Evolution of the gluon density in x with a running coupling constant

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.