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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 $f:X\to Y$ and $x,y\in X$ we denote

${D}_{n}f\left(x,y\right):=\sum _{i=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{i}\right){\left(-1\right)}^{n-i}f\left(ix+y\right)-n!f\left(x\right)·$

A solution $f$ of the functional equation ${D}_{n}f\left(x,y\right)=0$ ($x,y\in X$) is called a monomial function of degree $n$. The author proves the following stability results for the monomial equation. Suppose that for $n\ne p\ge 0$ a mapping $f:X\to Y$ satisfies

$\parallel {D}_{n}f\left(x,y\right)\parallel \le \epsilon \left({\parallel x\parallel }^{p}+{\parallel y\parallel }^{p}\right),\phantom{\rule{2.em}{0ex}}x,y\in X·\phantom{\rule{2.em}{0ex}}\left(1\right)$

Then there exists a unique monomial function $F:X\to Y$ of degree $n$ such that

$\parallel f\left(x\right)-F\left(x\right)\parallel \le M\frac{\epsilon }{{2}^{n}-{2}^{p}}{\parallel x\parallel }^{p},\phantom{\rule{2.em}{0ex}}x\in X$

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 $x,y\in X\setminus \left\{0\right\}$. 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 $n=1,2,3,\cdots$ one gets the (super)stability of additive, quadratic, cubic, ...mappings.

##### MSC:
 39B82 Stability, separation, extension, and related topics 39B52 Functional equations for functions with more general domains and/or ranges
##### Keywords:
stability; monomial functional equation