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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

${x}_{n+1}=\sum _{i=1}^{l}{A}_{{s}_{i}}{x}_{n-{s}_{i}}/\left(B+C\prod _{j=1}^{k}{x}_{n-{t}_{j}}\right)+D{x}_{n},\phantom{\rule{1.em}{0ex}}n=0,1,...,$

where the initial data ${x}_{-m},{x}_{-m+1},...,{x}_{-1},{x}_{0}\in {ℝ}^{+}$, $m=max\left\{{s}_{1},...,{s}_{l},{t}_{1},...,{t}_{k}\right\}$, ${s}_{1},...,{s}_{l}$, ${t}_{1},...,{t}_{k}$ 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 $\overline{x}=0$ 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:
 39A22 Growth, boundedness, comparison of solutions (difference equations)
##### References:
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