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On the difference equation \(X_{n+1} = \alpha + \frac{x_{n-1}}{x_n}\). (English) Zbl 1155.39305

Summary: We study the behavior of the solutions of the difference equation \[ x_{n+1} = \alpha + \frac{x_{n-1}}{x_n}\quad n=0,1,\dots \] where \(\alpha \) is a negative number. Included are results which considerably improve and correct those in the recently published paper [A.E. Hamza, J. Math. Anal. Appl. 322, No. 2, 668–674 (2006; Zbl 1105.39008)]. We also refute Conjecture 2 in [G. Ladas et al., J. Difference Equ. Appl. 7, No. 3, 477–482 (2001; Zbl 1081.39503)].

MSC:

39A23 Periodic solutions of difference equations
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[1] Amleh, A. M.; Georgiou, D. A.; Grove, E. A.; Ladas, G., On the recursive sequence \(x_{n + 1} = \alpha + x_{n - 1} / x_n\), J. Math. Anal. Appl., 233, 790-798 (1999) · Zbl 0962.39004
[2] Berenhaut, K.; Stević, S., The difference equation \(x_{n + 1} = \alpha + \frac{x_{n - k}}{\sum_{i = 0}^{k - 1} c_i x_{n - i}}\) has solutions converging to zero, J. Math. Anal. Appl., 326, 1466-1471 (2007) · Zbl 1113.39003
[3] Berg, L., Asymptotische Darstellungen und Entwicklungen (1968), Dt. Verlag Wiss: Dt. Verlag Wiss Berlin · Zbl 0165.36901
[4] Berg, L., On the asymptotics of nonlinear difference equations, Z. Anal. Anwendungen, 21, 4, 1061-1074 (2002) · Zbl 1030.39006
[5] Berg, L., Inclusion theorems for non-linear difference equations with applications, J. Difference. Equ. Appl., 10, 4, 399-408 (2004) · Zbl 1056.39003
[6] Berg, L., Corrections to “Inclusion theorems for non-linear difference equations with applications”, from [5], J. Difference. Equ. Appl., 11, 2, 181-182 (2005) · Zbl 1080.39002
[7] Berg, L.; Wolfersdorf, L.v., On a class of generalized autoconvolution equations of the third kind, Z. Anal. Anwendungen, 24, 2, 217-250 (2005) · Zbl 1104.45001
[8] Camouzis, E.; DeVault, R., The forbidden set of \(x_{n + 1} = p + \frac{x_{n - 1}}{x_n} \), J. Difference. Equ. Appl., 9, 8, 739-750 (2003) · Zbl 1049.39024
[9] DeVault, R.; Kent, C.; Kosmala, W., On the recursive sequence \(x_{n + 1} = p + \frac{x_{n - k}}{x_n} \), J. Difference Equ. Appl., 9, 8, 721-730 (2003) · Zbl 1049.39026
[10] Fisher, M. E.; Goh, B. S., Stability results for delayed-recruitment models in population dynamics, J. Math. Biol., 19, 147-156 (1984) · Zbl 0533.92017
[11] Gutnik, L.; Stević, S., On the behaviour of the solutions of a second order difference equation, Discrete Dyn. Nat. Soc., 2007 (2007), pages 14. Article ID 27562 · Zbl 1180.39002
[12] Hamza, A. E., On the recursive sequence \(x_{n + 1} = \alpha + \frac{x_{n - 1}}{x_n} \), J. Math. Anal. Appl., 322, 668-674 (2006) · Zbl 1105.39008
[13] He, W. S.; Li, W. T.; Yan, X. X., Global attractivity of the difference equation \(x_{n + 1} = \alpha + \frac{x_{n - k}}{x_n} \), Appl. Math. Comput., 17, 163-167 (2004)
[14] Hoag, J. T., Monotonicity of solutions converging to a saddle point equilibrium, J. Math. Anal. Appl., 295, 10-14 (2004) · Zbl 1055.39010
[15] Hritonenko, N.; Rodkina, A.; Yatsenko, Y., Stability analysis of stochastic Ricker population model, Discrete Dyn. Nat. Soc., 2006 (2006), pages 13. Article ID 64590 · Zbl 1099.92071
[16] Iričanin, B., A global convergence result for a higher-order difference equation, Discrete Dyn. Nat. Soc., 2007 (2007), 7 pages. Article ID 91292 · Zbl 1180.39003
[17] Karakostas, G. L., Asymptotic 2-periodic difference equations with diagonally self-invertible responses, J. Difference. Equ. Appl., 6, 329-335 (2000) · Zbl 0963.39020
[18] Karakostas, G. L.; Stević, S., On the recursive sequence \(x_{n + 1} = B + \frac{x_{n - k}}{\alpha_0 x_n + \cdots + \alpha_{k - 1} x_{n - k + 1} + \gamma} \), J. Difference. Equ. Appl., 10, 9, 809-815 (2004) · Zbl 1068.39012
[19] Kent, C. M., Convergence of solutions in a nonhyperbolic case, Nonlinear Anal., 47, 4651-4665 (2001) · Zbl 1042.39507
