Discrete Dynamics in Nature and Society

Volume 2010, Article ID 235808, 14 pages

http://dx.doi.org/10.1155/2010/235808

Research Article

## Quantitative Bounds for Positive Solutions of a Stević Difference Equation

College of Computer Science, Chongqing University, Chongqing 400044, China

Received 8 November 2009; Revised 5 March 2010; Accepted 7 April 2010

Academic Editor: Leonid Berezansky

Copyright © 2010 Wanping Liu and Xiaofan Yang. 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

- S. Stević, “Boundedness character of a max-type difference equation,” in
*Conference in Honour of Allan Peterson, Book of Abstracts*, p. 28, Novacella, Italy, July-August 2007. - S. Stević, “Boundedness character of two classes of higher order difference equations,” in
*Progress on Difference Equations, Book of Abstracts*, p. 31, Laufen, Germany, March 2008. - S. Stević, “On behavior of a class of difference equations with maximum,” in
*Mathematical Models in Engineering, Biology and Medicine. Conference on Boundary Value Problems. Book of Abstracts*, p. 35, Santiago de Compostela, Spain, September 2008. - R. M. Abu-Saris and R. DeVault, “Global stability of ${y}_{n}=A+{y}_{n}/{y}_{n-k}$,”
*Applied Mathematics Letters*, vol. 16, no. 2, pp. 173–178, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. M. Amleh, E. A. Grove, G. Ladas, and D. A. Georgiou, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}/{x}_{n}$,”
*Journal of Mathematical Analysis and Applications*, vol. 233, no. 2, pp. 790–798, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation ${y}_{n}=1+{({y}_{n-k}/{y}_{n-m})}^{p}$,”
*Proceedings of the American Mathematical Society*, vol. 136, no. 1, pp. 103–110, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut, J. D. Foley, and S. Stević, “The global attractivity of the rational difference equation ${y}_{n}=1+{y}_{n-k}/{y}_{n-m}$,”
*Proceedings of the American Mathematical Society*, vol. 135, no. 4, pp. 1133–1140, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. S. Berenhaut and S. Stević, “A note on positive non-oscillatory solutions of the difference equation ${x}_{n+1}=\alpha +{x}_{n-k}^{p}/{x}_{n}^{p}$,”
*Journal of Difference Equations and Applications*, vol. 12, no. 5, pp. 495–499, 2006. View at Publisher · 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 ${x}_{n+1}=A+{x}_{n-2}^{p}/{x}_{n-1}^{p}$,”
*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 - R. Devault, V. L. Kocic, and D. Stutson, “Global behavior of solutions of the nonlinear difference equation ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$,”
*Journal of Difference Equations and Applications*, vol. 11, no. 8, pp. 707–719, 2005. 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 ${x}_{n+1}=\alpha +{x}_{n-k}^{p}/{x}_{n}^{p}$,”
*Journal of Applied Mathematics & Computing*, vol. 12, no. 1-2, pp. 31–37, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Gutnik and S. Stević, “On the behaviour of the solutions of a second-order difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 27562, 14 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. E. Hamza and A. Morsy, “On the recursive sequence ${x}_{n+1}=A+{x}_{n-1}/{x}_{n}^{k}$,”
*Applied Mathematics Letters*, vol. 22, no. 1, pp. 91–95, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Iričanin and S. Stević, “On a class of third-order nonlinear difference equations,”
*Applied Mathematics and Computation*, vol. 213, no. 2, pp. 479–483, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. R. S. Kulenović, G. Ladas, and C. B. Overdeep, “On the dynamics of ${x}_{n+1}={p}_{n}+{x}_{n-1}/{x}_{n}$ with a period-two coefficient,”
*Journal of Difference Equations and Applications*, vol. 10, no. 10, pp. 905–914, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of a class of difference equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 70, no. 2, pp. 839–848, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the difference equation ${x}_{n+1}=\alpha +{x}_{n-1}/{x}_{n}$,”
*Computers & Mathematics with Applications*, vol. 56, no. 5, pp. 1159–1171, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\}$,”
