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continuation on page 733

Lecture Notes inMathematicsEdited by A. Dold and B. Eckmann

Ordinary and PartialDifferential EquationsProceedings of the Seventh ConferenceHeld at Dundee, Scotland, March 29 - April 2, 1982

Edited by W. N. Everitt and B.D. Sleeman

Springer-VerlagBerlin Heidelberg NewYork 1982

Editors

W.N. EverittDepartment of Mathematics, University of BirminghamPO. Box 363, Birmingham B15 2TI, England

B,D, Sleeman

Department of Mathematics, University of DundeeDundee DDl 4HN, Scotland

AMS Subject Classifications (1980): 34-06, 35-06

ISBN 3-540-11968-X Springer-Verlag Berlin Heidelberg New YorkISBN 0-387-11968-X Springer-Verlag New York Heidelberg Berlin

This work is subject to copyright. All rights are reserved, whether the whole orpart of the material is concerned, specifically those of translation, reprinting,re-use of illustrations, broadcasting, reproduction by photocopying machine orsimilar means, and storage in data banks. Under § 54 of the German CopyrightLaw where copies are made for other than private use, a fee is payable to"Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1982Printed in Germany

Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.2146/3140-543210

D E DIe A T ION

These Proceedings are dedicated to

The University of Dundee

on the occasion of the centenary celebrations

1882 - 1982

PRE F ACE

These Proceedings form a record of the lectures delivered at the seventh

Conference on Ordinary and Partial Differential Equations which was held at the

University of Dundee, Scotland during the week Monday 29 March to Friday 2 April

1982.

The Conference formed part of the centenary celebrations for the University of

Dundee 1882-1982. The University of Wurzburg was officially represented at the

Conference, for this purpose, by Professor Dr H WKnobloch.

The Conference was attended by 120 mathematicians from more than 25 countries.

The Conference was organised by a Committee with membership: E R Dawson,

WN Everitt, B D Sleeman.

Following again the tradition set by the earlier Dundee Conferences the

Organising Committee named as Honorary Presidents of the 1982 Conference

Professor Jean Mawhin (Belgium)

Professor Tung Chin-Chu (p R China).

The Committee expresses its gratitude to all mathematicians who took part in

the work of the Conference.

The Committee thanks : the University of Dundee for generously supporting the

Ce,nference; the Warden, Domestic Bursar and Staff of West Park Hall for all their

help in providing accommodation for the participants; the Department of Mathematical

Sciences for continuing support for the Conference; the Bursar of Residences, the

Accommodation Officer, the Information Officer and the Finance Officer of the

University.

The Committee offers special thanks to the Vice-Principal of the University,

Professor P D Griffiths, for representing the University at the opening ceremony,

and at the University reception held for all participants.

The Committee records with gratitude financial support, in the form of travel

and subsistence funds, from

The Royal Society of Londonand

The European Research Office of the United States Army.

VI

The Professors Everitt and Sleeman record special thanks to their colleague

Commander E R Dawson R N whose services, from retirement, were generously offered in

helping to organise the Conference, and who carried the main responsibility for the

accommodation of all participants in West Park Hall. Commander nawson also edited a

number of the manuscripts submitted for publication in these Proceedings.

Likewise special thanks are due to all the Secretaries in the Department of

Mathematical Sciences of the University; in particular to Mrs Caroline Peters for

her invaluable contribution.

WN Everitt B D Sleeman

CONTENTS

, V Atkinson:

)n the asymptotic behaviour of the Titchmarsh-Weyl m-coefficient and the:pectral function for scalar second-order differential expressions ... , .. ,.

, V Atkinson and C T Fulton:

,ome limit circle eigenvalue problems and asymptotic formulae for.i genvaLue s ...••.•..••..••.•.••.•.••.•.•.....••.•••.••••••....•.••.•....•. 28

, V Atkinson, J R Haddock and 0 J Staffans:

:ntegral inequalities and exponential convergence of solutions oflifferential equations with bounded delay .................•............... 56

) L Barrow and P WBates:

lifurcation of periodic travelling waves for a reaction-diffusion system.. 69

I Behncke:

Ule Dirac equation with an anomalous magnetic moment II . 77

I Benci, A Capozzi and D Fortunato:

'eriodic solutions of a class of Hamiltonian systems...................... 86

, J Browne:

