Oscillation Theory
- PDF / 5,455,791 Bytes
- 115 Pages / 483 x 720 pts Page_size
- 66 Downloads / 204 Views
324 Kurt Kreith University of California, Davis, CNUSA
Oscillation Theory
Springer-Verlag Berlin· Heidelberg· NewYork 1973
AMS Subject Classifications (1970): 34B25, 34ClO, 34G05, 35B05, 35015, 35L10
ISBN 3-540-06258-0 Springer-Verlag Berlin . Heidelberg . New York ISBN 0-387-06258-0 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 be determined by agreement with the publisher. © by Springer Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 73-79366. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachfBergstr.
PREFACE
These notes were written in conjunction with a series of lectures given at Chelsea College of the University of London in Fall, 1972. The author is indebted to Dr. M. S. P. Eastham for the invitation to spend a sabbatical year at Chelsea College and to the Science Research Council for its generous support.
Special thanks are also due to Mrs. Ida Orahood for her expert
typing and help in the preparation of the manuscript and to Professor Curtis Travis for a number of helpful suggestions.
CONTENTS
PARTIAL DIFFERENTIAL EQUATIONS ••••.•••.•...•••••..••.•••••..•••
17
1.
A Comparison Theorem for Elliptic Equations •.•••.••••.•..•.•.•.••.
17
2.
A Picone Identity for Elliptic Equations •••••••.••.•••.••.•.••...•
19
3.
Oscillation Theory for Elliptic Equations .•.•••••.••.••.•...•.•••.
21
4.
Hyperbolic Initial Boundary Value Problems ••..••••••••••••••••••••
24
5.
Hyperbolic Characteristic Initial Value Problems •••••••.•.••••••.•
26
6.
Oscillation Theory for Hyperbolic Equations ••••••..•.•••.•••••.•.•
28
RELATED TOPICS IN ANALYSIS .•••••.•....••..••..•..••.•...••••••.
30
Chapter 3.
Chapter 4. 1.
Calculus of Variations
30
2.
Bounds for Eigenvalues
33
3.
Green I S Functions ••.•.•••••••••.•••.•.•••••••••••.••.•••.••••••.•.
34
4.
An Ordering of Operators ••••••••••••••••••••••••••••••••••••.•••••
36
5.
Maximum Principles ••••••••••••••.••.•.•••••••••••••••••••.••.••••.
40
VI Chapter 5.
ABSTRACT OSCILLATION THEORY ••••••••••••••••••••••••••••••••••••
42
1-
Positive Operators .•••••••••••••••••••••••••••••••.•••••••••••••••
42
2.
An Abstract Comparison Theorem ••••••••.•••••••••••••••••••••••••••
43
3.
An Abstract Oscillation Theorem ••••••••••••••••••••••••••.••••••••
46
4.
An Abstract
Transformation •••••••••••••••••••••••••••••••••
49
5.
A Nontrigonometric Prufer Transformation .•••••••••••••••••••••••••
52
COMPLEX OSCILLATION THEORY •••••••••••••••••••••••••••••••••.•••
57
1.
A Physical Interpretation •••••••••••••••••••••••••••••••••••••••••
57
2.
Nonosci11ation Theorems •••••••••••••••
Data Loading...
