Phantom and non-phantom dark energy: The cosmological relevance of   non-locally corrected gravity
  In this paper we have investigated the cosmological dynamics of non-locally corrected gravity involving a function of the inverse d'Alembertian of the Ricci scalar, $f(\Box^{-1} R))$. Casting the dynamical equations into local form, we derive the fixed points of the dynamics and demonstrate the existence and stability of a one parameter family of dark energy solutions for a simple choice, $f(\Box^{-1} R)\sim \exp(\alpha \Box^{-1} R)$. The effective EoS parameter is given by, $w_{\rm eff}=({\alpha-1})/({3\alpha-1})$ and the stability of the solutions is guaranteed provided that $1/3<\alpha<2/3$. For $1/3<\alpha<1/2$ and $1/2<\alpha<2/3$, the underlying system exhibits phantom and non-phantom behavior respectively; the de Sitter solution corresponds to $\alpha=1/2$. For a wide range of initial conditions, the system mimics dust like behavior before reaching the stable fixed point. The late time phantom phase is achieved without involving negative kinetic energy fields. A brief discussion on the entropy of de Sitter space in non-local model is included. 
S. Jhingan, S. Nojiri, S. D. Odintsov, M. Sami, I Thongkool, S.   Zerbini