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Stability for the functional equation of cubic type

Authors: Chang, Ick-Soon; Jung, Yong-Soo;

Stability for the functional equation of cubic type

Abstract

The authors offer a (generalized) Hyers-Ulam [\textit{S. M. Ulam}, Problems in modern mathematics, Wiley, New York (1964; Zbl 0137.24201), Ulam originally posed the problem in 1940; \textit{D. H. Hyers}, Proc. Nat. Acad. Sci. USA. 27, 222--224 (1941; Zbl 0061.26403)] stability result for the functional equation \[ \begin{multlined} f(x_1+x_2+2x_3)+f(x_1+x_2-2x_3)+f(2x_1)+f(2x_2)+7[f(x_1)+f(-x_1))]\\ =2f(x_1+x_2) +4[f(x_1+x_3)+f(x_1-x_3)+f(x_2+x_3)+f(x_2-x_3)]\end{multlined} \] where \(x_j\) is orthogonal to \(x_k\;(j,k=1,2,3, j\neq k)\) in the sense of \textit{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 \textit{Th. M. Rassias} [Proc. Am. Math. Soc. 72, 297--300 (1978; Zbl 0398.47040)], for the result and proof cf., however, \textit{T. Aoki} [J. Math. Soc. Japan 2, 64--66 (1950; Zbl 0040.35501), not in the references]. For example also a paper by \textit{D. G. Bourgin} [Bull. Am. Math. Soc. 57, 223--237 (1951; Zbl 0043.32902)] and another paper by \textit{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 \textit{G.-L. Forti} [J. Math. Anal. Appl. 295, No. 1, 127--133 (2004; Zbl 1052.39031)].

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Keywords

Functional equations for real functions, orthogonally cubic functional equation, Cubic functional equation, Applied Mathematics, Stability, separation, extension, and related topics for functional equations, Functional equations for functions with more general domains and/or ranges, Hyers-Ulam stability, Stability, Analysis, Orthogonally cubic functional equation

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
6
Average
Average
Average
hybrid