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.

Autonomous linear neutral delay and, especially, (non-neutral) delay difference equations with continuous variable are considered, and some new results on the behavior of the solutions are established. The results are obtained by the use of appropriate positive roots of the corresponding characteristic equation.

The neutral delay difference equations of second order with positive and negative coefficients $\Delta ²(xₙ+pₙx_{n-\tau })+qₙx_{n-\sigma }-rₙx_{n-\lambda }=0$, n = 0,1,2,... studied sufficient condition existence positive solution equation obtained

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.

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\frac{{\alpha }_0{x}_n+{\alpha }_1{x}_{n-l}+{\alpha }_2{x}_{n-k}}{{\beta }_0{x}_n+{\beta }_1{x}_{n-l}+{\beta }_2{x}_{n-k}},n=0,1,2,\cdots$$ where the coefficients ${\alpha }_i,{\beta }_i\in (0,\infty )$ for $i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $x_{-k},\cdots ,x_{-l},\cdots ,x_{-1},x_0$ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.