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Asymptotic behavior of equilibrium point for a family of rational difference equations. (English) Zbl 1204.39011

Summary: This paper is concerned with the following nonlinear difference equation

x n+1 = i=1 l A s i x n-s i /(B+C j=1 k x n-t j )+Dx n ,n=0,1,...,

where the initial data x -m ,x -m+1 ,...,x -1 ,x 0 + , m=max{s 1 ,...,s l ,t 1 ,...,t k }, s 1 ,...,s l , t 1 ,...,t k are nonnegative integers, and A s i ,B,C and D are arbitrary positive real numbers. We give sufficient conditions under which the unique equilibrium x ¯=0 of this equation is globally asymptotically stable, which extends and includes corresponding results obtained in the work of C. Çinar [Appl. Math. Comput. 150, No. 1, 21–24 (2004; Zbl 1050.39005)], X. Yang et al. [Appl. Math. Comput. 162, No. 3, 1485–1497 (2005; Zbl 1068.39031)], and K. S. Berenhaut et al. [Appl. Math. Lett. 20, No. 1, 54–58 (2007; Zbl 1131.39006)]. In addition, some numerical simulations are also shown to support our analytic results.

MSC:
39A22Growth, boundedness, comparison of solutions (difference equations)
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