Sturmian Theory for Ordinary Differential Equations

574 Pages·1980·14.706 MB·English

by  William T. Reid (auth.)

Applied
Mathematical
Sciences
EDITORS
Fritz John Lawrence Sirovich Joseph P. LaSalle
Courant Institute of Division of Division of
Mathematical Sciences Applied Mathematics Applied Mathematics
New York University Brown University Lefschetz Center
New York, N.Y. 10012 Providence, R.1. 02912 for Dynamical Systems
Providence, R.1. 02912
ADVISORS
H. Cabannes University Paris-VI J. Marsden University of California at Berkeley
J.K. Hale Brown University G.B. Whitman California Institute of Technology
J. Keller Stanford University
EDITORIAL STATEMENT
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MANUSCRIPTS
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SPRINGER-VERLAG NEW YORK INC., 175 Fifth Avenue, New York, N. Y. 10010
Applied Mathematical Sciences I Volume 31
William T. Reid
Sturmian Theory
for Ordinary
Differential Equations
Springer-Verlag
New York Heidelberg Berlin
William T. Reid Prepared for publication by
formerly of the John Burns and Terry Herdman
Department of Mathematics Department of Mathematics
University of Oklahoma Virginia Polytechnic Institute
and State University
Blacksburg, Virginia 24061/USA
Calvin Ahlbrandt
Department of Mathematics
University of Missouri
Columbia, Missouri 65201/uSA
AMS Subject Classifications: 34-01, 34B25
Library of Congress Cataloging in Publication Data
Reid, William Thomas, 1907 (Oct. 4)-1977.
Sturmian theory for ordinary differential equations.
(Applied mathematical sciences; v. 31)
Bibliography: p.
Includes indexes.
I. Differential equations. I. Title. II. Series.
QAI.A647 vol. 31a [QA3721 5lOs [515.3'52180-23012
All rights reserved.
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag.
© 1980 by Springer-Verlag New York Inc.
9 8 7 6 543 2 1
ISBN-13: 978-0-387-90542-6 e-ISBN-13: 978-1-4612-6110-0
DOl: 10.1007/978-1-4612-6110-0
Dedicated to
DR. HYMAN J. ETTLINGER
Inspiring teacher, who introduced the author as
a graduate student to the wonderful world
of differential equations.
PREFACE
A major portion of the study of the qualitative nature
of solutions of differential equations may be traced to the
famous 1836 paper of Sturm [1), (here, as elsewhere throughout
this manuscript, numbers in square brackets refer to the
bibliography at the end of this volume), dealing with oscilla
tion and comparison theorems for linear homogeneous second
order ordinary differential equations. The associated work
of Liouville introduced a type of boundary problem known as
a "Sturm-Liouville problem", involving, in particular, an
introduction to the study of the asymptotic behavior of solu
tions of linear second order differential equations by the
use of integral equations.
In the quarter century following the 1891 Gottingen
dissertation [1) of Maxime Bacher (1867-1918), he was instru
mental in the elaboration and extension of the oscillation,
separation, and comparison theorems of Sturm, both in his
many papers on the subject and his lectures at the Sorbonne
in 1913-1914, which were subsequently published as his famous
Leaons sur Zes methodes de Sturm [7).
The basic work [1) of Hilbert (1862-1941) in the first
decade of the twentieth century was fundamental for the study
of boundary problems associated with self-adjoint differential
systems, both in regard to the development of the theory of
integral equations and in connection with the interrelations
between the calculus of variations and the characterization
of eigenvalues and eigensolutions of these systems. Moreover,
in subsequent years the significance of the calculus of
vii
viii
variations for such boundary problems was emphasized by
Gilbert A. Bliss (1876-1951) and Marston Morse (1892-1977).
In particular, Morse showed in his basic 1930 paper [1] in
the Mathematisahe AnnaZen that variational principles pro
vided an appropriate environment for the extension to self
adjoint differential systems of the classical Sturmian theory.
The prime purpose of the present monograph is the pres
entation of a historical and comprehensive survey of the
Sturmian theory for self-adjoint differential systems, and
for this purpose the classical Sturmian theory is but an im
portant special instance. On the othe~ hand, it is felt that
the Sturmian theory for a single real self-adjoint linear
homogeneous ordinary differential equation must be given
individual survey, for over the years it has continued to
grow and continually provide impetus to the expansion of the
subject for differential systems. There are many treatments
of the classical Sturmian theory, with varied methods of con
sideration, and in addition to Bacher [7] attention is di
rected to Ince [l-Chs. X, XI], Bieberbach [l-Ch. III, §§l-4],
Kamke [7-§6, especially Art. 25], Sansone [1, I-Ch. IV],
Coddington and Levinson [l-Chs. 7,8,11,12], Hartman [13-Ch.
XIl, Hille [2-Ch. 8], and Reid [35-Chs. 5,6].
In the present treatment there has been excluded work on
the extension of Sturmian theory to the areas of partial dif
