  For a large class of mass-varying massive gravity models, the graviton mass
cannot provide the late-time cosmic expansion of the universe due to its
vanishing at late time. In this work, we propose a new class of mass-varying
massive gravity in which the graviton mass varies according to a kinetic term
of a k-essence field. By using a more general form of the fiducial metric, we
found a solution such that a non-vanishing graviton mass can drive the
accelerated expansion of the universe at late time. We also perform dynamical
analyses of such model and found that without introducing the k-essence
Lagrangian, the graviton mass can be responsible for both dark contents of the
universe, namely dark energy that drives the accelerated expansion of the
universe and non-relativistic matter that plays the role of dark matter.
Moreover, by including the k-essence Lagrangian, we found that it is possible
to alleviate the so-called cosmic coincidence problem.
