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Asymptotic properties of solutions of the difference equation of the form $${\Delta }^m{x}_n=a_n\varphi (x_{{\tau }_1\left(n\right)},\cdots ,x_{{\tau }_k\left(n\right)})+b_n$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta }^m{y}_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained.

We show that if Y is the Hausdorffization of the primitive spectrum of a $C^{*}$-algebra $A$ then $A$ is $*$-isomorphic to the $C^{*}$-algebra of sections vanishing at infinity of the canonical $C^{*}$-bundle over $Y$.

The nonlinear difference equation $$x_{n+1}-x_n=a_n{\varphi }_n\left(x_{\sigma \left(n\right)}\right)+b_n,\left(\text{E}\right)$$ where $\left(a_n\right),\left(b_n\right)$ are real sequences, ${\varphi }_nℝ⟶ℝ$, $\left(\sigma \right(n\left)\right)$ is a sequence of integers and ${lim}_{n⟶\infty }\sigma \left(n\right)=\infty$, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation $y_{n+1}-y_n=b_n$ are given. Sufficient conditions under which for every real constant there exists a solution of equation () convergent to this constant are also obtained.

Asymptotic properties of solutions of difference equation of the form $${\Delta }^m{x}_n=a_n{\varphi }_n\left(x_{\sigma \left(n\right)}\right)+b_n$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta }^m{y}_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained. Moreover, the conditions under which every polynomial sequence of degree less than $m$ is asymptotically equivalent to some solution of the equation and every solution is asymptotically polynomial are obtained. The consequences of the existence of asymptotically...

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.