Lyapunov function for a difference equation

Summary

A Lyapunov function for a difference equation $x_{n+1} = f(x_n)$ is a function that is positive away from an equilibrium and does not increase along solutions. It is the discrete-time version of the Lyapunov function of a differential equation. Its existence proves stability of the equilibrium without solving the equation, and a strict decrease proves asymptotic stability. This is the main method for nonlinear difference equations whose linearization gives no answer.