# 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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topQian, 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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