Disconjugacy
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220
w. A. Coppel The Australian National University, Canberra/Australia
Disconjugacy
Springer-Verlag Berlin' Heidelberg' New York 1971
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich Series: Australian National University Advisers: L. G. Kovacs, B. H. Neumann, Hanna Neumann, M. F. Newman
220
w. A. Coppel The Australian National University, Canberra/Australia
Disconjugacy
Springer-Verlag Berlin' Heidelberg' New York 1971
AMS Subject Classifications (1970): 34C 10, 34A 30, 49B 10
ISBN 3-540-05584-3 Springer-Verlag Berlin . Heidelberg· New York ISBN 0-387-05584-3 Springer-Verlag New York· 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 he determined hy agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1971. Library ofCongress Catalog Card Number 70-172695. Printed in Germany.
Offsetdruck: Julius Beltz, Hemsbach.
PREFACE
Disconjugacy has assumed growing significance in recent years and for this reason seemed a worthwhile topic for the study group in differential equations at the Australian National University.
The group met almost once a week
throughout 1970 and covered most aspects of the subject.
I am
grateful to the other regular participants. Dr A. Howe. Mr G.C. O'Brien and Mr A.N. Stokes. for their collaboration and support.
The notes were prepared to provide a permanent
record of the lectures. chronological order. responsibility.
They follow a logical rather than a
For their preparation I accept full
In particular the decision to omit special
results for third and fourth order equations was mine.
I hope
that by making the notes available in their present form they may prove useful to a wider audience.
I thank Professor P.
Hartman for his comments on the first draft of these notes.
I
also thank Mrs Barbara Geary for her careful typing of the manuscript. and Professor B.H. Neumann and Dr M.F. Newman
for
their assistance with the proof-reading.
W.A. Coppel Department of Mathematics Institute of Advanced StUdies Australian National University
CONTENTS
Chapter O.
INTRODUCTION
1
Chapter 1.
SECOND ORDER EQUATIONS
4
1.
General properties .
4
2.
Comparison theorems
3.
Tests for oscillation
17
4.
'llie case
20
Chapter 2.
p(t) :: 1 •
LINEAR HAMILTONIAN SYSTEMS
9
34 34
1.
General properties •
2.
Principal solutions
39
3.
Conjugate points •
46
4. Matrix Riccati equations.
49
5.
Calculus of variations
58
6. 7.
The method of polar coordinates
67
Self-adjoint equations of higher order.
73
Chapter 3. 1.
EQUATIONS OF ARBITRARY ORDER.
General properties of ze
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