Discrete Dynamics in Nature and Society

Volume 2012, Article ID 406821, 17 pages

http://dx.doi.org/10.1155/2012/406821

Research Article

## The Form of the Solutions and Periodicity of Some Systems of Difference Equations

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

Received 20 May 2012; Accepted 6 July 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyright © 2012 M. Mansour 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*, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. - R. P. Agarwal and E. M. Elsayed, “On the solution of fourth-order rational recursive sequence,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 4, pp. 525–545, 2010. View at Google Scholar · View at Scopus - N. Battaloglu, C. Cinar, and I. Yalcinkaya, “The dynamics of the difference equation ${x}_{n+1}=(\alpha {x}_{n-m})/(\beta +\gamma {x}_{n-(k+1)}^{p})$,”
*Ars Combinatoria*, vol. 97, pp. 281–288, 2010. View at Google Scholar - E. Camouzis and G. Papaschinopoulos, “Global asymptotic behavior of positive solutions on the system of rational difference equations ${x}_{n+1}=1+1/{y}_{n-k},{y}_{n+1}={y}_{n}/{x}_{n-m}{y}_{n-m-k}$,”
*Applied Mathematics Letters*, vol. 17, no. 6, pp. 733–737, 2004. View at Publisher · View at Google Scholar - C. Çinar, “On the positive solutions of the difference equation system ${x}_{n+1}=1/{y}_{n},{y}_{n+1}={y}_{n}/{x}_{n-1}{y}_{n-1}$,”
*Applied Mathematics and Computation*, vol. 158, no. 2, pp. 303–305, 2004. View at Publisher · View at Google Scholar - C. Cinar, I. Yalçinkaya, and R. Karatas, “On the positive solutions of the difference equation system ${x}_{n+1}=m/{y}_{n},{y}_{n+1}=p{y}_{n}/{x}_{n-1}{y}_{n-1}$,”
*Journal of Institute of Mathematics and Computer Science*, vol. 18, pp. 135–136, 2005. View at Google Scholar - D. Clark and M. R. S. Kulenović, “A coupled system of rational difference equations,”
*Computers & Mathematics with Applications*, vol. 43, no. 6-7, pp. 849–867, 2002. View at Publisher · View at Google Scholar - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Global behavior of the solutions of difference equation,”
*Advances in Difference Equations*, vol. 2011, 28 pages, 2011. View at Google Scholar - E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, “Some properties and expressions of solutions for a class of nonlinear difference equation,”
*Utilitas Mathematica*, vol. 87, pp. 93–110, 2012. View at Google Scholar - E. M. Elabbasy and E. M. Elsayed, “Global attractivity and periodic nature of a difference equation,”
*World Applied Sciences Journal*, vol. 12, no. 1, pp. 39–47, 2011. View at Google Scholar - E. M. Elsayed, “On the solutions of a rational system of difference equations,”
*Polytechnica Posnaniensis*, no. 45, pp. 25–36, 2010. View at Google Scholar - E. M. Elsayed, “Dynamics of recursive sequence of order two,”
*Kyungpook Mathematical Journal*, vol. 50, no. 4, pp. 483–497, 2010. View at Publisher · View at Google Scholar - E. M. M. Elsayed, “Behavior of a rational recursive sequences,”
*Studia Universitatis Babeş-Bolyai Mathematica*, vol. 56, no. 1, pp. 27–42, 2011. View at Google Scholar - E. M. Elsayed, “Solution of a recursive sequence of order ten,”
*General Mathematics*, vol. 19, no. 1, pp. 145–162, 2011. View at Google Scholar - E. M. Elsayed, “Solution and attractivity for a rational recursive sequence,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 982309, 17 pages, 2011. View at Publisher · View at Google Scholar - E. M. Elsayed, “On the solution of some difference equations,”
*European Journal of Pure and Applied Mathematics*, vol. 4, no. 3, pp. 287–303, 2011. View at Google Scholar - E. M. Elsayed, “On the dynamics of a higher-order rational recursive sequence,”
*Communications in Mathematical Analysis*, vol. 12, no. 1, pp. 117–133, 2012. View at Google Scholar - E. M. Elsayed, “Solutions of rational difference system of order two,”
*Mathematical and Computer Modelling*, vol. 55, pp. 378–384, 2012. View at Publisher · View at Google Scholar - E. M. Elsayed, M. M. El-Dessoky, and A. Alotaibi, “On the solutions of a general system of difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2012, Article ID 892571, 12 pages, 2012. View at Publisher · View at Google Scholar - M. E. Erdogan, C. Cinar, and I. Yalcinkaya, “On the dynamics of the recursive sequence ${x}_{n+1}=(b{x}_{n-1})/(A+B{x}_{n}^{p}{x}_{n-2}^{q})$,”
