  We consider the process of double $J/\psi$ production in $pp$ collisions at
the LHC in the framework of $k_T$-factorization approach. We focus on the gluon
fragmentation mechanism which is related to multiple gluon emission in the
initial state. The initial state emission is treated according to
Catani-Ciafaloni-Fiorani-Marchesini evolution equation, and applies to both
single and double parton scattering cases. We show the importance of
fragmentation contributions to $J/\psi$ pair production and perform a
comparison between theoretical predictions and the latest ATLAS data collected
at $\sqrt{s} = 8$ TeV. We find that the effects of multiple gluon emission are
essential for both single- and multiparton interaction processes. Finally, we
highlight the problem of correct choice of the factorization scale with respect
to numerical stability of the calculations and the consistency with
non-collinear evolution equations.
