Abstract:The oscillation criteria are investigated for all solutions of second order nonlinear neutral delay differential equations. Our results extend and improve some results well known in the literature see ( [14] theorem 3.2.1 and theorem 3.2.2 pp.385-388). Some examples are given to illustrate our main results.

In this paper the asymptotic behavior for all nonoscillatory solutions of third order nonlinear neutral differential equations have been investigated, where some necessary and sufficient conditions are obtained to guarantee the convergence of these solutions to zero or tends to infinity as t → ∞. We introduced Lemma 2.1 and Lemma 2.2 which are a generalization of Lemma 1.5.2 [I.

In this paper some necessary and sufficient conditions of n – th order neutral differential equations are obtained to insure the convergence of all nonoscillatory solutions to zero or tends to infinity as t → ∞. Some examples are given to illustrate the main results.

The oscillation property of the second order half linear dynamic equation was studied, some sufficient conditions were obtained to ensure the oscillation of all solutions of the equation. The results are supported by illustrative examples.

Oscillatory properties of third-order half-linear neutral dynamic equations were investigated in this paper. Some conditions are obtained to ensure the oscillation of all solutions of a neutral dynamic equation with several coefficients and several variable delays, t ∈ [t0, ∞)𝕋, where γ > 0 is a quotient of odd positive integers, , and , and the results are supported by illustrative examples.