Abstract and Applied Analysis

On the Periodicity of Some Classes of Systems of Nonlinear Difference Equations

Stevo Stević, Mohammed A. Alghamdi, Dalal A. Maturi, and Naseer Shahzad

Full-text: Open access

Abstract

Some classes of systems of difference equations whose all well-defined solutions are periodic are presented in this note.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 982378, 6 pages.

Dates
First available in Project Euclid: 26 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1395858039

Digital Object Identifier
doi:10.1155/2014/982378

Mathematical Reviews number (MathSciNet)
MR3166677

Zentralblatt MATH identifier
07023450

Citation

Stević, Stevo; Alghamdi, Mohammed A.; Maturi, Dalal A.; Shahzad, Naseer. On the Periodicity of Some Classes of Systems of Nonlinear Difference Equations. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 982378, 6 pages. doi:10.1155/2014/982378. https://projecteuclid.org/euclid.aaa/1395858039


Export citation

References

  • G. Bastien and M. Rogalski, “Global behaviour of solutions of cyclic systems of $q$ order 2 or 3 generalized Lyness' difference equations and of other more general equations of higher order,” Journal of Difference Equations and Applications, vol. 17, no. 11, pp. 1651–1672, 2011.
  • E. M. Elsayed, B. Iričanin, and S. Stević, “On the max-type equation ${x}_{n+1}=\text{max}\{{A}_{n}/{x}_{n},{x}_{n-1}\}$,” Ars Combinatoria, vol. 95, pp.187–192, 2010.
  • E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, vol. 4 of Advances in Discrete Mathematics and Applications, Chapman & Hall, CRC Press, Boca Raton, Fla, USA,2005.
  • B. D. Iričanin and S. Stević, “Some systems of nonlinear difference equations of higher order with periodic solutions,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 13,no. 3-4, pp. 499–507, 2006.
  • B. D. Iričanin and N. Touafek, “On a second-order max-type system of difference equations,” Indian Journal of Mathematics, vol. 54, no. 1, pp. 119–142, 2012.
  • C. M. Kent, M. Kustesky, A. Q. Nguyen, and B. V. Nguyen, “Eventually periodic solutions of ${x}_{n+1}=\text{max}\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\}$ when the parameters are two cycles,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 1–3, pp. 33–49, 2003.
  • C. M. Kent and M. A. Radin, “On the boundedness nature of positive solutions of the difference equation ${x}_{n+1}=\text{max}\{{A}_{n}/{x}_{n},{B}_{n}/{x}_{n-1}\}$ with periodic parameters,” Dynamics of Continuous, Discrete & Impulsive Systems B, supplement, pp. 11–15, 2003.
  • W. Kosmala, “A period 5 difference equation,” International Journal of Nonlinear Analysis and Applications, vol. 2, no. 1, pp. 82–84, 2011.
  • G. Papaschinopoulos and C. J. Schinas, “On the behavior of the solutions of a system of two nonlinear difference equations,” Communications on Applied Nonlinear Analysis, vol. 5, no. 2, pp. 47–59, 1998.
  • G. Papaschinopoulos and C. J. Schinas, “Invariants for systems of two nonlinear difference equations,” Differential Equations and Dynamical Systems, vol. 7, no. 2, pp. 181–196, 1999.
  • G. Papaschinopoulos, J. Schinas, and V. Hatzifilippidis, “Global behavior of the solutions of a max-equation and of a system of two max-equations,” Journal of Computational Analysis and Applications, vol. 5, no. 2, pp. 237–254, 2003.
  • S. Stević, “A global convergence results with applications to periodic solutions,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 1, pp. 45–53, 2002.
  • S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha }_{n}+{x}_{n-1}/{x}_{n}$ II,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 911–916, 2003.
  • S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006.
  • S. Stević, “Global stability of a difference equation with maximum,” Applied Mathematics and Computation, vol. 210, no. 2, pp. 525–529, 2009.
  • S. Stević, “Periodicity of max difference equations,” Utilitas Mathematica, vol. 83, pp. 69–71, 2010.
  • S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.
  • S. Stević, “On some periodic systems of max-type difference equations,” Applied Mathematics and Computation, vol. 218, no. 23, pp. 11483–11487, 2012.
  • N. Touafek and E. M. Elsayed, “On the periodicity of some systems of nonlinear difference equations,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie. Nouvelle Série, vol. 55, no. 2, pp. 217–224, 2012.
  • H. Levy and F. Lessman, Finite Difference Equations, Macmillan, New York, NY, USA, 1961.
  • R. C. Lyness, “Notes 1581,” The Mathematical Gazette, vol. 26, p. 62, 1942.
  • R. C. Lyness, “Notes 1847,” The Mathematical Gazette, vol. 29, p. 231, 1945.
  • R. C. Lyness, “Notes 2952,” The Mathematical Gazette, vol. 45, p. 201, 1961. \endinput