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We prove that a generalized Fefferman-Phong type condition on a pair ofweights $u$ and $v$ is sufficient for the boundedness of the commutators ofpotential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We alsogive an improvement of this result in the sense that we not only consider avariable version of power bump conditions, but also weaker norms related toMusielak-Orlicz functions. We consider a wider class of symbols includingLipschitz symbols and some generalizations.

Commutators of potential type operators with Lipschitz symbols on variable Lebesgue spaces with different weights