Sun, Taixiang; Xi, Hongjian; Han, Caihong Stability of solutions for a family of nonlinear difference equations. (English) Zbl 1146.39020 Adv. Difference Equ. 2008, Article ID 238068, 6 p. (2008). Summary: We consider the family of nonlinear difference equations:\[ \begin{split} x_{n+1}=\left.\left(\sum^3_{i=1}f_i(x_n,\dots,x_{n-k})+ f_4(x_n,\dots,x_{n-k})f_5(x_n,\dots,x_{n-k})\right)\right/ \\ \left(f_1(x_n,\dots,x_{n-k})f_2(x_n,\dots,x_{n-k})+ \sum^5_{i=3}f_i(x_n,\dots,x_{n-k})\right),\quad n=0,1,\dots,\end{split} \]where \(f_i\in C((0,+\infty)^{k+1},(0,+\infty))\), for \(i\in \{1,2,4,5\}\), \(f_3\in C([0,+\infty)^{k+1}\), \((0,+\infty))\), \(k\in \{1,2,\dots\}\) and the initial values \(x_{-k},x_{-k+1},\dots,x_0\in (0,+\infty)\). We give sufficient conditions under which the unique equilibrium \(x=1\) of these equations is globally asymptotically stable, which extends and includes corresponding results obtained in the cited references. Cited in 3 Documents MSC: 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations Keywords:rational difference equation; global asymptotic stability; discrete dynamical system; Putnam difference equation; nonlinear difference equations; unique equilibrium PDFBibTeX XMLCite \textit{T. Sun} et al., Adv. Difference Equ. 2008, Article ID 238068, 6 p. (2008; Zbl 1146.39020) Full Text: DOI EuDML References: [1] doi:10.1016/j.jmaa.2004.02.039 · Zbl 1055.39017 · doi:10.1016/j.jmaa.2004.02.039 [2] doi:10.1006/jmaa.1999.6384 · Zbl 0933.37016 · doi:10.1006/jmaa.1999.6384 [3] doi:10.1016/j.jmaa.2006.02.087 · Zbl 1112.39002 · doi:10.1016/j.jmaa.2006.02.087 [4] doi:10.1016/j.aml.2006.02.022 · Zbl 1131.39006 · doi:10.1016/j.aml.2006.02.022 [5] doi:10.1016/j.aml.2005.09.009 · Zbl 1119.39004 · doi:10.1016/j.aml.2005.09.009 [6] doi:10.1016/j.aml.2005.10.014 · Zbl 1117.39005 · doi:10.1016/j.aml.2005.10.014 [7] doi:10.1080/1023619031000071303 · Zbl 1055.39014 · doi:10.1080/1023619031000071303 [8] doi:10.1016/j.jmaa.2005.03.097 · Zbl 1083.39007 · doi:10.1016/j.jmaa.2005.03.097 [9] doi:10.1016/j.jmaa.2005.02.063 · Zbl 1082.39004 · doi:10.1016/j.jmaa.2005.02.063 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.