  We present molecular dynamics simulations on the slow dynamics of a mixture
of big and small soft-spheres with a large size disparity. Dynamics are
investigated in a broad range of temperature and mixture composition. As a
consequence of large size disparity, big and small particles exhibit very
different relaxation times. As previously reported for simple models of
short-ranged attractive colloids and polymer blends, several anomalous dynamic
features are observed: i) sublinear behavior for mean squared displacements,
ii) concave-to-convex crossover for density-density correlators, by varying
temperature or wavevector, iii) logarithmic decay for specific wavevectors of
density-density correlators. These anomalous features are observed over time
intervals extending up to four decades, and strongly resemble predictions of
the Mode Coupling Theory (MCT) for state points close to higher-order MCT
transitions, which originate from the competition between different mechanisms
for dynamic arrest. For the big particles we suggest competition between
soft-sphere repulsion and depletion effects induced by neighboring small
particles. For the small particles we suggest competition between bulk-like
dynamics and confinement, respectively induced by neighboring small particles
and by the slow matrix of big particles. By increasing the size disparity, a
new relaxation scenario arises for the small particles. Self-correlators decay
to zero at temperatures where density-density correlations are frozen. The
behavior of the latters resembles features characteristic of type-A MCT
transitions, defined by a zero value of the critical non-ergodicity parameter.
