Abstract and Applied Analysis

Dynamics of a Family of Nonlinear Delay Difference Equations

Qiuli He, Taixiang Sun, and Hongjian Xi

Full-text: Open access

Abstract

We study the global asymptotic stability of the following difference equation: x n + 1 = f ( x n - k 1 , x n - k 2 , , x n - k s ; x n - m 1 , x n - m 2 , , x n - m t ) , n = 0,1 , , where 0 k 1 < k 2 < < k s and 0 m 1 < m 2 < < m t with { k 1 , k 2 , , k s } { m 1 , m 2 , , m t } = , the initial values are positive, and f C ( E s + t , ( 0 , + ) ) with E { ( 0 , + ) , [ 0 , + ) } . We give sufficient conditions under which the unique positive equilibrium x - of that equation is globally asymptotically stable.

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 456530, 4 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450355

Digital Object Identifier
doi:10.1155/2013/456530

Mathematical Reviews number (MathSciNet)
MR3055941

Zentralblatt MATH identifier
1272.39008

Citation

He, Qiuli; Sun, Taixiang; Xi, Hongjian. Dynamics of a Family of Nonlinear Delay Difference Equations. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 456530, 4 pages. doi:10.1155/2013/456530. https://projecteuclid.org/euclid.aaa/1393450355


Export citation

References

  • A. M. Amleh, D. A. Georgiou, E. A. Grove, and G. Ladas, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}$/${x}_{n}$,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 790–798, 1999.
  • K. S. Berenhaut and R. T. Guy, “Periodicity and boundedness for the integer solutions to a minimum-delay difference equation,” Journal of Difference Equations and Applications, vol. 16, no. 8, pp. 895–916, 2010.
  • K. S. Berenhaut, R. T. Guy, and C. L. Barrett, “Global asymptotic stability for minimum-delay difference equations,” Journal of Difference Equations and Applications, vol. 17, no. 11, pp. 1581–1590, 2011.
  • K. S. Berenhaut and A. H. Jones, “Asymptotic behaviour of solutions to difference equations involving ratios of elementary symmetric polynomials,” Journal of Difference Equations and Applications, vol. 18, no. 6, pp. 963–972, 2012.
  • E. Camouzis and G. Ladas, “When does local asymptotic stability imply global attractivity in rational equations?” Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 863–885, 2006.
  • R. Devault, V. L. Kocic, and D. Stutson, “Global behavior of solutions of the nonlinear difference equation ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$,” Journal of Difference Equations and Applications, vol. 11, no. 8, pp. 707–719, 2005.
  • H. El-Metwally, “Qualitative properties of some higher order difference equations,” Computers & Mathematics with Applications, vol. 58, no. 4, pp. 686–692, 2009.
  • Y. Fan, L. Wang, and W. Li, “Global behavior of a higher order nonlinear difference equation,” Journal of Mathematical Analysis and Applications, vol. 299, no. 1, pp. 113–126, 2004.
  • A. Gelişken, C. Çinar, and A. S. Kurbanli, “On the asymptotic behavior and periodic nature of a difference equation with maximum,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 898–902, 2010.
  • B. D. Iričanin, “Global stability of some classes of higher-order nonlinear difference equations,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1325–1328, 2010.
  • M. R. S. Kulenović, G. Ladas, and C. B. Overdeep, “On the dynamics of ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$ with a period-two coefficient,” Journal of Difference Equations and Applications, vol. 10, no. 10, pp. 905–914, 2004.
  • A. S. Kurbanl\i, C. Çinar, and G. Yalçinkaya, “On the behavior of positive solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}$/$({y}_{n}{x}_{n-1}+1)$, ${y}_{n+1}={y}_{n-1}$/$({x}_{n}{y}_{n-1}+1)$,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 1261–1267, 2011.
  • G. Ladas, “Open problems and conjecture,” Journal of Differential Equations and Applications, vol. 5, pp. 317–321, 1995.
  • G. Papaschinopoulos, M. A. Radin, and C. J. Schinas, “On the system of two difference equations of exponential form: ${x}_{n+1}=a+b{x}_{n-1}{e}^{-yn}$, ${y}_{n+1}=c+d{y}_{n-1}{e}^{-xn}$,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2969–2977, 2011.
  • G. Papaschinopoulos and C. J. Schinas, “On the dynamics of two exponential type systems of difference equations,” Computers & Mathematics with Applications, vol. 64, no. 7, pp. 2326–2334, 2012.
  • G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On the nonautonomous difference equation ${x}_{n+1}={A}_{n}+{x}_{n-1}^{p}$/${x}_{n}^{q}$,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5573–5580, 2011.
  • C. Qian, “Global attractivity of periodic solutions in a higher order difference equation,” Applied Mathematics Letters, vol. 26, pp. 578–583, 2013.
  • S. Stević, “Boundedness character of a class of difference equations,” Nonlinear Analysis. Theory, Methods & Applications A, vol. 70, no. 2, pp. 839–848, 2009.
  • S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.
  • T. Sun and H. Xi, “Global behavior of the nonlinear difference equation ${x}_{n+1}=f({x}_{n-s},{x}_{n-t})$,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 760–765, 2005.
  • T. Sun, H. Xi, and Q. He, “On boundedness of the difference equation ${x}_{n+1}={p}_{n}+{x}_{n-3s+1}$/${x}_{n-s+1}$ with period-k coefficients,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5994–5997, 2011.
  • N. Touafek and E. M. Elsayed, “On the solutions of systems of rational difference equations,” Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 1987–1997, 2012.