  Theoretical predictions show that at low values of Bjorken $x$ the spin
structure function, $g_1$ is influenced by large logarithmic corrections,
$ln^2(1/x)$, which may be predominant in this region. These corrections are
also partially contained in the NLO part of the standard DGLAP evolution. Here
we calculate the non-singlet component of the nucleon structure function,
$g_1^{NS}=g_1^p-g_1^n$, and its first moment, using a unified evolution
equation. This equation incorporates the terms describing the NLO DGLAP
evolution and the terms contributing to the $ln^2(1/x)$ resummation. In order
to avoid double counting in the overlapping regions of the phase-space, a
unique way of including the NLO terms into the unified evolution equation is
proposed. The scheme-independent results obtained from this unified evolution
are compared to the NLO fit to experimental data, GRSV'2000. Analysis of the
first moments of $g_1^{NS}$ shows that the unified evolution including the
$ln^2(1/x)$ resummation goes beyond the NLO DGLAP analysis. Corrections
generated by double logarithms at low $x$ influence the $Q^2$-dependence of the
first moments strongly.
