  The paper endeavours to solve the problem of the necessary and sufficient
conditions for testing asymptotic stability of the equilibrium state without
using a positive definite or semi-definite Lyapunov function for time-invariant
nonlinear discrete-time dynamical systems. The solution is based on the concept
of the $\bf G$-functions introduced in this paper. As a result, new necessary
and sufficient conditions for asymptotic stability of such systems and an
estimation or the exact determination of the asymptotic stability domain of the
state $x=0$ are established. Examples are worked out to illustrate the results.
