A partly sharp oscillation criterion for first-order delay differential equations

In this paper, a partly sharp oscillation criterion is established for the first-order delay differential equation x ′ ( t ) + p ( t ) x ( τ ( t ) ) = 0 , t ≥ T 0 , where p ( t ) is continuous and p ( t ) ≥ 0 for all t ≥ T 0 ; τ ( t ) is continuous, non-decreasing, τ ( t ) t for all t ≥ T 0 , and lim t → ∞ τ ( t ) = + ∞ . By improving techniques from previous researches, we show that when α = lim inf t → ∞ ∫ τ ( t ) t p ( s ) d s ∈ ( 0 , 1 e ] , all solutions of the equation are oscillatory if lim sup t → ∞ ∫ τ ( t ) t p ( s ) d s > 2 α + 2 λ 1 − 1 under no additional constraints, where λ 1 is the smaller root of the equation λ = e α λ . This result is proved to be weaker than previous ones, and sharp when α ∈ ( 0 , ln 2 2 ) by constructing a specific example.