  By employing a simplified nonlinear memory function proposed recently by the
present author, a universal equation for a collective-intermediate scattering
function derived based on the time-convolutionless mode-coupling theory is
numerically solved to study the dynamics of glass-forming liquids. The
numerical calculation is based on the simulation results performed on two types
of liquids, fragile liquids and strong liquids. The numerical solutions are
then shown to be uniquely determined by the long-time collective diffusion
coefficient $D(q_m)$, where $q_m$ is a first peak position of a static
structure factor for a whole system. It is confirmed based on four different
simulation results that the supercooled state consists of two substates, a
weakly supercooled state in which the nonlinear parameter $\mu$ in the memory
function increases as $D$ decreases and a deeply supercooled state in which
$\mu$ becomes constant up to the glass transition. Here $\mu$ is shown to be
constant in a liquid state, while it is shown to grow rapidly in a glass state.
The value of $\mu$ in each state is shown to be uniquely determined by $D$ and
also to depend on a type of liquids. Hence there exists such a universality
that there is only one solution for different liquids of a same type at a given
value of $D$. This may be consistent with the fact that strong liquids are
structurally quite different from fragile liquids. Thus, it is emphasized that
such a universality must be helpful to predict $q_m$ from experimental data.