*Computers & Mathematics with Applications*, vol. 61, no. 3, pp. 533–537, 2011. View at Publisher · View at Google Scholar - E. A. Grove, G. Ladas, L. C. McGrath, and C. T. Teixeira, “Existence and behavior of solutions of a rational system,”
*Communications on Applied Nonlinear Analysis*, vol. 8, no. 1, pp. 1–25, 2001. View at Google Scholar - A. S. Kurbanlı, C. Çinar, and I. 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. View at Publisher · View at Google Scholar - A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations: ${x}_{n+1}={x}_{n-1}/({y}_{n}{x}_{n-1}-1),{y}_{n+1}={y}_{n-1}/({x}_{n}{y}_{n-1}-1)$, and ${z}_{n+1}={z}_{n-1}/({y}_{n}{z}_{n-1}-1)$,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 932362, 12 pages, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. Y. Özban, “On the positive solutions of the system of rational difference equations ${x}_{n+1}=1/{y}_{n-k},{y}_{n+1}={y}_{n}/{x}_{n-m}{y}_{n-m-k}$,”
*Journal of Mathematical Analysis and Applications*, vol. 323, no. 1, pp. 26–32, 2006. View at Publisher · View at Google Scholar - A. Y. Özban, “On the system of rational difference equations ${x}_{n+1}=a/{y}_{n-3},{y}_{n+1}=b{y}_{n-3}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 833–837, 2007. View at Publisher · View at Google Scholar - I. Yalcinkaya, “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 - I. Yalcinkaya, “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 - X. Yang, Y. Liu, and S. Bai, “On the system of high order rational difference equations ${x}_{n}=a/{y}_{n-p},{y}_{n}=b{y}_{n-p}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 171, no. 2, pp. 853–856, 2005. View at Publisher · View at Google Scholar - C. J. Schinas, “Invariants for difference equations and systems of difference equations of rational form,”
*Journal of Mathematical Analysis and Applications*, vol. 216, no. 1, pp. 164–179, 1997. View at Publisher · View at Google Scholar - Y. Zhang, X. Yang, D. J. Evans, and C. Zhu, “On the nonlinear difference equation system ${x}_{n+1}=A+{y}_{n-m}/{x}_{n},{y}_{n+1}=A+{x}_{n-m}/{y}_{n}$,”
*Computers & Mathematics with Applications*, vol. 53, no. 10, pp. 1561–1566, 2007. View at Publisher · View at Google Scholar - D. Simsek, B. Demir, and C. Cinar, “On the solutions of the system of difference equations ${x}_{n+1}=max\{A/{x}_{n},{y}_{n}/{x}_{n}\},{y}_{n+1}=max\{A/{y}_{n},{x}_{n}/{y}_{n}\}$,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 325296, 11 pages, 2009. View at Google Scholar - S. Stević, “On a system of difference equations with period two coefficients,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4317–4324, 2011. View at Publisher · View at Google Scholar - 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 - S. Stević, “On some solvable systems of difference equations,”
*Applied Mathematics and Computation*, vol. 218, no. 9, pp. 5010–5018, 2012. View at Publisher · View at Google Scholar - N. Touafek and E. M. Elsayed, “On the solutions of systems of rational difference equations,”
*Mathematical and Computer Modelling*, vol. 55, pp. 1987–1997, 2012. View at Publisher · View at Google Scholar - I. Yalcinkaya, C. Cinar, and D. Simsek, “Global asymptotic stability of a system of difference equations,”
*Applicable Analysis*, vol. 87, no. 6, pp. 677–687, 2008. View at Publisher · View at Google Scholar - I. Yalcinkaya and C. Çinar, “Global asymptotic stability of a system of two nonlinear difference equations,”
*Fasciculi Mathematici*, no. 43, pp. 171–180, 2010. View at Google Scholar - 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 Google Scholar - X. Yang, “On the system of rational difference equations ${x}_{n}=A+{y}_{n-1}/{x}_{n-p}{y}_{n-q},{y}_{n}=A+{x}_{n-1}/{x}_{n-r}{y}_{n-s}$,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 1, pp. 305–311, 2005. View at Publisher · View at Google Scholar - 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 - Y. Zhang, X. Yang, G. M. Megson, and D. J. Evans, “On the system of rational difference equations ${x}_{n+1}=a/{y}_{n-3},{y}_{n+1}=b{y}_{n-3}/{x}_{n-q}{y}_{n-q}$,”
*Applied Mathematics and Computation*, vol. 176, no. 2, pp. 403–408, 2006. View at Publisher · View at Google Scholar