t ) ) # 0 on ]R t

p ( t ) : = b2(=) . [b2(=) + cosh 2 _~]-1 .

•T (L2(m+) )

C.£rol lary 5 . 1 - - Let a ,b ,c ,k be as above but a d d i t i o n a l l y

Then the second cond i t i on fo r the symbol is au tomat ica l l y f u l f i l l e d -~ ( l f 0 )

a two-sided regu la r i ze r R o

(5.43) R ° : = U3U2U I

where

(5.44) U I : = al - bS

(5.45) U 2 : = I - b (x ) .b (o ) [ l - t a n h ~] • Kq1-

(5.44)

where K qi

(5.45 a)

i s given by

(5.45 b)

1 I+S)Kk I ½(I-S) U 3 : = I + ~( + Kk2

• ~ c ~ 1 ; i = 1,2 ; and

q1( t ) : = [cosh 2 3_~ + b2(0)] - I

q2( t ) : = [cosh 2 ~ + b2(~)] - I

oo

~(0) : / k(x)dx = O.

and

Furthermore the index of W is given by

(5.46)

(c f . th. 1.4! ) where

a(O)+b(O) (5.47 a) a o = arg a(O)-b(O) = Co + 2~ m o

(5.47 b) - a = arg = - c - 2 ~ m

ind K : v(K) = ~ + ~ 1 ~ i n d l ~

mo+m ~ = -~ [arg G ( ~ ) ]

def ines mo,m ~

such tha t iCol, Ic I <

Coro l la ry 5.2 : ( [ 55 ] , th. 2.7), Let a, b, c, k be as above but b(o) = b(~) = O,

l e t a l + bS and I + ~K k be c ~ ( L 2 R + ) ) . Then eq. (5.39) is equ iva len t to an

eq. (l+V)m' = f ' where V~O v ia a c lass ica l Wiener-Hopf equation depending on the sign of ind(a l+bS).

Now, we shal l sketch the theory developed by GOHBERG & KRUPNIK [45] f o r s ingu la r

b(x)b(~) tanh • . Kq2

215

Cauchy-type integral equations involving piecewise continuous coef f ic ients. While

the whole theory has been developed for spaces LP(p;c), l<p<= , p(t) ~ O,

p , pl-qeLl(F) for I /p + I /q = 1 and FC~ , a system of Ljapounov arcs and/or

contours, we shall confine ourselves here to one closed contour rcC (or s = ~ ) and

L2(F). Ne are fol lowing mainly along the l ines of the i r paper in (1970) ~5 ].

Let PC(F) denote the algebra of a l l piecewise continuous functions a( t ) on F

being l e f t continuous a(to) = a(to-O ) = lim a(t ) and having l im i t ing values

from the r igh t a(to+O ) = lim a( t ) where ~ and ~ shall denote that the point t t ÷ t o t > t o

is before or behind t o , respect ively, in the sense of the or ientat ion on F .

G : = F~[O, I ] shall denote the cyl inder { ( t ,u ) : tEF , O~u~1} and M(t,p)#~2x2(F),

i . e . a 2x2-complex matrix whose entr ies O l l ( t , ~ ) , ml2(L,u), m21(t,~), and

~22(t,1-~)~C(G) where ~12(t,0) = ~21(t,0) = ~12(t,1) = ~21(t,1) = 0 for tCF . Now, the algebra ~ of a l l such function-matrices becomes a B-algebra by

introducing the norm (5.48) l l M ( t , ~ i i : = max Sl(M(t,~))

where s~(M(t,~)) denotes the largest eigenvalue of H(t ,~) .M*( t ,u) being

semi-posit ive de f in i te . They prove then the fol lowing theorems in several steps:

Theorem 5.4 : ( [45] , th . 0.2, p. 194). Let OL(PC(F)) be the smalle~subalgebra of

~(L2(F)) containing a l l operators of the form a ( t ) l + b(t)S F with coeff ic ients

a,bEPC(F) and ~ the ideal of compact operators on L2(F). Then

( i ) (~t(PC(s))/~ is isometric and isomorphic to the matrix-algebra ~ . This iso-

morphism transforms the operator C~mA = a , l + b-S F + V , Vc~, into the "symbol"

(5.49) aA(t '~)=~(t '~) := I (1-~)c( t )+~,c( t+O) , ~ [ d ( t + O ) - d ( t ) ] l

~ [ c ( t + O ) - c ( t ) ] , ~d(t)+(Z-~#)d(t+O) /

where c( t ) : = a ( t )+b( t ) , d( t ) : = c ( t ) - b ( t )

(5.50) ( i i ) l l~(t '~)l l = Vc~]inf IIA+V]I (L2(r))

Remark 5.1 : This may be generalized to the case of NxN-systems of integral

equations introducing then 2Nx2N-symbol matrices (cf . GOHBERG & KRUPNIK [45], § 1).

Since they show (th. 3.2) that for a matr ix-funct ion M(t,~) e ~ with

det M(t,~) ~ 0 on G (M(t ,~)) - IE ~ too, they can prove the fol lowing essential

(th. 4.2, p. 199!):

216

Let AE~(PC(F ) ) c~ (L2 (F ) ) , Then AE ~ ( L 2 ( ? ) ) or e~-_(L2(F)) i f f Theorem 5.5 :

det j ~ ( t , ~ ) ~ 0 on G~.

I f t h i s cond i t ion holds the func t i on

-1 (5.51) f A ( t , ~ ) : = det ~ ( t , u ) . [ ~ 2 2 ( t , O ) . ~ 2 2 ( t , l ) ] e C(G), G : = F ~ [ 0 , 1 ]

and AE~(L2(F) ) having the

(5.52) ind A = v(A) = - ~ [arg f A ( t , ~ ) ] m

Remarks : 5.2 : In t h e i r paper 1971 [46] they carry over t h e i r arguments to the - - - - m B k

spaces LP(p;?) , l<p<~, where p( t ) : = k~ 1 I t -Ckl w i th -1<6k<P-1 and points

Cker f i xed (c f . also t h e i r j o i n t book (1973)[47] , chap. IX ! ) .

5,3: SCHOPPEL (1973, '76) [79,80] t reated n o n - e l l i p t i c s ingu la r i n teg ra l operators

l i ke above in v~ 'P(p; r ) -spaces where d e t ~ ( t , ~ ) may have a f i n i t e number of

i so la ted zeros of f i n i t e order s m . The d iscussion is s t rong ly re la ted to PRUSS-

DORF's f o r closed curves F w i th continuous c o e f f i c i e n t s or f o r Wiener-Hopf equations

w i th n o n - e l l i p t i c symbols i -~ (~) on ~ (c f , e.g. PRUSSDORF's book (1974),

chapts, 5,6 ! ) .

5.4: GOHBERG & KRUPNIK discuss In sects. 6 and 7 of t h e i r book (1973) [47] the cases

of F = ]R and .IR+in LP(Po;? ), i< <=, and the weiaht func t ion Po(t ) ( l+x2) ~12 N 6 k N p ;I JX-Xkl ,- 1 < z Bk+B<P-1 or , p a r t i c u l a r l y in § 7, the operator

k=l k=l corresponding to the " two-par t composite s ingu la r convo lu t iona l equat ions" (c f . re-

mark 5) of ~hap. 3 be fo re ! ) :

I a I c~ fo r x < 0

(5.53) A = a.l+b.S]R where a = a 2 e~ fo r x > 0

I b l E C fo r x < 0

b b 2 E C fo r x > 0

in the spaces LP( I t l B, ~ ) , l<p<= , -1<B<p-1.

DUDU~AVA in (1973) [24] s ta r t s o f f w i th the t r a n s l a t i o n i n v a r i a n t operators

F-laFim ~j P+ .I , (5.54) Wal ~ : = and Wa : = P+Wa IR~P+) = x]R+

where P~I~(~) denotes the class of a l l func t ions on ]R of type n ^

a(~) = z ak(~)-×k(~ ) where the ak(~) = c k + gk (~ )E~( ]R) , the one-dimensional k=1

Wiener-a lgebra , and the x k being c h a r a c t e r i s t i c func t ions to i n t e r v a l s EkC IR

217

o n

such tha t EkFIE ~ J = (~ and U E k = IR. W a is then the unique con t inua t ion to a k=1

bounded t r a n s l a t i o n i nva r i an t operator on L2(IR) (c f , [24] , chap. 2, th. 2 and 1°!)

To each func t i on a(~) E P ~ ( ] R ) he associates the func t i on

{ a(~-O) [ l -u ]+a(~+O).u f o r I~I < ~ (5.55) a(2) (~,~) : = and O~u~l

a(+~) [1-~]+a(-~) "u f o r ~ = ±

having a closed curve cC as numerical range. This f unc t i on is ca l led " (2 - )nons in -

gu la r " i f inf__la£2)(~,~)l" " > 0. He proves then the

0 ~ I

Theorem 5.6 : ( [2~, th. 1, p. 1002 ) . Let a E P ~ ( R ) having the po in ts of jumps

at c I . . . . . c n. Then W a = P+F-IaFP+E~,(L2(~+) is E ~ + ( L 2 ( ~ + ~ or E ~ '_(L2(R+))

i f f a(2) (~,~) is non-s ingu la r . I f t h i s cond i t ion holds then W a is i n v e r t i b l e ,

l e f t i n v e r t i b l e , or r i g h t i n v e r t i b l e i f

v(Wa) : = Ind W a : = -K(a(2 ) (~ ,~ ) )

is zero, p o s i t i v e , or negat ive, respec t i ve l y . Here ~ denotes the winding number

n (5.56) < (a(2) (~ ,u ) ) : ~ [arg a(~)] + s ~ [arg a(2)(Ck,U)] I

-~ k=l ~=o

Remarks : 5.5 : In the case of cont inuous

r e s u l t ( c f . th . 2 .1 ) .

5 . 6 :

(5.57)

a E ~ ( ~ ) t h i s corresponds to KRE~N's

The whole theory may be general ized to LP(~+) by i n t r o d u c i n g

l a(~-O)[1-gp(~)]+a(~+O).gp(~) ; I~I < ~ a(P)(~,~) : = O~u~l

[a(+~) [1 -gq(~) ]+a( -=) .gq(U) ; ~ = ±

where now fo r r = p or q = p_-~l ~ 2 :

(5.58) gr(~) : _ s in [email protected] i~O I@ " @ : = ~ - 2 ~ r - l r sin @ .e

and ak(~ ) = c k + ~k(~ ) , g k ( X ) c L l ( ~ ) n L q ( ~ ) .

5.7 : He constructs an algebra of operators

r s (5.59) A = s ~ W a where •

k=l j = l kj akj E P~]~( ~R )

w i t h symbols given by

r s a(p) (5.60) oA(P)(~,~ ) : = ~ ~ (~,~) .

k=l j = l kj

218

Then a s im i l a r theorem ( [24 ] , th . 2, p. 1003 ) guarantees the Fredholm-Noether

character of A i f i n f ~(P) _K(~p)(~,~)) . r - - A (~'~)J > 0 holds. Then v(A) = ind A =

Os~l A deta i led representat ion is given by him (1975) [26 ].

5.8 : DUDU~AVA in (1974) [25] general izes his theory to include mu l t i p l e -pa r t

composite Wiener-Hopf equations with s t rongly s ingular t rans la t i on inva r ian t opera-

to rs invo lved, v i z .

N : = S a j ( x ) - (Wb j .c j ( y )m) (x ) (5.61) (Am)(x) j=1

where the b jeP?A)(~) as before, a j ( x ) and c j ( x ) •~ C ( R ) , the closure of the

piece-wise constant funct ions wi th a f i n i t e number of jumps in the operator norm of

I I ~ l l (Lp( ~ ~ , i . e. f o r p = 2 in L~(~)-~Jorm, since a l l L~-funct ions are )

m u l t i p l i e r s on L 2 ( ~ ) . The author constructs complicated 2 x 2-symbol matr ices along

the l ines of GOHBERG & KRUPNIK and formulates necessary and s u f f i c i e n t condit ions

fo r A to be e ~ ( L 2 ( ~ ) ) and calculates the index. The de ta i l s are too lengthy to

be wr i t ten down here!

In his paper (1976) [29] DUDU~AVA gives a deta i led account of the whole theory

extending i t to the quarter-plane case, permi t t ing Sobolev spaces, and systems as w~ l .

