Comparison theorems in delay differential equations in a critical state and applications

First, two comparison theorems are established between the first order delay differential equation x ' (t)+p(t)x(tau (t))= 0, t greater than or equal to t(0) (*) and two related second order ordinary differential equations in the case when f(tau (t))(t)p(s) ds greater than or equal to 1/e. (**) Next, by using these comparison theorems some new oscillation and non-oscillation criteria are given for (C) under condition (**). As a consequence, an open problem by Elbert and Stavroulakis is completely solved.