Sixth Mississippi State Conference on Differential Equations and Computational Simulations.
Electron. J. Diff. Eqns., Conference 15 (2007), pp. 229-238.

Global attractivity in a nonlinear difference equation

Chuanxi Qian, Yijun Sun

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 $f:[0,\infty)\to(0, \infty)$ 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.

Published February 28, 2007.
Math Subject Classifications: 39A10.
Key Words: Nonlinear difference equation; global attractor; unimodal function; positive equilibrium.

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Chuanxi Qian
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA
email: qian@math.msstate.edu
Yijun Sun
Department of Mathematics and Statistics
Mississippi State University
Mississippi State, MS 39762, USA
email: ys101@msstate.edu

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