1ultiparameter problems

, C Brown:

the las t decade ...........•........•............ 95

\n approach to the Dirichlet index for operators satisfying minimal:onditions •.•...............•......•................•..................•.. 110

J A Burns and E M Cliff and J M Amillo Gil

neutral functional differential equations in product spaces ..... 118

) A Clarkson and J B McLeod:

connection formula for the second Painleve transcendent .•...... .•....... 135

Colton:

rhe inverse scattering problem for acoustic waves ..•.... .•. .•. .... .... .... 143

J Dubois and P Morales:

)n the Hukuhara-Kneser property for some Cauchy problems in locally convextopological vector spaces •...•.•••.............•..........•.........•..... 162

X Dung:

ossential self-adjointness and self-adjointness for generalizedSch'ro dinge r operators ....•.••..•.•.....•..•..•.••..••.•.••........•....... 171

VIII

M S P Eastham:

Asymptotic theory for a critical class of fourth-order differentialequations •..••••••..•.....•......•......................•.........••...... 179

AElbert:

Oscillation and nonoscillation theorems for some nonlinear ordinarydifferential equations ...•••.....•.......................•............•... 187

D E Edmunds:

Entropy numbers, s-numbers and eigenvalues , 213

P C Fife and B Nicolaenko:

The singular perturbation approach to flame theory with chain andcompeting reactions .......•.....•.•.•...•....•.........•...•.....••....... 232

J Fleckinger:

On the singular values of non-self-adjoint operators of Schrodinger type.. 251

I M Gali:

Optimal control of systems governed by elliptic operator of infinite order 263

V B Goyal and P W Schaefer:

Comparison principles for some fourth order elliptic problems... 272

R C Grimmer and W Schappacher:

Weak solutions of integrodifferential equations and applications 280

D B Hinton and J K Shaw:

Titchmarsh's A-dependent boundary conditions for Hamiltonian systems 298

F A Howes:

Exterior boundary value problems for perturbed equations of elliptic type .

H G Kaper, C Gerrit Lekkerkerker and A Zettl:

318

Linear transport theory and an indefinite Sturm-Liouville problem 326

R M Kauffman:

Non-normalizable eigenfunction expansions for ordinary differentialoperators •...........•••.•..•••...•.•.............•..•.•..•.......•....... 362

H WKnobloch:

Some aspects and recent developments in linear control theory...... ... .... 368

I W Knowles:

Eigenvalue problems and the Riemann zeta-function ......••. •..... 388

IX

M A Kon and L A Raphael:

Superposition principles and pointwise evaluation of Sturm-Liouvilleeigenfunction expansions ....•.....•..........••...........................

K Kreith:

IQualitative theory of hyperbolic characteristic initial value problems

R T Lewis:

Applications of a comparison theorem for quasi-accretive operators in aHilbert space ..........••.................•......................•.....•..

L L Littlejohn and A M Krall:

A singular sixth order differential equation with orthogonal polynomialeigenfunctions .............•.................•.....•.....•........•.....•.

S-O Londen:

On the asymptotics of some Volterra equations with locally finitemeasures and large perturbations ..•................•.••....•.........•....

J Mawhin:

Periodic oscillations of forced pendulum-like equations

R C MacCamy and E Stephan:

406

416

422

435

445

458

A simple layer potential method for three-dimensional eddy current problems 477

A C McBride:

Index laws for some ordinary differential operators...... 485

P A McCoy:

Converse initial value problems for a class of heat equations

J R McLaughlin:

Higher order inverse eigenvalue problems ., ..........•..... , .

A B Mingarelli:

494

503

Indefinite Sturm-Liouville problems .................•..... ,. 519

SEA Mohammed:

The infinitesimal generator of a stochastic functional differentialequation ...•...•.......•.•.•.•...•......................•..•..•.....•..... 529

M Nakao:

On solutions to the initial-boundary problem for perturbed porous mediumequation .•...•....•••..••..•..•.••.................•...•.••......••....... 539

x

F Neuman:

A survey of global properties of linear differential equations of then-th order ....•.•••.•.•....••.•.•..•..•.•......................•..•....... 548

D Pascali:

Strongly nonlinear evolution variational inequalities .•.•.................

DRace:

564

A further result on the essential spectrum of linear 2nth orderdifferential expressions with complex coefficients .•. ••....•........... ... 569

R Rautmann:

On error bounds for nonstationary spectral Navier-Stokes approximations

T TRead:

Sectorial second order differential operators •.............•..............

