Averaging operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ algebras

The aim of this paper is to give some constructions results of averaging operators on Hom-Lie algebras. The homogeneous averaging operators on $q$-deformed Witt and $q$-deformed $W(2,2)$ Hom-algebras are classified. As applications, the induced Hom-Leibniz algebra structures are obtained and their multiplicativity conditions are also given.