OSCILLATION THEORY FOR DIFFERENCE AND FUNCTIONAL DIFFERENTIAL EQUATIONS

Transcription

1 OSCILLATION THEORY FOR DIFFERENCE AND FUNCTIONAL DIFFERENTIAL EQUATIONS

2 Oscillation Theory for Difference and Functional Differential Equations by Ravi P. Agarwal Department of Mathematics, National University l~lsingapore, Singapore Said R. Grace Department ld"engineering Mathematics, Cairo University, Orman, Gi::a, Egypt and Donal 0' Regan Department of" Mathematics, National Unil ersity of Ireland, Galway. Ireland SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

3 A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN ISBN (ebook) DOI / Printed on acid-free paper All Rights Reserved 2000 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2000 Softcover reprint of the hardcover 1st edition 2000 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

4 Contents Preface vii Chapter 1 Oscillation of Difference Equations 1.1. Introduction 1.2. Oscillation of Scalar Difference Equations 1.3. Oscillation of Orthogonal Polynomials 1.4. Oscillation of Functions Recurrence Equations 1.5. Oscillation in Ordered Sets 1.6. Oscillation in Linear Spaces 1.7. Oscillation in Archimedean Spaces 1.8. Oscillation of Partial Recurrence Equations 1.9. Oscillation of System of Equations Oscillation Between Sets Oscillation of Continuous-Discrete Recurrence Equations Second Order Quasilinear Difference Equations Oscillation of Even Order Difference Equations Oscillation of Odd Order Difference Equations Oscillation of Neutral Difference Equations Oscillation of Mixed Difference Equations Difference Equations Involving Quasi-differences Difference Equations with Distributed Deviating Arguments Oscillation of Systems of Higher Order Difference Equations Partial Difference Equations with Continuous Variables Chapter 2 Oscillation of Functional Differential Equations 2.1. Introduction 2.2. Definitions, Notations and Preliminaries 2.3. Ordinary Differential Equations

5 vi 2.4. Functional Differential Equations 2.5. Comparison of Equations of the Same Form 2.6. Comparison of Equations with Others of Lower Order 2.7. Further Comparison Results 2.8. Equations with Middle Term of Order (n- 1) 2.9. Forced Differential Equations Forced Equations with Middle Term of Order (n- 1) Superlinear Forced Equations Sublinear Forced Equations Perturbed Functional Equations Comparison of Neutral Equations with Nonneutral Equations Comparison of Neutral Equations with Equations of the Same Form Neutral Differential Equations of Mixed Type Functional Differential Equations Involving Quasi-derivatives Neutral and Damped Functional Differential Equations Involving Quasi-derivatives Forced Functional Differential Equations Involving Quasi-derivatives Systems of Higher Order Functional Differential Equations 309 References 318 Subject Index

6 Preface This monograph is devoted to a rapidly developing area of research of the qualitative theory of difference and functional differential equations. In fact, in the last 25 years Oscillation Theory of difference and functional differential equations has attracted many researchers. This has resulted in hundreds of research papers in every major mathematical journal, and several books. In the first chapter of this monograph, we address oscillation of solutions to difference equations of various types. Here we also offer several new fundamental concepts such as oscillation around a point, oscillation around a sequence, regular oscillation, periodic oscillation, point-wise oscillation of several orthogonal polynomials, global oscillation of sequences of realvalued functions, oscillation in ordered sets, (!, R, ~)-oscillate, oscillation in linear spaces, oscillation in Archimedean spaces, and oscillation across a family. These concepts are explained through examples and supported by interesting results. In the second chapter we present recent results pertaining to the oscillation of n-th order functional differential equations with deviating arguments, and functional differential equations of neutral type. We mainly deal with integral criteria for oscillation. While several results of this chapter were originally formulated for more complicated and/or more general differential equations, we discuss here a simplified version to elucidate the main ideas of the oscillation theory of functional differential equations. Further, from a large number of theorems presented in this chapter we have selected the proofs of only those results which we thought would best illustrate the various strategies and ideas involved. We hope this monograph will fill a vacuum in the oscillation theory of difference and functional differential equations and will be a stimulus to its further development. It is impossible to acknowledge individually colleagues and friends to whom we are indebted for assistance, inspiration and criticism in writing this monograph. We must, however, express our appreciation and thanks to Sadhna for her careful typing of the entire manuscript. Ravi P Agarwal Said R Grace Donal O'Regan