# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the asymptotic behavior and periodic nature of a difference equation with maximum. (English) Zbl 1189.39020

Summary: We investigate the asymptotic behavior and periodic nature of positive solutions of the difference equation

${x}_{n}=max\left\{\frac{A}{{x}_{n-1}},\frac{1}{{x}_{n-2}^{\alpha }}\right\},\phantom{\rule{3.33333pt}{0ex}}n\ge 0,$

where $A\ge 0$ and $0\le \alpha \le 1$. We prove that every positive solution of this difference equation approaches $\overline{x}=1$ or is eventually periodic with a period $2,3$ or 4.

##### MSC:
 39A23 Periodic solutions (difference equations) 39A22 Growth, boundedness, comparison of solutions (difference equations)
##### References:
 [1] Mishkis, A. D.: On some problems of the theory ofdifferential equations with deviating argument, Uspekhi mat. Nauk 32, No. 2, 173-202 (1977) · Zbl 0378.34052 · doi:doi:10.1070/RM1977v032n02ABEH001623 [2] Popov, E. P.: Automatic regulation and control, (1966) [3] Abu-Saris, R. M.; Allan, F. M.: Periodic and nonperiodic solutions of the difference equation xn+1=max$xn2,A}/xnxn-1$,,9-17\left(1997\right)$ · Zbl 0890.39012 [4] Abu-Saris, R.; Allan, F.: Rational recursive sequences involving the maximum function, Far east J. Math. 1, No. 3, 335-342 (1999) · Zbl 0932.39011 [5] Amleh, A. M.; Hoag, J.; Ladas, G.: A difference equation with eventually periodic solutions, Comput. math. Appl. 36, 401-404 (1998) · Zbl 0933.39030 · doi:doi:10.1016/S0898-1221(98)80040-0 [6] Berenhaut, K.; Foley, J.; Stevic, S.: Boundedness character of positive solutions of a MAX difference equation, J. difference equ. Appl. 12, No. 12, 1193-1199 (2006) · Zbl 1116.39001 · doi:doi:10.1080/10236190600949766 [7] Briden, W. J.; Grove, E. A.; Kent, C. M.; Ladas, G.: Eventually periodic solutions of xn+1=max{1/xn,An/xn-1}, Commun. appl. Nonlinear anal. 6, 31-34 (1999) [8] Briden, W. J.; Grove, E. A.; Ladas, G.; Mcgrath, L. C.: On the nonautonomous equation xn+1=max{An/xn,Bn/xn-1}, , 49-73 (1999) · Zbl 0938.39012 [9] Çinar, C.; Stević, S.; Yalçınkaya, I.: On positive solutions of a reciprocal difference equation with minimum, J. appl. Math. comput. 17, No. 1-2, 307-314 (2005) · Zbl 1074.39002 · doi:doi:10.1007/BF02936057 [10] Elsayed, E. M.; Stević, S.: On the MAX type equation, xn+1=max{A/xn,xn-2}, Nonlinear anal. TMA 71, No. 3–4, 910-922 (2009) [11] Feuer, J.: On the eventual periodicity of xn+1=max{1/xn,An/xn-1} with a period-four parameter, J. difference equ. Appl. 12, No. 5, 467-486 (2006) · Zbl 1095.39016 · doi:doi:10.1080/10236190600574002 [12] Gelişken, A.; Çinar, C.; Karataş, R.: A note on periodicity of the lyness MAX equation, Adv. differ. Equ. 2008 (2008) · Zbl 1149.39004 · doi:doi:10.1155/2008/651747 [13] Gelişken, A.; Çinar, C.; Yalçınkaya, I.: On the periodicity of a difference equation with maximum, Discrete dyn. Nat. soc. 2008 (2008) · Zbl 1149.39005 · doi:doi:10.1155/2008/820629 [14] Gelişken, A.; Çinar, C.: On the global attractivity of a MAX-type difference equation, Discrete dyn. Nat. soc. (2009) · Zbl 1178.39022 · doi:doi:10.1155/2009/812674 [15] A. Gelişken, C. Çinar, A note on a max-type difference equation, Indag. Math. (2009) (submitted for publication) [16] Grove, E. A.; Ladas, G.: Perodicities in nonlinear difference equations, (2005) [17] Grove, E. A.; Kent, C. M.; Ladas, G.; Radin, M. A.: On xn+1=max{1/xn,An/xn-1} with a period 3 parameter, Fields inst. Commun. 29 (2001) [18] Kent, C. M.; Radin, M. A.: On the boundedness nature of positive solutions of the difference equation xn+1=max{An/xn,Bn/xn-1} with periodic parameters, Dyn. contin. Discrete impuls. Syst. appl. Algorithms, 11-15 (2003) [19] Ladas, G.: On the recursive sequence xn=max{A1xn-1,A2xn-2,$\cdots$,Apxn-p}, open problems and conjectures, J. difference equ. Appl. 2, 339-341 (1996) [20] 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 · doi:doi:10.1016/S0898-1221(02)80010-4 [21] Patula, W. T.; Voulov, H. D.: On a MAX type recurrence relation with periodic coefficients, J. difference equ. Appl. 10, No. 3, 329-338 (2004) · Zbl 1050.39017 · doi:doi:10.1080/10236190310001659741 [22] Stević, S.: On the recursive sequence xn+1=a+xnpxn-1r, Discrete dyn. Nat. soc. 2007 (2007) [23] Stević, S.: Global stability of a difference equation with maximum, Appl. math. Comput. 210, No. 2, 524-529 (2009) · Zbl 1167.39007 · doi:doi:10.1016/j.amc.2009.01.050 [24] Stević, S.: On the recursive sequence xn+1=max$c,xnpxn-1p}$,Appl·math·Lett·21,No·8,791-796\left(2008\right)$ [25] Sun, F.: On the asymptotic behavior of a difference equation with maximum, Discrete dyn. Nat. soc. 2008 (2008) · Zbl 1155.39008 · doi:doi:10.1155/2008/243291 [26] Szalkai, I.: On the on the periodicity of the sequence xn+1=max\$A0xn,A1xn-1,...,Akxn-k}$,J·differenceequ·Appl·5,25-29\left(1999\right)$ [27] Voulov, H. D.: On the periodic character of some difference equations, J. difference equ. Appl. 8, No. 9, 799-810 (2002) · Zbl 1032.39004 · doi:doi:10.1080/1023619021000000780 [28] Yalçınkaya, I.; Irićanin, B. D.; Çinar, C.: On a MAX-type difference equation, Discrete dyn. Nat. soc. 2007 (2007) · Zbl 1152.39016 · doi:doi:10.1155/2007/47264 [29] Yang, X.; Liao, X.; Li, C.: On a difference equation with maximum, Appl. math. Comput. 181, 1-5 (2006) · Zbl 1148.39303 · doi:doi:10.1016/j.amc.2006.01.005