Fractional maximal operators on weighted variable Lebesgue spaces over the spaces of homogeneous type

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).