H Roh :

Spectral mapping theorems for dissipative Co-semigroup generators .

D A Sanchez:

Some preliminary results on periodic solutions of matrix Riccati equations

R Saxton:

Solitary and travelling waves in a rod ..

K Seitz:

576

584

591

598

603

Investigations in the theory of partial differential equations of infiniteorder .•.......•..•••....•.•.••.•..•................•.•..•................. 609

J K Shaw and D B Hinton:

Well-posed boundary problems for Hamiltonian systems of limit point orlimit circle type......................................................... 614

K Soni:

On uniform asymptotic expansion of a class of integral transforms .......•.

R L Sternberg, M J Goldstein and D Drinkard:

On a new numerical method for a new class of nonlinear partialdifferential equations arising in nonspherical geometrical optics

A Z-A M Tazali:

632

642

Local existence theorems for ordinary differential equations offrac tional order ..•...•.•.••..••......•......•......•.•................... 652

Tung Chin-Chu:

Application of the theorem of connectivity of M7 in astronomy . 666

XI

L Turyn:

Perturbations of periodic boundary conditions •••.. ...••.... .... ....•.. .•.. 675

A Vanderbauwhede:

Bifurcation problems with triangular symmetry............ .•..... ... ... .... 686

J Walter:

On the foundations of thermodynamics

A D Wood and R B Paris:

695

On some conjectures on the deficiency index for symmetric differentialoperators................................................................. 703

S D Wray:

On a second-order differential expression and its Dirichlet integral...... 710

E M E Zayed:

An inverse eigenvalue problem for the Laplace operator .... .•. .••.. .••. ..•. 718

Lectures given at the Conference which are not represented by

contributions to these Proceedings.

(Results may be presented elsewhere)

A U Afuwape:

On Lyapunov Razumikhin approach to certain third-order equations with delay

A H Azzam:

Boundary value problems for elliptic and parabolic equations in domainwith corners

I Bihari:

Distribution of the zeros of Bacher's pairs with respect to second orderhomogeneous differential equations

T R Blows:

Limit cycles of polynomial differential equations

L Brull:

A new abstract existence theory for non-linear Schredinger and wave equations

L Collatz:

Inclusion theorems for singular and free boundary value problems

P A Deift:

Some recent results ordinary differential equations and approximationtechniques

J Denig:

positive eigensolutions and the lower spectrum of Schrodinger operators

W D Evans:

On the distribution of eigenvalues of Schrodinger operators

W N Everitt:

Two examples of the Hardy-Littlewood type of integral inequalities

R E Kleinman:

Recent developments in modified Green's functions

M K Kwong:

Second order linear and nonlinear oscillation results

A GRamm:

Basisness property and asyrnptotics of spectrums of some nonselfadjointdifferential and pseudo-differential operators

XIII

B P Rynne:

Bloch waves and multiparameter spectral theory

B D Sleeman:

An abstract multiparameter spectral theory

R A Smith:

Poincare's index theorem and Bendixson's negative criterion for certa'differential equations of higher dimension

R Villella-Bress an:

1Functional equations of delay type in L -spaces

WL Walter:

Generalized Volterra prey-predator systems

Address list of authors and speakers

A U Afuwape:

J M Amil l o Gil:

F V Atkinson:

A H Azzam:

P W Bates:

H Behncke:

V Benci:

I Bihari:

T R Blows:

R C Brown:

P J Browne:

L BrulI:

J A Burns:

P A Clarkson:

E M Cliff:

L Collatz:

D L Colton:

P A Deift:

Department of Mathematics, University of Ife,

ILE-IFE, Nigeria

Universidad Politechnica De Madrid,

MADRID, Spain

Department of Mathematics, University of Toronto,

TORONTO, Ontario, Canada

Department of Mathematics, King Saud University,

RIYADH, Saudi Arabia

Department of Mathematics, Heriot-Watt University,

Riccarton, Currie, EDINBURGH, EH14 4AS, Scotland

Fachbereich Mathematik, University of Osnabruch,

45 OSNABRDcH, West Germany

Istituto de Matemati"a Applicata, Via Re David 200,

70125 BARI, Italy

Mathematical Institute, BUDAPEST, Realtanoda-U 13/15,

H-1053, Hungary

Pure Mathematics Department, University College Wales,

ABERYSTWYTH, Dyfed, SY23 3BZ, Wales

Department of Mathematics, University of Tennessee,

KNOXVILLE, Tennessee, U.S.A.

