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On the stability of the quadratic functional equation in topological spaces. (English) Zbl 1130.39021

Let $G$ be an abelian 2-divisible group and let $X$ be a sequentially complete locally convex linear topological Hausdorff space. It is shown that if mappings $f,g:G\to X$ approximately satisfy a generalized quadratic functional equation, namely if

$f\left(x+y\right)+f\left(x-y\right)-g\left(x\right)-g\left(y\right)\in B,\phantom{\rule{2.em}{0ex}}x,y\in G$

holds for a nonempty bounded $B\subset X$, then $f$ and $g$ are close to an exact solution of the quadratic equation

$Q\left(x+y\right)+Q\left(x-y\right)=2Q\left(x\right)+2Q\left(y\right),\phantom{\rule{2.em}{0ex}}x,y\in G·$

Actually, it is proved that there exists a unique quadratic mapping $Q:G\to X$ such that

$Q\left(x\right)+f\left(0\right)-f\left(x\right),Q\left(x\right)+g\left(0\right)-g\left(x\right)\in \frac{2}{3}\text{cl}\phantom{\rule{4.pt}{0ex}}\text{conv}\left(B-B\right),\phantom{\rule{2.em}{0ex}}x\in G·$

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