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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 present some new sufficient conditions for oscillation of even order nonlinear neutral difference equation of the form \[\Delta^m(x_n+ax_{n-\tau_1}+bx_{n+\tau_2})+p_nx_{n-\sigma_1}^{\alpha}+q_nx_{n+\sigma_2}^{\beta}=0,\quad n\geq n_0\gt0,\] where \(m\geq 2\) is an even integer, using arithmetic-geometric mean inequality. Examples are provided to illustrate the main results.

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 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