  We reconstruct $F(R)$ gravity models with exponential and power-law forms of
the scale factor in which bounce cosmology can be realized. We explore the
stability of the reconstructed models with analyzing the perturbations from the
background solutions. Furthermore, we study an $F(R)$ gravity model with a sum
of exponentials form of the scale factor, where the bounce in the early
universe as well as the late-time cosmic acceleration can be realized in a
unified manner. As a result, we build a second order polynomial type model in
terms of $R$ and show that it could be stable. Moreover, when the scale factor
is expressed by an exponential form, we derive $F(R)$ gravity models of a
polynomial type in case of the non-zero spatial curvature and that of a generic
type in that of the zero spatial curvature. In addition, for an exponential
form of the scale factor, an $F(R)$ bigravity model realizing the bouncing
behavior is reconstructed. It is found that in both the physical and reference
metrics the bouncing phenomenon can occur, although in general the contraction
and expansion rates are different each other.
