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.
