ABSTRACT
In general, the theory of neutral delay differential equations (NDDEs) is more complicated than the theory of delay differential equations without neutral terms. This chapter presents a systematic study for the oscillation theory of first order NDDEs, which contains some recent results and consequently is a useful source for researchers in this field. It considers the linear NDDEs with constant parameters, where the "characteristic equation" method plays an important role. The chapter presents some necessary and sufficient conditions for the characteristic equations to have no real roots. It also presents a comparison result for oscillation. The chapter discusses a class of sublinear NDDEs. The chapter is further concerned with the NDDEs with a nonlinear neutral term. The linearized oscillation result reveals that, under some assumptions, certain nonlinear differential equations have the same oscillatory behavior as an associated linear equation with constant coefficients.
