Singularity problem in f(R) model with non-minimal coupling
  We consider the non-minimal coupling between matter and the geometry in the
f(R) theory. In the new theory which we established, a new scalar $\psi$ has
been defined and we give it a certain stability condition. We intend to take a
closer look at the dark energy oscillating behavior in the de-Sitter universe
and the matter era, from which we derive the oscillating frequency, and the
oscillating condition. More importantly, we present the condition of coupling
form that the singularity can be solved. We discuss several specific coupling
forms, and find logarithmic coupling with an oscillating period $\Delta
T\sim\Delta z$ in the matter era $z>4$, can improve singularity in the early
universe. The result of numerical calculation verifies our theoretic
calculation about the oscillating frequency. Considering two toy models, we
find the cosmic evolution in the coupling model is nearly the same as that in
the normal f(R) theory when $lna>4$. We also discuss the local tests of the
non-minimal coupling f(R) model, and show the constraint on the coupling form.
