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.
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