  For higher-derivative f(R) gravity where R is the Ricci scalar, a class of
models is proposed which produce viable cosmology different from the LambdaCDM
one at recent times and satisfy cosmological, Solar system and laboratory
tests. These models have both flat and de Sitter space-times as particular
solutions in the absence of matter. Thus, a cosmological constant is zero in
flat space-time, but appears effectively in a curved one for sufficiently large
R. A 'smoking gun' for these models would be small discrepancy in values of the
slope of the primordial perturbation power spectrum determined from galaxy
surveys and CMB fluctuations. On the other hand, a new problem for dark energy
models based on f(R) gravity is pointed which is connected with possible
overproduction of new massive scalar particles (scalarons) arising in this
theory in the very early Universe.
