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

Let \((X,d,\mu )\) $(X,d,\mu )$ is a space of homogeneous type, we establish a new class of fractional-type variable weights \(A_{p(\cdot ), q(\cdot )}(X)\) $A_{p(·),q(·)}(X)$ . Then, we get the new weighted strong-type and weak-type characterizations for fractional maximal operators \(M_\eta \) $M_{\eta }$ on weighted variable Lebesgue spaces over \((X,d,\mu )\) $(X,d,\mu )$ . This study generalizes the results by Cruz-Uribe–Fiorenza–Neugebauer (J Math Anal Appl 64(394):744–760, 2012), Bernardis–Dalmasso–Pradolini (Ann Acad Sci Fenn-M 39:23-50, 2014), Cruz-Uribe–Shukla (Stud Math 242(2):109–139, 2018), and Cruz-Uribe–Cummings (Ann Fenn Math 47(1):457–488, 2022).