  An approximated solution for gluon distribution from DGLAP evolution
equations with NLO splitting function in the small-$x$ limit is presented. We
first obtain the simplified forms of LO and NLO splitting functions in the
small-$x$ limit. With these approximated splitting functions, we obtain the
analytical gluon distribution by using the Mellin transform. The free
parameters in the boundary conditions are obtained by fitting the CJ15 gluon
distribution data. We find that the asymptotic behavior of gluon distribution
are consistent with the CJ15 data, however, the NLO results with the
consideration of ``ladder'' structure of gluon emission are slightly better
than those from LO. These results indicate that the corrections from NLO is
significant and is necessary for a better description of the behavior of the
gluon distribution in small-$x$ region. In addition, we investigate the DGLAP
evolution of the proton structure function by using the analytical solution of
the gluon distribution. The differential structure function shows that our
results have a similar tendency with CJ15 at small-$x$.
