[en] 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 to considering models that are realistic ones, the Dolgov-Kawasaki instability will be investigated in each of them. (orig.)