The Dynamics of Solutions to the Equation $x_{n+1} • Journal title : Kyungpook mathematical journal • Volume 50, Issue 1, 2010, pp.153-164 • Publisher : Department of Mathematics, Kyungpook National University • DOI : 10.5666/KMJ.2010.50.1.153 Title & Authors The Dynamics of Solutions to the Equation$x_{n+1}
Xu, Xiaona; Li, Yongjin;

Abstract
We study the global asymptotic stability, the character of the semicycles, the periodic nature and oscillation of the positive solutions of the difference equation $x_{n+1} Keywords Difference equations;Asymptotic stability;Periodicity;Semicycle;Oscillation; Language English Cited by References 1. A. M. Amleh, E. A. Grove, D. A. Georgiou, On the recursive sequence$x_{n+1} = \alpha+x_{n-1}/x_{n}$, J. Math. Anal. Appl., 233(1999), no. 2, 790-798. 2. K. S. Berenhaut, J. D. Foley, S. Stevie, The global attractivity of the rational difference equation$y_{n} = 1 + \frac{y_{n-k}}{y_{n-m}}$, Proc. Amer. Math. Soc., 135(2007), no. 4, 1133-1140. 3. K. S. Berenhaut, K. M. Donadio, J. D. Foley, On the rational recursive sequence$y_{n} = A + \frac{y_{n-1}}{y_{n-m}}$for small A, Appl. Math. Lett., 21(2008), no. 9, 906-909. 4. R. DeVault, W. Kosmala, G. Ladas, S. W. Schultz, Global behavior of$y_{n+1 = \frac{p+y_{n-k}}{qy_{n}+y_{n-k}}}$, Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000). Nonlinear Anal. 47(2001), no. 7, 4743-4751. 5. H. M. El-Owaidy, A. M. Ahmed, M. S. Mousa, On asymptotic behaviour of the difference equation$x_{n+1} = \alpha + \frac{x_{n-k}}{x_{n}}$, Appl. Math. Comput., 147(2004), no. 1, 163-167. 6. E. A. Grove and G. Ladas, Periodicities in nonlinear difference equations, Advances in Discrete Mathematics and Applications, 4. Chapman and Hall/CRC, Boca Raton, FL, 2005. 7. D. Mehdi, R. Narges, On the global behavior of a high-order rational difference equation, Comput. Phys. Comm., (2009), In Press, Available online 7 December 2008. 8. S. Ozen, I. Ozturk, F. Bozkurt, On the recursive sequence$\frac{\alpha+y_{n-1}}{\beta+y_{n}} + \frac{y_{n-1}}{y_{n}}$, Appl. Math. Comput., 188(2007), no. 1, 180-188. 9. M. Saleh, M. Aloqeili, On the rational difference equation$A + \frac{y_{n-k}}{y_{n}}$, Appl. Math. Comput., 171(2005), no. 2, 862-869. 10. S. Stevic, Asymptotic periodicity of a higher-order difference equation, Discrete Dyn. Nat. Soc., 2007, Art. ID 13737, 9 pp. 11. T. Sun, H. Xi, The periodic character of the difference equation$x_{n+1} = f(x_{n-l+1}, x_{n-2k+1})\$, Adv. Difference Equ., 2008, Art. ID 143723, 6 pages.