  The generalized $f(R)$ gravity with curvature-matter coupling in
five-dimensional (5D) spacetime can be established by assuming a
hypersurface-orthogonal spacelike Killing vector field of 5D spacetime, and it
can be reduced to the 4D formulism of FRW universe. This theory is quite
general and can give the corresponding results to the Einstein gravity, $f(R)$
gravity with both no-coupling and non-minimal coupling in 5D spacetime as
special cases, that is, we would give the some new results besides previous
ones given by Ref.\cite{60}. Furthermore, in order to get some insight into the
effects of this theory on the 4D spacetime, by considering a specific type of
models with $f_{1}(R)=f_{2}(R)=\alpha R^{m}$ and $B(L_{m})=L_{m}=-\rho$, we not
only discuss the constraints on the model parameters $m$, $n$, but also
illustrate the evolutionary trajectories of the scale factor $a(t)$, the
deceleration parameter $q(t)$ and the scalar field $\epsilon(t)$, $\phi(t)$ in
the reduced 4D spacetime. The research results show that this type of $f(R)$
gravity models given by us could explain the current accelerated expansion of
our universe without introducing dark energy.

General Relativity and Quantum Cosmology