Nonoscillation and Oscillation: Theory for Functional Differential Equations
This book addresses the question of the existence of oscillatory solutions to ODE's with delays. Oscillatory solutions are solutions with an infinite number of zeros in a given interval. In most cases the author treats a single equation. The delays can depend on time and may even become negative (advanced equations). The book is organized as follows. Chapter 1 summarizes the preliminaries; in fact, not much introduction is required, which makes the book accessible to non-specialists and students. Chapters 2 and 3 deal with first order equations. Chapter 4 reviews oscillation theory for second order equations without delays. These results are extended in chapters 5 and 6 to delayed equations of second order and higher orders respectively. While the previous chapters deal with single equations, chapter 7 is devoted to systems of two equations. The final chapter 8 introduces the reader to a new and interesting area of differential equations at time scales. Roughly speaking, here one studies an evolutionary equation with time varying over an arbitrary subset T (the “time scale”) of real numbers.