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Zbl 1152.39023
Lee, Yang-Hi
On the stability of the monomial functional equation.
(English)
[J] Bull. Korean Math. Soc. 45, No. 2, 397-403 (2008). ISSN 1015-8634

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 {\it 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 $$\Vert D_nf(x,y)\Vert \leq\varepsilon\left(\Vert x\Vert ^p+\Vert y\Vert ^p\right),\qquad x,y\in X.\tag1$$ Then there exists a unique monomial function $F\colon X\to Y$ of degree $n$ such that $$\Vert f(x)-F(x)\Vert \leq M\frac{\varepsilon}{2^n-2^p}\Vert x\Vert ^p,\qquad x\in X$$ 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, \dots mappings.
[Jacek Chmieliński (Kraków)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: stability; monomial functional equation

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