Global attractivity and stability of a scalar nonlinear difference equation (Q1334567)

The author considers the autonomous scalar difference equation \[ x(t + 1) = f[x(t)],\qquad t = 0, 1, 2, 3, \dots \] where \(f\) is continuous and \(f(0) = 0\). If an appropriate Lyapunov function can be obtained, the equilibrium solution \(x = 0\) is globally asymptotically stable. In case \(f(x)\) has a discontinuity at \(x = 0\), it may be the case that the equilibrium solution is unstable but is a global attractor as the example \(x_{n+1} = {x \over 2}\) (if \(x \leq 0)\) and \(= - 1 - x\) (if \(x > 0)\) shows. General theorems for the equilibrium solution to be globally stable and for an unstable equilibrium to be a global attractor are given.