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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...

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 $Y$.

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.

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 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.