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Stability of a generalized quadratic functional equation with Jensen type. (English) Zbl 1074.39029
Assume that $$m$$ and $$n$$ are nonzero integers with $$m + 1 = 2n$$. For real vector spaces $$X$$ and $$Y$$, the author proves that a function $$f : X \to Y$$ is a solution to the functional equation ${m^2 f\!\left( \frac{x+y+z}{m} \right) + f(x) + f(y) + f(z)} = n^2 \!\left[ f\!\left( \frac{x+y}{n} \right) + f\!\left( \frac{y+z}{n} \right) + f\!\left( \frac{z+x}{n} \right) \right]$ if and only if $$f$$ is represented by $$f(x) = Q(x) + A(x) + B$$, where $$Q : X \to Y$$ is a quadratic function, $$A : X \to Y$$ is an additive function and $$B$$ is an element of $$Y$$ ($$B = 0$$ if $$m^2 + 3 \neq 3n^2$$). He also proves the Hyers-Ulam-Rassias stability of the above functional equation for integers $$m, n \not\in \{ -1, 0, 1 \}$$ with $$m + 1 = 2n$$ and for the functions $$f : X \to Y$$, where $$X$$ is a real normed space and $$Y$$ is a real Banach space.

##### 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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