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Global asymptotic stability of a higher order rational difference equation. (English) Zbl 1121.39008
The authors consider the following nonlinear difference equation $$x_{n+1}= {f(x_{n-r_1},\dots, x_{n-r_k}) g(x_{n-m_1},\dots, x_{n-m_l})+ 1\over f(x_{n-r_1},\dots, x_{n-r_k})+ g(x_{n-m_1},\dots x_{n-m_l})},\quad n= 0,1,\dots,$$ where $f\in C(\bbfR^k_+, \bbfR_+)$, $g\in C(\bbfR^l_+, \bbfR_+)$ with $k,l\in \{1,2,\dots\}$, $0\le r_k\le\cdots\le r_k$ and $0\le m_1\le\cdots\le m_l$, and the initial values are positive real numbers. The main result of this note gives sufficient conditions under which the unique equilibrium $x= 1$ of the above equation is globally asymptotically stable.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Generalized difference equations
##### Keywords:
nonlinear difference equations
Full Text:
##### References:
 [1] Ei-Owaidy, H. M.; Ahmed, A. M.; Mousa, M. S.: On the recursive sequences xn+1= - $\alpha$xn - $1\beta \pm$xn. Appl. math. Comput. 145, 747-753 (2003) · Zbl 1034.39004 [2] Cinar, C.: On the positive solutions of the difference equation xn+1=axn - 11+bxnxn - 1. Appl. math. Comput. 156, 587-590 (2004) [3] Ladas, G.: Open problems and conjectures. J. difference equ. Appl. 4, 497-499 (1998) [4] Nesemann, T.: Positive nonlinear difference equations: some results and applications. Nonlinear anal. 47, 4707-4717 (2001) · Zbl 1042.39510 [5] Papaschinopoulos, G.; Schinas, C. J.: Global asymptotic stability and oscillation of a family of difference equations. J. math. Anal. appl. 294, 614-620 (2004) · Zbl 1055.39017 [6] Li, X.: Global asymptotic stability in a rational equation. J. difference equ. Appl. 9, 833-839 (2003) · Zbl 1055.39014 [7] Li, X.: Global behavior for a fourth-order rational difference equation. J. math. Anal. appl. 312, 555-563 (2005) · Zbl 1083.39007 [8] Li, X.: Qualitative properties for a fourth-order rational difference equation. J. math. Anal. appl. 311, 103-111 (2005) · Zbl 1082.39004 [9] Berenhaut, K. S.; Stevic, S.: The global attractivity of a higher order rational difference equation. J. math. Anal. appl. 326, No. 2, 940-944 (2007) · Zbl 1112.39002