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On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings. (English) Zbl 1101.39017
The author states two lemmas on the stability of the single variable functional equation \(\varphi(2x)=3\varphi(x)+\varphi(-x) \quad (x \in G)\), where \(\varphi\) is a mapping from an abelian group \((G,+)\) into a Banach space \(X\); see G. L. Forti [J. Math. Anal. Appl. 295, No. 1, 127–133 (2004; Zbl 1052.39031)]. Then the author applies the lemmas to investigate the stability of functional equations \[ \begin{aligned}\varphi(x+y)+\varphi(x-y)&= 2\varphi(x)+\varphi(y)+\varphi(-y),\\ \varphi(x+y+z)+\varphi(x)+\varphi(y)+\varphi(z)&= \varphi(x+y)+\varphi(y+z)+\varphi(z+x), \end{aligned} \] which have solutions of the form \(\varphi=a+q\) where \(a\) is an additive mapping and \(q\) is a quadratic one. The obtained results may regard as complementary to those of S.-M. Jung and P. K. Sahoo [Aequationes Math. 64, No. 3, 263–273 (2002; Zbl 1022.39028)] and S.-M. Jung [J. Math. Anal. Appl. 222, No. 1, 126–137 (1998; Zbl 0928.39013)].

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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