ferential equations, and functional differential equations
with delayed argument. Also, for ordinary differential equa
tions the discussion and references on the asymptotic behavior
of solutions has been limited to a very small aspect that is
most intimately related to the oscillation theorems of the
classical Sturmian theory.
ix
For older literature on the subject the reader is
referred to the 1900 Enzyklopadie article by Bocher [4], and
his 1912 report to the Fifth International Congress of
Mathematicians on one-dimensional boundary problems [5].
For discussions of Becher's work and his influence on this
subject, attention is directed to the review of R. G. D.
Richardson [5] of Becher's Lecons sur Zes methodes de Sturm,
and the article by G. D. Birkhoff [4] on the scientific work
of Bocher. The account of subsequent literature prior to
1937 has been materially aided by the author's old report
[6]; in particular, not all of the Bibliography of that paper
has been reproduced in the set of references at the end of
this volume. For more recent literature the author has been
greatly helped by the survey articles in 1969 by Barrett [10]
and Willett [2]. Also, of special aid has been the report of
Buckley [1], which presented brief abstracts of many papers
dealing with the oscillation of solutions of scalar linear
homogeneous second order differential equations, and which
appeared in a number of journals, largely in the decade ending
with 1966.
Although the appended Bibliography is extensive, un
doubtedly the author has overlooked some very relevant papers
of which he is cognizant, and unfortunately others of which
he is not aware. To the authors of all such papers, regrets
are extended herewith and the hope expressed that they will
inform the author of the omission. Special regrets are ex
tended to the authors of papers written in the Russian lan
guage, for the author's inability to read the original papers
has necessitated his reliance upon translations and reviews.
x
In organization, most of the chapters contain a body
of material which might be described as textual, and which
presents concepts and/or methods that the author feels are
central for the considered topic. Such material is then
usually followed by a section with more detailed comments and
references to pertinent literature, and finally there is a
section on Topics and Exercises devoted to a variety of
examples of related results with references, and sometimes
comments on the principal ideas involved in derivation or
proof. Clearly such a selection involves a high order of sub-
jectivity on the part of the author, for which he assumes full
responsibility.
References to numbered theorems and formulas in a chap
ter other than the one in which the statement appears include
an adjoined Roman numeral indicating the chapter of reference,
while references to such items in the current chapter do not
contain the designating Roman numeral. For example, in
Chapter VI a reference to Theorem 6.4 or formula (4.6) of
Chapter V would be made by citing Theorem V.6.4 or formula
(V.4.6), whereas a reference to Theorem 1.2 or formula (3.15)
would mean the designated theorem or formula in Chapter VI.
Profound thanks are extended to the Administration of
the University of Oklahoma for support in providing secre
tarial help. The author is also deeply grateful to Mrs.
Debbie Franke for her typing of preliminary working papers
and the final version of this manuscript.
W. T. Reid
Norman, Oklahoma
September, 1975
ADDITION TO THE PREFACE
As indicated above, the main text of this book was com
pleted in September 1975. However, at the time of Professor
Reid's death (October 14, 1977) the manuscript was still in
the review process. In 1979 Calvin Ahlbrandt and I accepted
the responsibility for having the manuscript reviewed by
several publishers and an agreement for publication by
Springer-Verlag was completed. I agreed to undertake the
usual author's responsibility concerning proofreading, etc.
Therefore, I accept all responsibility for errors in the final
copy. I am certain that these errors would have been corrected
by Professor Reid had he lived to complete the publication of
the book. The main text of the present book is essentially
a faithful copy of Professor Reid's final manuscript except
for minor corrections and a few additions to the bibliography.
Many of the references were published after 1975 and these
references were updated wherever possible. However, we have
made no attempt to add references beyond those available to
Professor Reid in 1975.
I wish to express my sincere appreciation to Calvin
Ahlbrandt and Terry Herdman for their assistance in completing,
proofreading and publishing the manuscript. They devoted con
siderable time to the project and without their efforts it
would have been impossible to complete the book within any
reasonable time period. Also, I wish to thank Mrs. Kate
MacDougall for her excellent typing of the final camera-ready
copy for this volume.