The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type

Given a space of homogeneous type $(X,d,\mu )$, we present sufficient conditions on a variable exponent ${p(\cdot )}$ so that the fractional maximal operator $M_\eta $, $0\leq \eta \lt 1$, maps $L^{p(\cdot )}(X)$ to $L^{q(\cdot )}(X)$, where $1/{p(\cdot )