  The possibility to avoid the cosmic initial singularity as a consequence of
nonlinear effects on the Maxwell eletromagnetic theory is discussed. For a flat
FRW geometry we derive the general nonsingular solution supported by a magnetic
field plus a cosmic fluid and a nonvanishing vacuum energy density. The
nonsingular behavior of solutions with a time-dependent $\Lambda(t)$-term are
also examined. As a general result, it is found that the functional dependence
of $\Lambda(t)$ can uniquely be determined only if the magnetic field remains
constant. All these models are examples of bouncing universes which may exhibit
an inflationary dynamics driven by the nonlinear corrections of the magnetic
field.
