  In this paper on the basis of the generalized $f(R)$ gravity model with
arbitrary coupling between geometry and matter, four classes of $f(R)$ gravity
models with non minimal coupling between geometry and matter will be studied.
By means of conditions of power law expansion and the equation of state of
matter less than -1/3, the relationship among p, w and n, the conditions and
the candidate for late time cosmic accelerated expansion will be discussed in
the four classes of $f(R)$ gravity models with non minimal coupling.
Furthermore, in order to keep considering models to be realistic ones, the
Dolgov Kawasaki instability will be investigated in each of them.