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 791–796, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=A+{x}_{n}^{p}/{x}_{n-1}^{r}$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 40963, 9 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=A+{x}_{n-1}^{p}/{x}_{n}^{p}$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 34517, 9 pages, 2007. View at Publisher · 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 Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=\alpha +{x}_{n-1}/{x}_{n}$. II,”
*Dynamics of Continuous, Discrete & Impulsive Systems. Series A*, vol. 10, no. 6, pp. 911–916, 2003. View at Google Scholar · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}=A/{\prod}_{i=0}^{k}{x}_{n-i}+1/({\prod}_{j=k+2}^{2(k+1)}{x}_{n-j})$,”
*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 a nonlinear generalized max-type difference equation,” preprint, 2009.
- S. Stević, “On the recursive sequence ${x}_{n}=1+({\sum}_{i=1}^{k}{\alpha}_{i}{x}_{n-{p}_{i}}/{\sum}_{j=1}^{m}{\beta}_{j}{x}_{n-{q}_{j}})$,”
*Discrete Dynamics in Nature and Society*, vol. 2007, Article ID 39404, p. 7, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of a fourth order nonlinear difference equation,”
*Chaos, Solitons & Fractals*, vol. 40, no. 5, pp. 2364–2369, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Boundedness character of two classes of third-order difference equations,”
*Journal of Difference Equations and Applications*, vol. 15, no. 11-12, pp. 1193–1209, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On a class of higher-order difference equations,”
*Chaos, Solitons & Fractals*, vol. 42, no. 1, pp. 138–145, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On a generalized max-type difference equation from automatic control theory,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 3-4, pp. 1841–1849, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Sun, X. Yang, and C. Zhang, “On the recursive sequence ${x}_{n}=A+{x}_{n-k}^{p}/{x}_{n-1}^{r}$,”
*Discrete Dynamics in Nature and Society*, vol. 2009, Article ID 608976, 8 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet - T. Sun, B. Qin, H. Xi, and C. Han, “Global behavior of the max-type difference equation ${x}_{n+1}=\mathrm{max}\hspace{0.17em}\{1/{x}_{n},{A}_{n}/{x}_{n-1}\}$,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 152964, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - H. Xi, T. Sun, W. Yu, and J. Zhao, “On boundedness of solutions of the difference equation ${x}_{n+1}=(p{x}_{n}+q{x}_{n-1})/(1+{x}_{n})$ for $q>1+p>1$,”
*Advances in Difference Equations*, vol. 2009, Article ID 463169, 11 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus - T. Sun, H. Xi, and M. Xie, “Global stability for a delay difference equation,”
*Journal of Applied Mathematics and Computing*, vol. 29, no. 1-2, pp. 367–372, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, H. Xi, and W. Quan, “Existence of monotone solutions of a difference equation,”
*Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 917560, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun, H. Xi, and H. Wu, “On boundedness of the solutions of the difference equation ${x}_{n+1}={x}_{n-1}/(p+{x}_{n})$,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 20652, 7 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Sun and H. Xi, “On the global behavior of the nonlinear difference equation ${x}_{n+1}=f({p}_{n},{x}_{n-m},{x}_{n-t(k+1)+1})$,”
*Discrete Dynamics in Nature and Society*, vol. 2006, Article ID 90625, 12 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang, Y. Y. Tang, and J. Cao, “Global asymptotic stability of a family of difference equations,”
*Computers & Mathematics with Applications*, vol. 56, no. 10, pp. 2643–2649, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang, Y. Yang, and J. Luo, “On the difference equation ${x}_{n}=(p+{x}_{n-s})/(q{x}_{n-t}+{x}_{n-s})$,”
*Applied Mathematics and Computation*, vol. 189, no. 1, pp. 918–926, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet