×

zbMATH — the first resource for mathematics

On the max-type difference equation \(x_{n+1}=\max\{A/x_n,x_{n - 3}\}\). (English) Zbl 1188.39016
Summary: We show that every well-defined solution of the fourth-order difference equation
\[ x_{n+1}=\max\{A/x_n,x_{n-3}\}, \quad n\in \mathbb N_0, \] where parameter \(A\geq 0\), is eventually periodic with period four.

MSC:
39A23 Periodic solutions of difference equations
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] K. S. Berenhaut, J. D. Foley, and S. Stević, “Boundedness character of positive solutions of a max difference equation,” Journal of Difference Equations and Applications, vol. 12, no. 12, pp. 1193-1199, 2006. · Zbl 1116.39001 · doi:10.1080/10236190600949766
[2] C. \cCinar, S. Stević, and I. Yal\ccinkaya, “On positive solutions of a reciprocal difference equation with minimum,” Journal of Applied Mathematics & Computing, vol. 17, no. 1-2, pp. 307-314, 2005. · Zbl 1074.39002 · doi:10.1007/BF02936057
[3] E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the periodic nature of some max-type difference equations,” International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 14, pp. 2227-2239, 2005. · Zbl 1084.39004 · doi:10.1155/IJMMS.2005.2227 · eudml:52226
[4] E. M. Elsayed, B. Iri\vcanin, and S. Stević, “On the max-type equation xn+1=max An/xn,xn - 1,” to appear in Ars Combinatoria.
[5] E. M. Elsayed and S. Stević, “On the max-type equation xn+1=A/xn,xn - 2,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 910-922, 2009. · Zbl 1169.39003 · doi:10.1016/j.na.2008.11.016
[6] J. Feuer, “On the eventual periodicity of xn+1=max 1/xn,An/xn - 1 with a period-four parameter,” Journal of Difference Equations and Applications, vol. 12, no. 5, pp. 467-486, 2006. · Zbl 1095.39016 · doi:10.1080/10236190600574002
[7] E. A. Grove and G. Ladas, Periodicities in Nonlinear Ddifference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2005. · Zbl 1078.39009
[8] B. D. Iri\vcanin, The qualitative analysis of some classes of nonlinear difference equations, Ph.D. thesis, Prirodno-matemati\vcki fakultet, Univerzitet u Novom Sadu, Novi Sad, Serbia, 2009.
[9] C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation xn+1=max An/xn,Bn/xn - 1 with periodic parameters,” Dynamics of Continuous, Discrete & Impulsive Systems. Series B, vol. 2003, supplement, pp. 11-15, 2003.
[10] D. P. Mishev, W. T. Patula, and H. D. Voulov, “A reciprocal difference equation with maximum,” Computers & Mathematics with Applications, vol. 43, no. 8-9, pp. 1021-1026, 2002. · Zbl 1050.39015 · doi:10.1016/S0898-1221(02)80010-4
[11] D. P. Mishev, W. T. Patula, and H. D. Voulov, “Periodic coefficients in a reciprocal difference equation with maximum,” PanAmerican Mathematical Journal, vol. 13, no. 3, pp. 43-57, 2003. · Zbl 1050.39016
[12] W. T. Patula and H. D. Voulov, “On a max type recurrence relation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 10, no. 3, pp. 329-338, 2004. · Zbl 1050.39017 · doi:10.1080/10236190310001659741
[13] S. Stević, “Some open problems and conjectures on difference equations,” http://www.mi.sanu.ac.rs/colloquiums/mathcoll_programs/mathcoll.apr2004.htm.
[14] S. Stević, “Boundedness character of a max-type difference equation,” in Book of Abstracts, Conference in Honour of Allan Peterson, p. 28, Novacella, Italy, July-August 2007.
[15] S. Stević, “On the recursive sequence xn+1=A+(xnp/xn - 1r),” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 40963, 9 pages, 2007. · Zbl 1151.39011 · doi:10.1155/2007/40963
[16] S. Stević, “On the recursive sequence xn+1=max c,xnp/xn - 1p,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791-796, 2008. · Zbl 1152.39012 · doi:10.1016/j.aml.2007.08.008
[17] S. Stević, “Boundedness character of a class of difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 2, pp. 839-848, 2009. · Zbl 1162.39011 · doi:10.1016/j.na.2008.01.014
[18] S. Stević, “Boundedness character of two classes of third-order difference equations,” Journal of Difference Equations and Applications, vol. 15, no. 11-12, pp. 1193-1209, 2009. · Zbl 1182.39012 · doi:10.1080/10236190903022774
[19] S. Stević, “Global stability of a difference equation with maximum,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 525-529, 2009. · Zbl 1167.39007 · doi:10.1016/j.amc.2009.01.050
[20] S. Stević, “On a generalized max-type difference equation from automatic control theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1841-1849, 2010. · Zbl 1194.39007 · doi:10.1016/j.na.2009.09.025
[21] F. Sun, “On the asymptotic behavior of a difference equation with maximum,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 243291, 6 pages, 2008. · Zbl 1155.39008 · doi:10.1155/2008/243291 · eudml:129615
[22] I. Szalkai, “On the periodicity of the sequence xn+1=max A0/xn,A1/xn - 1,\cdots ,Ak/xn - k,” Journal of Difference Equations and Applications, vol. 5, no. 1, pp. 25-29, 1999. · Zbl 0930.39011 · doi:10.1080/10236199908808168
[23] H. D. Voulov, “Periodic solutions to a difference equation with maximum,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2155-2160, 2003. · Zbl 1019.39005 · doi:10.1090/S0002-9939-02-06890-9
[24] H. D. Voulov, “On the periodic nature of the solutions of the reciprocal difference equation with maximum,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 32-43, 2004. · Zbl 1053.39023 · doi:10.1016/j.jmaa.2004.02.054
[25] H. D. Voulov, “On a difference equation with periodic coefficients,” Journal of Difference Equations and Applications, vol. 13, no. 5, pp. 443-452, 2007. · Zbl 1121.39011 · doi:10.1080/10236190701264651
[26] I. Yal\ccinkaya, B. D. Iri\vcanin, and C. \cCinar, “On a max-type difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 47264, 10 pages, 2007. · Zbl 1152.39016 · doi:10.1155/2007/47264
[27] X. Yang, X. Liao, and C. Li, “On a difference equation wtih maximum,” Applied Mathematics and Computation, vol. 181, no. 1, pp. 1-5, 2006. · Zbl 1148.39303 · doi:10.1016/j.amc.2006.01.005
[28] E. P. Popov, Automatic Regulation and Control, Nauka, Moscow, Russia, 1966.
[29] K. S. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation xn=A+(xn - 2/xn - 1)p,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909-918, 2006. · Zbl 1111.39003 · doi:10.1080/10236190600836377
[30] L. Gutnik and S. Stević, “On the behaviour of the solutions of a second-order difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 27562, 14 pages, 2007. · Zbl 1180.39002 · doi:10.1155/2007/27562 · eudml:116979
[31] B. Iri\vcanin and S. Stević, “On a class of third-order nonlinear difference equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479-483, 2009. · Zbl 1178.39011 · doi:10.1016/j.amc.2009.03.039
[32] S. Stević, “On the recursive sequence xn+1=\alpha n+(xn - 1/xn). II,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A, vol. 10, no. 6, pp. 911-916, 2003.
[33] S. Stević, “On the recursive sequence xn+1=(A/\prod i=0kxn - i)+1/(\prod j=k+22(k+1)xn - j),” Taiwanese Journal of Mathematics, vol. 7, no. 2, pp. 249-259, 2003. · Zbl 1163.39306
[34] S. Stević, “A note on periodic character of a difference equation,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 929-932, 2004. · Zbl 1057.39005 · doi:10.1080/10236190412331272616
[35] S. Stević, “On the recursive sequence xn+1=\alpha +(xn - 1p/xnp),” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229-234, 2005. · Zbl 1100.39014 · doi:10.1007/BF02936567
[36] S. Stević, “On the difference equation xn+1=\alpha +(xn - 1/xn),” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159-1171, 2008. · Zbl 1155.39305 · doi:10.1016/j.camwa.2008.02.017
[37] S. Stević, “Boundedness character of a fourth order nonlinear difference equation,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2364-2369, 2009. · Zbl 1198.39020 · doi:10.1016/j.chaos.2007.10.030
[38] S. Stević, “On a class of higher-order difference equations,” Chaos, Solitons & Fractals, vol. 42, no. 1, pp. 138-145, 2009. · Zbl 1198.39021 · doi:10.1016/j.chaos.2008.11.012
[39] F. Balibrea and A. Linero, “On global periodicity of xn+2=f(xn+1,xn),” in Difference Equations, Special Functions and Orthogonal Polynomials. (Munich, July 25-30, 2005), pp. 41-50, World Scientific, Hackensack, NJ, USA, 2007. · Zbl 1130.39003 · doi:10.1080/10236190701388518
[40] F. Balibrea, A. Linero Bas, G. S. López, and S. Stević, “Global periodicity of xn+k+1=fk(xn+k) ... f1(xn+1),” Journal of Difference Equations and Applications, vol. 13, no. 10, pp. 901-910, 2007. · Zbl 1130.39003 · doi:10.1080/10236190701388518
[41] L. Berg and S. Stević, “Periodicity of some classes of holomorphic difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 827-835, 2006. · Zbl 1103.39004 · doi:10.1080/10236190600761575
[42] R. P. Kurshan and B. Gopinath, “Recursively generated periodic sequences,” Canadian Journal of Mathematics, vol. 26, pp. 1356-1371, 1974. · Zbl 0313.26019 · doi:10.4153/CJM-1974-129-6
[43] J. Rubió-Massegú and V. Mañosa, “Normal forms for rational difference equations with applications to the global periodicity problem,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 896-918, 2007. · Zbl 1121.39019 · doi:10.1016/j.jmaa.2006.10.061
[44] S. Stević, “On global periodicity of a class of difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 23503, 10 pages, 2007. · Zbl 1180.39005 · doi:10.1155/2007/23503 · eudml:116968
[45] S. Stević and K. S. Berenhaut, “The behaviour of the positive solutions of the difference equation xn=f(xn - 2)/g(xn - 1),” Abstract and Applied Analysis, vol. 2008, Article ID 53243, 9 pages, 2008.
[46] S. Stevich, “Nontrivial solutions of higher-order rational difference equations,” Matematicheskie Zametki, vol. 84, no. 5, pp. 772-780, 2008. · Zbl 1219.39007 · doi:10.1134/S0001434608110138
[47] B. Iri\vcanin and S. Stević, “Eventually constant solutions of a rational difference equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 854-856, 2009. · Zbl 1178.39012 · doi:10.1016/j.amc.2009.05.044
[48] S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60-68, 2006. · Zbl 1090.39009 · doi:10.1016/j.jmaa.2005.04.077
[49] S. Stević, “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 56813, 20 pages, 2007. · Zbl 1180.39009 · doi:10.1155/2007/56813 · eudml:116985
[50] S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28-31, 2007. · Zbl 1131.39009 · doi:10.1016/j.aml.2006.03.002
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.