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Our aim of this paper is to study a family of functional equation in vectorand Banach spaces with difference operators, where this family of functionalequation is a general mixed additive-quadratic-cubic-quartic functionalequations. We show that every function satisfies the our functional equation isa monomial function with a certain degree. Furthermore, we deal with thegeneralized Hyers-Ulam stability of this family of functional equations inBanach space.

A family of functional equations related to the monomial functions and its stability