  We study nonsingular cosmological scenarios in a general $D$-dimensional
string effective action of the dilaton-modulus-axion system in the presence of
the matter source. In the standard dilatonic Brans-Dicke parameter
($\omega=-1$) with radiation, we analytically obtain singularity-free bouncing
solutions where the universe starts out in a state with a finite curvature and
evolves toward the weakly coupled regime. We apply this analytic method to the
string-gas cosmology including the massive state in addition to the leading
massless state (radiation), with and without the axion. We numerically find
bouncing solutions which asymptotically approach an almost radiation-dominant
universe with a decreasing curvature irrespective of the presence of the axion,
implying that inclusion of the matter source is crucial for the existence of
such solutions for $\omega=-1$. In the theories with $\omega \ne -1$, it is
possible to obtain complete regular bouncing solutions with a finite dilaton
and curvature in both past and future asymptotics for the general dimension,
$D$. We also discuss the case where dilatonic higher-order corrections are
involved to the tree-level effective action and demonstrate that the presence
of axion/modulus fields and the matter source does not significantly affect the
dynamics of the dilaton-driven inflation and the subsequent graceful exit.
