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On the difference equation $x_{n+1}= \frac{px_{n-s} + x_{n-t}}{qx_{n-s} + x_{n-t}}$. (English) Zbl 1162.39015
The paper deals with the difference equation $$x_{n+1} = \frac{px_{n-s} + x_{n-t}}{qx_{n-s} + x_{n-t}}, \ n = 0,1,\dots \tag1$$ with positive initial conditions where $s, t$ are distinct nonnegative integers, $p > 0, q >0, p \not= q.$ The authors prove that the positive equilibrium of eq. (1) is globally asymptotically stable if one of the following two conditions is satisfied: (H1) Either $p > q \ge 1,$ or $1 \ge p > q,$ or $(1 + 3q) / (1 - q) \ge p > 1 > q.$ (H2) Either $q > p \ge 1,$ or $1 \ge q > p,$ or $(1 + 3p) / (1 - p) \ge q > 1 > p.$

39A11Stability of difference equations (MSC2000)
39A20Generalized difference equations
Full Text: DOI
[1] Dehghan, M.; Douraki, M. J.; Razzaghi, M.: Global stability of a higher order rational recursive sequence. Applied mathematics and computation 179, 161-174 (2006) · Zbl 1105.39004
[2] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S. W.: Global behavior of yn+1=(p+yn - k)/(qyn+yn - k). Nonlinear analysis 47, 4743-4751 (2001) · Zbl 1042.39523
[3] Graef, J. R.; Qian, C.; Spikes, P. W.: Stability in a population model. Applied mathematics and computation 89, 119-132 (1998) · Zbl 0904.39005
[4] Kocic, V. L.; Ladas, G.: Global behavior of nonlinear difference equations of higher order with applications. (1993) · Zbl 0787.39001
[5] Kruse, N.; Nesemann, T.: Global asymptotic stability in some discrete dynamical systems. Journal of mathematical analysis and applications 235, 151-158 (1999) · Zbl 0933.37016
[6] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: A rational difference equation. Computers and mathematics with applications 41, 671-687 (2001)
[7] Kulenovic, M. R. S.; Ladas, G.: Dynamics of second order rational difference equations with open problems and conjectures. (2002)
[8] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: The dynamics of $xn+1=\alpha +\beta $xnA+Bxn+Cxn-1 facts and conjectures. Computers and mathematics with applications 45, 1087-1099 (2003)
[9] Li, X.; Zhu, D.: Global asymptotic stability for two recursive difference equations. Applied mathematics and computation 150, 481-492 (2004) · Zbl 1044.39006
[10] Milton, G.; Belair, J.: Chaos, noise and extinction in models of population growth. Theoretical population biology 37, 273-290 (1990) · Zbl 0699.92020
[11] Saleh, M.; Abu-Baha, S.: Dynamics of a higher order rational difference equation. Applied mathematics and computation 181, 84-102 (2006) · Zbl 1158.39301
[12] Utida, S.: Population fluctuation an experimental and theoretical approach. Cold spring harbor symposium on quantitative biology 22, 139-151 (1957)
[13] Yang, X.: On the global asymptotic stability of the difference equation xn=(xn - 1xn - 2+xn - 3+a)/(xn - 1+xn - 2xn - 3+a). Applied mathematics and computation 171, 857-861 (2005)
[14] Yang, X.: Global asymptotic stability in a class of generalized Putnam equations. Journal of mathematical analysis and applications 322, 693-698 (2006) · Zbl 1104.39012
[15] Yang, X.; Chen, B.; Megson, G. M.; Evans, D. J.: Global attractivity in a recursive sequence. Applied mathematics and computation 158, 667-682 (2004) · Zbl 1063.39011
[16] Yang, X.; Lai, H.; Evans, D. J.; Megson, G. M.: Global asymptotic stability in a rational recursive sequence in a recursive sequence. Applied mathematics and computation 158, 703-716 (2004) · Zbl 1071.39018
[17] Yang, X.; Su, W.; Chen, B.; Megson, G. M.; Evans, D. J.: On the recursive sequence xn=(axn - 1+bxn - 2)/(c+dxn - 1xn - 2). Applied mathematics and computation 162, 1485-1497 (2005) · Zbl 1068.39031
[18] Yang, X.; Yang, Y.; Luo, J.: On the difference equation xn=(p+xn - s)/(qxn - t+xn - s). Applied mathematics and computation 189, 918-926 (2007) · Zbl 1128.39016