Little Rip, $\Lambda$CDM and singular dark energy cosmology from   Born-Infeld-$f(R)$ gravity
  We study late-time cosmic accelerating dynamics from Born-Infeld-$f(R)$ gravity in a simplified conformal approach. We find that a variety of cosmic efects such as Little Rip, $\Lambda$CDM universe and dark energy cosmology with finite-time future singularities may occur. Unlike the convenient Born-Infeld gravity where in the absence of matter only de Sitter expansion may emerge, apparentlly any FRW cosmology maybe reconstructed from this conformal version of the Born-Infeld-$f(R)$ theory. Despite the fact that the explicit form of $f(R)$ is fixed by the conformal ansatz, the relation between the two metrics in this approach may be changed so as to bring out any desired FRW cosmology. 
Andrey N. Makarenko, Sergei D. Odintsov, Gonzalo J. Olmo
gr-qc hep-th