  In this paper we study the occurrence of accelerating universe versus
decelerating universe between the F(R) gravity frame (Jordan frame) and
non-minimally coupled scalar field theory frame, and the minimally coupled
scalar field theory frame (Einstein frame) for various models. As we show, if
acceleration is imposed in one frame, it will not necessarily correspond to an
accelerating metric when transformed in another frame. As we will demonstrate,
this issue is model and frame-dependent but it seems there is no general scheme
which permits to classify such cases.
