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Fixed points, inner product spaces, and functional equations. (English) Zbl 1211.39020

Let \(X,Y\) be real vector spaces. A function \(Q: X\to Y\) is called a quadratic-type mapping iff
\[ Q(x-y)+Q(2x+y)+Q(x+2y)=3Q(x)+3Q(y)+3Q(x+y),\quad x,y\in X. \]
The author considers the conditional functional equation
\[ \sum_{i,j=1}^{n}f(x_i-x_j)=2n\sum_{i=1}^{n}f(x_i),\quad x_1,\dots,x_n\in X,\;\sum_{i=1}^{n}x_i=0\tag{1} \]
and shows that a solution \(f: X\to Y\) of (1) is a sum \(f=A+Q\) of an additive mapping \(A\) and a quadratic-type mapping \(Q\).
Assume now that \(X\) is a real normed space and \(Y\) a real Banach space. Using the fixed point technique, the stability of (1) is proved. If \(f: X\to Y\) satisfies \(f(0)=0\) and
\[ \left\|\sum_{i,j=1}^{n}f(x_i-x_j)-2n\sum_{i=1}^{n}f(x_i)\right\|\leq \varphi (x_1,\dots,x_n),\quad x_1,\dots,x_n\in X,\;\sum_{i=1}^{n}x_i=0 \]
with a suitable control mapping \(\varphi\), then for some additive mapping \(A: X\to Y\) and some quadratic-type mapping \(Q: X\to Y\), \(f\) can be approximated by \(A+Q\), i.e.,
\[ \|f(x)-A(x)-Q(x)\|\leq\Psi(x),\quad x\in X \]
where \(\Psi\) is expressed by \(\varphi\). In particular, for
\[ \varphi(x_1,\dots,x_n)=\theta\sum_{i=1}^{n}\|x_i\|^p \]
with \(p>2\) and \(\theta>0\), there is
\[ \Psi(x)=\left(\frac{1}{2^p-4}+\frac{2^p+2}{2n(2^p-2)}\right)\theta\|x\|^p \]
(similarly for \(p<1\)).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B55 Orthogonal additivity and other conditional functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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References:

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