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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.

In the paper sufficient conditions for the difference equation : Δx_n = Σ_i=0 ^r a_n ⁽ⁱ⁾ x_n+i to have a solution which tends to a constant, are given. Applying these conditions, an asymptotic formula for a solution of an m-th order equation is presented.

For the linear difference equation $$x_{n+1}-a_n{x}_n=\sum _{i=0}^r{a}_n^{\left(i\right)}{x}_{n+i},n\in N$$ sufficient conditions for the existence of an asymptotically periodic solutions are given.

We consider a second order nonlinear difference equation $${\Delta }^2{y}_n=a_n{y}_{n+1}+f(n,y_n,y_{n+1}),n\in N.\left(\text{E}\right)$$ The necessary conditions under which there exists a solution of equation (E) which can be written in the form $$y_{n+1}={\alpha }_n{u}_n+{\beta }_n{v}_n,\text{are}\text{given.}$$ Here $u$ and $v$ are two linearly independent solutions of equation $${\Delta }^2{y}_n=a_{n+1}{y}_{n+1},\left(\underset{n\to \infty }{lim}{\alpha }_n=\alpha <\infty \mathrm{and}\underset{n\to \infty }{lim}{\beta }_n=\beta <\infty \right).$$ A special case of equation (E) is also considered.