6. Convolutional in tegra l equations on the 9uadrant

In accordance with A ( i i i ) in Chap. 1 l e t us consider the fo l lowing "Wiener-Hopf

in tegra l equation on the quadrant"

(6.1) (W++ m)(x) : = m(x) - f k(x-y)m(y)dy = f ( x ) E L P ( ~ + )

2 where kcLI(R 2) and f are given, mcLP(~++) sought, ~ + ^ d e n o t i n g the f i r s t

quadrant = {x = (Xl ,X2)E~ 2 : x I ~ 0 , x 2~0 } . I f ~(~) : = 1-k(~) denotes the symbol

c I I I 0 ( ~ ) , the two-dimensional Wiener-algebra, and i f we assume i t to be # 0 on ~2

then we may fac to r i ze in to four continuous funct ions which are holomorphical ly ex-

+ ~< H ± such that tendable in to the four respect ive products of ha l f -p lanes H~i ~2

(6.2) ~(~) = ~++(~)~_+(~)~_~(~)~+_(~) where

(6.3) a±,±(~) = 1 + ~+,+(~) . . . . . = exp{Pt, +~ log ~(~)}

A

P±,± : = FX 2 . I .F -1 are the F-transformed projectors onto the four ±±

quadrants of ~ 2 . A f te r grouping the factors in the r i g h t way one recognizes that

(~_+~__)(~)(o++o+_)(~)corresponds to a f ac to r i za t i on of ~ (~ ) i n t o symbols belonging

219

to a WH problem fo r the l e f t and r i g h t ha l f -p lane of R 2 whi le the grouping

in to (~+j__)(~) . (~++o_+)(~) corresponds to one fo r the lower and upper half-plane~,

respec t i ve ly . Denoting the hal f -p lane WHOs by T D (W) and Tp (W) , respec t i ve l y , 1 ~m where W : = l - k ~ is the two-dimensional L -convolut ion on u ~L , we have the

inverse by GOLDENSTEIN & GOHBERG (c f . remark 2.1!) ex is t ing fo r aW(~ ) # 0 on ~2 as

(6.4) [TPr(W)]-I = F-I[~++~+_(~)] -1FP r F- I [o_+~__(~)] - IF

and a s im i l a r formula fo r [TPu(W)]-I . Due to a r esu l t by SIMONENKO (1967) [94]

assuming that XE.I.k,XGI is compact on Lp ( ~ 2 ) , l<p<~ , fo r quadrants E,Gc~ 2

ly ing opposite on the same diagonal we a r r i ve at the

Theorem 6.1 : (STRANG (1970)[ 99 ]) • Let kELI(R 2) and inf.~ I I -~ (~ ) I > O. Then 2 = the WHO W++EBm(LP(~++)), l<p<= , with ind W++ 0 with ~E~: the regu la r i ze r

(6.5) R = P++{[TPr(W)]-I + [TPu(W)]-I - W-I}P++

The present author and F.-O. SPECK [59] recent ly studied the WH in tegra l

equation with kELI(~ 3) and

(6.6) (WGm)(x) : = m(x) - } k(x-y)m(y)dy = f(x)EL2(G) G

where G = G(~) : = {x = (Xl,X2,X3) : x l + i x 2 = re i ~ , o ~ _ _ ~ , r ~0, x3E~}= S ~ denotes a wedge wi th x3-ax is as i t s edge and wi th an opening angle equal to ~ .

F i r s t they proved the

Theorem 6.2 : ( [59 ] , th. 1). Let Zc~ n , n ~ 2 , be a cy l i nd r i ca l region, i . e.

there ex is ts a vector hE~ n such that fo r a l l xcZ also x + phEZ for a l l p ~ O.

Let A be t rans la t i on i nva r i an t E ~ ( L 2 ( ~ n ) . Then Tp(A) : = PzAI~9~(pz)~(L2(Z))

i f f i t is i n v e r t i b l e on ~(P2) ~ L2(Z)"

Remarks : 6.1 : The same statement holds fo r LP(z) , 1~p~, fo r cer ta in Sobolev

spaces and spaces with a weight funct ion.

6.2 : The resu l t car r ies over to cones Fc~ n instead of Z admit t ing " d i l a t a t i o n

invar ian t " operators, e. g. pseudo-d i f fe rent ia l operators of order zero whose symbol

behaves l i ke ~A(p~) : ~A(~) fo r p > 0 and ~c~n- {o } .

Now, the fo l lowing resu l t is t rue

Theorem 6.3 : ( [59 ] , th. 2). Let W G be given as above. Then the fo l lowing resu l ts

are equivalent :

( i ) WGE~(L2(G)) is i n v e r t i b l e

( i i ) WGE ~(L2(G)) and

220

( i i i ) every operator W in the fami ly defined by F -1 ~( 1,2 " " ' x 3 ) F I , 2 ' ~,x3 X3E~, is inver t ib le as two-dimenslonaL WHO on the sector S~ with angle ~ at the

vertex. n While in the case of G = ~n or = ~+ , 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 su f f i c ien 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<p<~, for general semi - in f in i te GcR n not

so much is known but GERLACH & LATZ proved 1977

Theorem 6.4 : Let A = l - k , , kELI(Rn), E9 r be a measurable subset in ~n con-

taining a cone. Then for Tp(A) : = PEAI~(pF) to be inver t ib le on "~.(PE) ~ L2(E)

e l l i p t i c i t y is necessary and "strong e l l i p t ~ c i t y " , i . e.

(6.7) i n f n ( [ Z - ~ ( ~ ) ] . e i ~ ) a ~ > 0 ,

for sui table ~E[0,2=) and ~ > 0 , is su f f i c ien t .

C or011ary 6.1 : ( [59] , Corollary 1). For the WHO W G of eq. (6.6) to be inver t ib le

i t is a) necessary that l -k(~) be e l l i p t i c and b) su f f i c ien t that i t be point-

wise strongly e l l i p t i c

(6.8) in f {Re e i~[ l -~(~ l ,~2,x3 )] : (~I,~2)ER 2 } = ~(~,x3) > 0

In order to admit strongly singular convolutional operators on quadrants depending

even on the variables Xl,X 2 we introduce the theory of operators of "b i - local type"

by PILIDI (1971) [66] which has a predecessor in SEELEY's paper (1965) [82] Chap.13~,

in a sense (see also DOUGLAS & HOWE (1971)[23]!).

Def in i t ion 6.1 : Let ~1' ~2 denote two B-spaces of Lp-functions on ~m and ~n ,

respect ively. Ajc ~(~ j ) the operators of local type on Xj • ~j the set of opera-

tors which are t ranslat ion invar iant c ~ ( ~ j ) , "~j the subalgebra and ~ the

ideals of the Fredholm and compact operators on ~ , respect ively. Then denoting

by BI®B 2 the topological (and algebraica]) tensor product of two B-spaces B I and

B 2 we have:

a) A,BEA : = AI~ A 2 are cal led " i -equ iva lent " , A~B , i . f f A-BE~I@A 2

b) AEA is called "l-Fredholm-Noether", AE~ 1, i f f there ex is t R~,RrEA such that

R~A ~ A R r ~ I o m

c) A,BCA are called " loca l l y I -equivalent at XlE~ " i f f for a l l ~ > 0 there

exists neighborhood ~ (Xo)C~ m and ~uEC~(~ m) with ~L(x) ~ 1 on ~ (Xo)

such that

(6.9) in f II (A -B) (~u ' I I~ I2 ) -T I ~ < TE~A 2

221

d) AEA is called " local ly 1-Fredholm-Noether at x~E~ m'' i f f there is a neighborhood 1~L(x~)c~ m ~ "m , a~Lcc ( ~ ) as above, and R~,x~ , Rr,x~ E A such that

(6.10) R~ o a(o~Iz® 12)-~i ,~1 12 ~,x I (~ l l~12)ARr ,x~ ~ I i ~

Remark 6.3 : Analogous notations hold with respect to the second component.

Theorem 6.5 : (PILIDI (1971)[66]) : AEA = A1(~(1)®A2(~2) is c~(JEI~)E2) i f f A is local ly l-Fredholm for al l x~c]R m and is local ly 2-Fredholm for a l l x~E]R n.

Theorem 6.6 : (PILIDI (1971)[66]): Let A,BEA and A be local ly l-equivalent to B at xOER m . Then A and B are at the same time local ly l-Fredholm at x~ or

1 not.

These oheorems are applied to "bisingular Cauchy-type integral equations on ~2 or ~m~n , , and to Wiener-Hopf integral equations on the quadrant.

Theorem 6.7 : Let a (Xl,X2)EC((~)2), ~ = 0,1,2,12, and the bi-singular Cauchy-

type operator be defined by

(6.11) (Lm)(Xl,X 2) : = ao(Xl,X2)~(Xl,X2)+al(Xl,X2).(Sl~)(Xl,X2)

+a2(xl,x2)(S2~)(x1,x2)+a12(Xl,X2)(S12m)(Xl,X2 )

where • (Yl,X2)dY 1

(6.12) (Sl~)(Xl 'X2) : = ~ i f Yl - Xl

(6.13) (S2m)(Xl'X2) : = ~ i f

~2

~(xl,Y2)dY 2

Y2 - x2

denote the "part ia l Cauchy transforms" and

i S f (6.14) (S12~)(Xz,X2) : = (Sz(S2~))(Xz,X2) = ~ ]RI JR2

Then the following statements are equivalent: LE~(L2(IR2)) or

m(Yl,Y2)dY2dY 1

(Y2-X2){Yz-Xl) "

(6.15 a) [ao(Zl,X2)+al(Zl,X2).sign ~ i ] . I + a2(zl,x2)+a12(Zl,X2)sign ~i ] S 2

for al l ZlE~ I and f ixed ~i = ±I and

(6.15 b) [ao(Xl,Z2)+a2(xl,z2)-sign C2]-I+ a1(xl,z2)+a12(x1,z2)sign ~2 ] S I

for al l z2E~ 2 and fixed C2 = ±1 are inver t ib le one-dimendional singular operators

with respect to x 2 and x I , or the symbol of L :

222

(6.16) aL(Z1'Z2;~l '~2) : = ao(Z1'Z2)+al(Zl 'Z2)sign ~1 +

+ a2(z1'z2) 's ign ~2+a12(z1'z2 )sign ~I "sign ~2

is e l l i p t i c i . e. # 0 on I~ 2 ~ { - 1 , 1 } 2 and add i t iona l ly

]~ ; = ±I (6.17 a) [arg aL(Zl,Z2;~l,~2) = 0 for a l l Z lC~ l ~i '~2

z2=-~ and

(6.17 b) [arg aL(Zl ,Z2;~l ,~2)] = = 0 for al l z2c~ 2 ; ~i,~2 = ±1 Zl=-~

Remarks : 6.3 : In case of avEC(~2 ) the last two conditions are superfluous!

6.4 : This theorem holds also for R I ~ ~2 replaced by Ljapounov-curves r l , r2cC

and B-spaces LPl(pl;F1) and LP2(p2;F2) instead of L2(~1 ) and L2(~2 )" The

coef f ic ients a may even be piecewise continuous only or systems of equations with

matrices a may be involved. (cf. mainly PILIDI & SAZANOV (1974)[67] and

DUDU{AVA (1975) [27,28]. They calculated a two-sided regular izer and the index of

the operator L.

Corollary 6.2 : Let LEA = AI(LPl(p l ;FI ) )~A2(Lp2(p2;~) ) be a Fredholm-Noether

operator. Let R 1 and R 2 denote I - and 2- par t ia l regular izers, respect ively. Then

a regular izer R for L is given by

(6.18) R : = R 1 + R 2 - RILR 2

and the index L by

(6.19) Ind L = [<l (a++)-<l (a _)] . [<2(a+_)-<2(a_+)]

where Kj, j = 1,2 , denote the winding numbers with respect to

functions

xjeFj of the

(6.20) a±±(Xl,X2) : = (ao±al±a2±al2)(Xl,X2)

Remarks : 6.5 : Applying the two-dimensional F-transformation to the Wiener-Hopf

equation on the quadrant ~ + - or more general: studying four-part-composite

equations

223

(6.21) (Am)(Xl'X2) = ~ ( X l ' X 2 ) - L kI(Xl-YI 'X2-Y2)m(YI'Y2)d(Yl 'Y2) ~++'~

N2.~(Yl,Y2)d(YI,Y2)-S N3-~(Yl,Y2)d(YI,Y 2) IR 2 ]R 2

- ~ r = -

k4.~(yl,Y2)d(Y1,Y2) = f(x] ,x2)EkP(m 2) , l<p<~ , ~2

4 - -

where ~ = ~++E$ ~ leads to Riemann boundary value problems for two complex variables

in the four products H ±± of half-spaces in C . This is then equivalent to an

equation l ike (6.11) with operator L . The problems have been thoroughly discussed

mainly by DUDU~AVA in 1976 [ 29] §§ 2,3, in Sobolev spaces (Hs'P(~ 2 ))m . mE~" ' + + ' ,

i . e. including systems.

6.6 : PILIDI & SAZANOV (1971,'74) [66,67] treated also operators A of the bi-singu-

lar CMO-type where AI~XAxEQm®A2(~ n) for a l l x ~ m and A2'~YBy~AI(~m)®Qn for

a l l yE~ n, where the Qm' Qn denote mul t ip l i ca t ion by homogeneous functions of

degree zero Ec(~m-{o}) and Ec (~n - {o } ) , respect ively.

