Global Attractivity of the Equilibrium of a Difference Equation: An Elementary Proof Assisted by Computer Algebra System

TL;DR

This paper proves that solutions to a specific difference equation converge to a positive equilibrium for all positive parameters, completing a conjecture by Ladas with an elementary proof aided by computer algebra.

Contribution

It provides an elementary proof, assisted by computer algebra, establishing the global attractivity of the equilibrium for all positive parameters, confirming a longstanding conjecture.

Findings

Solutions converge to equilibrium when q < p.

The equilibrium is globally attractive for all positive parameters.

The proof completes the conjecture of Ladas.

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.