A necessary and sufficient condition for a discrete dynamical system to be globally stable and plus-globally stable are first established in Section 2. The V-condition is introduced and Theorems 3.5 and 3.7 are presented in Section 3. The two theorems link the V-condition to the most relevant properties of globally stable and plus-globally stable discrete dynamical systems. In Section 4 we provide a simple application to a convergence problem for forward neural networks.