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 ΛCDM 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.Comment: LaTeX 11 pages, no figure, typos correcte