2 6.7 : KREHER (1976) [56] studied non-e l l i p t i c Wiener-Hopf operators on ~++ where

the symbol may have a f i n i t e number of zero l ines of f i n i t e order in each of the two

variables ~i,~2.

7. Concluding Remarks

SHINBROT in 1964 [ 89] , DEVINATZ & SHINBROT in 1969 [16], REEDER in 1973 [76], and

PELLEGRINI in 1973 [65], jus t to mention a few papers, were involved into the study

where AE~(~) and P = P2E ~(~.) of general WHOs on Hi lber t spaces ~ ; Tp(A) i~(p )

a projector. They derived c r i t e r i a for the i n v e r t i b i l i t y of Tp(A) on ~(P) and

the connection to fac tor iza t ion . Concerning e l l i p t i c systems of singular integral

operators having symbol matrices being (~+l)-t imes continuously d i f fe ren t iab le n pos i t ive ly homogeneous functions of ~ : ( 61 . . . . ~n) , ~ > ~ , the i n v e r t i b i l i t y of the

n corresponding WHOs on ~+ has been discussed by SHAMIR in 1966,'67 ~7,88] in

wS'P(~) -spaces. He derives a pr io r i estimates and makes use of the fac tor iza t ion

of matrices according to GOHBERG & KRE~N (1958)[43]. See also the paper by ~HUBIN

in 1971 [90]! These problems have been displayed and treated by the Banach f i x e d -

point pr inc ip le for strongly e l l i p t i c t ranslat ion invar iant A by HEISTER & SPECK

(1977/78) [58]. They also discussed the more general problem of an "N-part conjposite WHO"

N ~JN : = z Aj,Pj on LP(~ n) , l<p<~,

j= l

224

where the AjE~(LP(IRn)) are inver t ib le and PjPk = 6jkPk E~(Lp(IRn)) with N z P . = I .

j =1J

In th is paper one may f ind also a couple of examples of mathematical d i f f rac t ion

theory leading to such operators (cf . [58], chap. 2!).

The present research centers around the fol lowing questions, at least in the mind

of the author:

(1) Which are the necessary conditions for an AE~(3C) and a given continuous

projector P on ~ for Tp(A)_ _1~l~,p) to be inver t ib le? The same question applies N

to ~ AjPj to be inver t ib le on ~. j= l

In order to allow x-dependent symbols ~A(X,~) of generalized t ranslat ion

invar iant operators or pseudo-dif ferent ial operators one wants to know:

(2) Which are the necessary conditions on ~A(~) EL=(~ n) such that A = F-I~A(~)F

is of local type, i . e. [~I,A] is compact on LP(IR n ) , l<p<~, for a l l ~EC(]R n) or

what are the conditions on ~,aEL=(IR n) - or subspaces - so that ~(x).F-I~(~)F is

compact on ws'P(~ n) scIR , l<p<~, fixed? In the case of a posi t ive answer one may

obtain a Fredholm c r i te r ion for this more general class of operators. Some informa-

t ion concerning the las t question has been given recently by CORDES in 1975 [13] and

by SPECK in (1976,'77') [97,98] generalizing the results by RAKOVSH#IK ( loc. c i t . )

Most questions on mixed boundary value and transmission problems for e l l i p t i c

par t ia l or pseudo-dif ferent ial equations may be put into a generalized Wiener-Hopf

set t ing, as the work by ESKIN, DIKANSKII, RABINOVI~, SHAMIR, SEELEY, BOUTET DE MONVEL

and many others show, but the deta i ls concerning the question above s t i l l have to be

worked out.

Prof. Dr. Erhard Meister

Fachbereich Mathematik Technische Hochschule Darmstadt Schlo~gartenstr. 7

D 6100 - Darmstadt

225

References

1 Agranovi~, M.S. E l l i p t i c singular integro-di f ferent ia l operators. Russ. Math. Surveys 20, 1-121 (1965)

2 Atkinson, F.V. The normal so lub i l i ty of l inear equations in normed spaces (russ.). Hat. Sbornik N.S. 28 (70), 3-14 (1951)

3 Bancuri, R.D. Integro-di f ferent ia l equation on the half-axis with kernels depending on the difference of arguments (russ.), Sakharth. SSR Mecn. Akad. Moambe 54, 17-20 (1969)

4 Betz, A. TragflUgeltheorie . Ber.u. Abhandlq. d. Wiss. Ges. f. Luft fahrt , Heft 2 (1920)

5 Boutet deMon~el, L. Op~rateurs pseudo di f f~rent ie ls analytiques et probl@mes aux limites el l ipt iques . Ann.lnst. Fourier 19, 163-268 (1969)

6 Boutet de Monvel, L. Boundary problems for pseudo--d-ifferential operators Acta Math.126, 11-51 (1971)

7 Calder~n, A~. Singular integrals , B u l l . Amer. Math, Soc. 72, 427-465 (1966)

8 Calder6n, A.P., Zygmund, A. On singular integrals. Amer. J. Math. 78 289-309 (1956) --~

9 Calder~n, A.P., Zygmund, A. Singular integral operators and d i f ferent ia l equations. Amer. J. Hath. 79, 901-921 (1957)

I0 Colella, P., Cordes, H.O. ~ e C~-algebr~ of the e l l i p t i c boundary problem. SFB 72, preprint no.160, Univ. Bonn 1977

II Cordes, H.O. Pseudo-differential operators on a ha l f - l i ne . Journ. Math. Mech. 18, 893-908 (1969)

12 CordesT-H.O. Lecture notes on Banach algebra methodsin part ial d i f ferent ia l equations. Univ. Lund 1970/71

13 Cordes, H.O. On compactness of commutators of mult ipl ications and convolutions, and boundedness of pseudo-differenti~al operators. Journ. Funct.Anal. 18, 115-13! (1975)

14 C-ordes, H.O. A global parametrix for pseudo-differential operators over ~n, with applications. SFB 72 preprint no.90, Univ. Bonn 1976

15 Cordes, H.O., Herman, E. Singular integral operators On a ha l f - l ine . Proc. Nat. Acad. Sci. USA 56, 1668-1673 (1966)

16 Devinatz, A., Shinbrot, M.--General Wiener-Hopf operators . Trans. Amer. Math. Soc. 145, 467-494 (1969~

17 Dikanski~, ~ . Problems adjoint to e l l i p t i c pseudodifferential boundary value problems, Soviet Math. Dokl. 12, 1520-1525 (1971)

18 Dikanski~, A.S Conjugate problems~f e l l i p t i c d i f ferent ia l and pseudo- d i f ferent ia l boundary value problems in a bounded domain. Math. USSR Sbornik 20, 67-83 (1973)

19 ~ n i g , J. Zur Theorie einer Klasse e l l ip t ischer singul~rer Integro-Diffe- rentialoperatoren in Grund- und Distributionenr~umen. Diss. Univ. TUbingen 1973

20 Donig, J. Das Cauchy-Problem fur eine parabolische Gleichung mit singul~ren Koeffizienten. Vortr. Tagg. Math. Meth. d. Phys., qberwolfach 1974

21 Donig, J. 'A priori'Absch~t~un~en fur eine Klasse e l l ip t ischer Pseudo-Diffe- rentialoperatoren im Raum Lv(~"), Function theoretic methods for part ial d i f ferent ia l equations: Proc. Int. Symp., Darmstadt, 171-191, Springer Lec- ture Notes 561, Berlin -Heidelberg 1976

22 Donig, J. Ober ein Transmissionsproblem mit Integro-Differentialbedingun- gen. manuscripta math. 23, 1 -18 (1977)

23 Douglas, R.G., Howe, R.--On the Cm-algebra of Toeplitz operators on the quarter-plane, Trans. Amer. Math. Soc. 158, 203-217 (1971)

24 Dudu~ava, R.V. Wiener-Hopf integral operators with discontinuous symbols. Sov.~ath. DoKI. 14, 1001-1005 (1973)

25 Duducava, R.V. O~convolution integral operators with discontinuous coeff i - cients, Sov. Math. Do~l. 15, 1302-1306 (1974)

26 Dudu~ava, R.V. On Wiener~opf integral operators (russ.).Hathem. Nachr. 65, 59-82 (1975)

226

27 Duduc~ava, R.V. On bisingular integral operatorsandconvolution operators on a quadrant, Sov. Hath. Dokl. 16, 330-334 (1975)

28 Dudu~ava, R.V. Bisingular integral operators and boundary value problems of the theory of analytic functions in spaces of generalized functions. Sov. rlath. Dohl. 15, 1324-1328 (1975)

V 29 Dud ucava, R.W Convolution integral operators on a quadrant with discontinuous sj~nbols. Math. USSR Izvest i ja I0, 371-392 (1976)

30 Eskin, G.I. Boundary value problems a~nd the parametrix for systems of e l l i p - t i c pseudo-differential equations. Trans. Hoscow Hath. Soc. 28, 74-115(1973)

31 Eskin, G. Boundary problems for e l l i p t i c pseudo-differentiaT-operators (russ.). 'Nauka', Hoscow 1973

32 Feldman, I.A. An equation of radiation energy transfer and Wiener-Hopf operator equations (russ.). Functional. Anal. i . Prilo~en.5, I06-I08 (1971)

33 Feldman, I.A. On an i terat ion method for the equation oTradiant energy transfer, Sov.Hath.Do~l. 12, 1034-1038 (1971)

34 Feldman, I.A. Operator ~J~ner-Hopf equations and their applications to the transport equation (russ.). Matem. Issled. 6,115-132 (1971)

35 Friedman, A. Partial d i f ferent ia l equations. Holt, Rinehart & Winston, New York 1969

36 Friedrichs, K.O. Pseudo-differential operators, an introduction. Lecture Notes, Courant Inst. Hath. Sci, New York Univ. 1970

37 Gakhov, F.D. Boundary value problems. Pergamon Press, Oxford et al. 1966 38 Gerlach, E. Z~ Theorie einer ~ von Integro-Differentialgleichungen.

Diss. TU Berlin 1969 39 Gerlach, E., Kremer, M. Singul~re Integraloperatoren in LP-R~umen. Zeitschr.

Angew. rlath. Mech. 53, T 158-159 (1973) 40 Gerlach, E., Kremer,--M. Singul~re Integraloperatoren in LP-R~umen. Math.

Ann. 204, 285-304 (1973) 41 Gerlach, E., Latz, N. Zur Spektraltheorie bei Faltungsoperatoren. Z.Angew.

Math. Hech. 57, T 231-232 (1977) 42 Gohberg, I.Z-T., Kre~n, M.G. The basic propositions on defect numbers, root

numbers and indices of l inear operators, Uspehi Mat. Nauk 12 (74), 44-118 (1957) (russ.). AMS Transl. (2) 13, 185-264 (1960)

43 Gohberg, I .Z. , Kre~n,H.G. Systems of integral equations on a ha l f - l ine with kernels depending on the difference of arguments. AHS Transl.(2) I_44, 217-287 (1960)

44 Gohberg, I .Z . , Krupnik, N.A. The spectrum of one-dimensional singular integral o~erators with piece-wise continuous coeff icients. Matem. Issled. 3 16-30 (1968) (russ.). AMS transl . , ser. 2, 103, 181-193 (1974)

45 Gohberg I .Z. , Krupnik, ~.A. Algebra gener~d,by one-dlmens~onal singular integral operators with piece-wise continuous coeff ic ients. Funct. Anal, and i ts Applic. 4, 193-201 (1970)

46 Gohberg I .Z. , Krupnik, N.A. Singular integral operators with piece-wise continuous coeff icients and their s~bols. Math. USSR Izvest i ja 5, 955-979 (1971)

47 Gohber~, I .Z. , Krupnik, N.A. Introduction to the theory of one-dimensional singular integral operators (russ.). ~ t i inc ie , Kishinjow 1973 . . . . . . . .

