Global attractivity of the equilibrium of a difference equation: An elementary proof assisted by computer algebra system

Abstract

Let p and q be arbitrary positive numbers. It is shown that if , then all solutions to the difference equation

converge to the positive equilibrium . The above result, taken together with the 1993 result of Kocić and Ladas for equation (E) with , gives global attractivity of the positive equilibrium of (E) for all positive values of the parameters, thus completing the proof of a conjecture of Ladas.

Keywords:

• difference equation
• rational
• second order
• global attractivity
• equilibrium

AMS 2000 Mathematics Subject Classification::

• 39A11

Acknowledgements

The author thanks M. Kulenović for helpful discussions and for suggesting the change of variables (1) which established a direct connection of the main difference equation to Lyness' equation, and J. Montgomery for suggesting a simplification of the proof. Also thanks to G. Bastien and M. Rogalski, who read a preliminary version and suggested improvements, and to two anonymous referees for their careful reading of the paper and for their valuable suggestions.