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On the recursive sequence \(x_{n+1} = \max \left\{ c, \frac {x_n^p}{x_{n-1}^p} \right\}\). (English) Zbl 1152.39012
This work studies the boundedness and global attractivity for the posivitive solutions of the difference equation \(x_{n+1}= \max\{c,{x^p_n\over x^p_{n-1}}\}\), \(n\in\mathbb{N}_0\) with \(p,c\in(0,+\infty)\).
It is shown that: (a) there exist unbounded solutions whenever \(p\geq 4\); (b) all positive solutions are bounded when \(p\in(0, 4)\); (c) every positive solution is eventually equal to 1 when \(p\in(0, 4)\) and \(c\geq 1\); (d) all positive solutions converge to 1 whenever \(p,c\in(0,1)\).

MSC:
39A11 Stability of difference equations (MSC2000)
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[1] Amleh, A.M.; Hoag, J.; Ladas, G., A difference equation with eventually periodic solutions, Comput. math. appl., 36, 10-12, 401-404, (1998) · Zbl 0933.39030
[2] Cinar, C.; Stević, S.; Yalçinkaya, I., On positive solutions of a reciprocal difference equation with minimum, J. appl. math. comput., 17, 1-2, 307-314, (2005) · Zbl 1074.39002
[3] Feuer, J., On the eventual periodicity of \(x_{n + 1} = \max \{\frac{1}{x_n}, \frac{A_n}{x_{n - 1}} \}\) with a period-four parameter, J. difference equ. appl., 12, 5, 467-486, (2006) · Zbl 1095.39016
[4] Kent, C.M.; Kustesky, M.; Nguyen, A.Q.; Nguyen, B.V., Eventually periodic solutions of \(x_{n + 1} = \max \{A_n / x_n, B_n / x_{n - 1} \}\) when the parameters are two cycles, Dyn. contin. discrete impuls. syst. ser. A math. anal., 10, 1-3, 33-49, (2003) · Zbl 1038.39006
[5] Kent, C.M.; Radin, M.A., On the boundedness nature of positive solutions of the difference equation \(x_{n + 1} = \max \{A_n / x_n, B_n / x_{n - 1} \}\) with periodic parameters, Dyn. contin. discrete impuls. syst. ser. B appl. algorithms, Suppl., 11-15, (2003)
[6] Ladas, G., Open problems and conjectures, J. difference equ. appl., 2, 339-341, (1996)
[7] Mishev, D.P.; Patula, W.T.; Voulov, H.D., A reciprocal difference equation with maximum, Comput. math. appl., 43, 1021-1026, (2002) · Zbl 1050.39015
[8] Mishkis, A.D., On some problems of the theory of differential equations with deviating argument, Uspekhi mat. nauk, 32:2, 194, 173-202, (1977)
[9] E.P. Popov, Automatic Regulation and Control, Moscow, 1966 (in Russian)
[10] Stević, S., On the recursive sequence \(x_{n + 1} = \frac{A}{\prod_{i = 0}^k x_{n - i}} + \frac{1}{\prod_{j = k + 2}^{2(k + 1)} x_{n - j}}\), Taiwanese J. math., 7, 2, 249-259, (2003)
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