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Asymptotic behavior of solutions of nonlinear difference equations. (English) Zbl 1080.39501
Summary: The nonlinear difference equation $x_{n+1}-x_n=a_n\varphi _n(x_{\sigma (n)})+b_n, \tag{$$\text{E}$$}$ where $$(a_n), (b_n)$$ are real sequences, $$\varphi _n\: \mathbb R\rightarrow \mathbb R$$, $$(\sigma (n))$$ is a sequence of integers and $$\displaystyle\lim _{n\rightarrow \infty }\sigma (n)=\infty$$, 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 (E) convergent to this constant are also obtained.