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

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

### Asymptotic properties of the solutions of the second order difference equation

Archivum Mathematicum

Asymptotic properties of the solutions of the second order nonlinear difference equation (with perturbed arguments) of the form ${\Delta }^{2}{x}_{n}={a}_{n}\varphi \left({x}_{n+k}\right)$ are studied.

### On unstable neutral difference equations with maxima''

Mathematica Slovaca

### 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.

### On the oscillation of solutions of third order linear difference equations of neutral type

Mathematica Bohemica

In this note we consider the third order linear difference equations of neutral type ${\Delta }^{3}\left[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)\right]+\delta q\left(n\right)x\left(\tau \left(n\right)\right)=0,\phantom{\rule{1.0em}{0ex}}n\in N\left({n}_{0}\right),\phantom{\rule{2.0em}{0ex}}\left(\mathrm{E}\right)$ where $\delta =±1$, $p,q\phantom{\rule{0.222222em}{0ex}}N\left({n}_{0}\right)\to {ℝ}_{+};$ $\sigma ,\tau \phantom{\rule{0.222222em}{0ex}}N\left({n}_{0}\right)\to ℕ$, ${lim}_{n\to \infty }\sigma \left(n\right)=\underset{n\to \infty }{lim}\tau \left(n\right)=\infty .$ We examine the following two cases: $\begin{array}{ccc}\hfill \left\{01,\phantom{\rule{4pt}{0ex}}\sigma \left(n\right)=n-k,\phantom{\rule{4pt}{0ex}}\tau \left(n\right)=n-l\right\},\end{array}$ where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

### Existence of nonoscillatory solutions of some higher order difference equations.

Applied Mathematics E-Notes [electronic only]

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