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On the generalized Hyers-Ulam-Rassias stability of quadratic functional equations. (English) Zbl 1167.39014

Summary: We achieve the general solution and the generalized Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stabilities for quadratic functional equations
\[ \begin{split} f(ax+by)+f(ax - by)\\ =(b(a+b)/2)f(x+y)+(b(a+b)/2)f(x - y)+(2a^{2} - ab - b^{2})f(x)+(b^{2} - ab)f(y)\end{split} \]
where \(a, b\) are nonzero fixed integers with \(b\neq \pm a, - 3a\), and \[ f(ax+by)+f(ax - by)=2a^{2}f(x)+2b^{2}f(y) \]
for fixed integers \(a, b\) with \(a,b\neq 0\) and \(a\pm b\neq 0\).

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B22 Functional equations for real functions
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References:

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