The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator ${\left(a\left(\iota \right){\left[\right(b\left(\iota \right){\left[x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )\right]}^{\prime })}^{\prime }\right]}^{\beta }{)}^{\prime }$$+{\int }_c^{d}q(\iota ,\mu )x^{\beta }\left(\sigma (\iota ,\mu )\right)d\mu =0,$ where $\iota \ge {\iota }_0$ and $w\left(\iota \right):=x\left(\iota \right)+p\left(\iota \right)x(\iota -\tau )$. New oscillation results are established by using the generalized Riccati technique under the assumption of ${\int }_{{\iota }_0}^{\iota }{a}^{-1/\beta }\left(s\right)ds$<${\int }_{{\iota }_0}^{\iota }\frac{1}{b\left(s\right)}ds=\infty \mathrm{as}\iota \to \infty$. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.

New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems.

The achieved sufficient conditions for nonexistence of so-called Kneser solutions are based on the new comparison principles, which help us decrease the problem of the wavering between the third and first-order equations. Examples are given to prove the significance of new theorems.

The present paper focuses on the oscillation and asymptotic properties of the third-order nonlinear neutral differential equations with variable delay arguments. By applying the Riccati transformation and the integral averaging technique, we give an analytical method for the estimation of Riccati differential inequality to establish several oscillation criteria for the discussed equation, which show that any solution either oscillates or converges to zero. We give several theorems and related examples to prove the significance of new theorems.