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In this paper, we study the oscillation of even-order nonlinear neutral differential equations with both retarded and advanced arguments. The arithmetic-geometric mean inequality is employed. Examples are provided to illustrate the main results.

In this paper, we establish some sufficient conditions which ensure that the solutions of the third order delay difference equation with a negative middle term $$ \Delta \bigl(a_{n}\Delta (\Delta w_{n})^{\alpha } \bigr)-p_{n}(\Delta w_{n+1})^{ \alpha }-q_{n}h(w_{n-l})=0,\quad n\geq n_{0}, $$ Δ ( a n Δ ( Δ w n ) α ) − p n ( Δ w n + 1 ) α − q n h ( w n − l ) = 0 , n ≥ n 0 , are oscillatory. Moreover, we study the asymptotic behavior of the nonoscillatory solutions. Two illustrative examples are included for illustration.

We examine properties of positive solutions of the third order differential equation with advanced argument in neutral term of the form $$(a(t)(b(t)(z'(t))^{\alpha})')'+q(t)y^{\alpha}(t)=0,$$ where $${z(t)=y(t)+p(t)y(\sigma(t))}$$ . They show that the positive solutions are in fact Kneser type solutions and then provide lower and upper estimate that yield the rate of convergence to zero for such solutions. An example is provided to illustrate the importance of the main result.

In this paper, we present some oscillation criteria for second order nonlinear delay difference equation with non-positive neutral term of the form $$\Delta (a_n (\Delta z_n )^\alpha ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;n \ge n_0 > 0,$$ where zn = xn − pnxn−τ, and α is a ratio of odd positive integers. Examples are provided to illustrate the results. The results obtained in this paper improve and complement to some of the existing results.

In this paper we study the oscillation criteria for the second order nonlinear difierential equation with delay and advanced arguments of the form ((x(t) + a(t)x(t i ae1) + b(t)x(t + ae2)) fi ) 00 + q(t)x fl (t i ?1) + p(t)x ∞ (t + ?2) = 0; t ‚ t0 where ae1; ae2; ?1 and ?2 are nonnegative constants and fi; fl and ∞ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

In this paper, we obtain some sufficient conditions for the oscillation of all solutions of the third order nonlinear neutral difference equation with mixed arguments of the form $$\begin{aligned} \Delta ^3(x_n+b_n x_{n-\tau _1}+c_nx_{n+\tau _2})^{\alpha }+q_nx_{n-\sigma _1}^{\beta }+p_n x_{n+\sigma _2}^{\gamma }=0, n\ge n_0, \end{aligned}$$ where \(\alpha \), \(\beta \) and \(\gamma \) are the ratios of odd positive integers. Some examples are provided to substantiate our results.

In this paper, some new sufficient conditions are established for the oscillation of all solutions of the second-order neutral differential equation with mixed neutral terms of the formfor all

Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form \begin {equation*} \Delta (a_n(\Delta ^2(x_n+b_{n}x_{n-\delta }))^\alpha )+q_{n}x^{\alpha }_{n+1-\tau }=0 \end {equation*} and \begin {equation*} \Delta (a_n(\Delta ^2(x_n-b_nx_{n-\delta }))^\alpha )+q_nx^{\alpha }_{n+1-\tau }=0 \end {equation*} are established. Some examples are presented to illustrate the main results.

The aim of this paper is to investigate the oscillatory and asymptotic behavior of solutions of a third-order delay difference equation. By using comparison theorems, we deduce oscillation of the difference equation from its relation to certain associated first-order delay difference equations or inequalities. Examples are given to illustrate the main results.