  In an open Friedmann-Robertson-Walker (FRW) space background, we study the
classical and quantum cosmological models in the framework of the recently
proposed nonlinear massive gravity theory. Although the constraints which are
present in this theory prevent it from admitting the flat and closed FRW models
as its cosmological solutions, for the open FRW universe, it is not the case.
We have shown that, either in the absence of matter or in the presence of a
perfect fluid, the classical field equations of such a theory adopt physical
solutions for the open FRW model, in which the mass term shows itself as a
cosmological constant. These classical solutions consist of two distinguishable
branches: One is a contacting universe which tends to a future singularity with
zero size, while another is an expanding universe having a past singularity
from which it begins its evolution. A classically forbidden region separates
these two branches from each other. We then employ the familiar canonical
quantization procedure in the given cosmological setting to find the
cosmological wave functions. We use the resulting wave function to investigate
the possibility of the avoidance of classical singularities due to quantum
effects. It is shown that the quantum expectation values of the scale factor,
although they have either contracting or expanding phases like their classical
counterparts, are not disconnected from each other. Indeed, the classically
forbidden region may be replaced by a bouncing period in which the scale factor
bounces from the contraction to its expansion eras. Using the Bohmian approach
of quantum mechanics, we also compute the Bohmian trajectory and the quantum
potential related to the system, which their analysis shows are the direct
effects of the mass term on the dynamics of the universe.