[20] Kocic, V. L.; Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Application (1993), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0787.39001
[21] Kosmala, W.; Teixeira, C., More on the difference equation \(y_{n + 1} = (p + y_n) /(q y_n + y_{n - 1})\), Appl. Anal., 81, 143-151 (2003) · Zbl 1022.39005
[22] Kulenović, M. R.S.; Ladas, G., Dynamics of Second Order Rational Difference Equations (2002), Chapman & Hall: Chapman & Hall CRC · Zbl 0981.39011
[23] Ladas, G., Open problems and conjectures, J. Difference. Equ. Appl., 7, 2, 477-482 (2001) · Zbl 1081.39503
[24] Pielou, E. C., Population and Community Ecology (1974), Gordon and Breach · Zbl 0349.92024
[25] Stević, S., Asymptotic behaviour of a sequence defined by iteration, Mat. Vesnik, 48, 3-4, 99-105 (1996) · Zbl 1032.40002
[26] Stević, S., Asymptotic behaviour of a sequence defined by iteration with applications, Colloq. Math., 93, 2, 267-276 (2002) · Zbl 1029.39006
[27] Stević, S., Asymptotic behaviour of a sequence defined by a recurrence formula II, Austral. Math. Soc. Gaz., 29, 4, 209-215 (2002) · Zbl 1051.39013
[28] Stević, S., On the recursive sequence \(x_{n + 1} = x_{n - 1} / g(x_n)\), Taiwanese J. Math., 6, 3, 405-414 (2002) · Zbl 1019.39010
[29] Stević, S., Asymptotic behaviour of a nonlinear difference equation, Indian J. Pure Appl. Math., 34, 12, 1681-1687 (2003) · Zbl 1049.39012
[30] Stević, S., On the recursive sequence \(x_{n + 1} = \alpha_n + \frac{x_{n - 1}}{x_n} \), Int. J. Math. Sci., 2, 2, 237-243 (2004) · Zbl 1063.39010
[31] Stević, S., On the recursive sequence \(x_{n + 1} = \alpha_n + \frac{x_{n - 1}}{x_n}\) II, Dyn. Contin. Discrete Impuls. Syst, 10a, 6, 911-917 (2003) · Zbl 1051.39012
[32] Stević, S., Periodic character of a class of difference equation, J. Difference. Equ. Appl., 10, 6, 615-619 (2004) · Zbl 1054.39009
[33] Stević, S., Periodic character of a difference equation, Rostock. Math. Kolloq., 59, 3-10 (2004) · Zbl 1083.39011
[34] Stević, S., On the recursive sequence \(x_{n + 1} = \alpha + \frac{x_{n - 1}^p}{x_n^p} \), J. Appl. Math Comput., 18, 1-2, 229-234 (2005) · Zbl 1078.39013
[35] Stević, S., Global stability and asymptotics of some classes of rational difference equations, J. Math. Anal. Appl., 316, 60-68 (2006) · Zbl 1090.39009
[36] Stević, S., On positive solutions of a \((k + 1)\)-th order difference equation, Appl. Math. Lett., 19, 5, 427-431 (2006) · Zbl 1095.39010
[37] Stević, S., Existence of nontrivial solutions of a rational difference equation, Appl. Math. Lett., 20, 28-31 (2007) · Zbl 1131.39009
[38] Stević, S., Asymptotic behavior of a class of nonlinear difference equations, Discrete Dyn. Nat. Soc. (2006), 10 pages. Article ID 47156 · Zbl 1121.39006
[39] Stević, S., Asymptotics of some classes of higher order difference equations, Discrete Dyn. Nat. Soc., 2007 (2007), 20 pages. Article ID 56813 · Zbl 1180.39009
[40] Stević, S., On the recursive sequence \(x_n = 1 + \frac{\sum_{i = 1}^k \alpha_i x_{n - p_i}}{\sum_{j = 1}^m \beta_j x_{n - q_j}} \), Discrete Dyn. Nat. Soc., 2007 (2007), 7pages. Article ID 39404 · Zbl 1180.39006
[41] Stević, S., Asymptotic periodicity of a higher order difference equation, Discrete Dyn. Nat. Soc., 2007 (2007), 9 pages. Article ID 13737 · Zbl 1152.39011
[42] Sun, T.; Xi, H.; Wu, H., On boundedness of the solutions of the difference equation \(x_{n + 1} = x_{n - 1} /(p + x_n)\), Discrete Dyn. Nat. Soc., 2006 (2006), 7 pages. Article ID 20652 · Zbl 1149.39301
[43] Voulov, H. D., Existence of monotone solutions of some difference equations with unstable equilibrium, J. Math. Anal. Appl., 272, 2, 555-564 (2002) · Zbl 1010.39002
[44] Yan, X. X.; Li, W. T.; Zhao, Z., On the recursive sequence \(x_{n + 1} = \alpha -(x_n / x_{n - 1})\), J. Appl. Math. Comput., 17, 1, 269-282 (2005) · Zbl 1068.39030
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