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Sturmian Theory for Ordinary Differential Equations PDF

574 Pages·1980·14.706 MB·English
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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 The mathematization of all sciences, the fading of traditional scientific bounda ries, the impact of computer technology, the growing importance of mathematical computer modelling and the necessity of scientific planning all create the need both in education and research for books that are introductory to and abreast of these developments. The purpose of this series is to provide such books, suitable for the user of mathematics, the mathematician interested in applications, and the student scientist. In particular, this series will provide an outlet for material less formally presented and more anticipatory of needs than finished texts or monographs, yet of immediate in terest because of the novelty of its treatment of an application or of mathematics being applied or lying close to applications. The aim of the series is, through rapid publication in an attractive but inexpen sive format, to make material of current interest widely accessible. This implies the absence of excessive generality and abstraction, and unrealistic idealization, but with quality of exposition as a goal. Many of the books will originate out of and will stimulate the development of new undergraduate and graduate courses in the applications of mathematics. Some of the books will present introductions to new areas of research, new applications and act as signposts for new directions in the mathematical sciences. This series will often serve as an intermediate stage of the publication of material which, through exposure here, will be further developed and refined. These will appear in conven tional format and in hard cover. MANUSCRIPTS The Editors welcome all inquiries regarding the submission of manuscripts for the series. Final preparation of all manuscripts will take place in the editorial offices of the series in the Division of Applied Mathematics, Brown University, Providence, Rhode Island. 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.

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