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### Oscillation of nonlinear three-dimensional difference systems with delays

Mathematica Bohemica

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.

### On the asymptotic behavior of solutions of linear difference equations.

Publicacions Matemàtiques

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

### Some properties of solutions of a class of nonlinear difference equations

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

### On the asymptotically periodic solution of some linear difference equations

Archivum Mathematicum

For the linear difference equation ${x}_{n+1}-{a}_{n}{x}_{n}=\sum _{i=0}^{r}{a}_{n}^{\left(i\right)}{x}_{n+i},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}n\in N$ sufficient conditions for the existence of an asymptotically periodic solutions are given.

### On the existence of solutions of some second order nonlinear difference equations

Archivum Mathematicum

We consider a second order nonlinear difference equation ${\Delta }^{2}{y}_{n}={a}_{n}{y}_{n+1}+f\left(n,{y}_{n},{y}_{n+1}\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n\in N\phantom{\rule{0.166667em}{0ex}}.\phantom{\rule{2.0em}{0ex}}\left(\text{E}\right)$ The necessary conditions under which there exists a solution of equation (E) which can be written in the form ${y}_{n+1}={\alpha }_{n}{u}_{n}+{\beta }_{n}{v}_{n}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\text{are}\phantom{\rule{4.0pt}{0ex}}\text{given.}$ Here $u$ and $v$ are two linearly independent solutions of equation ${\Delta }^{2}{y}_{n}={a}_{n+1}{y}_{n+1}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}\left(\underset{n\to \infty }{lim}{\alpha }_{n}=\alpha <\infty \phantom{\rule{1.0em}{0ex}}\mathrm{and}\phantom{\rule{1.0em}{0ex}}\underset{n\to \infty }{lim}{\beta }_{n}=\beta <\infty \right)\phantom{\rule{0.166667em}{0ex}}.$ A special case of equation (E) is also considered.

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