## 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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