Department of Mathematics & Statistics, University of

Calgary, CALGARY, Alberta, T2N lN4, Canada

Mathematisches Institut der Universitat zu Kaln,

5000 41, Weyertal 86, West Germany

Department of Mathematics, Virginia Tech University,

BLACKSBURG, VA 24061, U.S.A.

The Queen's College, OXFORD, OXI 4AW, England



Eulenkrugstrasse 84, 2000 HAMBURG 67, Germany

Department of Mathematical Sciences, University of

Delaware, NEWARK, Delaware 191711, U.S.A.

Courant Institute of Mathematical Sciences,

251 Mercer Street, NEW YORK, NY 10012, U.S.A.

J Donig:

N X Dung:

M S P Eastham:

AElbert:

D E Edmunds:

W D Evans:

WN Everitt:

P C Fife:

J Fleckinger:

C T Fulton:

I M Gali:

R C Grimmer:

D B Hinton:

F A Howes:

H G Kaper:

R M Kauffman:

R E Kleinman:

xv

Technische Hochschule Darmstadt, Fachbereich Mathematik,

Schlossgartenstrasse 7, D6100 DARMSTADT, West Germany

Texas Tech University, Department of Mathematics,

LUBBOCK, Texas 79409, U.S.A.

Department of Mathematics, Chelsea College,

Manresa Road, LONDON, SW3 6LX, England

Mathematical Institute, BUDAPEST, Realtanoda-U 13/15,

H-1053, Hungary

School of Mathematical & Physical Sciences, University

of Sussex, FALMER, Brighton, BNl 9QH, England

Department of Pure Mathematics, University College,

POBox 78, CARDIFF, CFl lXL, Wales

Department of Mathematics, University of Birmingham,

P.O. Box 363, Birmingham Bl5 2TT, England

Department of Mathematics, University of Arizona,

TUCSON, Arizona 85721, U.S.A.

Universite Paul Sabatier, UER MIG, 118 Route de Narbonne,

31062 TOULOUSE, Cedex, France

Florida Institute of Technology, Department of

Mathematical Sciences, MELBOURNE, Florida 32901, U.S.A.

Mathematics Department, Qatar University, POBox 2713,

DOHA, Qatar, Arabian Gulf

Institut fur Mathematik, Elisabethstrasse 11,

A-8010 GRAZ, Austria

Mathematics Department, University of Tennessee,

KNOXVILLE, Tenn. 37916, U.S.A.

Department of Mathematics, University of California,

DAVIS, CA 95616, U.S.A.

Applied Mathematics Division, Argonne National

Laboratory, ARGONNE, III 60439, U.S.A.

Department of Mathematics, Western Washington University,

BELLINGHAM, WA 98225, U.S.A.

Department of Mathematics, University of Strathclyde,

Livingstone Tower, 26 Richmond Street, GLASGOW, Gl lXH,

Scotland

H WKnobloch:

I WKnowles:

M A Kon:

A MKrall:

K Kreith:

M K Kwong:

R T Lewis:

L L Littlejohn:

S-O Londen:

J Mawhin:

A C McBride:

P A McCoy:

J R McLaughlin:

J B McLeod:

A B Mingarelli:

SEA Mohammed:

XVI

Mathematisches Institut, Am Hubland, D-8700 WURZBURG,

Germany

Department of Mathematics, University of Alabama in

Birmingham, BIRMINGHAM, Alabama 35294, U.S.A.

Department of Mathematics, Boston University, BOSTON,

MA 02215, U.S.A.

McAllister Building, Pennsylvania State University,

UNIVERSITY PARK, PA 16802, U.S.A.

Mathematics Department, University of California,

DAVIS, CA 95616, U.S.A.

Department of Mathematics, Northern Illinois University,

DEKALB, Illinois, U.S.A.

Department of Mathematics, University of Alabama in

Birmingham, BIRMINGHAM, Alabama U.S.A.

Department of Mathematics, Computer Science & Systems

Design, The University of Texas at San Antonio,

SAN ANTONIO, Texas 78285, U.S.A.

