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Global attractivity of a higher-order nonlinear difference equation. (English) Zbl 1201.39006

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

MSC:

39A20 Multiplicative and other generalized difference equations
39A30 Stability theory for difference equations
39A22 Growth, boundedness, comparison of solutions to difference equations
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[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
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