Discrete Dynamics in Nature and Society

Volume 2007 (2007), Article ID 13737, 9 pages

http://dx.doi.org/10.1155/2007/13737

Research Article

## Asymptotic Periodicity of a Higher-Order Difference Equation

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 35/I, Beograd 11000 , Serbia

Received 27 April 2007; Accepted 13 September 2007

Copyright © 2007 Stevo Stević. 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

- K. S. Berenhaut, J. D. Foley, and S. Stević, “Quantitative bounds for the recursive sequence ${y}_{n+1}=A+{y}_{n}/{y}_{n-k}$,”
*Applied Mathematics Letters*, vol. 19, no. 9, pp. 983–989, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Berg, “On the asymptotics of nonlinear difference equations,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 21, no. 4, pp. 1061–1074, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Bibby, “Axiomatisations of the average and a further generalisation of monotonic sequences,”
*Glasgow Mathematical Journal*, vol. 15, pp. 63–65, 1974. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-C. Chang and D.-M. Nhieu, “A difference equation arising from logistic population growth,”
*Applicable Analysis*, vol. 83, no. 6, pp. 579–598, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. T. Copson, “On a generalisation of monotonic sequences,”
*Proceedings of the Edinburgh Mathematical Society*, vol. 17, pp. 159–164, 1970/1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. DeVault, C. Kent, and W. Kosmala, “On the recursive sequence ${x}_{n+1}=p+{x}_{n-k}/{x}_{n}$,”
*Journal of Difference Equations and Applications*, vol. 9, no. 8, pp. 721–730, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. M. El-Owaidy, A. M. Ahmed, and M. S. Mousa, “On asymptotic behaviour of the difference equation ${x}_{n+1}=\alpha +{x}_{n-1}^{p}/{x}_{n}^{p}$,”
*Journal of Applied Mathematics & Computing*, vol. 12, no. 1-2, pp. 31–37, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Karakostas, “Convergence of a difference equation via the full limiting sequences method,”
*Differential Equations and Dynamical Systems*, vol. 1, no. 4, pp. 289–294, 1993. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Karakostas, “Asymptotic 2-periodic difference equations with diagonally self-invertible responses,”
*Journal of Difference Equations and Applications*, vol. 6, no. 3, pp. 329–335, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Kosmala and C. Teixeira, “More on the difference equation ${y}_{n+1}=\left(p+{y}_{n-1}\right)/\left(q{y}_{n}+{y}_{n-1}\right)$,”
*Applicable Analysis*, vol. 81, no. 1, pp. 143–151, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Niven and H. S. Zuckerman,
*An Introduction to the Theory of Numbers*, 2nd edition, John Wiley & Sons, New York, NY, USA, 1991. - S. Stević, “A note on bounded sequences satisfying linear inequalities,”
*Indian Journal of Mathematics*, vol. 43, no. 2, pp. 223–230, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “A generalization of the Copson's theorem concerning sequences which satisfy a linear inequality,”
*Indian Journal of Mathematics*, vol. 43, no. 3, pp. 277–282, 2001. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “A global convergence result,”
*Indian Journal of Mathematics*, vol. 44, no. 3, pp. 361–368, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “A note on the difference equation ${x}_{n+1}={\displaystyle {\sum}_{i=0}^{k}{\alpha}_{i}/{x}_{n-i}^{{p}_{i}}}$,”
*Journal of Difference Equations and Applications*, vol. 8, no. 7, pp. 641–647, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “A global convergence results with applications to periodic solutions,”
*Indian Journal of Pure and Applied Mathematics*, vol. 33, no. 1, pp. 45–53, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic behavior of a sequence defined by iteration with applications,”
*Colloquium Mathematicum*, vol. 93, no. 2, pp. 267–276, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=A/{\prod}_{i=0}^{k}{x}_{n-i}+1/{\prod}_{j=k+2}^{2\left(k+1\right)}{x}_{n-j}$,”
*Taiwanese Journal of Mathematics*, vol. 7, no. 2, pp. 249–259, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={\alpha}_{n}+{x}_{n-1}/{x}_{n}$. II,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 10, no. 6, pp. 911–916, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Periodic character of a class of difference equation,”
*Journal of Difference Equations and Applications*, vol. 10, no. 6, pp. 615–619, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\left(\alpha +\beta {x}_{n-k}\right)/f\left({x}_{n},\dots ,{x}_{n-k+1}\right)$,”
*Taiwanese Journal of Mathematics*, vol. 9, no. 4, pp. 583–593, 2005. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic behavior of a class of nonlinear difference equations,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 47156, p. 10, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n}=1+{\displaystyle {\sum}_{i=1}^{k}{\alpha}_{i}{x}_{n-{p}_{i}}}/{\displaystyle {\sum}_{j=1}^{m}{\beta}_{j}{x}_{n-{q}_{j}}}$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 39404, p. 7, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation ${x}_{n+1}={x}_{n-1}/\left(p+{x}_{n}\right)$,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 20652, p. 7, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - S.-E. Takahasi, Y. Miura, and T. Miura, “On convergence of a recursive sequence ${x}_{n+1}=f\left({x}_{n-1},{x}_{n}\right)$,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 3, pp. 631–638, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. Wang, F.-P. Zeng, G.-R. Zhang, and X.-H. Liu, “Dynamics of the difference equation ${x}_{n+1}=\left(\alpha +{\beta}_{1}{x}_{n-1}+{B}_{3}{x}_{n-3}+\cdots +{B}_{2k+1}{x}_{n-2k-1}\right)/\left(A+{B}_{0}{x}_{n}+{B}_{2}{x}_{n-2}+\cdots +{B}_{2k}{x}_{n-2k}\right)$,”
*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 399–417, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet