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.

Inhalt

Preface. 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. 1.10. Oscillation Between Sets. 1.11. Oscillation of Continuous-Discrete Recurrence Equations. 1.12. Second Order Quasilinear Difference Equations. 1.13. Oscillation of Even Order Difference Equations. 1.14. Oscillation of Odd Order Difference Equations. 1.15. Oscillation of Neutral Difference Equations. 1.16. Oscillation of Mixed Difference Equations. 1.17. Difference Equations Involving Quasi-differences. 1.18. Difference Equations with Distributed Deviating Arguments. 1.19. Oscillation of Systems of Higher Order Difference Equations. 1.20. Partial Difference Equations with Continuous Variables. 2. Oscillation of Functional Differential Equations. 2.1. Introduction. 2.2. Definitions, Notations and Preliminaries. 2.3. Ordinary Difference Equations. 2.4. Functional Difference 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. 2.10.Forced Equations with Middle Term of Order (n - 1). 2.11. Superlinear Forced Equations. 2.12. Sublinear Forced Equations. 2.13. Perturbed Functional Equations. 2.14. Comparison of Neutral Equations with Nonneutral Equations. 2.15 Comparison of Neutral Equations with Equations of the Same Form. 2.16. Neutral Differential Equations of Mixed Type. 2.17. Functional Differential Equations Involving Quasi-derivatives. 2.18. Neutral and Damped Functional Differential Equations Involving Quasi-derivatives. 2.19. Forced Functional Differential Equations Involving Quasi-derivatives. 2.20. Systems of Higher Order Functional Differential Equations. References. Subject Index.