  Let $(X,d,\mu)$ is a space of homogeneous type, we establish a new class of
fractional-type variable weights $A_{p(\cdot), q(\cdot)}(X)$. Then, we get the
new weighted strong-type and weak-type characterizations for fractional maximal
operators $M_\eta$ on weighted variable Lebesgue spaces over $(X,d,\mu)$. This
study generalizes the results by Cruz-Uribe-Fiorenza-Neugebauer (2012),
Bernardis-Dalmasso-Pradolini (2014), Cruz-Uribe-Shukla (2018), and
Cruz-Uribe-Cummings (2022).
