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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 \[ f(x+y)+f(x-y)-g(x)-g(y)\in B,\qquad x,y\in G \] 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 \[ Q(x)+f(0)-f(x), Q(x)+g(0)-g(x) \in\tfrac{2}{3}\text{cl conv}(B-B),\qquad x\in G. \]

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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