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(cnnrinued after index)
Po-Fang Hsieh Yasutaka Sibuya
Basic Theory of Ordinary
Differential Equations
With 114 Illustrations
Springer
Po-Fang Hsieh |
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Yasutaka Sibuya |
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Department of Mathematics |
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School of Mathematics |
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and Statistics |
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University of Minnesota |
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Western Michigan University |
206 Church Street SE |
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Kalamazoo, MI 49008 |
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Minneapolis, MN 55455 |
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USA |
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USA |
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philip.hsieh@wmich.edu |
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sibuya 0 math. umn.edu |
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Editorial Board |
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(North America): |
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|
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S. Axler |
F.W. Gehring |
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K.A. Ribet |
Mathematics Department |
Mathematics Department |
Department of |
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San Francisco State |
East Hall |
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Mathematics |
University |
University of Michigan |
University of California |
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San Francisco, CA 94132 |
Ann Arbor. Ml 48109- |
at Berkeley |
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USA |
1109 |
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Berkeley, CA 94720-3840 |
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USA |
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USA |
Mathematics Subject Classification (1991): 34-01
Library of Congress Cataloging-in-Publication Data
Hsieh, Po-Fang.
Basic theory of ordinary differential equations I Po-Fang Hsieh,
Yasutaka Sibuya.
p.cm. - (Unrversitext)
Includes bibliographical references and index.
ISBN 0-387-98699-5 (alk. paper)
1. Differential equations. I. Sibuya. Yasutaka. 1930-
II.Title. III. Series
OA372_H84 1999 |
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515'.35-dc2l |
99-18392 |
Printed on acid-free paper.
r 1999 Spnnger-Verlag New York, Inc.
All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc.. 175 Fifth Avenue, New
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Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.
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Production managed by MaryAnn Cotton: manufacturing supervised by Jeffrey Taub. Photocomposed copy prepared from the authors' A5-71EX 2.1 files.
Printed and bound by R.R. Donnelley and Sons, Harrisonburg. VA. Printed in the United States of America.
9 8 7 6 5 4 3 2 1
ISBN 0-387-98699-5 Springer-Verlag New York Berlin Heidelberg SPIN 10707353
To Emmy and Yasuko
PREFACE
This graduate level textbook is developed from courses in ordinary differential equations taught by the authors in several universities in the past 40 years or so. Prerequisite of this book is a knowledge of elementary linear algebra, real multivariable calculus, and elementary manipulation with power series in several complex variables. It is hoped that this book would provide the reader with the very basic knowledge necessary to begin research on ordinary differential equations.
To this purpose, materials are selected so that this book would provide the reader with methods and results which are applicable to many problems in various fields.
In order to accomplish this purpose, the book Theory of Differential Equations by
E. A. Coddington and Norman Levinson is used as a role model. Also, the teaching of Masuo Hukuhara and Mitio Nagumo can be found either explicitly or in spirit in many chapters. This book is useful for both pure mathematician and user of mathematics.
This book may be divided into four parts. The first part consists of Chapters I,
II, and III and covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part consists of Chapters IV, VI, and VII and covers the basic results concerning linear differential equations. The third part consists of Chapters VIII, IX, and X and covers nonlinear differential equations. Finally, Chapters V, XI, XII, and XIII cover the basic results concerning power series solutions.
The particular contents of each chapter are as follows. The fundamental exis- tence and uniqueness theorems and smoothness in data of an initial problem are explained in Chapters I and II, whereas the results concerning nonuniqueness are explained in Chapter III. Topics in Chapter III include the Kneser theorem and maximal and minimal solutions. Also, utilizing comparison theorems, some suf- ficient conditions for uniqueness are studied. In Chapter IV, the basic theorems concerning linear differential equations are explained. In particular, systems with constant or periodic coefficients are treated in detail. In this study, the S-N decomposition of a matrix is used instead of the Jordan canonical form. The S-N decomposition is equivalent to the block-diagonalization separating distinct eigen- values. Computation of the S-N decomposition is easier than that of the Jordan canonical form. A detailed explanation of linear Hamiltonian systems with constant or periodic coefficients is also given. In Chapter V, formal power series solutions and their convergence are explained. The main topic is singularities of the first kind. The convergence of formal power series solutions is proven for nonlinear systems. Also, the transformation of a linear system to a standard form at a singular point of the first kind is explained as the S-N decomposition of a linear differential operator. The main idea is originally due to R. Gerard and A. H. M. Levelt. The Gerard-Levelt theorem is presented as the S-N decomposition of a matrix of infinite order. At the end of Chapter V, the classification of the singu-
vii
viii |
PREFACE |
larities of linear differential equations is given. In Chapter VI, the main topics are the basic results concerning boundary-value problems of the second-order linear differential equations. The comparison theorems, oscillation and nonoscillation of solutions, eigenvalue problems for the Sturm-Liouville boundary conditions, scattering problems (in the case of reflectionless potentials), and periodic potentials are studied. The authors learned much about the scattering problems from the book by S. Tanaka and E. Date [TD]. In Chapter VII, asymptotic behaviors of solu- tions of linear systems as the independent variable approaches infinity are treated.
Topics include the Liapounoff numbers and the Levinson theorem together with its various improvements. In Chapter VIII, some fundamental theorems concern- ing stability, asymptotic stability, and perturbations of 2 x 2 linear systems are explained, whereas in Chapter IX, results on autonomous systems which include the LaSalle-Lefschetz theorem concerning behavior of solutions (or orbits) as the independent variable tends to infinity, the basic properties of limit-invariant sets including the Poincar6-Bendixson theorem, and applications of indices of simple closed curves are studied. Those theorems are applied to some nonlinear oscillation problems in Chapter X. In particular, the van der Pot equation is treated as both a problem of regular perturbations and a problem of singular perturbations. In
Chapters XII and XIII, asymptotic solutions of nonlinear differential equations as a parameter or the independent variable tends to its singularity are explained. In these chapters, the asymptotic expansions in the sense of Poincare are used most of time. However, asymptotic solutions in the sense of the Gevrey asymptotics are explained briefly. The basic properties of asymptotic expansions in the sense of
Poincare as well as of the Gevrey asymptotics are explained in Chapter XI.
At the beginning of each chapter, the contents and their history are discussed briefly. Also, at the end of each chapter, many problems are given as exercises. The purposes of the exercises are (i) to help the reader to understand the materials in each chapter, (ii) to encourage the reader to read research papers, and (iii) to help the reader to develop his (or her) ability to do research. Hints and comments for many exercises are provided.
The authors are indebted to many colleagues and former students for their valuable suggestions, corrections, and assistance at the various stages of writing this book. In particular, the authors express their sincere gratitude to Mrs. Susan Coddington and Mrs. Zipporah Levinson for allowing the authors to use the materials in the book Theory of Differential Equations by E. A. Coddington and Norman
Levinson.
Finally, the authors could not have carried out their work all these years without the support of their wives and children. Their contribution is immeasurable. We thank them wholeheartedly.
PFH
YS
March, 1999
CONTENTS
Preface |
vii |
|
Chapter I. Fundamental Theorems of Ordinary Differential Equations |
1 |
|
I-1. Existence and uniqueness with the Lipschitz condition |
1 |
|
1-2. Existence without the Lipschitz condition |
8 |
|
1-3. |
Some global properties of solutions |
15 |
1-4. |
Analytic differential equations |
20 |
Exercises I |
23 |
|
Chapter II. Dependence on Data |
28 |
II-1. Continuity with respect to initial data and parameters |
28 |
11-2. Differentiability |
32 |
Exercises II |
35 |
Chapter III. Nonuniqueness |
41 |
III-1. Examples |
41 |
111-2. The Kneser theorem |
45 |
111-3. Solution curves on the boundary of R(A) |
49 |
111-4. Maximal and minimal solutions |
52 |
111-5. A comparison theorem |
58 |
111-6. Sufficient conditions for uniqueness |
61 |
Exercises III |
66 |
Chapter IV. General Theory of Linear Systems |
69 |
IV-1. Some basic results concerning matrices |
69 |
IV-2. Homogeneous systems of linear differential equations |
78 |
IV-3. Homogeneous systems with constant coefficients |
81 |
IV-4. Systems with periodic coefficients |
87 |
IV-5. Linear Hamiltonian systems with periodic coefficients |
90 |
IV-6. Nonhomogeneous equations |
96 |
IV-7. Higher-order scalar equations |
98 |
Exercises IV |
102 |
Chapter V. Singularities of the First Kind |
108 |
V-1. Formal solutions of an algebraic differential equation |
109 |
V-2. Convergence of formal solutions of a system of the first kind |
113 |
V-3. The S-N decomposition of a matrix of infinite order |
118 |
V-4. The S-N decomposition of a differential operator |
120 |
V-5. A normal form of a differential operator |
121 |
V-6. Calculation of the normal form of a differential operator |
130 |
V-7. Classification of singularities of homogeneous linear systems |
132 |
Exercises V |
137 |
ix