Introduction to Oscillation Theory

The first chapter starts with an introduction to the theory of autonomous delay differential equations: existence and uniqueness, representation of solutions which uses basic notions and methods (no knowledge of advanced analysis is required to understand this chapter). However, many of the ideas which will be used later, such as the characteristic equations, characteristic inequalities, the study of the sign of the fundamental function, properties which are equivalent to nonoscillation, application of the solution representations, are introduced in this chapter using autonomous equations. A simple example demonstrates that the properties of equations with coefficients of different signs are more complicated than of autonomous equations with positive coefficients. The basic linearization principles are presented for nonlinear autonomous models, and several important equations of mathematical biology are introduced. These applied models include not only Hutchinson’s, Lasota-Wazewska, Nicholson’s blowflies and Mackey-Glass equations, but also the autonomous analogues of their generalizations which will later be studied in the monograph. For a linear autonomous impulsive equation, the idea of the construction of a non-impulsive equation with discontinuous coefficients such that oscillation properties of solutions are preserved, is introduced using a simple example where an analytic solution can be constructed. Nevertheless, this idea will be used later to analyze more complicated impulsive models. Finally, autonomous versions of all the other equations which will be studied in the monograph (integrodifferential and mixed equations, as well as equations with a distributed delay) are introduced. Some of the theorems are presented without proofs, with the references to the next chapters where more general results will be justified. Overall, the material of the chapter should be easily understood by undergraduate students who completed a basic course in the ordinary differential equations.