  In the theories of generalized modified gravity, the acceleration equation is
generally fourth order. So it is hard to analyze the evolution of the Universe.
In this paper, we present a class of generalized modified gravity theories
which have the acceleration equation of second order derivative. Then both the
cosmic evolution and the weak-field limit of the theories are easily
investigated. We find that not only the Big-bang singularity problem but also
the current cosmic acceleration problem could be easily dealt with.
