  This is a draft describing the calculation of the evolution of the gluon
density in $x$ from an initial value $x=x_{0}=0.01$ to smaller values, up to
$x=10^{-8}$ in the hard pomeron formalism with a running coupling introduced on
the basis of the bootstrap equation. The obtained gluon density is used to
calculate the singlet part of the proton structure function. Comparison with
experiment and the results following from the fixed coupling evolution is made.
