Discrete Dynamics in Nature and Society

Volume 2013, Article ID 495838, 6 pages

http://dx.doi.org/10.1155/2013/495838

Research Article

## Qualitative Behavior of Rational Difference Equation of Big Order

^{1}Mathematics Department, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia^{2}Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 4 February 2013; Accepted 20 April 2013

Academic Editor: Cengiz Çinar

Copyright © 2013 M. M. El-Dessoky. 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. Abu-Saris, C. Çinar, and I. Yalçinkaya, “On the asymptotic stability of ${x}_{n+1}=a+{x}_{n}{x}_{n-k}/\left({x}_{n}+{x}_{n-k}\right)$,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1172–1175, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal,
*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 1st edition, 1992. View at MathSciNet - R. P. Agarwal,
*Difference Equations and Inequalities*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. View at MathSciNet - R. P. Agarwal and E. M. Elsayed, “Periodicity and stability of solutions of higher order rational difference equation,”
*Advanced Studies in Contemporary Mathematics*, vol. 17, no. 2, pp. 181–201, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Aloqeili, “Dynamics of a rational difference equation,”
*Applied Mathematics and Computation*, vol. 176, no. 2, pp. 768–774, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Battaloglu, C. Cinar, and I. Yalçınkaya, “The dynamics of the difference equation,”
*Ars Combinatoria*, vol. 97, pp. 281–288, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Çinar, “On the positive solutions of the difference equation ${x}_{n+1}={ax}_{n-1}/\left(1+{bx}_{n}{x}_{n-1}\right)$,”
*Applied Mathematics and Computation*, vol. 156, no. 2, pp. 587–590, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation ${x}_{n+1}={ax}_{n}-\left({bx}_{n}/{cx}_{n}-{dx}_{n-1}\right)$,”
*Advances in Difference Equations*, vol. 2006, Article ID 82579, 10 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “On the difference equation ${x}_{n+1}={ax}_{n-k}/\left(\beta +\gamma {\prod}_{i=0}^{k}{x}_{n-i}\right)$,”
*Journal of Concrete and Applicable Mathematics*, vol. 5, no. 2, pp. 101–113, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Qualitative behavior of higher order difference equation,”
*Soochow Journal of Mathematics*, vol. 33, no. 4, pp. 861–873, 2007. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Kang and B. Shi, “Periodic solutions for a system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 760328, 9 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Y. Őzban, “On the system of rational difference equations ${x}_{n}=a/{y}_{n-3}$, ${y}_{n}={by}_{n-3}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 833–837, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Camouzis and G. Papaschinopoulos, “Global asymptotic behavior of positive solutions on the system of rational difference equations ${x}_{n+1}=1+{x}_{n}/{y}_{n-m}$, ${y}_{n+1}=1+{y}_{n}/{x}_{n-m}$,”
*Applied Mathematics Letters*, vol. 17, no. 6, pp. 733–737, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elabbasy and E. M. Elsayed, “Global asymptotic behavior attractivity and periodic nature of a difference equation,”
*World Applied Sciences Journal*, vol. 12, no. 1, pp. 39–47, 2011. View at Google Scholar - V. L. Kocić and G. Ladas,
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications*, Kluwer Academic, Dordrecht, The Netherlands, 1993. View at MathSciNet - M. R. S. Kulenović and G. Ladas,
*Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2001. View at MathSciNet - I. Yalçinkaya, “On the global asymptotic stability of a second-order system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 860152, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. El-Metwally, “Global behavior of an economic model,”
*Chaos, Solitons & Fractals*, vol. 33, no. 3, pp. 994–1005, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, “On the global asymptotic behavior of a system of two nonlinear difference equations,”
*Ars Combinatoria*, vol. 95, pp. 151–159, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, C. Çinar, and M. Atalay, “On the solutions of systems of difference equations,”
