# Asymptotic behavior of solutions of nonlinear difference equations

Mathematica Bohemica (2004)

• Volume: 129, Issue: 4, page 349-359
• ISSN: 0862-7959

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## Abstract

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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, ${\varphi }_{n}\phantom{\rule{0.222222em}{0ex}}ℝ⟶ℝ$, $\left(\sigma \left(n\right)\right)$ 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.

## How to cite

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Migda, Janusz. "Asymptotic behavior of solutions of nonlinear difference equations." Mathematica Bohemica 129.4 (2004): 349-359. <http://eudml.org/doc/249394>.

@article{Migda2004,
abstract = {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.},
author = {Migda, Janusz},
journal = {Mathematica Bohemica},
keywords = {difference equation; asymptotic behavior; difference equation; asymptotic behavior},
language = {eng},
number = {4},
pages = {349-359},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behavior of solutions of nonlinear difference equations},
url = {http://eudml.org/doc/249394},
volume = {129},
year = {2004},
}

TY - JOUR
AU - Migda, Janusz
TI - Asymptotic behavior of solutions of nonlinear difference equations
JO - Mathematica Bohemica
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 129
IS - 4
SP - 349
EP - 359
AB - 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.
LA - eng
KW - difference equation; asymptotic behavior; difference equation; asymptotic behavior
UR - http://eudml.org/doc/249394
ER -

## References

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