Çinar, Cengiz; Stević, Stevo; Yalçinkaya, Ibrahim A note on global asymptotic stability of a family of rational equations. (English) Zbl 1083.39003 Rostocker Math. Kolloq. 59, 41-49 (2005). The authors prove that all positive solutions of the difference equations \[ x(n+1)= \frac{1+x_n\sum_{i=1}^kx_{n-i}}{x_n+x_{n-1}+x_n\sum_{i=2}^kx_{n-i}}, \;\;n=0,1,\dots, \] where \(k\in N\), converge to the positive equilibrium \(\overline{x}=1\). The result generalizes the main theorem in the paper by X. Li and D. Zhu [J. Difference Equ. Appl. 9, No. 9, 833–839 (2003; Zbl 1055.39014)]. The authors present a very short proof of the theorem. At the same time, the authors find the asymptotics of some of the positive solutions. Reviewer: Wei Nian Li (Binzhou) Cited in 1 ReviewCited in 7 Documents MSC: 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations Keywords:rational difference equation; global asymptotic stability; equilibrium point; positive solution; asymptotics Citations:Zbl 1055.39014 PDFBibTeX XMLCite \textit{C. Çinar} et al., Rostocker Math. Kolloq. 59, 41--49 (2005; Zbl 1083.39003)