48 G--~-Idenstein, L.S.,Gohberg, I.Z. On a multidimensional integral equation on a half-space whose kernel is a function of the difference of the arguments and on a discrete analogue of this equation. Soy. Hath.Dokl.~,173-176 (1960)

49 GrabmUller, H. On l inear theory of heat conduction in materials with memory, Existence and uniqueness theorems for the f inal value problem o Proc, Roy, Soc.Edinburgh 76A, II0-137 (1976)

50 GrabmUller, H. Asymptotic behavior of solutions of l inear integro-di f feren~ ial equations. J. Integral Equs. and Op.Theory l , 28-56 (1978)

51 H~rmander, L. Estimates for translation invar~ant operators in LP-spaces. Acta Hath. I04,93-140 (1960)

52 Kohn, J.J.,--N-{renberg, L. An algebra of pseudo-differential operators. Comm.Pure Appl.Math. 18, 269-305 (1965)

53 Koppelman, W., Pincus-~-J.D. Spectral representations for f i n i t e Hi lbert transformations, Math. Zeitschr. 71, 399-407 (1959)

227

54 Kre~n, M.G. Integral equations on a ha l f - l ine with a kernel depending upon the difference of the arguments. AMS Transl. 22, 163-288 (1962)

55 Kremer, M. Ober eine Klasse singul~rer Integ~Igleichungen vom Faltungstyp. DisS. TU Berlin 1969

56 Kremer, M. Ober eine Algebra 'nicht normaler' Wiener-Hopf-Operatoren I & II. Math. Ann. 220, 77-86 & 87-95 (1976)

57 Kumano-Go, H. Pseudo-differential operators ' (japan.), lwanami-Shoten Publish., Tokyo 1974

58 Meister, E., Speck, F.-O. Some multi-dimensional Wiener-Hopf equations with applications. FB Hathem., TH Darmstadt, preprint Nr. 373 (1977); to appear in Proc. 2nd Sympos. on Trends in Appl. Pure Math. Mech. Kozubnik, Sept. 1977 Pitman Monographs, vol. 2 (1978)

59 Meister, E., Speck, F.-O. Wiener-Hopf operators onthree-dimensional wedge- shaped regions, FB Mathematik, TH Darmstadt, preprint Nr. 389 (1978)~

( to appear in Applicable Analysis. 60 Michlin, S.G. Singular integral equations, Uspehi mat. Nauk (N~,) 3,

29-112 (1948) (russ.),~IS transl. 24, 84-198 (1950) 61 Michlin, S.G. Multidimensional s-Tngular integrals and integral equations.

Pergamon Press, Oxfor~ et al. 1965 62 Mushkelishvil i, N.I. Singular integral equations. Noordhoff, Groningen 1953 63 Nickel, K. L~sung eines Integ-r'algleichUngssystems aus der TragflUgeltheorie,

Math. Zeitschr. 54, 81-96 (1951) 64 Peetre, J. M ix~ problems for higher order e l l i p t i c equations in two va-

riables I. Annali di Scuola norm. Pisa (3) 15, 337-353 (1961) 65 Pel legr in i , V. General Wiener-Hopf operators and the numerical range of an

operator. Proc. Amer. Math. Soc. 38, 141 - 146 (1973) 66 P i l i d i , V.S. On multidimensional~isingular operators, Sov. Math. Dokl.l_~2,

1723-1726 (1971) 67 P i l i d i , V.S., Sazanov, L. I . A pr ior i estimates for characterist ic bisingular

integral operators. Sov. Math. Dokl. 15, 1064-1067 (1974) 68 Pr~ssdorf, S. Operators admitting unfunded regularization (russ.). Vestn.

Leningrad Univ. 20, 59-67 (1965) 69 Pr~ssdorf, S. Ei~imensionale singul~re Integralgleichungen und Faltungs-

91eichungen ni%-ht normalen Typs in lokalkonvexen R~umen-~.~abil.schrif{,- Karl-Marx-S'tadt (1967) "

70 Pr~ssdorf, S. Zur Theorie der Faltungsgleichungen nicht-normalen Typs. Math. Nachr. 4_22, I03-131 (1969)

71 Pr~ssdorf, S. Einige Klassen singul~rer Gleichungen. Birkh~user Verlag, Basel-Stuttgart 1974 . . . . . . . . . . . . .

72 Rabinovi~, V.S. Pseudo-differential equations in unbounded regions with conical structure at i n f i n i t y . ~lath. USSR Sbornik 9, 73-92 (1969)

73 Rabinovi~, V.S. Pseudo-differential equations in unbounded megi'ons. Soy. Math. Dokl. 12, 452-456 (1971)

74 Rabinovi~, V.S. Pseudo-differential operators on a class of non-compact manifolds. Math. USSR Sbornik 18, 45-59 (1972)

75 Rakovsh~ik, L.S. On the theormof integral equations of convolution type (russ.). Uspehi Matem.Nauk 18, 171-178 (1963)

76 Reeder, J. On the i n v e r t i b i ~ t y of general Wiener-Hopf operators. Proc.Amer. Math. Soc. 27, 72-76 (1971)

77 ~ahbagjan, R-~.L. Convolution equations in a half-space, AMS Translat.,Ser. 2, 75, I17-148 (1968)

78 Samko, S.G The general singular equation in the exceptional case. Di f f . Equat. l , 867-874 (1965)

79 SchUppeT, B. Regularisierung singul~rer nicht e l l ip t ischer Integralglei- chungen mit unstetigen Koeffizienten, Diss. Univ. MUnchen 1973

80 SchUppel, B. Regularisierung singul~rer Integralgleichungen vom nichtnor- malen Typ mit stUckweise stetigen Koeffizienteno 'Function theoretic methods for partial d i f ferent ia l equations': Proc. Int. Symp., Darmstadt; 4 3 0 - T , Springer Lecture Notes 561, Berlin-Heidelberg 1976

228

81 Schwartz, J.T. Some resul ts on the spectra and spectral resolutions of a class of singular integral operators. Comm. Pure Appl. ~,lath. 15, 75-90 (1962)

82 Seeley, R.T. In tegro-d i f fe ren t ia l equations on vector bundles. Trans. Amer. Math. Soc. 117, 167-204 (1965)

83 Seeley, R.T. Topics in pseudo-dif ferent ial operators. Pseudo-Diff.Opera- tors (C.l.M.E.,Stresa 1968). Edizioni Cremonese, Roma, 169-305 (1969)

84 ~ i r , E. Evaluation dansW~,Ppour des probl~mes au l imi tes e l l i p t iques mixtes dans le plan. C.R. Acad. Sci. Paris 254~ 3621-3623 (1962)

85 Shamir, E. Mixed boundary value problems for e l l i p t i c equations in the plane. The L p theory, Annali Scuola Norm. Sup. Pisa (3) !7, I17-139 {1963)

86 Shamir, E. Reduced Hi lber t transforms and singular in~-~gral equations. Journ. Analyse Hath. (Jerusalem)f2, 277-304 (1964)

87 Shamir, E. ~Jiener-Hopf type pr~lems for e l l i p t i c systems of singular in te- gral equations. Bull.Amer.Math.Soc 72, 501-503 (1966)

88 Shamir, E. E l l i p t i c systems of singular integral operators, l . The ha l f - space case. Trans. Amer. Hath. Soc. ]_~_7_, 107-124 (1967)

89 Shinbrot, M. On singular integral operators. Journ. Math. Mech. I_33, ~95-406 (1964)

90 Subin, M.A. Factorization of matrices depending on a parameter and e l l i p - t i c equations in a half-space. Math. USSR Sbo~nik 14,65-84 (1971)

91 Simonenko, I.B. A new general method of investigat~-n-nq l inear operator equations of the type of singular integral equations. Sov. Math. Dokl. 5, 1323-1326 (1964)

92 Simonenko, I.B. A new general method of invest igat ing l inear integral ope- ratorequations of the type of singular equations. ( russ.) . Izvest. Akad., Nauk, Ser. Matem., 29 567-586 (1965)

93 Simonenko, I.B. Convolution operators in cones. Sov. Math. Dokl. 8, 1320-1323 (1967)

94 S~monenko, I.B. Operators of convolution type in cones. Math. USSR Sbornik 3, 279-293 (1967)

95 S~hngen, H. Zur Theorie der endlichen Hilbert-Transformation. Math. Zei t - schr. ~ 31-51 (1954)

96 Speck, F.-O. Ober verallgemeinerte Faltungsoperatoren und eine Klasse von Integrodi f ferent ialgleichungen. Dissert. TH Darmstadt 1974

97 Speck, F.-O. Ober verallgemeinerte Faltungsoperatoren und ihre Symbole. Function theoret ic methods for par t ia l d i f f e ren t i a l equations: Proc. In t . S-~p., Darmstadt;459-471, Springer L e ~ r e Notes 561 '~, Berl~nJHeidelberg 1976

98 Speck, F.-O. Eine Erweiterung des Satzes von Rak--~vsh~ik und ihre Anwendung in der Simonenko-Theorie. Math. Ann. 228, 93-100 (1977)

99 Strang, G. Toepl i tz operators in a quarter-plane. Bul l . Amer. Math. Soc. 76, 1303-1307 (1970)

I00 Talent i , G. Sulle equazioni in tegra l i di Wiener-Hopf, Boi l . Un. Mat. I t a l . ~, Suppl. fasc. ~, 18-118 (1973) 2n d

I01 Titchmarsh, E.C. Theory of Fourier In tegra ls . ed. , Clarendon Press., Oxford 19~8

102 Tricomi, F. On the f i n i t e Hi lbert- t ransformat ion. Quart. J. Math. (2), 2 199-211 (1951) ,

103 Vishik, M. I . , Eskin, G.I. Equations in convolutions in a bounded region. Russ. Math. Surveys 20, 85-151 (1965)

104 Vishik, M. I . , ~skin~--G.l. Normally solvable problems for e l l i p t i c systems of equations in convolutions ( russ.) . Mat. Sbornik 74, (116), 326-356(1967)

105 Vishik, M. I . , Eskin, G.I. E l l i p t i c equations in co~olut ion in a bounded domain and the i r appl icat ions. Russ. Math. Surveys 22, 13-75 (1967)

106 Widom, H. Singular integral equations in L p. Trans'-. Amer. Math. Soc. 9_7_7, 131-160 (1960)

107 Wiener, N., Hopf, E. Ober eine Klasse singul~rer Integralgleichungen. Si tzg.-ber. Preuss. Akad. Wiss.; Phys.-Math. KI.30-32, 696-706 (1931)

108 Zabreyko, p.p. et a l . Integral Equations - a reference text . Noordhoff Intern. Publ., Leyden 197-~ . . . . . . .

MULTIPARAMETER PERIODIC DIFFERENTIAL EQUATIONS

B. D. Sleeman

Dedicated to the memory of Professor Arthur Erd~lyi

§I. Introduction.

The most widely used method of solving boundary value problems for linear

elliptic partial differential equations is the classic separation of variables

technique. Usually this approach leads to the study of eigenvalue problems

for linear ordinary differential equations containing a single spectral parameter,

namely the separation constant. Indeed it is fair to say that this is perhaps

the prime motivation for the study of spectral problems associated with linear

ordinary differential expressions. In many cases, however, when the separation

of variables method is effected it is found that the separation Constants cannot

themselves be decoupled and several may appear as spectral parameters in each of

the attendant ordinary differential equations.

A classic example of this is the problem governing the vibrations of an

elliptic membrane with fixed boundary. An application of the separation of

variables technique leads to the study of solutions of the following pair of

linked eigenvalue problems for the Mathieu's equations.

d2yl dx~ + (-Xl + x2c°sh2xl)y I = 0, 0 ~ x I ~ ~,

Yl(O) - regular, yl(~) = 0,

d2y2 dx~ + (%1 - %2 cos2x2)Y 2 = O, 0 ~ x 2 N 2~,

Y2(X2) - periodic of period ~ or 2~.

In this example %1 and %2 are the spectral parameters. Another example, recently

considered in [ 15], is the problem of diffraction by a plane angular sector. App-

lying separation of variables in this case results in the study of the following

pair of Lame equations.

230

d2y 1 + {%1 - %2 k2 sn2xl}yl = 0, dx]

yl(-K) = Yl(K) = 0,

d2y2

dx~ + {%1 - %2 k2 sn2x2}Y2 = O,

= '(K + 2iK') = 0. Y2 (K) Y2

-K -< x I -< K,

x 2 K + 2i~, 0 < ~ < K' = _ -- ,

Once again %1' %2 are spectral parameters and k, the modulus of the Jacobian

elliptic function sn~ is related to the semi-apex angle ~ of the sector by

k = sin ~.

One can cite a wide range of problems leading to linked systems of ordinary

differential equations containing 2, 3 or more spectral parameters.

During the early decades of this century the study of multiparameter eigen-

value problems gave way to some extent to the study of the "special functions"

generated by such problems. Thus in the 1930~s and 40's there was much interest

in the properties of Mathieu functions, Lame functions, ellipsoidal wave functions,

Heun functions and the like. The leading instigator of these researches was

undoubtedly Arthur Erd~lyi who not only advanced our knowledge of these functions

to the degree that we know them today, but also gave an encyclopaedic account of

them in the well known volumes on higher transcendental functions [12] which

arose from the Bateman manuscript project. These special functions are being

actively studied today and apart from their obvious interest to the mathematical

physicist they do arise in other areas as well. Let us consider for the moment

one example which leads to an as yet unsolved problem.

