Table of Contents Author Guidelines Submit a Manuscript
International Journal of Analysis
Volume 2013, Article ID 486357, 12 pages
http://dx.doi.org/10.1155/2013/486357
Research Article

Global Attractivity Results on Complete Ordered Metric Spaces for Third-Order Difference Equations

1Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood road, Pretoria 0002, South Africa
2University of Tunisia, Tunis College of Sciences and Techniques, 5 Avenue Taha Hussein, BP 56, Bab Manara, Tunis, Tunisia

Received 24 October 2012; Accepted 31 January 2013

Academic Editor: Jacques Liandrat

Copyright © 2013 Mujahid Abbas and Maher Berzig. 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. M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002, With open problems and conjectures. View at MathSciNet
  2. E. Camouzis and G. Ladas, Dynamics of third-order rational difference equations with open problems and conjectures, vol. 5 of Advances in Discrete Mathematics and Applications, Chapman Hall/CRC, Boca Raton, Fla, USA, 2008. View at MathSciNet
  3. M. R. S. Kulenović and O. Merino, “A global attractivity result for maps with invariant boxes,” Discrete and Continuous Dynamical Systems B, vol. 6, no. 1, pp. 97–110, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. R. D. Nussbaum, “Global stability, two conjectures and Maple,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 5, pp. 1064–1090, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. L. Smith, “The discrete dynamics of monotonically decomposable maps,” Journal of Mathematical Biology, vol. 53, no. 4, pp. 747–758, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. L. Smith, “Global stability for mixed monotone systems,” Journal of Difference Equations and Applications, vol. 14, no. 10-11, pp. 1159–1164, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. I. Altun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,” Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4889–4897, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Gnana Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1379–1393, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Brett and M. R. S. Kulenović, “Basins of attraction of equilibrium points of monotone difference equations,” Sarajevo Journal of Mathematics, vol. 5, no. 2, pp. 211–233, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. Burgić, S. Kalabušić, and M. R. S. Kulenović, “Period-two trichotomies of a difference equation of order higher than two,” Sarajevo Journal of Mathematics, vol. 4, no. 1, pp. 73–90, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. L. Cirić, N. Cakić, M. Rajović, and J. S. Ume, “Monotone generalized nonlinear contractions in partially ordered metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 131294, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. E. Karapınar, “Couple fixed point theorems for nonlinear contractions in cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3656–3668, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. V. Lakshmikantham and L. Cirić, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 12, pp. 4341–4349, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. J. J. Nieto and R. Rodríguez-López, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 12, pp. 4508–4517, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  20. D. Burgić, S. Kalabušić, and M. R. S. Kulenović, “Global attractivity results for mixed-monotone mappings in partially ordered complete metric spaces,” Fixed Point Theory and Applications, vol. 2009, Article ID 762478, 17 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Bhatia, Matrix Analysis, vol. 169 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  22. T. Ando, “Limit of iterates of cascade addition of matrices,” Numerical Functional Analysis and Optimization, vol. 2, no. 7-8, pp. 579–289, 1980. View at Publisher · View at Google Scholar · View at MathSciNet
  23. W. N. Anderson, T. D. Morley, and G. E. Trapp, “Ladder networks, fixed points and the geometric mean,” Circuits, Systems and Signal Processing, vol. 2, no. 3, pp. 259–268, 1983. View at Google Scholar
  24. J. C. Engwerda, “On the existence of a positive definite solution of the matrix equation X+ATX-1A=I,” Linear Algebra and its Applications, vol. 194, pp. 91–108, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. W. Pusz and S. L. Woronowicz, “Functional calculus for sesquilinear forms and the purification map,” Reports on Mathematical Physics, vol. 8, no. 2, pp. 159–170, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. B. L. Buzbee, G. H. Golub, and C. W. Nielson, “On direct methods for solving Poisson's equations,” SIAM Journal on Numerical Analysis, vol. 7, pp. 627–656, 1970. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. W. L. Green and E. W. Kamen, “Stabilizability of linear systems over a commutative normed algebra with applications to spatially-distributed and parameter-dependent systems,” SIAM Journal on Control and Optimization, vol. 23, no. 1, pp. 1–18, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet