Global attractivity in a nonlinear difference equation

Abstract

In this paper, we study the asymptotic behavior of positive solutions of the nonlinear difference equation $$ x_{n+1}=x_n f(x_{n-k}), $$ where $f:[0,infty)o(0, infty)$ is a unimodal function, and $k$ is a nonnegative integer. Sufficient conditions for the positive equilibrium to be a global attractor of all positive solutions are established. Our results can be applied to to some difference equations derived from mathematical biology.

Keywords

• Nonlinear difference equation
• global attractor
• unimodal function
• positive equilibrium.