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

Conditions and instability in $f(R)$ gravity with non-minimal coupling between matter and geometry