Some new oscillation and nonoscillation criteria for the second order neutral delay difference equation $$\Delta \left(c_n\Delta (y_n+p_n{y}_{n-k})\right)+q_n{y}_{n+1-m}^{\beta }=0,n\ge n_0$$ where $k$, $m$ are positive integers and $\beta$ is a ratio of odd positive integers are established, under the condition ${\sum }_{n=n_0}^{\infty }\frac{1}{c_n}<\infty .$

Consider the difference equation $$\Delta x\left(n\right)+\sum _{i=1}^m{p}_i\left(n\right)x\left({\tau }_i\left(n\right)\right)=0,n\ge 0\left[\nabla x\left(n\right)-\sum _{i=1}^m{p}_i\left(n\right)x\left({\sigma }_i\left(n\right)\right)=0,n\ge 1\right],$$ where $\left(p_i\left(n\right)\right)$, $1\le i\le m$ are sequences of nonnegative real numbers, ${\tau }_i\left(n\right)$ [${\sigma }_i\left(n\right)$], $1\le i\le m$ are general retarded (advanced) arguments and $\Delta$ [$\nabla$] denotes the forward (backward) difference operator $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ [$\nabla x\left(n\right)=x\left(n\right)-x(n-1)$]. New oscillation criteria are established when the well-known oscillation conditions $$\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=\tau \left(n\right)}^n{p}_i\left(j\right)>1\left[\underset{n\to \infty }{lim\; sup}\sum _{i=1}^m\sum _{j=n}^{\sigma \left(n\right)}{p}_i\left(j\right)>1\right]$$ and $$\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j={\tau }_i\left(n\right)}^{n-1}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\left[\underset{n\to \infty }{lim\; inf}\sum _{i=1}^m\sum _{j=n+1}^{{\sigma }_i\left(n\right)}{p}_i\left(j\right)>\frac{1}{\mathrm{e}}\right]$$ are not satisfied. Here $\tau \left(n\right)={max}_{1\le i\le m}{\tau }_i\left(n\right)$$[\sigma ...$

2000 Mathematics Subject Classification: 39A10.The oscillatory and nonoscillatory behaviour of solutions of the second order quasi linear neutral delay difference equation Δ(an | Δ(xn+pnxn-τ)|α-1 Δ(xn+pnxn-τ) + qnf(xn-σ)g(Δxn) = 0 where n ∈ N(n0), α > 0, τ, σ are fixed non negative integers, {an}, {pn}, {qn} are real sequences and f and g real valued continuous functions are studied. Our results generalize and improve some known results of neutral delay difference equations.

Some new criteria for the oscillation of difference equations of the form $${\Delta }^2{x}_n-p_n\Delta x_{n-h}+q_n{\left|x_{g_n}\right|}^c\mathrm{s}gn{x}_{g_n}=0$$ and $${\Delta }^i{x}_n+p_n{\Delta }^{i-1}{x}_{n-h}+q_n{\left|x_{g_n}\right|}^c\mathrm{s}gn{x}_{g_n}=0,i=2,3,$$ are established.

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor's Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality...

Necessary and sufficient conditions are obtained for every solution of $$\Delta (y_n+p_n{y}_{n-m})\pm q_{n}G\left(y_{n-k}\right)=f_n$$ to oscillate or tend to zero as $n\to \infty$, where $p_n$, $q_n$ and $f_n$ are sequences of real numbers such that $q_n\ge 0$. Different ranges for $p_n$ are considered.

In this paper the three-dimensional nonlinear difference system $$\begin{array}{cc}\hfill \Delta x_n& =a_{n}f\left(y_{n-l}\right),\hfill \\ \hfill \Delta y_n& =b_{n}g\left(z_{n-m}\right),\hfill \\ \hfill \Delta z_n& =\delta c_{n}h\left(x_{n-k}\right),\hfill \end{array}$$ is investigated. Sufficient conditions under which the system is oscillatory or almost oscillatory are presented.