On the global behavior of the nonlinear difference equation \(x_{n+1}=f(p_{n},x_{n - m},x_{n - t(k+1)+1})\). (English) Zbl 1118.39004

Sun, Taixiang; Xi, Hongjian

Discrete Dyn. Nat. Soc. 2006, No. 3, Article ID 90625, 12 p. (2006).

Let \(k\) and \(t\) be natural numbers, \(m\) be a nonnegative integer such that \(0\leq m<t(k+1)-1\), and \(p_n\) be a positive sequence of period \(k+1\). The authors establish sufficient conditions under which every positive solution of the nonlinear difference equation \[ x_{n+1}=f\left(p_n,x_{n-m},x_{n-t(k+1)}\right),\quad n=0,1,2,\dots \] converges to a period \(k+1\) solution.

Reviewer: Raghib Abu-Saris (Sharjah)

MSC:

39A11

Stability of difference equations (MSC2000)

Keywords:

periodic solution, convergence; positive solution