Jia, Xiu-Mei; Hu, Lin-Xia Global attractivity of a higher-order nonlinear difference equation. (English) Zbl 1201.39006 Appl. Math. Comput. 216, No. 3, 857-861 (2010). Authors’ abstract: The main goal of this paper is to investigate the locally asymptotically stable, period-two solutions, invariant intervals and global attractivity of all negative solutions of the nonlinear difference equation \[ x_{n+1} = \frac{1-x_n}{A+x_{n-k}}, \quad n=0,1,\dots , \] where \(A\in (-\infty ,-1),k\) is a positive integer and initial conditions \(x_{-k},\dots , x_0\in (-\infty ,0]\). It is shown that the unique negative equilibrium of the equation is a global attractor with a basin that depends on certain conditions of the coefficient Reviewer: Miloš Čanak (Beograd) MSC: 39A20 Multiplicative and other generalized difference equations 39A30 Stability theory for difference equations 39A22 Growth, boundedness, comparison of solutions to difference equations Keywords:period-two solution; invariant interval; global attractor; local asymptotic stability; rational difference equation; negative solutions PDF BibTeX XML Cite \textit{X.-M. Jia} and \textit{L.-X. Hu}, Appl. Math. Comput. 216, No. 3, 857--861 (2010; Zbl 1201.39006) Full Text: DOI OpenURL References: [1] Cunningham, K.C.; Kulenovic, M.R.S.; Ladas, G.; Valicenti, S.V., On the recursive sequence \(x_{n + 1} = (\alpha + \beta x_n) /(\mathit{Bx}_n + \mathit{Cx}_{n - 1})\), Nonlinear. anal. TMA, 47, 4603-4614, (2001) · Zbl 1042.39522 [2] Hamza, A.E., On the recursive sequence \(x_n = \alpha + x_{n - 1} / x_n\), J. math. anal. appl., 322, 668-674, (2006) · Zbl 1105.39008 [3] He, W.S.; Hu, L.X.; Li, W.T., Global attractivity in a higher order nonlinear difference equation, Pure appl. math., 20, 213-218, (2004) · Zbl 1125.39303 [4] Hu, L.X.; Li, W.T.; Stević, S., Global asymptotic stability of a second order rational difference equation, J. difference equ. appl., 8, 779-797, (2008) · Zbl 1153.39015 [5] Hu, L.X.; Li, W.T., Global asymptotic stability of a second order rational difference equation, Comput. math. appl., 54, 1260-1266, (2007) · Zbl 1148.39004 [6] Kocic, V.L.; Ladas, G., Global behavior of nonlinear difference equations of higher order with application, (1993), Kiuwer Academic Publishers Dordrecht · Zbl 0787.39001 [7] Kulenonvić, M.R.S.; Ladas, G., Dynamics of second order rational difference equations with open problem and conjectures, (2002), Chapman & Hall/CRC Boca Raton [8] Li, W.T.; Zhang, Y.H.; Su, Y.H., Global attractivity in a class of higher-order nonlinear difference equation, Acta math. sci., 25, 59-66, (2005) · Zbl 1168.39301 [9] Li, W.T.; Sun, H.R., Global attractiveness in a rational recursive equation, Dyn. syst. appl., 11, 339-345, (2002) [10] Stević, S., On the difference equation \(x_{n + 1} = \alpha + x_{n - 1} / x_n\), Comput. math. appl., 56, 1159-1171, (2008) · Zbl 1155.39305 [11] Su, Y.H.; Li, W.T.; stević, S., Dynamics of a higher order nonlinear rational difference equation, J. difference equ. appl., 11, 133-150, (2005) · Zbl 1071.39017 [12] Su, Y.H.; Li, W.T., Global attractivity of a higher order nonlinear difference equation, J. difference equ. appl., 11, 947-958, (2005) · Zbl 1081.39005 [13] Yan, X.X.; Li, W.T.; Sun, H.R., Global attractivity in a higher order nonlinear difference equation, Appl. math. E-notes, 2, 51-58, (2002) · Zbl 1004.39010 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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.