  In present article we consider an axion F(R) gravity model and described with
the help of holographic principle the cosmological models of viscous dark fluid
coupled with axion matter in a spatially flat Friedmann-Robertson-Walker (FRW)
universe. This description based on generalized infrared-cutoff holographic
dark energy, proposed by Nojiri and Odintsov. We explored the Little Rip, the
Pseudo Rip, and the power-law bounce cosmological models in terms of the
parameters of the inhomogeneous equation of the state of viscous dark fluid and
calculated the infrared cutoffs analytically. We represented the energy
conservation equation for the dark fluid from a holographic point of view and
showed a correspondence between the cosmology of a viscous fluid and
holographic cosmology. We analyzed the autonomous dynamic system. In the
absence of interaction between fluids, solutions are obtained corresponding to
two cases. In the first case, dark energy is missing and the extension
describes the component of dark matter. The second case corresponds to
cosmological models with an extension due to dark energy. The solutions
obtained are investigated for stability. For a cosmological model with the
interaction of a special type, the stability of solutions of the dynamic system
is also investigated.
