  The problem of stability of saturated and non-saturated ferromagnetism in the
Hubbard model is considered in terms of the one-particle Green's functions.
Approximations by Edwards and Hertz and some versions of the self-consistent
approximations based on the 1/z-expansion are considered. The account of
longitudinal fluctuations turns out to be essential for description of the
non-saturated state. The corresponding pictures of density of states are
obtained. "Kondo" density-of-states singularities owing to spin-flip processes
are analyzed. The critical electron concentrations for instabilities of
saturated ferromagnetism and paramagnetic state are calculated for various
lattices. Drawbacks of various approximations are discussed. A comparison with
the results of previous works is performed.