Institute of Mathematics, Helsinki University of

Technology, ESPOO 15, Finland

Institut Mathematique, Universite de Louvain,

Chemin du Cyclotron 2, B-1348 LOUVAIN LA NEUVE, Belgium

Department of Mathematics, University of Strathclyde,

Livingstone Tower, 26 Richmond Street, GLASGOW Gl lXH,

Scotland

Department of Mathematics, U S Naval Academy, ANNAPOLIS,

Maryland 21402, U.S.A.

Department of Mathematical Sciences, Rensselaer

Polytechnic Institute, TROY, New York 12181, U.S.A.

Wadham College, Oxford University, 24-29 St Giles,

OXFORD, OXI 3LB, England

Department of Mathematics, University of Ottawa,

OTTAWA, Ontario, KIN 9B4, Canada

Department of Pure Mathematics, University of Khartoum,

Khartoum, the Sudan

P A Morales:

M N M Nakao:

F Neuman:

D Pascali:

DRace:

A GRamm:

L A Raphael:

K A R Rautmann:

T TRead:

H Roh:

B P Rynne:

D A Sanchez:

R A Saxton:

P W Schaefer:

K Seitz:

J K Shaw:

B D Sleeman:

XVII

Departement de Mathematiques et d'Informatique,

Universite de Sherbrooke, SHERBROOKE, Quebec, JIK 2Rl,

Canada



Mathematical Institute, Czechoslovak Academy of Sciences,

Janackovo Nam 2a, 66295 BRNO, Czechoslovakia

Fachbereich Mathematik, Technische Hochschule Darmstadt,

Schlossgartenstrasse 7, D-6l00 DARMSTADT, West Germany

Mathematics Department, Witwatersrand University,

Jan Smuts Avenue, JOHANNESBURG, South Africa

Department of Mathematics, Kansas State University,

MANHATTAN, Kansas 66506, U.S.A.

Howard University, Department of Mathematics, WASHINGTON,

DC 20059, U.S.A.

Department of Mathematics, The University, Warburger

Strasse 100, D-4790 PADERBORN, West Germany

Department of Mathematics, Western Washington University,

BELLINGHAM, Washington 98225, U.S.A.



Department of Mathematical Sciences, The University,

DUNDEE, DDI 4HN, Scotland

Department of Mathematics, University of New Mexico,

ALBUQUERQUE, NM 87131, U.S.A.

Department of Mathematics, BruneI University, UXBRIDGE,

Middlesex, UB8 3PH, England



Technical University of Budapest, 1111 Muegyetem,

RAKPART 9, H. EP. V EM. 5, Hungary



Department of Mathematical Sciences, The University,

DUNDEE, DDI 4HN, Scotland

R A Smith:

K Soni:

E Stephan:

R L Sternberg:

A Z-A M Ta2ali:

Tung Chin-Chu:

L Turyn:

A Vanderbauwhede:

R Villella-Bressan:

J Walter:

WL Walter:

A D Wood:

S D Wray:

E ME Zayed:

A Zettl:

XVIII

Department of Mathematics, University of Durham,

South Road, DURHAM, DHI 3LE, England



rachbereich Mathematik, Technische HochEchule,

Schlossgartenstrasse 7, D-6l00 DARMSTADT, West Germany

Office of Naval Research, BId 114 Section D,

666 Summer Street, BOSTON, Mass 02210, U.S.A.

The University of Mosul, College of Science, Department

of Mathematics, MOSUL, Iraq

Graduate School, Chinese Academy of Sciences,

POBox 3908, BEIJING, P R China

Department of Mathematics & Statistics, University of

Calgary, CALGARY, Alberta, T2N lN4, Canada

Instituut voor Theoretische Mechanica, Rijksuniversiteit

Gent, Krijgslaan 28l-S9, B-9000 GENT, Belgium

Istituto de Analisi E Meccanica, Universita di Padova,

Via Be120ni 7, 35100 PADOVA, Italy

Institut fur }lathematik, Templergraben 55, D-5l00

AACHEN, Germany

Mathematisches Institut I, Universitat Karlsruhe,

Kaiserstrasse 12, D-7500 KARLSRUHE, Germany

School of Mathematical Sciences, N.I.H.E., Ballymun Road,

DUBLIN 9, Ireland

Department of Mathematics, Royal Roads Military College,

F.M.O., VICTORIA, British Columbia, V05 lBO, Canada

Department of Mathematics, Zaga2ig University, Faculty

of Science, ZAGAZIG, Egypt

Department of Mathematics, Northern Illinois University,

DEKALB, Illinois, U.S.A.

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