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Zbl 1130.39021
On the stability of the quadratic functional equation in topological spaces.
(English)
[J] Banach J. Math. Anal. 1, No. 2, 245-251, electronic only (2007). ISSN 1735-8787/e

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 $$f(x+y)+f(x-y)-g(x)-g(y)\in B,\qquad x,y\in G$$ holds for a nonempty bounded $B\subset X$, then $f$ and $g$ are close to an exact solution of the quadratic equation $$Q(x+y)+Q(x-y)=2Q(x)+2Q(y),\qquad x,y\in G.$$ Actually, it is proved that there exists a unique quadratic mapping $Q:G\to X$ such that $$Q(x)+f(0)-f(x), Q(x)+g(0)-g(x) \in\tfrac{2}{3}\text{cl conv}(B-B),\qquad x\in G.$$
[Jacek Chmielinski (Kraków)]
MSC 2000:
*39B82 Stability, separation, extension, and related topics
39B52 Functional equations for functions with more general domains

Keywords: stability of functional equations; abelian 2-divisible group; sequentially complete locally convex linear topological Hausdorff space

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