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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.

In this note we consider the third order linear difference equations of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+\delta q\left(n\right)x\left(\tau \left(n\right)\right)=0,n\in N\left(n_0\right),\left(\mathrm{E}\right)$$ where $\delta =\pm 1$, $p,qN\left(n_0\right)\to {ℝ}_{+};$ $\sigma ,\tau N\left(n_0\right)\to \mathbb{N}$, ${lim}_{n\to \infty }\sigma \left(n\right)=\underset{n\to \infty }{lim}\tau \left(n\right)=\infty .$ We examine the following two cases: $$\begin{array}{ccc}\hfill \{0<p(n)& \le 1,\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l\},\{p\left(n\right)\hfill & \hfill >1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l\},\end{array}$$ where $k$, $l$ are positive integers and we obtain sufficient conditions under which all solutions of the above equations are oscillatory.

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.