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difference equation; asymptotic behavior
The nonlinear difference equation \[ x_{n+1}-x_n=a_n\varphi _n(x_{\sigma (n)})+b_n, \qquad \mathrm{(\text{E})}\] where $(a_n), (b_n)$ are real sequences, $\varphi _n\: \mathbb{R}\longrightarrow \mathbb{R}$, $(\sigma (n))$ is a sequence of integers and $\lim _{n\longrightarrow \infty }\sigma (n)=\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.
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