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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 \(f\colon X\to Y\) 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
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