On Converses to the Stability Theorems for Difference Equations

In this paper, converses to a number of the Lyapunov-type stability theorems for difference equations are obtained. The basic vector difference equation considered is \[(1) X(n + 1) = f(n,X(n))\] subject to the initial condition X(n₀ ) = x₀. It is first shown that if the equilibrium $X = 0$ of the difference equation (1) is stable and if all solutions through any point $(n,X)$ in the domain considered can be uniquely extended back to an initial value at time n₀, then there exists a positive definite discrete Lyapunov function $V(n,X)$ whose total difference \[Δ V(n,X) = V(n + 1,f(n,X(n))) - V(n,X(n))\] is negative semidefinite along the discrete trajectories represented by the solutions. Moreover, if the equilibrium $X = 0$ of the linear difference equation $X(n + 1) = A(n)X(n)$, where $A(n)$ is a nonsingular matrix for all n, is asymptotically stable, then it is proved that there exists a positive definite Lyapunov function $V(n,X)$ whose total difference is negative definite along trajectories. Finally, if the equilibrium $X = 0$ of the difference equation (1) is uniformly asymptotically stable, then there exists a positive definite, decrescent, locally Lipschitzian Lyapunov function $V(n,X)$ whose total difference is negative definite along trajectories.