## On the stability of the monomial functional equation.(English)Zbl 1152.39023

Let $$X$$ be a normed space, $$Y$$ a Banach space and $$n$$ a positive integer. For a mapping $$f\colon X\to Y$$ and $$x,y\in X$$ we denote
$D_nf(x,y):=\sum_{i=0}^n {n\choose i} (-1)^{n-i}f(ix+y)-n!f(x).$
A solution $$f$$ of the functional equation $$D_nf(x,y)=0$$ ($$x,y\in X$$) is called a monomial function of degree $$n$$. The author proves the following stability results for the monomial equation. Suppose that for $$n\neq p\geq 0$$ a mapping $$f\colon X\to Y$$ satisfies
$\| D_nf(x,y)\| \leq\varepsilon\left(\| x\| ^p+\| y\| ^p\right),\qquad x,y\in X.\tag{1}$
Then there exists a unique monomial function $$F\colon X\to Y$$ of degree $$n$$ such that
$\| f(x)-F(x)\| \leq M\frac{\varepsilon}{2^n-2^p}\| x\| ^p,\qquad x\in X$
with some explicitly given constant $$M$$ depending on $$n$$ and $$p$$. A similar stability result can be obtained if $$f$$ satisfies the inequality (1) with some $$p<0$$ and for all $$x,y\in X\setminus\{0\}$$. However, the author proves that in this case the mapping $$f$$ itself has to be a monomial function of degree $$n$$, i.e., surprisingly the superstability phenomenon appears. Applying the above results for $$n=1,2,3,\dots$$ one gets the (super)stability of additive, quadratic, cubic, …mappings.

### MSC:

 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges

### Keywords:

stability; monomial functional equation
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