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 $\overline{x}=1/2(q-1+\sqrt{(q-1{)}^2+4p})$. \medskip The above result, taken together with the 1993 result of Kocić and Ladas for equation (E) with $q\ge p$, 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.

10 pages