  We consider the superconducting pairing induced by spin waves exchange in a
ferromagnet with both conduction and localized electrons, the latter being
described as spins. We use the microscopic Eliashberg theory to describe the
pairing of conducting electrons and the RPA approach to treat the localized
spins assuming an exchange coupling between the conducting electrons and spins.
In the framework of non relativistic Hamiltonian twe found that he spin wave
exchange results in equal spin electron pairing described by the two components
of the order parameter, $\Delta^{\uparrow}$ (both spins up) and
$\Delta^{\downarrow}$ (both spins down). Due to the conservation of total spin
projection on the axis of the spontaneous ferromagnetic moment, the spin wave
exchange at low temperatures includes an emission of magnons and an absorption
of thermal magnons by the conduction electrons. The absorption and emission
processes depend differently on the temperature, with the absorption being
progressively suppressed as the temperature drops. As a result, the
superconducting pairing exists only if the electron-spin wave exchange
parameter $g$ exceeds some critical value $g_c$. At $g>g_c$ pairing vanishes if
the temperature drops below the lowest point $T_{cl}$ or increases above the
upper critical point $T_{ch} \approx T_m$ (the Curie temperature) where the
spin waves cease to exist. This behavior inherent to the spin carrying glue is
in an obvious disagreement with the results of conventional BCS approach which
assumes that the effective electron-electron attraction is simply proportional
to the static magnetic susceptibility.
