  We consider a class of nonlinear differential equations that arises in the
study of chemical reaction systems that are known to be locally asymptotically
stable and prove that they are in fact globally asymptotically stable. More
specifically, we will consider chemical reaction systems that are weakly
reversible, have a deficiency of zero, and are equipped with mass action
kinetics. We show that if for each $c \in \R_{> 0}^m$ the intersection of the
stoichiometric compatibility class $c + S$ with the subsets on the boundary
that could potentially contain equilibria, $L_W$, are at most discrete, then
global asymptotic stability follows. Previous global stability results for the
systems considered in this paper required $(c + S) \cap L_W = \emptyset$ for
each $c \in \R^m_{> 0}$, and so this paper can be viewed as an extension of
those works.
