  The possibility of obtaining singularity free cosmological solutions in four
dimensional effective actions motivated by string theory is investigated. In
these effective actions, in addition to the Einstein-Hilbert term, the
dilatonic and the axionic fields are also considered as well as terms coming
from the Ramond-Ramond sector. A radiation fluid is coupled to the field
equations, which appears as a consequence of the Maxwellian terms in the
Ramond-Ramond sector. Singularity free bouncing solutions in which the dilaton
is finite and strictly positive are obtained for models with flat or negative
curvature spatial sections when the dilatonic coupling constant is such that
$\omega < - 3/2$, which may appear in the so called $F$ theory in 12
dimensions. These bouncing phases are smoothly connected to the radiation
dominated expansion phase of the standard cosmological model, and the
asymptotic pasts correspond to very large flat spacetimes.
