# Existence and global attractivity of periodic solutions in a higher order difference equation

Archivum Mathematicum (2018)

• Volume: 054, Issue: 2, page 91-110
• ISSN: 0044-8753

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## Abstract

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Consider the following higher order difference equation $x\left(n+1\right)=f\left(n,x\left(n\right)\right)+g\left(n,x\left(n-k\right)\right)\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{1.0em}{0ex}}n=0,1,\cdots$ where $f\left(n,x\right)$ and $g\left(n,x\right):\left\{0,1,\cdots \right\}×\left[0,\infty \right)\to \left[0,\infty \right)$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\left\{\stackrel{˜}{x}\left(n\right)\right\}$ under certain conditions, and then establish a sufficient condition for $\left\{\stackrel{˜}{x}\left(n\right)\right\}$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given.

## How to cite

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Qian, Chuanxi, and Smith, Justin. "Existence and global attractivity of periodic solutions in a higher order difference equation." Archivum Mathematicum 054.2 (2018): 91-110. <http://eudml.org/doc/294193>.

@article{Qian2018,
abstract = {Consider the following higher order difference equation \begin\{equation*\} x(n+1)= f\big (n,x(n)\big )+ g\big (n, x(n-k)\big )\,, \quad n=0, 1, \dots \end\{equation*\} where $f(n,x)$ and $g(n,x)\colon \lbrace 0, 1, \dots \rbrace \times [0, \infty ) \rightarrow [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\lbrace \tilde\{x\}(n)\rbrace$ under certain conditions, and then establish a sufficient condition for $\lbrace \tilde\{x\}(n)\rbrace$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given.},
author = {Qian, Chuanxi, Smith, Justin},
journal = {Archivum Mathematicum},
keywords = {higher order difference equation; periodic solution; global attractivity; Riccati difference equation; population model},
language = {eng},
number = {2},
pages = {91-110},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Existence and global attractivity of periodic solutions in a higher order difference equation},
url = {http://eudml.org/doc/294193},
volume = {054},
year = {2018},
}

TY - JOUR
AU - Qian, Chuanxi
AU - Smith, Justin
TI - Existence and global attractivity of periodic solutions in a higher order difference equation
JO - Archivum Mathematicum
PY - 2018
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 054
IS - 2
SP - 91
EP - 110
AB - Consider the following higher order difference equation \begin{equation*} x(n+1)= f\big (n,x(n)\big )+ g\big (n, x(n-k)\big )\,, \quad n=0, 1, \dots \end{equation*} where $f(n,x)$ and $g(n,x)\colon \lbrace 0, 1, \dots \rbrace \times [0, \infty ) \rightarrow [0,\infty )$ are continuous functions in $x$ and periodic functions in $n$ with period $p$, and $k$ is a nonnegative integer. We show the existence of a periodic solution $\lbrace \tilde{x}(n)\rbrace$ under certain conditions, and then establish a sufficient condition for $\lbrace \tilde{x}(n)\rbrace$ to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given.
LA - eng
KW - higher order difference equation; periodic solution; global attractivity; Riccati difference equation; population model
UR - http://eudml.org/doc/294193
ER -

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