Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2011, Article ID 671967, 24 pages
http://dx.doi.org/10.1155/2011/671967
Research Article

Asymptotic Behavior of Solutions of Delayed Difference Equations

1Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, University of Technology, 602 00 Brno, Czech Republic
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 616 00 Brno, Czech Republic

Received 24 January 2011; Accepted 9 May 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 J. Diblík and I. Hlavičková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. Diblík, “Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique,” Computers and Mathematics with Applications, vol. 45, no. 6–9, pp. 1041–1057, 2003, Advances in difference equations, I. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. Diblík, “Discrete retract principle for systems of discrete equations,” Computers and Mathematics with Applications, vol. 42, no. 3–5, pp. 515–528, 2001, Advances in difference equations, II. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. Diblík, “Anti-Lyapunov method for systems of discrete equations,” Nonlinear Analysis, vol. 57, no. 7–8, pp. 1043–1057, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. J. Diblík, M. Migda, and E. Schmeidel, “Bounded solutions of nonlinear discrete equations,” Nonlinear Analysis, vol. 65, no. 4, pp. 845–853, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. J. Diblík, I. Růžičková, and M. Růžičková, “A general version of the retract method for discrete equations,” Acta Mathematica Sinica, vol. 23, no. 2, pp. 341–348, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. J. Diblík and I. Růžičková, “Compulsory asymptotic behavior of solutions of two-dimensional systems of difference equations,” in Proceedings of the 9th International Conference on Difference Equations and Discrete Dynamical Systems, pp. 35–49, World Scientific Publishing, University of Southern California, Los Angeles, Calif, USA, 2005.
  7. J. Diblík and I. Hlavičková, “Asymptotic properties of solutions of the discrete analogue of the Emden-Fowler equation,” in Advances in Discrete Dynamical Systems, vol. 53 of Advanced Studies in Pure Mathematics, pp. 23–32, Mathematical Society of Japan, Tokyo, Japan, 2009. View at Google Scholar · View at Zentralblatt MATH
  8. G. Ladas, C. G. Philos, and Y. G. Sficas, “Sharp conditions for the oscillation of delay difference equations,” Journal of Applied Mathematics and Simulation, vol. 2, no. 2, pp. 101–111, 1989. View at Google Scholar · View at Zentralblatt MATH
  9. I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations, Clarendon Press, Alderley, UK, 1991.
  10. J. Baštinec and J. Diblík, “Subdominant positive solutions of the discrete equation Δu(k+n)=p(k)u(k),” Abstract and Applied Analysis, no. 6, pp. 461–470, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. J. Baštinec and J. Diblík, “Remark on positive solutions of discrete equation Δu(k+n)=p(k)u(k),” Nonlinear Analysis, vol. 63, no. 5-7, pp. e2145–e2151, 2004. View at Google Scholar
  12. J. Baštinec, J. Diblík, and B. Zhang, “Existence of bounded solutions of discrete delayed equations,” in Proceedings of the Sixth International Conference on Difference Equations and Applications, pp. 359–366, CRC, Boca Raton, Fla, USA, 2004. View at Google Scholar · View at Zentralblatt MATH
  13. R. P. Agarwal, M. Bohner, and W.-T. Li, Nonoscillation and Oscillation: Theory for Functional Differential Equations, Marcel Dekker, New York, NY, USA, 2004.
  14. L. Berezansky and E. Braverman, “On existence of positive solutions for linear difference equations with several delays,” Advances in Dynamical Systems and Applications, vol. 1, no. 1, pp. 29–47, 2006. View at Google Scholar · View at Zentralblatt MATH
  15. G. E. Chatzarakis and I. P. Stavroulakis, “Oscillations of first order linear delay difference equations,” The Australian Journal of Mathematical Analysis and Applications, vol. 3, no. 1, article 14, 2006. View at Google Scholar · View at Zentralblatt MATH
  16. D. Chengjun and S. Qiankun, “Boundedness and stability for discrete-time delayed neural network with complex-valued linear threshold neurons,” Discrete Dynamics in Nature and Society, vol. 2010, article 368379, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. J. Čermák, “Asymptotic bounds for linear difference systems,” Advances in Difference Equations, vol. 2010, article 182696, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. I. Györi and M. Pituk, “Asymptotic formulae for the solutions of a linear delay difference equation,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 376–392, 1995. View at Publisher · View at Google Scholar
  19. P. Karajani and I. P. Stavroulakis, “Oscillation criteria for second-order delay and functional equations,” Studies of the University of Žilina. Mathematical Series, vol. 18, no. 1, pp. 17–26, 2004. View at Google Scholar · View at Zentralblatt MATH
  20. L. K. Kikina and I. P. Stavroulakis, “A survey on the oscillation of solutions of first order delay difference equations,” CUBO, A Mathematical Journal, vol. 7, no. 2, pp. 223–236, 2005. View at Google Scholar · View at Zentralblatt MATH
  21. L. K. Kikina and I. P. Stavroulakis, “Oscillation criteria for second-order delay, difference, and functional equations,” International Journal of Differential Equations, vol. 2010, article 598068, 2010. View at Google Scholar · View at Zentralblatt MATH
  22. M. Kipnis and D. Komissarova, “Stability of a delay difference system,” Advances in Difference Equations, vol. 2006, article 31409, 2006. View at Google Scholar · View at Zentralblatt MATH
  23. E. Liz, “Local stability implies global stability in some one-dimensional discrete single-species models,” Discrete and Continuous Dynamical Systems. Series B, vol. 7, no. 1, pp. 191–199, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  24. R. Medina and M. Pituk, “Nonoscillatory solutions of a second-order difference equation of Poincaré type,” Applied Mathematics Letters, vol. 22, no. 5, pp. 679–683, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. I. P. Stavroulakis, “Oscillation criteria for first order delay difference equations,” Mediterranean Journal of Mathematics, vol. 1, no. 2, pp. 231–240, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  26. J. Baštinec, J. Diblík, and Z. Šmarda, “Existence of positive solutions of discrete linear equations with a single delay,” Journal of Difference Equations and Applications, vol. 16, no. 9, pp. 1047–1056, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH