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This work is concerned with oscillation of a class of fourth-order nonlinear delay dynamic equations on a time scale. A new comparison theorem is established that improves related results reported in the literature.

In this paper new oscillation criteria for the second order neutral differential equations of the form \begin{equation*} \left(r(t)\left[x(t)+p(t)x(\tau(t))\right]'\right)'+q(t)x(\sigma(t))+v(t)x(\eta(t))=0 \tag{$E$}\end{equation*} are presented. Gained results are based on the new comparison theorems, that enable us to reduce the problem of the oscillation of the second order equation to the oscillation of the first order equation. Obtained comparison principles essentially simplify the examination of the studied equations. We cover all possible cases when arguments are delayed, advanced or mixed.

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form $$\Delta(a_{n}(\Delta^{2}(x_{n} + p_{n}x_{n-\delta}))^{\alpha}) + q_{n} {x^{\alpha}_{n-\tau}} = 0,\quad n\geq 0, \tag{E}$$ where $\alpha > 0, 0\leq p_{n}\leq p < \infty. $ By using Riccati transformation we estabilish some new sufficient conditions which ensure that every solution of equation (E) is either oscillatory or converges to zero. These results improve some known results in the literature. Examples are provided to illustrate the main results.

In this paper, by defining a class of functions, we establish some oscillation criteria for the second order nonlinear dynamic equations with forced term $$ x^{\Delta\Delta}(t)+a(t)f(x(q(t)))=e(t) $$ on a time scale $\mathbb{T}.$ Our results unify the oscillation of the second order forced differential equation and the second order forced difference equation. An example is considered to illustrate the main results.

In this paper, the authors study the oscillatory properties of third order quasilinear neutral difference equation of the form Δ(an(Δ² (xn + pnxn-δ)) α) + qnxn-Τα = 0, n ≥ 0, (E) where α > 0, qn ≥ 0, 0 ≤ pn ≤ p < ∞. By using Riccati transformation we establish some new suffcient conditions which ensure that every solution of equation (E) is either oscillatory or converges to zero. These results improve some known results in the literature. Examples are provided to illustrate the main results. [ABSTRACT FROM AUTHOR]