Let be a normed space, a Banach space and a positive integer. For a mapping and we denote
A solution of the functional equation () is called a monomial function of degree . The author proves the following stability results for the monomial equation. Suppose that for a mapping satisfies
Then there exists a unique monomial function of degree such that
with some explicitly given constant depending on and . A similar stability result can be obtained if satisfies the inequality (1) with some and for all . However, the author proves that in this case the mapping itself has to be a monomial function of degree , i.e., surprisingly the superstability phenomenon appears. Applying the above results for one gets the (super)stability of additive, quadratic, cubic, ...mappings.