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 $${x}_{n+1}-{x}_{n}={a}_{n}{\varphi}_{n}\left({x}_{\sigma \left(n\right)}\right)+{b}_{n},\phantom{\rule{2.0em}{0ex}}\left(\text{E}\right)$$
where $\left({a}_{n}\right),\left({b}_{n}\right)$ are real sequences, ${\varphi}_{n}\phantom{\rule{0.222222em}{0ex}}\mathbb{R}\u27f6\mathbb{R}$, $\left(\sigma \right(n\left)\right)$ is a sequence of integers and ${lim}_{n\u27f6\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.

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