  We discuss how the modifications of the standard Einstein's equations needed
to support the cosmological bounce can at the same time lead to vanishing of
the well known cosmological constant problem, while also studying the effects
of spacetime fluctuations. We first concentrate on the case of the classical
FRWL spacetime and show that in a bouncing cosmology, where the modification of
the Einstein-Hilbert action can be represented by the most general function
needed to support the bounce, the cosmological constant problem is absent if
this function at late cosmological times approaches a sufficiently large value.
We show that this result is general and does not depend on the details of
modifications needed to support the cosmological bounce. Therefore, the
bouncing cosmologies could potentially at the same time solve the problem of
singularity and cosmological constant. Motivated by the recent studies of
cosmological constant problem in the context of fluctuations of the metric, we
then generalize our study to incorporate the effects of spacetime fluctuations.
We show that some problems of the recent earlier proposals, like singularities
and negative values of the scale factor, are also naturally resolved in the
approach proposed here.
