  Motivated by Horava-Lifshitz gravity theory, we propose and investigate two
kinds of modified gravity theories, the f(R) kind and the K-essence kind, in
the Arnowitt-Deser-Misner (ADM) formalism. The f(R) kind includes one
ultraviolet (UV) term and one infrared (IR) term together with the
Einstein-Hilbert action. We find that these two terms naturally present the
ultraviolet and infrared modifications to the Friedmann equation. The UV and IR
modifications can avoid the past Big-Bang singularity and the future Big-Rip
singularity, respectively. Furthermore, the IR modification can naturally
account for the current acceleration of the Universe. The Lagrangian of
K-essence kind modified gravity is made up of the three dimensional Ricci
scalar and an arbitrary function of the extrinsic curvature term. We find the
cosmic acceleration can also be naturally interpreted without invoking any kind
of dark energy. The static, spherically symmetry and vacuum solutions of both
theories are Schwarzschild or Schwarzschild-de Sitter solution. Thus these
modified gravity theories are viable for solar system tests.
