  We propose the reconstruction of $F(R)$ gravity in such a way that
corresponding theory admits cyclic and ekpyrotic universe solutions. The number
of explicit examples of such $F(R)$ model is found. The comparison with the
reconstructed scalar-tensor theory is made. We also present $F(R)$ gravity
which provides the little rip evolution and gives the realistic gravitational
alternative for $\Lambda$CDM cosmology. The time for little rip dissolution of
bound structures in such theory is estimated. We demonstrate that transformed
little rip $F(R)$ solution becomes qualitatively different cosmological
solution with Big Bang type singularity in Einstein frame.
