In the paper sufficient conditions for the difference equation :Δxn = Σi=0r an(i) xn+ito 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.

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.

We investigate the behaviour of the solutions of a $k$-dimensional cyclic system of difference equations with maximum. More precisely, we study the existence and the number of the equilibria in the case when $k$ is an odd or an even positive integer, but also for the various values of the exponents of the terms of the difference equations of this system. In addition, we find invariant intervals for our system and we invistegate the convergence of the solutions to the unique positive equilibrium. Finally,...

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.