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

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