On the global attractivity and oscillations in a class of second order difference equations from macroeconomics

TL;DR

This paper develops new criteria for the global attractivity and oscillation of a class of second order difference equations in macroeconomics, using Lyapunov-like methods and comparison techniques.

Contribution

It introduces novel global attractivity criteria and oscillation conditions for a specific second order difference equation, supporting recent conjectures.

Findings

New global attractivity criteria derived using Lyapunov-like methods.

A necessary and sufficient condition for oscillation established.

Results support and extend recent conjectures in the field.

Abstract

New global attractivity criteria are obtained for the second order difference equation \[ x_{n+1}=cx_{n}+f(x_{n}-x_{n-1}),\quad n=1, 2, ... \] via a Lyapunov-like method. Some of these results are sharp and support recent related conjectures. Also, a necessary and sufficient condition for the oscillation of this equation is obtained using comparison with a second order linear difference equation with positive coefficients.