  We study flat Friedmann-Robertson-Walker (FRW) models with a perfect fluid
matter source and a scalar field non minimally coupled to matter having a
double exponential potential. It is shown that the scalar field almost always
diverges to infinity. Under conditions on the parameter space, we show that the
model is able to give an acceptable cosmological history of our universe, that
is, a transient matter era followed by an accelerating future attractor. It is
found that only a very weak coupling can lead to viable cosmology. We study in
the Einstein frame, the cosmological viability of the asymptotic form of a
class of f(R) theories predicting acceleration. The role of the coupling
constant is briefly discussed.
