Abstract: Using the fixed point theorem in Banach space, and by introducing parameter function and the generalized Riccati transformation, a new nonoscillation criteria for the equation was obtained. In addition, a sufficient condition for oscillation of the equation was proposed. These criteria can improve the restriction of the conditions for the equation. Some existed results in the literatures have been further improved and extended.

Key words: neutral functional differential equation, nonlinear, oscillation and nonoscillation, Riccati transformation, positive and negative coefficient, neutral functional differential equation, nonlinear, oscillation and nonoscillation, Riccati transformation