  Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The
nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is
given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we
mainly give some necessary and sufficient conditions for the boundedness of
$[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or
$BMO(\rn)$ spaces, by which some new characterizations for certain subclasses
of Lipschitz and $BMO(\rn)$ spaces are obtained.
