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 (τ( t ))=0, t ⩾ t 0 (*) and two related second order ordinary differential equations in the case when∫ τ ( t ) t p ( s ) d s ⩾ 1 e .( * * )Next, by using these comparison theorems some new oscillation and non‐oscillation criteria are given for (*) under condition (**). As a consequence, an open problem by Elbert and Stavroulakis is completely solved.