  The resummation of $O(\alpha_s^{l+1} \ln^{2l} x)$ terms in the evolution
kernels of non--singlet combinations of unpolarized and polarized structure
functions is investigated. The agreement with complete calculations up to order
$\alpha_s^2$ is demonstrated, and the leading small-$x$ contributions to the
three--loop non--singlet splitting functions $P^{\pm}$ are derived. The
additional contributions due to the resummed terms are studied numerically for
the most important non--singlet structure functions. They are found to be about
$1\,\%$ or smaller in the kinematical regions accessible at present and in the
foreseeable future.
