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 $q < p$, then all solutions to the difference equation \tag{E} x_{n+1} = \frac{p+q x_n}{1+x_{n-1}}, \quad n=0,1,2,..., \quad x_{-1}>0, x_0>0 converge to the positive equilibrium $\overline{x} = {1/2}(q-1 + \sqrt{(q-1)^2 + 4 p})$. \medskip The above result, taken together with the 1993 result of Koci\'c and Ladas for equation (E) with $q \geq p$, 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.