Oscillation theory for difference and functional differential equations

This book reviews material from more than three hundred publications on the oscillation theory of difference and functional differential equations of various types. For difference equations, a large number of new concepts are explained and supported by interesting theoretical developments. For differential equations, simplified versions of several new integral criteria for oscillations are presented. Proofs which illustrate the various strategies and ideas involved are given. This book should be a stimulus to the further development of the theory. Audience: This work will be of interest to mathematicians and graduate students in the disciplines of theoretical and applied mathematics. Front cover......Page 1 Title page......Page 3 Date-line......Page 4 Contents......Page 5 Preface......Page 7 1.1. Introduction......Page 9 1.2. Oscillation of Scalar Difference Equations......Page 10 1.3. Oscillation of Orthogonal Polynomials......Page 16 1.4. Oscillation of Functions Recurrence Equations......Page 22 1.5. Oscillation in Ordered Sets......Page 27 1.6. Oscillation in Linear Spaces......Page 30 1.7. Oscillation in Archimedean Spaces......Page 32 1.8. Oscillation of Partial Recurrence Equations......Page 35 1.9. Oscillation of System of Equations......Page 40 1.10. Oscillation Between Sets......Page 44 1.11. Oscillation of Continuous-Discrete Recurrence Equations......Page 47 1.12. Second Order Quasilinear Difference Equations......Page 50 1.13. Oscillation of Even Order Difference Equations......Page 64 1.14. Oscillation of Odd Order Difference Equations......Page 73 1.15. Oscillation of Neutral Difference Equations......Page 81 1.16. Oscillation of Mixed Difference Equations......Page 87 1.17. Difference Equations Involving Quasi-differences......Page 102 1.18. Difference Equations with Distributed Deviating Arguments......Page 125 1.19. Oscillation of Systems of Higher Order Difference Equations......Page 151 1.20. Partial Difference Equations with Continuous Variables......Page 157 2.1. Introduction......Page 174 2.2. Definitions, Notations and Preliminaries......Page 175 2.3. Ordinary Differential Equations......Page 181 2.4. Functional Differential Equations......Page 188 2.5. Comparison of Equations of the Same Form......Page 207 2.6. Comparison of Equations with Others of Lower Order......Page 213 2.7. Further Comparison Results......Page 216 2.8. Equations with Middle Term of Order $(n-1)$......Page 233 2.9. Forced Differential Equations......Page 243 2.10. Forced Equations with Middle Term of Order $(n-1)$......Page 250 2.11. Superlinear Forced Equations......Page 252 2.12. Sublinear Forced Equations......Page 255 2.13. Perturbed Functional Equations......Page 257 2.14. Comparison of Neutral Equations with Nonneutral Equations......Page 260 2.15. Comparison of Neutral Equations with Equations of the Same Form......Page 269 2.16. Neutral Differential Equations of Mixed Type......Page 273 2.17. Functional Differential Equations Involving Quasi-derivatives......Page 283 2.18. Neutral and Damped Functional Differential Equations Involving Quasi-derivatives......Page 294 2.19. Forced Functional Differential Equations Involving Quasi-derivatives......Page 299 2.20. Systems of Higher Order Functional Differential Equations......Page 317 References......Page 326 Subject Index......Page 344 Back cover......Page 346 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 real­ valued 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 oscil­ lation of n-th order functional differential equations with deviating argu­ ments, 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 differ­ ential 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. The theory of difference equations (recurrence relations), the methods used in their solutions and their wide applications have advanced beyond their adolescent stage to occupy a central position in Applicable Analysis.