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Stability for the functional equation of cubic type. (English) Zbl 1120.39026

The authors offer a (generalized) Hyers-Ulam [S. M. Ulam, Problems in modern mathematics, Wiley, New York (1964; Zbl 0137.24201), Ulam originally posed the problem in 1940; D. H. Hyers, Proc. Nat. Acad. Sci. USA. 27, 222–224 (1941; Zbl 0061.26403)] stability result for the functional equation \[ \begin{split} f(x_1+x_2+2x_3)+f(x_1+x_2-2x_3)+f(2x_1)+f(2x_2)+7[f(x_1)+f(-x_1))]\\ =2f(x_1+x_2) +4[f(x_1+x_3)+f(x_1-x_3)+f(x_2+x_3)+f(x_2-x_3)]\end{split} \] where \(x_j\) is orthogonal to \(x_k\;(j,k=1,2,3, j\neq k)\) in the sense of J. Rätz [Aequationes Math. 28, 35–49 (1985; Zbl 0569.39006)] in real vector spaces and for its generalization to \(n\geq 3\) variables. Structure and language is somewhat unorthodox but the meaning gets through.
The paper starts with a survey on Hyers-Ulam stability. Among the 32 references only one is from between Hyers’s 1941 paper and a 1978 paper by Th. M. Rassias [Proc. Am. Math. Soc. 72, 297–300 (1978; Zbl 0398.47040)], for the result and proof cf., however, T. Aoki [J. Math. Soc. Japan 2, 64–66 (1950; Zbl 0040.35501), not in the references]. For example also a paper by D. G. Bourgin [Bull. Am. Math. Soc. 57, 223–237 (1951; Zbl 0043.32902)] and another paper by D. H. Hyers [Pac. J. Math. 11, 591–602 (1961; Zbl 0099.10501)] could have been quoted. For the core of the direct method for proving Hyers-Ulam stability, see G.-L. Forti [J. Math. Anal. Appl. 295, No. 1, 127–133 (2004; Zbl 1052.39031)].

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B22 Functional equations for real functions
39B52 Functional equations for functions with more general domains and/or ranges
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