  An extremely fast exponential expansion of the Universe is typical for the
stable version of the inflationary model, based on the anomaly-induced action
of gravity. The total amount of exponential $e$-folds could be very large,
before the transition to the unstable version and the beginning of the
Starobinsky inflation. Thus, the stable exponential expansion can be seen as a
pre-inflationary semiclassical cosmological solution. We explore whether this
stable phase could follow after the bounce, subsequent to the contraction of
the Universe. Extending the previous consideration of the bounce, we explore
both stable expansion and the bounce solutions in the models with non-zero
cosmological constant and the presence of background radiation. The critical
part of the analysis concerns stability for small perturbations of the Hubble
parameter. It is shown that the stability is possible for the variations in the
bounce region, but not in the sufficiently distant past in the contraction
phase.
