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.

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.

We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $$\Delta \left(p\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)+\eta \left(n\right)y^{\beta }(n-k)=0$$ under the condition ${\sum }_{n=n_0}^{\infty }{p}^{-\frac{1}{\alpha }}\left(n\right)<\infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form $$\Delta \left(a_n{\left({\Delta }^2(x_n+b_n{x}_{n-\delta })\right)}^{\alpha }\right)+q_n{x}_{n+1-\tau }^{\alpha }=0$$ and $$\Delta \left(a_n{\left({\Delta }^2(x_n-b_n{x}_{n-\delta })\right)}^{\alpha }\right)+q_n{x}_{n+1-\tau }^{\alpha }=0$$ are established. Some examples are presented to illustrate the main results.

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^q{a}_k\left(n\right)x(n+k)=0,n>n_0,$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{a_k\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

We obtain some new sufficient conditions for the oscillation of the solutions of the second-order quasilinear difference equations with delay and advanced neutral terms. The results established in this paper are applicable to equations whose neutral coefficients are unbounded. Thus, the results obtained here are new and complement some known results reported in the literature. Examples are also given to illustrate the applicability and strength of the obtained conditions over the known ones.

Consider the first order linear difference equation with general advanced argument and variable coefficients of the form $$\nabla x\left(n\right)-p\left(n\right)x\left(\tau \right(n\left)\right)=0,n⩾1,$$ where p(n) is a sequence of nonnegative real numbers, τ(n) is a sequence of positive integers such that $$\tau \left(n\right)⩾n+1,n⩾1,$$ and ▿ denotes the backward difference operator ▿x(n) = x(n) − x(n − 1). Sufficient conditions which guarantee that all solutions oscillate are established. Examples illustrating the results are given.

This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.

In this paper, sufficient conditions are obtained for oscillation of all solutions of third order difference equations of the form $$y_{n+3}+r_n{y}_{n+2}+q_n{y}_{n+1}+p_n{y}_n=0,n\ge 0.$$ These results are generalization of the results concerning difference equations with constant coefficients $$y_{n+3}+r{y}_{n+2}+q{y}_{n+1}+p{y}_n=0,n\ge 0.$$ Oscillation, nonoscillation and disconjugacy of a certain class of linear third order difference equations are discussed with help of a class of linear second order difference equations.