A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay

We consider linear differential equations with variable delay of the form x ′ ( t ) + p ( t ) x ( t − τ ( t ) ) = 0 , t ≥ t 0 , where p : [ t 0 , ∞ ) → [ 0 , ∞ ) and τ : [ t 0 , ∞ ) → ( 0 , ∞ ) are continuous functions, such that t − τ ( t ) → ∞ (as t → ∞ ). It is well-known that, for the oscillation of all solutions, it is necessary that B : = lim sup t → ∞ A ( t ) ≥ 1 e holds , where A : = ( t ) ∫ t − τ ( t ) t p ( s ) d s . Our main result shows that, if the function A is slowly varying at infinity (in additive form), then under mild additional assumptions on p and τ , condition B > 1 / e implies that all solutions of the above delay differential equation are oscillatory.