The problem we have in mind is of importance in the study of quasi-conformal

I mappings[5] and may be described as follows. In Lam~'s equation let %2 = -

and consider the equation

d~ (h + 1 k 2 dx 2 + ~ sn 2 x)y = 0

and denote two linearly independent solutions by ~I and ~2- Classical

231

uniformisation theory asserts that for each value of the modulus k and consequently

there for each fundamental pair of periods of the Jacobian elliptic function sn x,

is precisely one value h such that

n_l n 2

However the

we ask; for

n~ n 2

C l e a r l y one

(C - periods of sn x) = ~ a simply covered disc}

L or a half plane J

actual value of h is known in only a few special cases.

which values of h is

~ simply covered 1 (C - periods of snx) = ? L Jordan domain

In general

method of attacking this problem is to investigate thoroughly the

solutions nl and N2" In the slightly more general situation in which %2 = V(V + I)

1 where V = ~ + m (m an integer) there are results due to Halphen LI4j, Ince [16J

and quite recently by Pearman [|8j. Indeed Ince [16j observed that when k 2 = ~ 2

some of the eigenvalues of Lam~'s equation giving rise to periodic solutions of

periods 8K, 8iK' were rational and conjectured that this may be true for infinitely

1 many values of v = ~ + m. We may confirm this conjecture by observing that when

l ! l = ~ + m, %1 = ~ ~(~ + l) and k 2 = ~. Lam~'s equation has solutions of the form

n l ' ~2 = ( s n x ± d n x / k ) 1 / 2 ] 1 3. 1 2FI (- ~m,~m +I; ~, ~ (snx ± dnx_k.2../) ~

This can be verified by direct substitution. Clearly if m is even and positive

then there are infinitely many rational values of %1 each giving rise to solutions

of periods 8K, 8iK'. There must be a large number of results of this kind relevant

to the problems above.

An important feature of the multiparameter eigenvalue problems which arise in

mathematical physics is that the differential equations often have either singly or

doubly periodic coefficients and it is to such systems that we now turn.

§2. The Problem

Consider the linked multiparameter system of periodic ordinary linear differ-

ential equations of the form

232

2 (Xr) d Yr n

+ qr(Xr)Yr(Xr) + ~ %sars(Xr)Yr(Xr ) = O, dx 2 s=l

r

r = 1 .... , n. (2.1)

Here x r £ [0, ~0rj , (0 < 0o r < ~) and the coefficients ars, qr are continuous

real valued functions with period ~r" We make the basic assumption that the

determinant

det{ars(Xr)} n > O. r,s=! (2.2)

for all x r E [0, ~r j, r = I, ..., n. Several other structural hypotheses may be

applied, see for example [20] or [4J.

In addition to (2.2) we impose periodic boundary conditions of the form

(P) Yr(0) = yr(~r), (2.3)

y'(0) = Yi(~ ), r = l .... n. r Lr

or semi-periodic boundary conditions of the form

(S-P) Yr(0) = -yr(~r),

(2.4)

'(0) = -y~(~r ), r = l, Yr ..., n.

As is now standard we say that the n-tuple of necessarily real numbers

% = (%1 ..... %n ) is an eigenvalue of the system (2.1, 2.2, 2.3) (or 2.4) if this

system has a non-trivial set of solutions Yr(Xr; %), r = l ..... n, for this

particular n-tuple. The algebraic product Yl(Xl; %) ... Yn(Xn; %) is called the

corresponding eigenfunction.

The study of the above system and similar Sturm-Liouville problems has

captured the interest of several authors recently. Indeed such problems may be

formulated in terms of linear operators in Hilbert space. The book of Atkinson

[4] and the monograph of Sleeman [20J are suitable references to this and other

matters. From these works it may be readily verified that eigenfunctions and

eigenvalues do exist for the above problems and that the eigenfunctions form a

complete orthonormal set in the weighted Hilbert space of Lebesque measurable

functions on

233

n

E0, ~] ~ x [0, ~r ], r=l

the Cartesian product of the n compact intervals [0, ~r j, r = I, ..., n. The

norm in this space is given by

]lull 2 = f [ul 2 det{ars}dX dx (2.5) [0,~] 1 "'" n"

In the following section we show how a classical argument well known in the

one parameter case may be used to assert the existence of eigenvalues and eigen-

functions.

A question of particular interest is to ask whether the eigenvalues of system

(2.1, 2.2, 2.3) or (2.1, 2.2, 2.4) or any combination of these problems enjoy any

interlacing properties thus generalising the results well known in the one-

parameter periodic case. In the same way one would like to discuss the stability

of eigenfunctions. Recently Browne [8j has studied the interlacing problem and

Browne and Sleeman [9j have obtained some results concerning the stability of

eigenfunctions. To describe these results we shall need to introduce some

notation.

Let o = (oi, ..., ~n ) £ ~n, Or = 0 or I r = I, ..., n be an index set. We

can then speak of boundary conditions of type o applied to the system (2.1) to

mean that periodic conditions are to be applied to the r-th equation if O = 0 and r

semi-periodic conditions if o = I. Thus for each fixed o we have formulated a r

mnltiparameter eigenvlaue problem consisting of (2.1) (2.2) with boundary

conditions (P) or (S-P) as indicated by O. In this fashion it is appropriate

to label the eigenvalues as

i i

X~(O) = {%r(O)[ r = | ..... n},

where ~i = (i 1, ..., in) is a multl-index. That is each ir,* r = I, ..., n, is a

non-negative integer. This multi-index provides an ordering of the eigenvalues.

Notice that there are 2 n possible spectra corresponding to the 2 n different types

of boundary conditions. Thus to begin with we would like to know how these

spectra interlace each other.

234

~n Consider vectors a, b c which are partially ordered by a ~ b if a

~ ~ ~ r

r = I, ..., n. Now for a collection of functions Yr(Xr) ~ L2(O, ~r ), with

llyrl I = ! we denote by V(y) the n × n matrix whose entries are

w

i r 12dXr, 0 ars(Xr)lYr(Xr) r, s = l, ..., n.

Next introduce the set C c ~n defined by

-<b r

C = {a e i~ n I V(y)a -< 0 for some Yr £ L2(O' C°r)' IIYrll = |, r = l ..... n}.

Clearly C defines a cone and, as demonstrated by Binding and Browne [6], each of

the 2 n possible spectra is ordered by C.

Finally we define the set

i = {~~(~) l i-multi-index; o = 0 or I, r = 1 ..... n}.

~ ~ ~r

Thus I is the collection of all eigenvalues from all possible problems periodic

or semi-periodic.

The main interlacing theorem of Browne LSJ can now be stated;

Theorem 2.].

Let T ~ ~n be such that ~ = 0 or |, r = |, ..., n. r

multi-indices with 0 N e N (l,|, ..., l). the set

i i+e (~~(~) + c ) n (~~ ~(~) - c ) n ~,

J contains any eigenvalue X~(~) for which

Then if i and e are

(i) i -< j -< i + e and

(ii) O r = T r except that if e r = ; and (ir, T r) is (even, 0) or (odd, |)

then ~ may be 0 or l, r = ;, ..., n. r

To illustrate this result consider the following elementary example for n = 2.

Let x l, x 2 e [O, lj and take Sl, s2, ql' q2 to be real valued continuous periodic

functions on EO, IJ with period one. We further assume s I > O, s 2 > 0 on [O,l].

The two parameter system is

235

d2yl(x 1 )

dx~ + q l (x l )Yl (Xl ) + (-%1 + %2)sI(xl)Yl = O,

d2y2(x2 )

dx~ + q2(x2)Y2(X2 ) + (-~'1 - %2)s2(x2)Y2 = O.

The definiteness condition (2.2) is satisfied since

det -Sl(X 1) Sl(X 1)

-S2(X 2) - s2 (x 2)

= 2Sl(Xl)S2(x 2) > O.

For this example the cone C is given by

~2 C = {(a,b) ~ 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 ~~. ~ ~ i ~

Finally if o = (1,1) 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 problem.

Definition 2.1

(i) 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 (-~,~), r = I, ..., n.

(ii) A point % E ~ n is said to be a point of conditional stability for the

system (2.1) if each of the equations has a non-trivial solution bounded over

(-~,~).

(iii) A point % £ ~n which does not satisfy either (i) or (ii) is said to

be a point of instability for the system (2.1).

236

/ '

",.

i

r s

1

• ~ 0 ~

1 /

i >,

i s

I s

O ~

p i

z i

r i

o~

n

¢e\

%

%

\

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

~n L 2 (0, ~ ) such that points a E for which there are non-zero points fr r

287

[V(f)a]r; the r-th element of the column vector V(f)a, satisfies

[V(f)a] r~ ~ 0 if ir is even,

0 if i is odd. r

This defines 2 n cones, i.e. 2 n-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 i S ¢ u [{X~(0) + C(i)} n {X~(1) - C(i)}], (2.6)

I

w h e r e I = ( 1 , | , l ) e

This result is illustrated in the case of the above example by the shaded reg-

ions 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, (3.1)

~'(0; ~) = 0, ~'(0; ~) = i, r r

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

~r(mr; X) = I,

Cr(mr; X) = 0,

238

~r(~r; X) = 0,

~'(~ ; X ) = 1. r r

(3.3)

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, (3.4)

T !

~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) = ±2, (3.5)

r = | ~ ..., n.

The problem of existence of eigenvalues is thus reduced to the question of

solvability of the system (3.5).

method we find

Dr(X) X) + ~X

S

r r 2

JO {~r(T; %)~r(~r;

By a standard use of the variation of parameters

+ ~r (T; X)~r(T; %)[~$(~r; X) - ~r(~r; %)] (3.6)

- ~$(T; %)¢~(mr; X)}ars(T)dT,

r~s = I, ...~ n.

Now if D (%) = i2 then the term in { } is a perfect square and since r

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 Hessian matrix constructed from the second partial

derivatives of D and make use of the inverse function theorem again. r

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 for each r = I, ..., n to arrive at the multiparameter analogue r

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

(4.1)

-I < t ~ I, r = 1,2, together with the definiteness condition (2.2). r

For this problem the existence of a countably infinite set of real eigen-

values 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 functions the boundary

conditions (4.2) may be rewritten in the form

that

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~,

al2(X I) < 0 on [0, ~i j, (4.4)

a21(x 2) > 0 on [0, ~2 j.

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 of the system (4.1)

(4.2) are simultaneous eigenvalues and eigenfunctions of the following periodic

r = 1,2, x e [0, ~ J r r

y$(Wr) = y$(O)exp i~ t r,

yr(mr) = Yr(0)exp i~t r, (4.2)

240

problems for partial differential equations; viz.:

~2y ~2y

- al2(X I) ~ + a22(x 2) ~x 2 ~x!

r = 1,2.

+ La12(x])q2(x 2) - a22(x2)ql(Xl)~Y

Y(x + ~r ) = Y(x)expi~ t r,

= % det{a }Y (4.5) 1 rs

(4.6

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).

defined as

In this condition the vector ~ is ~r

a12

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). We also have the completeness n ~

theorem.

~l = (el' 0), ~2 = (0, w2).

It should be noted that because of the assumed positivity conditions on

and a21 the left hand side of (4.5) is elliptic and it is to this equation that

Theorem 4.1.

- °-th roots For each j = 1,2 let exp(i~tr ) (r~a = 0, I .... , k~ I) be the kj

J < I. Let ~R (I < R < klk 2) denote the pair (trl, tr2) of unity where -I < tr. - - -

J

in any order. Then the set of all functions ~n(X; tR ) where n ~ 0 and

241

1 ~ R ~ klk 2 is a complete set of eigenfunctions for the periodic problem over

the rectangle

Xl, x 2 E [0, WlklJ x [0, ~2k2 j.

Again completeness here means completeness in the sense of functions

Lebesque measurable on [0, ~Ikl j × [0, ~2k2 ] with respect to the weight det{ars}.

The proof of this theorem is a simple modification of the proof of Theorem

6.2.1 in Ill]. With the aid of theorem 4.1 we can now argue as in [19j to prove

that the eigenfunctions of the given problem defined by (4.1) (4.2) are complete

in the same sense.

Now let

Q(Xl , x2) = a l2 (Xl )q2 (x2 ) - a 2 2 ( x 2 ) q l ( X l ) , (4 .8)

and let F be the set of complex valued functions f(x) which are continuous in

A ~ [0, ~i j x [0, w 2] and have continuous first order partial derivatives in A.

Define the Dirichlet integral

fA{a22(x2) ~u ~V D(u,v) = ~x I ~x 1

for all u,v ~ F.

~u ~

a l2 (X l ) ~x2 ~ x 2

+ Q(x 1, x2)u(x)V(x)}dx I dx 2 (4.9)

This Dirichlet integral forms the basis for a variational approach to the

multiparameter periodic problem. It is discussed, at least for SchrSdinger

operators, in Eastham [I|] and its application to multiparameter Sturm-Liouville

problems is considered in [19, 20].

Suppose we impose Dirichlet conditions on the boundary of A, i.e.

Y(x) = 0 on ~A,

then for the associated Dirichlet problem we have corresponding eigenfunctions

and eigenvalues which may be denoted by @n(X; 6n) and 6n respectively. In a

similar way, if we impose Neumannconditions on ~A then the associated Neumann

problem will give rise to corresponding eigenfunctions ~(x; Nn ) and eigenvalues ~.

