Discrete Dynamics in Nature and Society

Volume 2006 (2006), Article ID 47156, 10 pages

http://dx.doi.org/10.1155/DDNS/2006/47156

## Asymptotic behavior of a class of nonlinear difference equations

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

Received 7 April 2006; Accepted 28 May 2006

Copyright © 2006 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 and E. G. Goedhart, “Explicit bounds for second-order difference equations and a solution to a question of Stević,”
*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 1–10, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Berg,
*Asymptotische Darstellungen und Entwicklungen*, Hochschulbücher für Mathematik, Band 66, VEB Deutscher Verlag der Wissenschaften, Berlin, 1968. 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 - L. Berg, “Corrections to “Inclusion theorems for non-linear difference equations with
applictions”,”
*Journal of Difference Equations and Applications*, vol. 11, no. 2, pp. 181–182, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Berg and L. von Wolfersdorf, “On a class of generalized autoconvolution equations of the third kind,”
*Zeitschrift für Analysis und ihre Anwendungen*, vol. 24, no. 2, pp. 217–250, 2005. View at Google Scholar · View at MathSciNet - R. J. Beverton and S. J. Holt,
*On the Dynamics of Exploited Fish Populations*, vol. 19, Fish. Invest., London, 1957. - 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. Borwein, “Convergence criteria for bounded sequences,”
*Proceedings of the Edinburgh Mathematical Society. Series II*, vol. 18, pp. 99–103, 1972/1973. 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. Series II*, vol. 17, pp. 159–164, 1970/1971. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Cox, “Problem E 3034,”
*American Mathematical Monthly*, vol. 91, p. 58, 1984. View at Google Scholar - N. G. de Bruijn,
*Asymptotic Methods in Analysis*, vol. 4 of*Bibliotheca Mathematica*, North-Holland, Amsterdam, 1958. View at Zentralblatt MATH · View at MathSciNet - R. DeVault, G. Dial, V. L. Kocic, and G. Ladas, “Global behavior of solutions of ${x}_{n+1}=a{x}_{n}+f({x}_{n},{x}_{n-1})$,”
*Journal of Difference Equations and Applications*, vol. 3, no. 3-4, pp. 311–330, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ya. Dorogovcev, “Problem MS'88 no. 6,”
*Matematika Segodnya*, vol. 6, pp. 176–177, 1990 (Russian). View at Google Scholar - A. W. Goodman, “Problem 6610,”
*American Mathematical Monthly*, vol. 96, p. 774, 1989. View at Google Scholar - E. A. Grove, C. M. Kent, G. Ladas, S. Valicenti, and R. Levins, “Global stability in some population models,” in
*Communications in Difference Equations. Proceedings of the 4th International Conference on Difference Equations (Poznan, 1998)*, pp. 149–176, Gordon and Breach, Amsterdam, 2000. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H.-F. Huo and W.-T. Li, “Permanence and global stability of positive solutions of a nonautonomous discrete ratio-dependent predator-prey model,”
*Discrete Dynamics in Nature and Society*, vol. 2005, no. 2, pp. 135–144, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Kuang and J. M. Cushing, “Global stability in a nonlinear difference-delay equation model of flour beetle population growth,”
*Journal of Difference Equations and Applications*, vol. 2, no. 1, pp. 31–37, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. G. Pakes, “Asymptotic behaviour of monotone weakly-nonlinear recurrences,”
*The Australian Mathematical Society. Gazette*, vol. 29, no. 2, pp. 91–98, 2002. View at Google Scholar · View at MathSciNet - E. C. Pielou,
*Population and Community Ecology*, Gordon and Breach, New York, 1974. - G. Polya and G. Szegö,
*Aufgaben und lehrsätze aus der analysis*, Verlag von Julius, Berlin, 1925. - D. C. Russell, “On bounded sequences satisfying a linear inequality,”
*Proceedings of the Edinburgh Mathematical Society. Series II*, vol. 19, pp. 11–16, 1974/1975. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic behaviour of a sequence defined by iteration,”
*Matematichki Vesnik*, vol. 48, no. 3-4, pp. 99–105, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Behavior of the positive solutions of the generalized Beddington-Holt equation,”
*Panamerican Mathematical Journal*, vol. 10, no. 4, pp. 77–85, 2000. 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, “On the recursive sequence ${x}_{n+1}=-\frac{1}{{x}_{n}}+\frac{A}{{x}_{n-1}}$,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 27, no. 1, pp. 1–6, 2001. View at Publisher · 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 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, “Asymptotic behaviour of a sequence defined by a recurrence formula. II,”
*The Australian Mathematical Society. Gazette*, vol. 29, no. 4, pp. 209–215, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={x}_{n-1}/g({x}_{n})$,”
*Taiwanese Journal of Mathematics*, vol. 6, no. 3, pp. 405–414, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “Asymptotic behavior of a nonlinear difference equation,”
*Indian Journal of Pure and Applied Mathematics*, vol. 34, no. 12, pp. 1681–1687, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Stević, “On the recursive sequence ${x}_{n+1}={x}_{n}+\frac{{x}_{n}^{\alpha}}{{n}^{\beta}}$,”
*Bulletin of the Calcutta Mathematical Society*, vol. 95, no. 1, pp. 39–46, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at MathSciNet - S. Stević, “On positive solutions of a $(k+1)$-th order difference equation,”
*Applied Mathematics Letters*, vol. 19, no. 5, pp. 427–431, 2006. View at Publisher · View at Google Scholar - International student competition in mathematics, Belgrade 1984.