ABSTRACT
This chapter describes some of the recent developments in the oscillation theory of first order delay differential equations. This theory is interesting from the theoretical as well as the practical point of view. It is well known that homogeneous ordinary differential equations (ODEs) of first order do not possess oscillatory solutions. But the presence of deviating arguments can cause the oscillation of solutions. The chapter shows various techniques used in the oscillation and nonoscillation theory of differential equations with deviating arguments. It presents some criteria for oscillation, for the existence of positive solutions and results in the distribution of zeros of oscillatory solutions of DDEs of first order. The chapter estimates the distances between adjacent zeros of oscillatory solutions of the first order delay differential equation. It discusses the oscillation and nonoscillation behavior of some basic ecological delay equations.
