Stević, Stevo On the recursive sequence \(x_{n+1} = \max \left\{ c, \frac {x_n^p}{x_{n-1}^p} \right\}\). (English) Zbl 1152.39012 Appl. Math. Lett. 21, No. 8, 791-796 (2008). 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 \(p,c\in(0,1)\). Reviewer: Stefan Balint (Timişoara) Cited in 81 Documents MSC: 39A11 Stability of difference equations (MSC2000) Keywords:max type difference equations; boundedness; difference equation; global attractivity PDF BibTeX XML Cite \textit{S. Stević}, Appl. Math. Lett. 21, No. 8, 791--796 (2008; Zbl 1152.39012) Full Text: DOI OpenURL References: [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) 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.