`Abstract and Applied AnalysisVolume 2010, Article ID 237129, 6 pageshttp://dx.doi.org/10.1155/2010/237129`
Research Article

## Global Behavior of the Difference Equation

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China
2Department of Mathematics, Guangxi College of Finance and Economics, Nanning 530003, China

Received 31 March 2010; Revised 17 April 2010; Accepted 30 April 2010

Copyright © 2010 Taixiang Sun 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.

1. 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, 2002.
2. C. H. Gibbons, M. R. S. Kulenovic, and G. Ladas, “On the recursive sequence ${x}_{n+1}=\left(\alpha +\beta {x}_{n-1}\right)/\left(\gamma +{x}_{n}\right)$,” Mathematical Sciences Research Hot-Line, vol. 4, no. 2, pp. 1–11, 2000.
3. L. Berg, “On the asymptotics of nonlinear difference equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 21, no. 4, pp. 1061–1074, 2002.
4. S. Stević, “On the recursive sequence ${x}_{n+1}={x}_{n-1}/g\left({x}_{n}\right)$,” Taiwanese Journal of Mathematics, vol. 6, no. 3, pp. 405–414, 2002.
5. L. Berg, “Inclusion theorems for non-linear difference equations with applications,” Journal of Difference Equations and Applications, vol. 10, no. 4, pp. 399–408, 2004.
6. L. Berg, “On the asymptotics of the difference equation ${x}_{n-3}={x}_{n}\left(1+{x}_{n}{{}_{-}}_{1}{x}_{n-2}\right)$,” Journal of Difference Equations and Applications, vol. 14, no. 1, pp. 105–108, 2008.
7. S. Stević, “Asymptotic behavior of a class of nonlinear difference equations,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 47156, 10 pages, 2006.
8. S. Stević, “Global stability and asymptotics of some classes of rational difference equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 1, pp. 60–68, 2006.
9. S. Stević, “On monotone solutions of some classes of difference equations,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 53890, 9 pages, 2006.
10. S. Stević, “On positive solutions of a $\left(k+1\right)$th order difference equation,” Applied Mathematics Letters, vol. 19, no. 5, pp. 427–431, 2006.
11. S. Stević, “Asymptotics of some classes of higher-order difference equations,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 56813, 20 pages, 2007.
12. S. Stević, “Asymptotic periodicity of a higher-order difference equation,” Discrete Dynamics in Nature and Society, vol. 2007, Article ID 13737, 9 pages, 2007.
13. S. Stević, “Existence of nontrivial solutions of a rational difference equation,” Applied Mathematics Letters, vol. 20, no. 1, pp. 28–31, 2007.
14. S. Stević, “Nontrivial solutions of higher-order rational difference equations,” Matematicheskie Zametki, vol. 84, no. 5, pp. 772–780, 2008.
15. B. Iričanin and S. Stević, “Eventually constant solutions of a rational difference equation,” Applied Mathematics and Computation, vol. 215, no. 2, pp. 854–856, 2009.
16. C. M. Kent, “Convergence of solutions in a nonhyperbolic case,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 7, pp. 4651–4665, 2001.
17. T. Sun, “On non-oscillatory solutions of the recursive sequence ${x}_{n+1}=p+{x}_{n-k}/{x}_{n}$,” Journal of Difference Equations and Applications, vol. 11, no. 6, pp. 483–485, 2005.
18. T. Sun and H. Xi, “On the solutions of a class of difference equations,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 766–770, 2005.
19. T. Sun and H. Xi, “Existence of monotone solutions of a difference equation,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 917560, 8 pages, 2008.
20. S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha }_{n}+{x}_{n-1}/{x}_{n}$ II,” Dynamics of Continuous, Discrete & Impulsive Systems A, vol. 10, no. 6, pp. 911–916, 2003.
21. K. Berenhaut and S. Stević, “The behaviour of the positive solutions of the difference equation ${x}_{n}=A+{\left({x}_{n-2}/{x}_{n-1}\right)}^{p}$,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909–918, 2006.
22. S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha }_{n}+{x}_{n-1}/{x}_{n}$,” International Journal of Mathematical Sciences, vol. 2, no. 2, pp. 237–243, 2004.
23. S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}^{p}/{x}_{n}^{p}$,” Journal of Applied Mathematics & Computing, vol. 18, no. 1-2, pp. 229–234, 2005.
24. S. Stević, “On the difference equation ${x}_{n+1}={\alpha }_{n}+{x}_{n-1}/{x}_{n}$,” Computers & Mathematics with Applications, vol. 56, no. 5, pp. 1159–1171, 2008.
25. S. Stević, “On a class of higher-order difference equations,” Chaos Solitons and Fractals, vol. 42, no. 1, pp. 138–145, 2009.
26. S. Stević and K. Berenhaut, “The behavior of positive solutions of a nonlinear second-order difference equation ${x}_{n}=f\left({x}_{n-2}\right)/g\left({x}_{n-1}\right)$,” Journal of Difference Equations and Applications, vol. 12, no. 9, pp. 909–918, 2006.
27. E. Camouzis and G. Ladas, “When does local asymptotic stability imply global attractivity in rational equations?” Journal of Difference Equations and Applications, vol. 12, no. 8, pp. 863–885, 2006.