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On the asymptotic stability of x n+1 =(a+x n x n-k )/(x n +x n-k ). (English) Zbl 1155.39300

Summary: We prove that the equilibrium solution of the rational difference equation

x n+1 =a+x n x n-k x n +x n-k n=0,1,2,

where k is a nonnegative integer, a0, and x -k ,,x 0 >0, is globally asymptotically stable.

MSC:
39A11Stability of difference equations (MSC2000)
References:
[1]Sedaghat, H.: The impossibility of unstable, globally attracting fixed points for continuous mappings of the line, Amer. math. Monthly 104, 356-358 (1997) · Zbl 0876.58027 · doi:10.2307/2974585
[2]Xianyi, L.; Deming, Z.: Global asymptotic stability in a rational difference equation, J. difference eqs. Appl. 9, No. 9, 833-839 (2003) · Zbl 1055.39014 · doi:10.1080/1023619031000071303
[3]Xianyi, L.; Deming, Z.: Two rational recursive sequences, Comput. math. Appl. 47, No. 10–11, 1487-1494 (2004) · Zbl 1072.39008 · doi:10.1016/j.camwa.2004.06.001
[4]Kocic, V.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications, (1993)