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.

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The objective of this paper is to study the oscillatory and asymptotic properties of third order mixed neutral differential equation of the form

 $$ (a(t) [x(t) + b(t) x(t - au_{1}) + c(t) x(t + au_{2})]'')' + q(t) x^{alpha}(t - sigma_{1}) + p(t) x^{eta}(t + sigma_{2}) = 0 $$

where $a(t), b(t), c(t), q(t)$ and $p(t)$ are positive continuous functions, $alpha$ and $eta$ are ratios of odd positive integers, $au_{1}, au_{2}, sigma_{1}$ and $sigma_{2}$ are positive constants. We establish some sufficient conditions which ensure that all solutions are either oscillatory or converge to zero. Some examples are provided to illustrate the main results.

Some oscillation results are obtained for the third order nonlinear mixed type neutral differential equations of the form$$\left(\left(x(t)+b(t) x\left(t-\tau_1\right)+c(t) x\left(t+\tau_2\right)\right)^\alpha\right)^{\prime \prime \prime}=q(t) x^\beta\left(t-\sigma_1\right)+p(t) x^\gamma\left(t+\sigma_2\right), t \geq t_0$$where $\alpha, \beta$ and $\gamma$ are ratios of odd positive integers $\tau_1, \tau_2, \sigma_1$ and $\sigma_2$ are positive constants.