Open Access

Global Asymptotic Behavior of

Advances in Difference Equations20082007:041541

DOI: 10.1155/2007/41541

Received: 9 July 2007

Accepted: 19 November 2007

Published: 17 January 2008

Abstract

We investigate the global stability character of the equilibrium points and the period-two solutions of , with positive parameters and nonnegative initial conditions. We show that every solution of the equation in the title converges to either the zero equilibrium, the positive equilibrium, or the period-two solution, for all values of parameters outside of a specific set defined in the paper. In the case when the equilibrium points and period-two solution coexist, we give a precise description of the basins of attraction of all points. Our results give an affirmative answer to Conjecture 9.5.6 and the complete answer to Open Problem 9.5.7 of Kulenović and Ladas, 2002.

[1234567891011121314151617]

Authors’ Affiliations

(1)
Department of Mathematics, University of Rhode Island

References

  1. Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2002:xii+218.MATHGoogle Scholar
  2. Kocić VL, Ladas G, Rodrigues IW: On rational recursive sequences. Journal of Mathematical Analysis and Applications 1993,173(1):127–157. 10.1006/jmaa.1993.1057MATHMathSciNetView ArticleGoogle Scholar
  3. Gibbons CH, Kulenović MRS, Ladas G: On the recursive sequence . Mathematical Sciences Research Hot-Line 2000,4(2):1–11.MATHMathSciNetGoogle Scholar
  4. Kulenović MRS, Ladas G, Prokup NR: On the recursive sequence . Journal of Difference Equations and Applications 2000,6(5):563–576. 10.1080/10236190008808246MATHMathSciNetView ArticleGoogle Scholar
  5. Kulenović MRS, Ladas G, Prokup NR: A rational difference equation. Computers & Mathematics with Applications 2001,41(5–6):671–678. 10.1016/S0898-1221(00)00311-4MATHMathSciNetView ArticleGoogle Scholar
  6. Kulenović MRS, Ladas G, Sizer WS: On the recursive sequence . Mathematical Sciences Research Hot-Line 1998,2(5):1–16.MATHMathSciNetGoogle Scholar
  7. Kulenović MRS, Merino O: Convergence to a period-two solution for a class of second order rational difference equations. In Proceedings of the 10th International Conference on Difference Equations, July 2007, Munich, Germany. World Scientific; 344–353.Google Scholar
  8. Kulenović MRS, Merino O: Global attractivity of the equilibrium of for . Journal of Difference Equations and Applications 2006,12(1):101–108. 10.1080/10236190500410109MATHMathSciNetView ArticleGoogle Scholar
  9. Nussbaum RD: Global stability, two conjectures and Maple. Nonlinear Analysis: Theory, Methods & Applications 2007,66(5):1064–1090. 10.1016/j.na.2006.01.005MATHMathSciNetView ArticleGoogle Scholar
  10. Camouzis E, Ladas G: When does local asymptotic stability imply global attractivity in rational equations? Journal of Difference Equations and Applications 2006,12(8):863–885. 10.1080/10236190600772663MATHMathSciNetView ArticleGoogle Scholar
  11. Enciso GA, Sontag ED: Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete and Continuous Dynamical Systems. Series A 2006,14(3):549–578.MATHMathSciNetGoogle Scholar
  12. Kulenović MRS, Yakubu A-A: Compensatory versus overcompensatory dynamics in density-dependent Leslie models. Journal of Difference Equations and Applications 2004,10(13–15):1251–1265.MATHMathSciNetView ArticleGoogle Scholar
  13. Smith HL: The discrete dynamics of monotonically decomposable maps. Journal of Mathematical Biology 2006,53(4):747–758. 10.1007/s00285-006-0004-3MATHMathSciNetView ArticleGoogle Scholar
  14. Kulenović MRS, Merino O: A global attractivity result for maps with invariant boxes. Discrete and Continuous Dynamical Systems. Series B 2006,6(1):97–110.MATHMathSciNetGoogle Scholar
  15. Janowski EJ, Kulenović MRS: Attractivity and global stability for linearizable difference equations.Google Scholar
  16. Kulenović MRS, Merino O: Competitive-exclusion versus competitive-coexistence for systems in the plane. Discrete and Continuous Dynamical Systems. Series B 2006,6(5):1141–1156.MATHMathSciNetView ArticleGoogle Scholar
  17. Smith HL: Planar competitive and cooperative difference equations. Journal of Difference Equations and Applications 1998,3(5–6):335–357. 10.1080/10236199708808108MATHMathSciNetView ArticleGoogle Scholar

Copyright

© A. Brett and M. R. S. Kulenović 2007

This article is published under license to BioMed Central Ltd. 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.