242

Now, arguing as in Eastham [II, p 101] we have the interlacing theorem

Theorem 4.2.

For n e 0 and all t

~n ~ An(i) ~ Nn" (4.10)

This simple interlacing result may be employed to give a further interlacing i i

theorem for the two-parameter problem as follows. Let (%7(t), ~%(t)) be an eigen- i

j ~

value for (4.1) (4.2) then since the %7(t) form a subset of the An(i) theorem

(4.2) gives, in an obvious notation, the result

i i i

for all t. ~ i

Substituting %7(t)into the first member of the system (4.1) we have the clas-

sic one parameter problem.

d2y I i dx~ + ql(Xl)Yl+ a l l (Xl)X~(t )Yl + X2al2(Xl)Yl = O,

~ (4.12)

Yl(X! + e l) = Y l ( X l ) e x p ( i g t l ) ,

where of course al2 < 0.

If we let the eigenvalues associated with (4.12) and the conditions

yl(wl) = yl(0) = 0,

i be denoted by 62 and those associated with (4.12) and the conditions

y~(~l ) = y~(0) = O,

i be denoted by n2 then again appealing to Ell, p 39] we have, for those eigenvalues i

%~(t) which are simultaneous eigenvalues of (4.1) with r = 2,

i i i

(4.13)

These arguments may be summarised in

Theorem 4.3.

Let the eigenvalues for the Dirichlet problem over A for (4.5) be denoted by

243

i i ~ ~ and the eigenvalues for the associated Neumann problem be denoted by n~,

i i i where i = (i|, i 2) is a multi-index. If k~(t) = (%~(t), %2(t)) is an eigenvalue

for the two-parameter t-periodic problem then

i i i 8 ~ ~ k~(t)~ ~ ~n ~.

i For this value of k~(~) we also have

i i i

i i where B2, 62 are the Neum~nn and Dirichlet eigenvalues for the one parameter

equation (4.12).

There are a number of results to be obtained via the variational approach

all of which extend in one way or another the results known for periodic

Schr~dinger equations and promise interesting applications to the multiparameter

periodic problem. We close this section with the remark that hidden in the

variational approach outlined above is the fact that the partial differential

operator appearing in (4.5) is elliptic. For n ~ 3 we cannot always arrange for

any of the associated partial differential operators to be elliptic in general.

In this case the arguments have to be modified and for details at least in the

abstract case we refer to the monograph [20].

§5. The Alsebraie Approach

In the study of periodic differential equations some useful insights are to

be gained by the study of various algebraic forms of the associated periodic

differential equations. There is a strong connection here with the study of

Monodromic groups, the famous Riemann problem arising from Hilbert's 21st problem

(see the article by Nicholas Katz in [7]) and also the study of quasi-conformal

mappings, the subject raised in the first section of this paper.

Here we outline some of the results which are to be obtained in this setting

and in particular the case of differential equations, like Lam~'s equation,

which have doubly-periodic coefficients. For a background to this work we cite

[17, I, 2] as suitable references. The essential feature to be noticed is that

when expressed in algebraic form a singly-periodic differential equation is

244

distinguished by having only two finite singularities whereas a doubly-periodic

equation has three.

The basic concept is that of a solution which is multiplicative for a given

path in the complex plane. That is a solution which when continued analytically

along a closed path is merely multiplied by a constant. Consider the differential

equation

_ _ dw d2w + p(t) ~-~ + q(t)w = 0, (5.1) dt 2

where t E ¢ and p(t), q(t) are rational functions of t. Let the singularities

(not necessarily regular) of (5.1), i.e. points where p(t), q(t) or both are

singular, be denoted by tl, t2, ..., tn, ~. If t o is an ordinary point of (5.1)

then we denote by F i a simple closed path from t o encircling the singularity t. i

oncepositively (clockwise) and enclosing no other singularity of (5.1). We

further denote by rij the path consisting of r i, rj described successively in that

order and similarly for rij k etc. The path which is the same as F i but described

in the negative (anti-clockwise) direction is denoted by -F i. Observe that

provided the singularities tl, ... t are labelled in an appropriate order then ~ n

• is effectively a path making a negative circuit about infinity. That is FI2 " "n

FI2 ... n is equivalent to the circuit -r .

Since p(t), q(t) are rational functions of t the singularities of equation

(5.1) are isolated. Hence if y(t) is a solution valid in a neighbonrhood of the

ordinary point t o it can be continued analytically along r i back to the neighbour-

hood of t o giving a solution y*(t). Symbolically we write

y(t) ÷ (Fi)Y*(t). (5.2)

In the case that y*(t) is a constant multiple s i of y(t), i.e. y*(t) = siY(t) we

can write

y(t) + (Fi)siY(t). (5.3)

In this case we say that y(t) is multiplieativ e for the path r i and s i is the path

factor. It is easy to prove that there exists at least one multiplicative solution

of (5.1) for the path F.. If (5.3) holds then also I

245

l y(t) ~ (-r i) Y7 y(t).

l

A pa th F i f o r which t h e r e e x i s t two l i n e a r l y ,,independent m u l t i p l i c a t i v e s o l u t i o n s

i s c a l l e d n o n - d e g e n e r a t e . I f only one such s o l u t i o n e x i s t s the pa th i s d~generat ,~

Thus for a non-degenerate path r. there are linearly independent multiplicative l

(1)(t), y~2)(t) which can be combined into a solution vector solutions Yi

(2) (t) } such that Yi (t) = {Y l)(t)' Yi

Yi + (ri)siYi (5.4)

where S. is the constant diagonal matrix <s! l), s! 2)>. l i i

Let us now consider the case in which (5.1) has two finite singularities

tl, t 2 in the complex t-plane and a singularity at infinity. We wish to consider

the behaviour of solutions when continued analytically around a composite path

such as FI2. To this end we introduce the following notation.

2Pi = s(1)z" + s(2)'i 2qi = s(1)i - s.i(2),

r. = s! I) s! 2), i = 1,2, i i i

M(012) = cos @12 sin Ol2 I '

sin Ol2 -cos 012

where el2 £ C in general and is called the "link" parameter for the path FI2.

Consider now a solutiQn vector YI such that

YI ÷ (FI)S]YI" (5.5)

The analytic continuation of YI about r 2 yields a solution vector of (5.1) so

there is a constant matrix L = (%ij) such that

Y| ÷ (r2)LY | .

The eigenvalues of L must be the path factors s~ I), s~ 2) for F 2 and so

(1) s~2) trace (L) = s 2 + = 2P2

(I) s~2) = r2" det (L) = s 2

This then shows that for some constant ~ the entries in L must be such that

246

%11 = P2 + ~' %22 = P2 - ~'

2 2 %12%21 = q2 - ~ "

As is demonstrated in [2j there is no loss in choosing

= q2 cosel2 with -~ ~ Re ~12 ~ ~

and taking L to be

LI2 = p21 + q2M(Ol2 ),

where I is the 2 × 2 identity matrix.

To summarise these results we have

Theorem 5.1.

Let F I, r 2 be non-degenerate paths. Then there exists a link parameter el2

uniquely determined in the region

R = {O I {0 < Re 0 < ~} u {Re 0 = 0, Im e < 0} u {Re e = ~, Im e < 0}

such that there is a unique vector Yl2 determined up to a constant scalar multiplier

with the properties

Yl2 + (FI)SIYI2

Y12 ÷ (F2)LI2YI2 °

From this result we deduce

Theorem 5.2.

The path factors for FI2 are the roots of the equation

2 + qlq2 cos + r = O. (5.6) s - 2(plp 2 e12)s ir2

Since (5.1) in this case has only 2-finite singularities r12 is equivalent

to -F and so the path factors for -r must be precisely the path factors

determined from (5.6). Indeed as we shall show this provides a means of deter-

mining the link parameter el2 or the characteristic exponents of 5.1 at infinity.

Consider for example Hill's equation in algebraic form, i.e.

__ 1 | ~ dw ~(t) w _ 0, (5.7) dXw + 2 {~ + } d-t + t(t l) dt 2

247

where ~(t) is an integral function of t. In this case it is known that t = 0,I

are regular singularities while the singularity at infinity is irregular. It

turns out that with t I = 0, t 2 = I

Pl = P2 = O, ql = q2 = 1, r I = r 2 = -l,

Here (5.6) reduces to

with roots s = exp(±ie).

sll) = s~ I) = 1, s12) = s~ 2) = -I.

2 s - 2s cosel2 + I = o,

But since FI2 is equivalent to -r it follows that the

characteristic exponents ± ~ at infinity, in the sense of Erd~lyi [13] are

precisely ± el2. In another way it follows that el2 is related to the number t

in the definition of the t-periodic problem (see (4.1) (4.2) above), or if t = 0

to the descriminant D discussed in section 3. In this way the determination of

characteristic exponents at infinity is related to the solvability of the equation

D = 2 cos ~t.

This interplay between characteristic exponents and the descriminant D has been

exploited considerably in the study of such special functions as Whittaker functians,

Mathieu functions, spheroidal wave functions etc. and in the older literature to

the study of the so-called Hill determinants. See for example the book of

Arscott Ill. How successful these ideas are in the study of general multiparameter

periodic problems is yet to be investigated.

It should be noted that the assertion that there is at least one multiplicative

solution of (5.7) for FI2 is simply the algebraic form of Floquets theorem.

If (5.1) has three singularities in the finite part of the plane then the

situation becomes much more complicated. However the results can be simply stated

and have a nice geometrical interpretation, see [2].

Theorem 5.3.

The path factors for FI23 are determined as the roots of the equation

s2 - 2s{PlP2P 3 + I qlq2P3 cos012 - 2iqlq2q 3 sin012 sin @23 sine3l} 1,2,3

+ rlr2r 3 = O, (5.8)

248

where cos e23 = cos 012 cos 813 + sin el2 sin 013 cos~23. (5.9)

In this result (5.9) has, at least for real 8ij, the following geometrical

interpretation. Suppose e12, 013, e23 are the three sides of a spherical triangle,

then ~23 is the angle between the sides 012, 613. Equation (5.8) appears to lack

the symmetry one would expect between the indices I, 2, 3. However returning to

our geometrical analogy we observe that

N = sin~23 sin @31 sin @12

is simply the "norm" of the our spherical triangle and so is also given by

N 2 = sins sin(s - Ol2)sin(s - 023)sin(s - e31) ,

where 2s = 812 + 023 + 031.

The syn~netry of N and so of (5.8) is now obvious.

As an illustration of theorem 5.3 consider the algebraic form of Lam~'s

equation, i.e.

_ _ I I I I dw h - ~ (V + 1 ) k 2 t d2w +~ {t + + t _--~-- + w = 0. dt 2 t- ] - dt t(t- l)(t- k -2)

( 5 . 1 0 )

-2 Here we take t I = 0, t~ = I, t 3 = k and find that

s (I)~. = I, s (2)~. = -I, Pi = O, qi = I, r.~ = -1, i = 1,2,3.

Thus in this case (5.8) reduces to

2 s - 4iNs - I = O.

But FI23 is equivalent to -F and so the path factors of FI23 must be exp i~,

exp-i(v + l)n. Consequently

I N = ~sinv~. (5.11)

Thus if two of the link parameters O., are known then the third is determined by lj

( 5 . 1 1 ) .

Using ideas related to the above Arscott and Wright [3] have essentially made

a study of the link parameters when ~ is rational and have discussed the uniformi~

of the resulting solutions. This in a sense brings us full circle as these ideas

249

and results may have direct bearing on the outstanding problem of quasi-conformal

mappings introduced at the beginning of this paper.

References

[l] F. M. Arscott, Periodic Differential Equations. Pergamon Press, London

(1964).

[23 F. M. Arscott and B. D. Sleeman, Multiplicative solutions of linear differ-

ential equations. J. London Math. Soe. 43 (1968), 263-270.

[33 F. M. Arscott and G. P. Wright, Floquet theory for doubly-periodic differ-

ential equations. Spisy P~irodov Fak. Univ. J. E. Purkyn~ V. Brn~

(]969), I]]-124.

[43 F. V. Atkinson, Multiparameter eigenvalue Problems: Matrices an d Compact

Operators. Academic Press, New York and London 1972.

[5] Lipman Bers, Quasi conformal mappings with applications to differential

equations, function theory and topology. Bull. Amer. Math. Soc. 83 (1977)

1083-1100.

[6] P. Binding and P. J. Browne, A variational approach to multiparameter

eigenvalue problems in Hilbert space. SIAM J. Math. Anal. (1978) (to

appear).

[7] F. E. Browder (Ed.), ~thematicalDevelopments arisin$ from Hilbert Problems.

Proceedings of Symposia in Pure Mathematics, Vol. 28 Part 2. American

Math. Soc. Providence R.I. (1976).

[8] P. J. Browne, The interlacing of eigenvalues of periodic multi-parameter

problems. Proc. Roy. Soc. Edin. A (]978) (to appear).

[9] P. J. Browne and B. D. Sleeman, Stability regions for multi-parameter

systems of periodic second-order ordinary differential equations (submitted),

[I0] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations.

McGraw-Hill, New York (]955).

25O

[11] M. S. P. Eastham, Th e Spectral Theory of Periodic Differential Equations

Scottish Academic Press, Edinburgh and London, (1973).

[12] A. Erd~lyi et al, Higher Transcendental Functions Vol 3. Mc Graw-Hill,

New York (1955)

[13] ~Erd~lyi, Asymptotic Expansions. Dover (1956).

• Y

[14] G. H. Halphen, Tralte des Fonctions Ell iptiques Vol 2. 1888.

[15] B. A. Hargrave and B. D. Sleeman, The numerical solution of two-parameter

eigenvalue problems with an application to the problem of diffraction by a

plane angular sector. J. Inst. Maths. Applics. 14 (1974) 9-22.

[16] E. L. Ince. Periodic Lame functions. Proc. Roy. Soc. Edinbursh A 60

(1940) 47-63.

[17] E. L. Inee, Ordinary Differential Equations. Dover (1956).

[18] A. E. Pearman, Lam~ functions in scattering problems with particular

emPhasis on the elliptic cone. PhD Thesis, Dundee (1974).

[19] B. D. Sleeman, Completeness and expansion theorems for a two-parameter

eigenvalue problem in ordinary differential equations using variational

principles. J. London Math. Soc. (2) 6 (1973) 705-712.

[20] B. D. Sleeman, Multiparameter Spectral Theory in Hilbert Space.

Pitman Press, London (1978).

Department of Mathematics

The University

DUNDEE DDI 4HN

Scotland UK.

UNIFORM SCALE FUNCTIONS AND THE ASYMPTOTIC

EXPANSION OF INTEGRALS

by

Jet Wimp

Drexel University, Philadelphia

and

University of Strathclyde, Glasgow.

NOTATION

S, sector : S E S(~l,~2.z o) = {z ~ ~ i~ I < arg(z-z o) ~ ~2' z # Zo} ,

S A = S(-7 + A, ~- A, 0), 0 < A < ~ .

UR(Zo) = interior of circle, radius R, centre Zo," UR(O) = UR; U 1 = U.

CR(Z o) = circumference of UR(Z o) with clockwise orientation; CR(O)

= CR; C 1 = C.

~R(Zo) = UR(Z o) U CR(Zo). H 2 = Hardy class.

n,z = asymptotic parameters, n ÷ ~ in J+, z + ~ or z ° in S.

~(A) = class of functions analytic in A.

P = {(~l,a2 ..... ap) = ~laj e ~}.

i o ~(z) = e -zt f(t)dt (Laplace transform). o

252

I. Introduction

Asymptotic series are today one of the next important tools in mathematical an-

alysis. In their original form they were discovered almost simultaneously by Poin- J

care (1886) and Stieltjes (1886). It is possible these writers were simply giving

embodiment to pre-existing - though vague and non-rigorous - ideas already floating

about the mathematical community, and which can be traced back at least to Euler

and his free-wheeling use of divergent series.

Asymptotic series - in fact, asymptotic analysis in general - is a subject that

has never quite managed to escape the taint of mystery. Even recently, asymptotic

series have not always been spoken of clearly. I feel that Arthur Erd~lyi did much

to clear the dead wood of heuristic thinking from this subject. As Erd~lyi himself

pointed out, his ideas were not new (what mathematical idea is ?). His idea of an

asymptotic scale goes back at least to Schmidt (1936), but Erd~lyi was the first to

exploit the idea on a systematic basis.

In no other area of mathematics has intuition been both so vitalizing and so

crippling. In the early 1900's it was clear that in asymptotic analysis the good

had become the enemy of the best. What were needed - among other things - were reas-

onable definitions. And Erd~lyi's definitions of asymptotic equivalence and scale

were very reasonable and useful ones. Erd~lyi was one of those who recognized that

the asymptotic properties of an expansion shouldn't be judged by the magnitude of

the individual terms, let's call them fk(z,~), where ~ is in some parameter set

c ~P and z E S. For certain values of z individual terms could be O, thus

conveying a false sense of the precision of the expansion. The asymptotic measures

needed were functions - call them pk(Z,~) - which "closely" bounded the fk but which

could not become arbitrarily small on compact (z,~) sets. Fortunately, the essential

idea turned out to be easier to define and implement than this clumsy formulation

suggests. The effect of these definitions was to greatly enlarge our concept of a

permissible term - a base function - in an asymptotic expansion. Now the fk(z,~)

were allowed to be members of a much larger class of mathematical functions than the

simple inverse powers of z occurring in the classical Poincare theory. In fact, the

larger the parameter space GL and the greater the requirements of uniformity of the

expansion in~, the more arcane will he the functions fk(z,~). One does not have to

travel far in Erd~lyi's general theory - see our discussion of Darboux's method and

its application to functions having the unit circle as a natural boundary - to en-

. i Stieltjes would have recognized counter asymptotic series that neither Polncare nor

Inevitably, the behaviour and properties of these new base functions became sub-

jects of mathematical interest in their own right. In this paper, I wish to discuss

how general asymptotic expansions arise, and to describe some of the special base

functions involved.

253

2. .... Poincar~ asymptotic series

We differentiate between two kinds of Pozncare

Definition

asymptotic series.

( I )

(i) Let f be defined in S. co

-k f ~ E akz ,

k=O

By

z->= in S,

we m~an K

If - ~ akz-k I = o(z-K), z ÷ = in S,

6=0 K = 0,1,2, ....

we mean

(ii) Let

f~

f be defined in S with vertex z . By O co

a (Z-Zo)-k , z ÷ z in S, E k-O o

K

I f - 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~, or better yet, a modern treatment, such as Olver C1974)o

This definition is good, as far as it goes• 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 an-- [(l-xm)n], n = 0,1,2,.... Pn(X) 2 n nl dx n

where z is real, z = n e J+, n ÷ ~. I will write

(n - ~ - ¼)~} _, cos{(n-k 2~0 + k (2) Pn(C°S 8) ~ ( )½ ~ ( k)(

k=O (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

t were called inverse central factorial series

totic-like in their properties.

, and were both convergent and asymp~

One example he gave was

- i _ I) (a 2 _ ~) 1 I (a 2 ~) (a 2 9 = _ + - - +

(3) E k2 a2 k---n+~ - 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. This is a feature

of certain Poincar~ asymptotic expansions, see Knopp (1948). 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 i E

k=O (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) in the functions

I= t.2k (2k)~ P(z-k) e -zt (2 sinh ~) dt. £(z+k+l)

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. For a given problem

both sequences will be defined for Izl > R, or IZ-Zol < ~ in some sector S withvertex

z . (This allows us to combine z ÷ ~ and z + z in one definition). In addition, o o

~k and ~k may depend on ~ s ~ c[P.

t Factorial series ak/(Z+l)(z+2)...(z+k) had already been discussed by many writers

see Norlund (1954) and his references - but not from the point of view of asymptotic series.

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, k = 0,1,2 .....

(iii) ~k and ~k are equivalent if each dominates the other.

(iv) #k is an asymptotic scale if ~k+l = °(~k)' k = 0,1,2,....

(v) the series

fk k=O

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 in~.

(vii) the series

(5) f ~ E Ck ~k ; {~k } ' k=O

• J

is called a Polncare asymptotic series. (Then the term on the right

is usually deleted.)

For the basic properties of asymptotic sequences and expansions, see Erd~lyi

(1956), (1961), and particularly Erd~lyi and Wyman (1963). Because of the generality

of these definitions, 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. This, in prac-

tice, does not seem to be a drawback. 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 scale~-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 prob-

lem easy. I wish to find an asymptotic expansion for the coefficients a n in

256

co

r(l+t) = Z (-I) n a n t n, Itl < i.

n=0

I have

= b + Cn, an n

I 1 I (-l)n e -t (-l)n e -t (Zn t) n dt. bn = n' (£n t) n dt, c n = ~ i

o

-t . Expanding e in its

gives

b = E (-l)k

n k= 0 k~(k+l)n+l

But this is also an asymptotic series, with scale ~k = (k+l)-n since

K (-i) k 1 e - E ~ - o~K+I) -n] b n

k=O k!(k+l) n+l I (K+2) n+l

Taylor series in the first integral and integrating termwise

In the integral for c, using the fact that

2t ½ n ~n t ~--~- , i < t ~ ~,

shows

C n

and so

a ~ Z (-l)k -n ; { (k+l) } ,

n k=O k: (k+l) n+l

the same result obtained by Riekstins (1974) from a general theory.

It is interesting that the series converges (but not to Cn~).

Another approach is to use Laplace's method, see section 5, on the integral

(-I) n e-t an = n! (~n t) n dt.

O

The location of the critical point t* depends on the large parameter, and is the root

of a transcendental 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 scales ; Laplace transforms

The result called Watson's lemma is historically the first of a large number of

generalized initial and final value theorems for the Laplace transform.

Theorem: (Watson's lena)

Let ~ > 0, ReB > -i, and k

f ~ Z a k t ~ k=0

t *O + .

Let f exist for some z.

Then

akr(~ ~ + B +I) % Z k , z ~ = in S h.

k=O --+ B + I z ~

The applications of Watson's lemma are many. For a discussion, see Olver (1974).

For example, if

I(z) = [ e zh(t) g(t)dt , ~F

where g,h c~(B), F c B, then the method of steepest descents supposes that the value

of I for large z is determined primarily by the values of h near those points in B

where h' (t) = O, called critical points. Assume h has just one critical point, I

will not give conditions, which are difficult (again, see Olver (1974)) but the gen-

eral idea is to make the substitution

h(t) - h(t*) = (t-t*) 2 2 h''(t*) + ... = -w ,

2

h''(t*) < O.

This transformation is at least locally invertible, so in a neighbourhood of 0

t = t* + [-2/h''(t*)] ½ w + ....

One would expect the major contribution to I to occur at w = O. Thus

l(z) % e zh(t*) [--2/h''(t*)] ~ I~ e-zw2 u(w)dw,

where

e zh(t*) [-2/h' ' (t*)] ½ 0o

-k-I Z C2k F(k+½)z 2,

k=O

Z -> oo in S A ,

u(w) = Co+ClW+C2W2 + .... , lul < 6.

258

In his first paper in the area of asymptotics (1947), Erdelyl introduces the con-

cept of an asymptotic scale to handle other initial and final value theorems for the

Laplace transform, including the result mysteriously alluded to in the Aitken paper.

This work is sun, ned up and vastly extended in his 1961 paper. Generally speaking, +

if {#k } is an asymptotic sequence as t ÷ 0 , then {$k } is an asymptotic sequence as

k } O + z ÷ ~ in SA and vice versa. And if ($ is an asymptotic sequence as z ~ in R,

then {~n } is an asymptotic sequence as t ÷ ~ in R and vice versa. In the latter

paper, Erd~lyi states 2 theorems showing when this is true, and correcting a result

in the earlier paper. Such a relationship, of course, induces a correspondence bet-

ween asymptotic expansions for f and for f.

We quote two of the main results:

Theorem: (generalized initial value theorem)

Let O < Re~ ° < Re~ I < .... and

~k-I 0 +. (6) f ~ E fk ; {t } , t ÷

k=O

Let f, fk' k = 0,1',2,..., exist for some z.

Then

-~k (7) ~ ~ E fl& ; {z } , z ÷ ~ in some S A.

k=O

Theorem: (generalized final value theorem)

Let Re% ° > Re% I > ... > O and

(8) f ~ Z fk ; {t k-l} , t ÷ ~ in R +. k=O

Let f, fk' k = O,1,2,..., exist for each z > O.

Then

-~k (9) f % Z ~k ; {z } , z ÷ 0 in some S A.

k=O

The word "some" preceding each S A is a bother. But Erd~lyi gives an alternative

formulation in terms of the scale { (Re z) k} for which (8) and (9) hold in any S A.

Specific examples are:

(io)

f ~ E Ck(l-e-t)k ; {tk},t 0 + ÷

k=O

E k~Ck/Z(z+l)...(z+k); {z -k} , k=O

(this is a factorial series;

z -~ ~ in some S A

see the footnote following equation (2))

259

(II) i f~

f~

E ck(et-l)k • {tk}, t + 0 + ,

k=O

E k:Ck/Z(z-l)...(z-k); {z -k-1 } z + k=O

in some SA

t 2k , O + f ~ I Ck(2 sinh 7) ; {t 2k} t + ; k=O

(12) -2k-l~

~ E (2k) l Ck/(Z-k)(z-k+l)...(z+k) ; {z ~, z ÷ ~ in some S A. k=O

Next, Erd~lyi gives a general theorem, similar to (6) - (7), for asymptotic ex-

pansions with respect to the scales {(in t) Bk t ~k-l} and {(in z) ~k z -~k} . This the-

orem generalizes a number of results given in Doetsch (1950-1956).

Many authors have discussed other generalizatio~of Watson's lermna. The books by

Olver (1974), Bleistein and Handelsman (1975) and the Doetsch volumes referenced

above provide much information. Of special interest are two early papers by van der

Corput (1934, 1938) where integrals of the form

ib x h(t)-yt e (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). It is no loss of generality here to assume f e ~(U) at least, so

(13) f = Z fn tn' Itl < I. n=O

An important problem is:how does f behave as n÷~? 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 [ f (t) = ~ dt, (14) fn ~ -r t

where F c U is homotopic to C. 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 in an example given by Olver (1974,

p. 329) where f(t) = exp[et]. P~lya (1922) gave the lead term for fn when

f = Pe Q, P,Q polynomials. The case f = e Q, Q a polynomial, was more fully treated by

Moser and Wyman (1956, 1957), who give references to earlier work. Other examples

have been given by Rubin (1967) and Harris and Schoenfeld (1968). Often the asymptotic

formulasobtained 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

I~-~ I 1 reiO) 12d0}' M(f,r) = I ~ f( 2, 0 < r < I.

Definition:

If

then we say f ~ H 2

lira M(f,r) < o~ r÷l

(the Hardy class H2).

Example:

Let

then f g H 2.

f = h(~-t) O, ~ e C, Re o > -I, h E ~(U);

Definition:

Let f, g g ~(U) and for a fixed m = 0,1,2,..., f(m) - g(m)

a comparison function of order m (to f).

In what follows let

g = E gn tn" n=O

Theorem: (Darboux's method)

Let g be a comparison function of order m to f.

Then

(15) fn = gn + °(n-m)' n ÷ ~.

h = f-g,

Proof: I may write

I I h(m)(t) dt (16) fn-g n 2~i(n_m+l) m CR tn+l.m ,

H 2 . E Then g is called

O<R<I,

261

But the radial limit of h (m) exists almost everywhere and eL2(C), see Rudin (1966,

p.366). Thus the integral on the right of(16) maybe expressed as an integral around C

(again, see Rudin).

The use of the Riemann-Lebesgue lena then gives the theorem.

A simple but important case is when the singularities of f on C are finite and

algebraic in nature, so

Br+kY r (17) f Z ak(r) (I t) = - ' ~ ~ C,

r k=O r

r = 1,2,...,R; Re m r > O, t near ~r ,t and the ~r are distinct.

Then the function formed by adding together the first (K+I) terms of each of the R

series (17) will be a comparison function of order m provided

(18) m < ½ + min {(K+I) Rear + Re8 r} r

I have thus demonstrated Szeg~'s result (1959, p.205) it.

Theorem: (Darboux's method for algebraic singularities)

Let f be as in (17). Then

K R ~r)~r+~Tr)(_~r)-n + o(n-m) (19) fn = Z

k=O r=l

for all m satisfying (18).

It is easy to show this formula is uniform for ~ = (el,~2 .... ,=k) e ~[whenever ~is

a Cartesian product of compact disjoint subsets of ~.

The general term of the sum in (19) is

[( r)] -k~r-Br-I Br + ~Y = O(n ),

0 n

• JV but we have no way of interpreting the formula in terms of Pozncare s definition of an

asymptotic series. However when K -~ ~ the series yields an asymptotic expansion in -Pk

Erdedlyi's mare generalized sense with respectto the scale {n }~Pk = l+min{k ReYr+ReBr}. r

(The reader will verify this is, in fact, an asymptotic scale, since Rey r > O).

Formula (19) has been very fruitful in classical analysis. It provides asymptotic

expansions for the Jacobi polynomials (SzegS, (1959)), the generalized Bernoulli

polynomials, and even for more exotic polynomials, such as the Pollaczek polynomials,

t For this expansion, we make the branch cut along B =[~r' ~r]"

t s Ud(~r), t ~ B.

tt There is a minor error in this reference.

Thus (17) holds for

262

P% (x; a,b), Szeg~ (1959, p.390). Darboux himself (1878) originally applied his n

method to the Legendre polynomials.

Darboux's method says if we can solve the problem of finding an asymptotic formula

for the Taylor coefficients of a comparison function, we can find, to within algebraic

terms, the Taylor coefficients for f. This sometimes suffices to describe f n

completely.

An enormous amount of work has been done on deriving asymptotic formula for gn for

certain typical g. One of the simplest functions having an essential singularity on

C is exp {%/(l-t)}. Perron (1914), in fact, considered the function (l-t)-aexp{%/(l-t)}

and gave the leading term of gn" Wright (1932) gave the complete asymptotic develop-

ment.

Perron generalized his own work in (1920) to find that if

(l-t) -a ~Qa;c l--%t) = ~ n n 0 gnt ' % > O,

then I c % c 3 F(c) %4 2 2 a - ~ - ~ 2 %~n

gn = ~ e n e

1

X[I + O(n- ")j.

Faber (1922), Ha~sler (1930), Wright (1933, 1949) continued this kind of research,

for the very general the last reference giving the leading term of gn

g(t) = (l-t)-a[£n(l-t)] b e P(t) h(t),

where h ~ ~(U), h(1) # O, and

M c m e(t) =

m=l (l-t) dm

Recently, Wong and Wyman (1974) have done work on functions with logarithmic singu ~

larities on the circle of convergence.

Suprisingly, mathematicians have obtained results even in cases where f has singu-

larities which are dense on C, i.e. when the unit circle is a natural boundary for f,

Such cases are of great interest to number theoreticians. Uspensky (1920), Rademacher

(1937), Ingham (1941), Szekeres (1953), and Bender (1974) are some of the workers who

have contributed to this research. In particular, I want to talk about Rademacher's

work, since many analysts seem to be unfamiliar with it, though it is one of the

triumphs of analytic number theory and a tour de force of complex analysis.

Let Pn be the number of ways n can be written as a sum of positive integers $ n,

Po = I, Pl = I. Then the infinite product I write below converges for It] < 1

and

263

n f(t) = H (l-tJ) -I = E pn t , Itl < i.

j =i n=O

see Rademacher (1973). Clearly, C is a natural boundary for f. Rademacher extends

the previous ,~ork of Hardy and Ramanujan (1918). He integrates Cauchy's integral

(14) over a doubly indexed sequence of so-called Farey arcs adjoining a circle which

approaches the unit circle from the inside. His analysis involves the transformation

theory of modular functions and a lot of bone breaking estimates of integrals, but

the result is worth it. He finds

I k½ d -I (20) Pn = ~ kZl A~,n sinh )',_ C = , N = (n - ~i ~

The Ak, n are bounded functions of n,

-2~in/k = ~ e

Ak'n h mod k ~h,k

(h,k) = i

where ~h,k is a 24k th root of unity.

The series (20) is asymptotic in n (perhaps the reader can verify a scale is

N-2e cN/k) and also convergent, a very unusual feature in series derived this way, It

can thus be used to compute Pn" In fact Rademacher computes exact values of P599 and

P721 and shows they agree with earlier estimates baked on the work of Hardy and

Ramanuj an.

What about uniform asymptotic expansions? In other words, if f E f(t,~),

~ ~C ~P, can Darboux's method be made to yield expansions which are uniform for

all interesting choices of~? A typical question is, what happens if

~[i = ~2 = "'" =~r = C

in (19)? As the reader may suspect, such questions are very difficult to answer, and

the requirement of uniformity of the expansion in a parameter set invariably raises

the hierarchy of the base functions in the expansion for f . See Olver's very nice n

(1975) discussion of this "you-can't-get-something-for-nothing" principle of asymptotic

analysis. The only effort I know in this area is the substantial work of Fields ~968),

who treats the case when f in (19) has two singularities which are allowed to coalesce.

Finally, can anything at all be said when f belongs to some general class of func-

tions more interesting than just those with algebraic singularities? Surprisingly,

yes. Hayman (1956), Wyman (1959), and Harris and Schoenfeld (1968) have all worked on

the problem of defining what general properties of classes of functions ~, are

adequate to enable one to make asymptotic statements about fn for f g ~. The authors

call the elements of ~ admissible functions. In any case, the properties of ~-f are

not easy to describe, but generally they involve restrictions on the growth indicators

of the elements.

264

6. Higher transcendental scale s

Let us return to the problem of estimating

(21) I(z) = [ e zh(t'~) g(t)dt, ~ e~C~ P. ~p

It would seem that if I could do the analysis for certain simple representative choices

of h then I have really solved the problem for a wide class of integrals, namely, those

which can be reduced to the representative form by a change of variable, just as the

method of steepest descents enabled me to reduce an integral with one stationary

critical point to an integral which could be handled by Watson's lemma (h(t) = t).

The simplest function having a movable critical point is

h(t,~) = st + t 2,

and the representative integral is

i =o e_Z (~t+t2) I(z) = g(t)dt. O

Assume L~. = [O,r] for some r > O. If g can be expanded in a series

co

g= E gk tk, Itl <~, k=O

then termwise integration generates the expansion

co

(22) E gk fk' k=O

I ~ e_ z (at+t 2) (23) fk = t k dt.

O

St seems fk cannot itself be uniformly estimated in ~Lby simpler functions. But

this is to be expected: as the requirements of uniformity of the approximation or

the dimensionality of the parameter space ~ p increase, so will the complexity of the

base functions involved in the asymptotic expansion.

Nevertheless, the functions f k can be considered known. They can be expressed in

terms of parabolic cylinder functions. Integration by parts shows they satisfy a 3-

term recurrence relation, and they can be very easily generated on a computer by

applying the Miller algorithm,(see Wimp (1970) and the references given there.) We

have

2zfk+ 2 + z~fk+ 1 - (k+l)f k = O, k = O,1,2, .....

A possible normalization relationship for the application of the Miller algorithm

is

265

Z

k=0

k z f2k I

(2k)~ z~ ~ > 0,

Furthermore aN

F(k_k_k_k~l)e ,,7 2

fk - k+l ~I + O(k-½)] , k ÷ ~,

2 2z

and using this I can show if ~ E (0, ~) the Miller algorithm for the computation of

fk will converge for z e S A- • J

One would expect the expansion (22) to be of Polncare type since

Theorem:

+ f k i s a u n i f o r m a s y m p t o t i c s c a l e i n a a s z ÷ ~ i n R .

Pf : Let t = t*(l+u) where t ~ satisfies 2zt ~2 + ~zt*-k = 0,

or

I [ / / 2 8k ~] t* =~ +--- .

4 z

I get O

Ik = ~k l+u) e du, -I

-~zt*-zt .2 ~k = (t*)k+l e

= 4k/E(a2z+4k) + ~z½ Ja2z + 8k], 0~<o,< i.

Thus the integral above may be bounded and bounded away from zero uniformly in ~ and

z, so it follows that

fk+l z ~ ~ > O,

fk 7 '

independent of a and z.

• J

Rather than show for which conditions the expansion C22) is a Polncare expansion, I

will use an asymptotic scale simpler than fk (but still uniform ink). The following

result is in Erdeflyi (1970).

Theorem:

Let g e ~(0) and let I exist for some z ° > 0 and all ~ e ~ then

I ~ ~ ~g~ fk ; {(~z + 2/~z) -k} , z + ~ in R +. k--O

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

will show.

-(z-z o)(~t+t 2) it -z (~u+u 2) I e G(t)dt, G(t) = e o g(u) du

e o

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 space~ 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 2

Ai(cz ~) Ai'(cz ~)

2k ' 2k Z 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). For an

exposition of this method, see the survey by Jones (1972) orOlver (1974). 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 cal-

culated 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 + ... + ~ t p + t p+I, ) , p ~ = (~i,~2 .... ,~p

then one will have to live with incomplete results; justifying the reduction of

(25) to the representative integral

f=

J o e-Zh(t'~) g(t,~)dt

involves difficult-to-verify hypotheses, and some of the work is only formal. Authors

who have treated this problem are Bleistein (1966, 1967) and Ursell (1972). In this

case the base functions are called generalized Airy functions. They satisfy a

differential equation of order p+l (see Bleisten (1967)), possess an asymptotic ex-

pansion in z (Levey and Felsen (1969)) and the techniques Wimp uses on similar in-

tegrals (1969) will work to show the functions satisfy a(p+2) term recursion relation-

ship 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 just-

ifying 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. For integrals such as these, there are a large number of

special results available. 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. See also the

book (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 3 - that would

make the impossible easy.

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.

• I!

Doetsch, G. (1950-56). 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). Higher transcendental functions, 3v. McGraw-Hill, New York.

J .

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.

Riekstins,

Springer-Verlag, Berlin.

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

E. (1974). Asymptotic expansions of integra! s, v. I. (Russian) Riga.

E., et. al. (1965-75). Latvian Math. Yearbook.

Riekstins,

Riekstins,

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. Rep. ARL69 - O186, Aerospace Res. Lab. Wright-Patterson AFB, Ohio.

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.

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Ordinary and Partial Differential Equations

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