*Advances in Difference Equations*, vol. 2008, Article ID 143943, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Diblík, B. Iricanin, S. Stevic, and Z. Šmarda, “On some symmetric systems of difference equations,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 246723, 7 pages, 2013. View at Publisher · View at Google Scholar - E. M. Elsayed, “Dynamics of a recursive sequence of higher order,”
*Communications on Applied Nonlinear Analysis*, vol. 16, no. 2, pp. 37–50, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Saleh and M. Aloqeili, “On the difference equation ${y}_{n+1}=A+\left({y}_{n}/{y}_{n-k}\right)$ with $A<0$,”
*Applied Mathematics and Computation*, vol. 176, no. 1, pp. 359–363, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Wang, S. Wang, Z. Wang, H. Gong, and R. Wang, “Asymptotic stability for a class of nonlinear difference equation,”
*Discrete Dynamics in Natural and Society*, vol. 2010, Article ID 791610, 10 pages, 2010. View at Publisher · View at Google Scholar - C.-y. Wang, Q.-h. Shi, and S. Wang, “Asymptotic behavior of equilibrium point for a family of rational difference equations,”
*Advances in Difference Equations*, vol. 2010, Article ID 505906, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Yalçinkaya, “On the difference equation ${x}_{n+1}=\alpha +\left({x}_{n-m}/{x}_{n}^{k}\right)$,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 805460, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed and M. A. El-Moneam, “On the rational recursive sequence ,”
*Communications on Applied Nonlinear Analysis*, vol. 15, no. 2, pp. 47–57, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. E. Zayed and M. A. EL-Moneam, “On the rational recursive sequence ${x}_{n+1}=\alpha +\beta {x}_{n}+\gamma {x}_{n-1}/\left(A+B{x}_{n}+C{x}_{n-1}\right)$,”
*Communications on Applied Nonlinear Analysis*, vol. 12, no. 4, pp. 15–28, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. M. Elsayed and M. M. El-Dessoky, “Dynamics and behavior of a higher order rational recursive sequence,”
*Advances in Difference Equations*, pp. 2012–69, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - D. Simsek, B. Demir, and C. Cinar, “On the solutions of the system of difference equations ${x}_{n+1}=\text{max}\{A/{x}_{n},{y}_{n}/{x}_{n}\}$, ${y}_{n+1}=\text{max}\{A/{y}_{n},{x}_{n}/{y}_{n}\}$,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 325296, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Mansour, M. M. El-Dessoky, and E. M. Elsayed, “The form of the solutions and periodicity of some systems of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 406821, 17 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Gelisken, C. Cinar, and I. Yalcinkaya, “On a max-type difference equation,”
*Advances in Difference Equations*, vol. 2010, Article ID 584890, 6 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Wang, S. Wang, L. Li, and Q. Shi, “Asymptotic behavior of equilibrium point for a class of nonlinear difference equation,”
*Advances in Difference Equations*, vol. 2009, Article ID 214309, 8 pages, 2009. View at Publisher · View at Google Scholar - C.-Y. Wang, S. Wang, Z.-w. Wang, F. Gong, and R.-f. Wang, “Asymptotic stability for a class of nonlinear difference equations,”
*Discrete Dynamics in Nature and Society. An International Multidisciplinary Research and Review Journal*, vol. 2010, Article ID 791610, 10 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Zhang, L. Liu, Y. Wu, and Y. Lu, “The iterative solutions of nonlinear fractional differential equations,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4680–4691, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations: ${x}_{n+1}={x}_{n-1}/\left({y}_{n}{x}_{n-1}-1\right)$, ${x}_{n+1}={y}_{n-1}/\left({x}_{n}{y}_{n-1}-1\right)$ and ${z}_{n+1}={z}_{n-1}/\left({y}_{n}{z}_{n-1}-1\right)$,”
*Discrete Dynamics in Nature and Society. An International Multidisciplinary Research and Review Journal*, vol. 2011, Article ID 932362, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional systems of rational difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 178483, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - S. Stević, “On a system of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 7, pp. 3372–3378, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet