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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.

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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