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