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

Qian, Chuanxi ; Smith, Justin

Archivum mathematicum, Tome 54 (2018) no. 2, pp. 91-110.

Résumé

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.

Détail

DOI : 10.5817/AM2018-2-91

Classification : 39A10, 92D25
Keywords: higher order difference equation; periodic solution; global attractivity; Riccati difference equation; population model