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经典教材：差分方程基本教程- Elaydi.pdf

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    Undergraduate Texts in Mathematics Editors S. Axler F.W. Gehring K.A. Ribet
    Saber Elaydi An Introduction to Difference Equations Third Edition
    Saber Elaydi Department of Mathematics Trinity University San Antonio, Texas 78212 USA Editorial Board S. Axler Mathematics Department Mathematics Department Department of San Francisco State Mathematics F.W. Gehring K.A. Ribet University San Francisco, CA 94132 USA East Hall University of Michigan Ann Arbor, MI 48109 USA University of California at Berkeley Berkeley, CA 94720-3840 USA Mathematics Subject Classification (2000): 12031 Library of Congress Cataloging-in-Publication Data Elaydi, Saber, 1943– An introduction to difference equations / Saver Elaydi.—3rd ed. p. cm. — (Undergraduate texts in mathematics) Includes bibliographical references and index. ISBN 0-387-23059-9 (acid-free paper) 1. Difference equations. I. Title. II. Series. QA431.E43 2005 515′.625—dc22 2004058916 Printed on acid-free paper. ISBN 0-387-23059-9 © 2005 Springer Science+Business Media, 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 Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connec- tion with reviews or scholarly analysis. Use in connection with any form of informa- tion storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. (MV) 9 8 7 6 5 4 3 2 1 springeronline.com SPIN 10950678
    Preface to the Third Edition In contemplating the third edition, I have had multiple objectives to achieve. The first and foremost important objective is to maintain the ac- cessibility and readability of the book to a broad readership with varying mathematical backgrounds and sophistication. More proofs, more graphs, more explanations, and more applications are provided in this edition. The second objective is to update the contents of the book so that the reader stays abreast of new developments in this vital area of mathematics. Recent results on local and global stability of one-dimensional maps are included in Chapters 1, 4, and Appendices A and C. An extension of the Hartman–Grobman Theorem to noninvertible maps is stated in Appendix D. A whole new section on various notions of the asymptoticity of solutions and a recent extension of Perron’s Second Theorem are added to Chapter 8. In Appendix E a detailed proof of the Levin–May Theorem is presented. In Chapters 4 and 5, the reader will find the latest results on the larval– pupal–adult flour beetle model. The third and final objective is to better serve the broad readership of this book by including most, but certainly not all, of the research areas in difference equations. As more work is being published in the Journal of Difference Equations and Applications and elsewhere, it became apparent that a whole chapter needed to be dedicated to this enterprise. With the prodding and encouragement of Gerry Ladas, the new Chapter 5 was born. Major revisions of this chapter were made by Fozi Dannan, who diligently and painstakingly rewrote part of the material and caught several errors and typos. His impact on this edition, particularly in Chapters 1, 4, and Chapter 8 is immeasurable and I am greatly indebted to him. My thanks v
    vi Preface to the Third Edition go to Shandelle Henson, who wrote a thorough review of the book and suggested the inclusion of an extension of the Hartman–Groman Theorem, and to Julio Lopez and his student Alex Sepulveda for their comments and discussions about the second edition. I am grateful to all the participants of the AbiTuMath Program and to its coordinator Andreas Ruffing for using the second edition as the main reference in their activities and for their valuable comments and dis- cussions. Special thanks go to Sebastian Pancratz of AbiTuMath whose suggestions improved parts of Chapters 1 and 2. I benefited from comments and discussions with Raghib Abu-Saris, Bernd Aulbach, Martin Bohner, Luis Carvahlo, Jim Cushing, Malgorzata Guzowska, Sophia Jang, Klara Janglajew, Nader Kouhestani, Ulrich Krause, Ronald Mickens, Robert Sacker, Hassan Sedaghat, and Abdul-Aziz Yakubu. It is a pleasure to thank Ina Lindemann, the editor at Springer-Verlag for her advice and support during the writing of this edition. Finally, I would like to express my deep appreciation to Denise Wilson who spent many weekends typing various drafts of the manuscript. Not only did she correct many glitches, typos, and awkward sentences, but she even caught some mathematical errors. I hope you enjoy this edition and if you have any comments or questions, please do not hesitate to contact me at selaydi@trinity.edu. San Antonio, Texas April 2004 Suggestions for instructors using this book. Saber N. Elaydi The book may be used for two one-semester courses. A first course may include one of the following options but should include the bulk of the first four chapters: 1. If one is mainly interested in stability theory, then the choice would be Chapters 1–5. 2. One may choose Chapters 1–4, and Chapter 8 if the interest is to get to asymptotic theory. 3. Those interested in oscillation theory may choose Chapters 1, 2, 3, 5, and 7. 4. A course emphasizing control theory may include Chapters 1–3, 6, and 10.
    Preface to the Third Edition vii Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 7 Chapter 6 Chapter 5 Chapter 6 Chapter 9 Chapter 7 The diagram above depicts the dependency among the chapters.
    Preface to the Second Edition The second edition has greatly benefited from a sizable number of com- ments and suggestions I received from users of the first edition. I hope that I have corrected all the errors and misprints in the book. Important revisions were made in Chapters 1 and 4. In Chapter 1, I added two ap- pendices (Global Stability and Periodic Solutions). In Chapter 4, I added a section on applications to mathematical biology. Influenced by a friendly and some not so friendly comments about Chapter 8 (previously Chapter 7: Asymptotic Behavior of Difference Equations), I rewrote the chapter with additional material on Birkhoff’s theory. Also, due to popular demand, a new chapter (Chapter 9) under the title “Applications to Continued Frac- tions and Orthogonal Polynomials” has been added. This chapter gives a rather thorough presentation of continued fractions and orthogonal poly- nomials and their intimate connection to second-order difference equations. Chapter 8 (Oscillation Theory) has now become Chapter 7. Accordingly, the new revised suggestions for using the text are as follows. The book may be used with considerable flexibility. For a one-semester course, one may choose one of the following options: (i) If you want a course that emphasizes stability and control, then you may select Chapters 1, 2, and 3, and parts of Chapters 4, 5, and 6. This is perhaps appropriate for a class populated by mathematics, physics, and engineering majors. (ii) If the focus is on the applications of difference equations to orthogonal polynomials and continued fractions, then you may select Chapters 1, 2, 3, 8, and 9. ix
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