Bounded oscillation for second-order nonlinear delay differential equation in a critical state

In this paper we establish sufficient conditions for bounded oscillation of the second-order nonlinear delay differential equation x ″ ( t ) = ∑ i = 1 m p i x ( t - τ i ) + f ( t , x ( t - σ 1 ( t ) ) , … , x ( t - σ n ( t ) ) ) under a critical state by the second-order ordinary differential equation y ″ ( t ) + ( 1 - ε ) e - λ 0 t f ( t , e λ 0 ( t - σ 1 ( t ) ) y ( t ) , … , e λ 0 ( t - σ n ( t ) ) y ( t ) ) = 0 . At the same time, necessary conditions for the above bounded oscillation equation are obtained.