Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 969813, 11 pages

http://dx.doi.org/10.1155/2012/969813

Research Article

## Local Stability of Period Two Cycles of Second Order Rational Difference Equation

^{1}School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, Malaysia^{2}Department of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi Arabia

Received 1 September 2012; Accepted 11 October 2012

Academic Editor: Mustafa Kulenovic

Copyright © 2012 S. Atawna et al. 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

- R. P. Agarwal,
*Difference Equations and Inequalities: Theory, Methods, and Applications*, vol. 228, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at Zentralblatt MATH - S. N. Elaydi,
*An introduction to Difference Equations*, Springer, New York, NY, USA, 2nd edition, 1999. - W. G. Kelley and A. C. Peterson,
*Difference Equations: An Introduction with Applications*, Harcour Academic, New York, NY, USA, 2nd edition, 2001. - A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Basu, “Global behaviour of solutions to a class of second-order rational difference equations when prime period-two solutions exist,”
*Journal of Difference Equations and Applications*. In press. - S. Basu and O. Merino, “Global behavior of solutions to two classes of second-order rational difference equations,”
*Advances in Difference Equations*, vol. 2009, Article ID 128602, 27 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. Brett and M. R. S. Kulenović, “Global asymptotic behavior of ${y}_{n+1}=(p{y}_{n}+{y}_{n-1})/(r+q{y}_{n}+{y}_{n-1})$,”
*Advances in Difference Equations*, Article ID 41541, 22 pages, 2007. View at Google Scholar · View at Zentralblatt MATH - K. Cunningham, M. R. S. Kulenović, G. Ladas, and S. V. Valicenti, “On the recursive sequence ${x}_{n+1}=$$(\alpha +\beta {x}_{n-1})/(B{x}_{n}+C{x}_{n-1})$,”
*Nonlinear Analysis: Theory, Methods & Applications A*, vol. 47, no. 7, pp. 4603–4614, 2001. View at Publisher · View at Google Scholar - M. Dehghan, M. J. Douraki, and M. J. Douraki, “Dynamics of a rational difference equation using both theoretical and computational approaches,”
*Applied Mathematics and Computation*, vol. 168, no. 2, pp. 756–775, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - C. H. Gibbons, M. R. S. Kulenovic, and G. Ladas, “On the recursive sequence ${x}_{n+1}=(\alpha +\beta {x}_{n-1})/(\gamma +{x}_{n})$,”
*Mathematical Sciences Research Hot-Line*, vol. 4, no. 2, pp. 1–11, 2000. View at Google Scholar · View at Zentralblatt MATH - C. H. Gibbons, M. R. S. Kulenović, G. Ladas, and H. D. Voulov, “On the trichotomy character of ${x}_{n+1}=(\alpha +\beta {x}_{n}+\gamma {x}_{n-1})/(A+{x}_{n})$,”
*Journal of Difference Equations and Applications*, vol. 8, no. 1, pp. 75–92, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-X. Hu and W.-T. Li, “Global stability of a rational difference equation,”
*Applied Mathematics and Computation*, vol. 190, no. 2, pp. 1322–1327, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-X. Hu, W.-T. Li, and S. Stević, “Global asymptotic stability of a second order rational difference equation,”
*Journal of Difference Equations and Applications*, vol. 14, no. 8, pp. 779–797, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - L.-X. Hu, W.-T. Li, and H.-W. Xu, “Global asymptotical stability of a second order rational difference equation,”
*Computers & Mathematics with Applications*, vol. 54, no. 9-10, pp. 1260–1266, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Y. S. Huang and P. M. Knopf, “Boundedness of positive solutions of second-order rational difference equations,”
*Journal of Difference Equations and Applications*, vol. 10, no. 11, pp. 935–940, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Karatas and A. Gelişken, “Qualitative behavior of a rational difference equation,”
*Ars Combinatoria*, vol. 100, pp. 321–326, 2011. View at Google Scholar - W. A. Kosmala, M. R. S. Kulenović, G. Ladas, and C. T. Teixeira, “On the recursive sequence ${y}_{n+1}=(p+{y}_{n-1})/(q{y}_{n}+{y}_{n-1})$,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 2, pp. 571–586, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. R. S. Kulenovic and G. Ladas, “Open problems and conjectures: On period two solutions of ${x}_{n+1}=(\alpha +\beta {x}_{n}+\gamma {x}_{n-1})/(a+b{x}_{n}+c{x}_{n-1})$,”
*Journal of Difference Equations and Applications*, vol. 6, no. 5, pp. 641–646, 2000. View at Google Scholar - M. R. S. Kulenović, G. Ladas, and N. R. Prokup, “On the recursive sequence ${x}_{n+1}=(\alpha {x}_{n}+\beta {x}_{n-1})/(1+{x}_{n})$,”
*Journal of Difference Equations and Applications*, vol. 6, no. 5, pp. 563–576, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. R. S. Kulenović, G. Ladas, and W. S. Sizer, “On the recursive sequence ${x}_{n+1}=(\alpha {x}_{n}+\beta {x}_{n-1})/(\gamma {x}_{n}+c{x}_{n-1})$,”
*Mathematical Sciences Research Hot-Line*, vol. 2, no. 5, pp. 1–16, 1998. View at Google Scholar · View at Zentralblatt MATH - G. Ladas, “Open problems on the boundedness of some difference equations,”
*Journal of Difference Equations and Applications*, vol. 1, no. 3, pp. 317–321, 1995. View at Google Scholar - M. Saleh and S. Abu-Baha, “Dynamics of a higher order rational difference equation,”
*Applied Mathematics and Computation*, vol. 181, no. 1, pp. 84–102, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequence ${x}_{n+1}=a{x}_{n}-b{x}_{n}/(c{x}_{n}-d{x}_{n-k})$,”
*Communications on Applied Nonlinear Analysis*, vol. 15, no. 2, pp. 47–57, 2008. View at Google Scholar · View at Zentralblatt MATH - E. M. E. Zayed, A. B. Shamardan, and T. A. Nofal, “On the rational recursive sequence ${x}_{n+1}=(\alpha -\beta {x}_{n})/(\gamma -\delta {x}_{n}-{x}_{n-k})$,”
*International Journal of Mathematics and Mathematical Sciences*, Article ID 391265, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, vol. 256, Kluwer Academic, Dordrecht, The Netherlands, 1993. - M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations with Open Problem and Conjectures*, Chapman & Hall, Boca Raton, Fla, USA, 2002. - E. Camouzis and G. Ladas,
*Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures*, vol. 5, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2008.