An oscillation theorem for a class of second-order forced neutral delay differential equations with mixed nonlinearities

A class of second-order forced neutral delay differential equation with mixed nonlinearities ( r ( t ) | x ′ ( t ) + p x ′ ( t − σ ) | α − 1 ( x ′ ( t ) + p x ′ ( t − σ ) ) ) ′ + q 0 ( t ) | x ( τ 0 ( t ) ) | α − 1 x ( τ 0 ( t ) ) + ∑ i = 1 n q i ( t ) | x ( τ i ( t ) ) | β i − 1 x ( τ i ( t ) ) = e ( t ) sgn ( x ( t ) ) is investigated in this paper. Using a new method, we obtain some new sufficient conditions for the oscillation of the above equation, and some known oscillation criteria have been extended.